ABSTRACT: The proposal discusses motivation for developing a robust model-based clustering approach. We will present a general framework in which a rough idea with regard to the thesis research is proposed. Relevant literature concerning clustering analysis especially statistical model-based clustering techniques is reviewed.
ABSTRACT: A three-dimensional random vector $(X_0,X_1,X_3)$ with
independent components is transformed into a bivariate vector
$(Y_1,Y_2)=(\phi_1(X_0,X_1),\phi_2(X_0,X_2))$. We want to identify the
distributions of $X_i$'s having observed $Y_i$'s. For coding functions
$\phi_i$'s falling into a semigroup scheme described in Kotlarski and
Sasvari (1992) such an identification is possible even in quite abstract
settings. A thorough review is given in Prakasa Rao's (1992) monograph.
Here we consider a new coding method - independent random choices (with
the same unknown probability): $X_0$ or $X_1$ for $Y_1$ and $X_0$ or $X_2$
for $Y_2$. It appears that in this case the full identification of the
model is possible. Also this new approach will be combined with standard
coding functions used earlier.
A somewhat related question of identification of a finite bivariate
mixtures has been treated recently in Hall and Zhou (2001) (earlier
studied also by Luboi\'nska and Niemiro (1991)).
ABSTRACT: Mathematical techniques of great importance, involving elements of the calculus, were developed between the 14th and 16th centuries in Kerala, India. Kerala during that time was a region which had been in continuous contact with the outside world, with China amongst others to the East and with Arabia to the West. And after the pioneering voyage of Vasco da Gama in 1499, there was a direct conduit to Europe. Despite these communication routes Kerala mathematics, according to current knowledge, lay localised in Kerala until an Englishman, Charles Whish, ‘re-discovered' it in the 19th century. The talk is based on some of the findings of an ongoing research project which examines the epistemology of the calculus of the Kerala school, its transmission to Europe and the consequential educational implications.
ABSTRACT: I will prove Gelfand's Theorem, which states that a C*-algebra
is isomorphic to the set of continuous functions over a locally compact
Hausdorff space.
Seminar requirement for Masters students.
Reminder: Master's Mathematics students are expected to attend the talks
of other students. Documented evidence at 6 such talks is expected. Attendance
sheets can be picked up from N519 Ross.
ABSTRACT: Crossover is synonymous with recombination--the process
by which positive features of two parents are combined into a "super"
offspring. However, this effect is a minority occurrence. The uncommon
components of two parents which are available to be combined are more
likely to be less-positive features. Thus, the likely purpose of
crossover (and its majority occurrence) is the preservation of common
components. This alternate perspective leads to new models for the design
of crossover operators, for population-based search, and for perhaps even
evolution.
A specific example will be presented for feature subset selection where
a commonality-based crossover operator performs better in the traditional
binary-string search space than the best traditional genetic
algorithm.
ABSTRACT: Steel bars traveling on a moving belt with an initial
temperature of 800C are cooled by water to an target value of 500C within
a distance of 100m. The water is provided by shower heads arranged in
rows in two groups. The number of the shower heads can be adjusted so
that at the exit point the target temperature is reached.
(objective): Derive a mathematical model for the problem; Use numerical
simulations (by solving the model equation with C or Matlab) to verify the
analysis.
Seminar requirement for Masters students.
Reminder: Master's Mathematics students are expected to attend the
talks of other students. Documented evidence at six such talks is
expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: Crossed modules were introduced by Whitehead in 1946 to
study higher homotopy groups. They have been used successfully in various
areas by many other authors.
We present the category of crossed modules and use them to generalize
some well-known exact sequences of abelian groups to a class of nonabelian
groups. A kernel-cokernel sequence is associated to a short exact sequence
of crossed modules.
We study homology of crossed chain complexes and obtain a long exact
homology sequence. Also homotopy can be defined for crossed chain maps in
such a way that homotopic crossed chain maps induce the same homology
maps.
With every crossed module we associate a categorical group. The
kernel-cokernel sequence for crossed modules will be interpreted in this
context.
ABSTRACT: Version of Rota and Taylor's classical umbral method is introduced. An generalization of multivariate umbral method on the vector space is given. As an application, the notation and calculation of Bezier curves and surfaces can be greatly simplified. New results can be relatively easier to get. Calculation of Grassmann's weighted points and vectors will also be briefly introduced.
ABSTRACT: In joint work with B. Allison and A. Pianzola we have developed a theory of covering algebras. We can obtain new EALA's from old ones using this theory as well as obtaining a vast generalization of V. Kac's early work on covering algebras of the finite dimensional simple Lie algberas. This theory, together with applications, will be presented in this talk.
ABSTRACT: Let Hom(V) be the set of quivers V_0 -> V_1 -> ... -> V_n. A quiver cycle is a subset O_r of Hom(V) where the ranks of the composite maps V_i -> V_j are bounded above by specified integers r=(r_{ij}) for i < j. Our goal is to compute the equivariant cohomology class [O_r]. As a special case one obtains Fulton's universal Schubert polynomials. Buch and Fulton expressed [O_r] in terms of Schur functions, and conjectured a combinatorial formula for the coefficients. In particular, they conjectured that the coefficients, which directly generalize the Littlewood-Richardson coefficients, are positive. In this ongoing project, we construct a flat family whose general fiber is isomorphic to O_r, and whose special fiber has components that are direct products of matrix Schubert varieties. This proves that [O_r] is a sum of products of Stanley symmetric functions (stable double Schubert polynomials) where each summand is indexed by a list w of permutations. Our formula is obviously positive for geometric reasons and immediately implies the positivity of the Buch-Fulton formula. We conjecture that the special fiber is generically reduced, so that each list of permutations w occurs with multiplicity 1. We propose a simple nonrecursive combinatorial characterization of which lists w appear. This is joint work with Allan Knutson and Ezra Miller.
ABSTRACT: In this talk I will discuss some inference methods based on statistics defined by estimating equations. In particular, I will considerstatistics determined by quasi-likelihood estimating equations (which are based on first and second moments of the response variable) and robust regression estimates (which also require weak distribution assumptions on the response). I will describe a bootstrap method that is asymptotically correct and significantly faster than the classical bootstrap. For quasi-likelihood estimates I will compare confidence intervals built with this method with those obtained using the "sandwich" variance estimate. Part of this work is joint work with Ruben H. Zamar, Department of Statistics, University of British Columbia.
ABSTRACT: In this talk, We investigate whether abstract differential operator with the symbol polynomial generates integrated semigroup and regularized semigroup by making different conditions. The application to partial differential operators with constant coefficients can be obtained immediately on several different function spaces. In particular, more exact results are obtained for pseudodifferential operator on certain function space. Finally, the results are applied to partial differential equations and compared by the size of their initial value spaces. It turns out that the regularized semigroup is an appropriate tool for non-elliptic partial differential operators and is far superior to the integrated semigroup approach. Similarly, corresponding results are obtained for integrated cosine function and regularized cosine function.
ABSTRACT: Vago-sympathetic balance was proven critical for
electrophysiological stability and robustness vis-à-vis of physiological
challenges in both diseased and normal heart. As found in both clinical
studies and animal models, supra-normal sympathetic drive is
arrhythmogenic and life-threatening, as opposed to anti-arrhythmic
protection due to normal vagal tone. For a few years researchers got
insight upon atrial versus ventricular effects of autonomic drive by using
comparative RR (heart period) & QT (ventricular re-polarization period)
beat-by-beat variability, analysed in frequency domain. While RR spectrum
offers marks of both vagal (by power within HF band: 0.15 to 0.4 Hz) or
sympathetic (LF power: 0.04 to 0.15 Hz, or LF/HF power ratio) traffic
along sinoatrial neurites, LF power-cluster in the QT (auto)spectra or in
RR*QT cross-spectra has been only recently "read" in terms of sympathetic
neural control to ventricles (SDV). Unlike other non-invasive pointers to
ventricular function (as systolic time intervals), spectral QT message is
derived from a single low-noise ECG signal digitized with high temporal
resolution (1 to 2 ms). Further on, requirements of performant
QT-detection software are largely paid by getting quasi-continuous access
(via Holter or telemetry) to ventricles' response to real-life
psycho-social stresses, chronic exacerbation of which was related to
sudden death in both ardiac patients and apparently normal subjects.
A thoracic ECG lead was collected at 1 ms resolution (Codas, Dataq
Instr., USA ), and beat-by-beat RR and QT time series (ms) were derived
using detection software yielding QTs within ± 1.12 ms of tangent method.
After resampling at 500 ms, the most stationary 3 min epochs have been
Fourier-transformed and LF spectral powers (ms2) were summed over 0.04 to
0.15 Hz band. To compute mean square coherence (MSC) I used a 9-line
moving-triangle smoothing of its cross- and quadrature- spectral terms. To
get "idioventricular" fraction of QT-LF I corrected each QT-LF power line
by multiplying with 1-MSC at that frequency. Other methods like
autoregressive modeling, wavelet analysis or T-wave alternans are
explored.
Results:
1) Work done in a group of mid-age to elder subjects with normal hearts
showed in both, that QT-LF (and RR-LF) on stress-interview increases
versus relaxed rest, to a similar extent as sustained bicycle exercise
does (not true for RR-LF).
Conclusion: QT-LF mainly derives from RR-LF under relaxed rest but an
RR-independent factor is activated under both mental stress and exercise.
Using RR*QT coherence spectrum, the RR-dependent component can be removed
to enlighten QT-LF's SDV-only fraction, disclosure of which puts Holter
high-resolution ECG in an ideal position to ascertain ventricular
arrhythmia and sudden cardiac death risk.
2) In young normal subjects, right atrial pacing to stabilize RR
clearly spoils QT-LF when relaxed, but superimposing mental stress
significantly resuscitates QT-LF despite RR's invariance.
3) The idio-ventricular fraction of QT-LF (got by subtraction of mean
squared RR* QT coherence over LF band) was consistently higher under
mental stress versus lower sympathetic-profile settings (as propranolol
blockade or relaxed RAP), proving effective removal of distorting
influences from RR-LF.
ABSTRACT: Cuneiform tablets recovered from the sites of Babylon and other cities in what is now Iraq provide extensive material for studying Babylonian astronomy. These texts mainly date to the first millennium BC and attest to widespread astronomical activity by Babylonian scribes. This included both a regular programme of astronomical observation that apparently continued uninterupted from circa 750 BC to the first century AD, and the development of arithmetical techniques for calculating planetary and lunar phenomena such as first visibilities and eclipses of the sun and moon. In this talk I will provide an overview of Babylonian astronomy and the theoretical developments which led to the world's first astronomical models. I will then present some examples of recent research on the techniques of Babylonian mathematical astronomy, and what it may have been used for in Babylonian society.
ABSTRACT: The Landau-Pollak-Slepian operator in signal analysis has prompted the study of wavelet multipliers, which are in fact localization operators associated to modulations on the additive group $\Rn$. Such a wavelet multiplier is defined in terms of one admissible wavelet and its spectral properties such as the trace and the trace class norm inequality have been studied in detail. Recent works have been focussed on two-wavelet localization operators on locally compact and Hausdorff groups. We give in this talk sharp lower and upper estimates for the trace class norms of two-wavelet multipliers.
ABSTRACT: We consider (B,S)-securities market with a standard
riskless asset (bond) and a risky asset (stock) with stochastic volatility
depending on time and the history of stock price.
We state some results on option pricing in such market and its
completeness. A continuous-time analogue of GARCH model for our stochastic
volatility is proposed.
We then show that the equation for the expected squared volatility
under risk-neutral measure is a deterministic delay differential equation,
and we construct the solutions for such an equation. We also derive the
partial integro-differential equation for the evaluation function with
boundary conditions defined by the option final payoff function.
And, finally, we propose numerical and estimation procedures for above
model and show the comparison of numerical results.
ABSTRACT: The current high-throughput (HTP) revolution is rapidly
changing the way we formulate and test biological hypotheses. Advances in
gene expression profiling by microarrays and protein profiling by MS have
suggested the potential to simultaneously view all genes expressed, all
subsequent protein products, and all the interacting partners of each
individual protein within a biological system. Such views are already
having an impact on our understanding of human disease, particularly in
the realm of cancer biology. However, it is challenging to get all useful
information out of these studies, provide effective focus for further
research, and deliver measurable clinical impact. To make biologically
relevant inferences from massive data sets is a key aspect of
computational biology, which draws on techniques from data warehousing and
knowledge discovery, statistics, stochastic processes, pattern
recognition, computer vision, as well as other machine learning and
artificial intelligence approaches.
The focus of this talk is to overview specific challenges, introduce
machine-learning solutions, and present some evidence of their utility in
HTP biological domains. I will use three areas of HTP biology as an
illustrative example: protein crystallization, microarray analysis, and
protein-protein interaction data analysis.
ABSTRACT: Give a topological space X, denote by CL(X) the set of all nonempty closed subsets of X. A "hyperspace" of X is a subfamily H(X) of CL(X) endowed with some "natural topology" such as those of Vietoris, Wijsman or Hausdorff. A continuous selection for a hyperspace H(X) is a continuous function from H(X) into X which assigns to each closed subset C in H(X) a point of C. The idea of selecting a point from each element of the family H(X) is a special case of the more classical one of selecting a point in the image of each C under the multifunction H(X) => X : C -> C. We investigate conditions under which selections and epsilon approximations to these selections exist for the Vietoris and other hyperspace topologies because we can define a suitable linear ordering of the base space; and the possibility that suitable completeness of the base space can allow such orderings to be defined. We thus obtain interpolations for, and strengthenings of, such well-known and classical results as that of Michael on the existence of continuous selections for lower semicontinuous multifunctions.
ABSTRACT: Our inability to quantitatively predict stem cell fate limits the use of these cells in a variety of therapeutic applications. While significant inroads have been made in investigating the native or plastic potential of stem cells from adult, embryonic and fetal origins, strategies to control or direct the differentiation of these cells to produce large numbers of purified stem or differentiated cells is at a relatively early stage. A systematic and computational approach would assist in the identification and characterization of parameters that regulate stem cell responses; these parameters can then be further engineered to produce desired cells or cell products. Most computational models of stem cell determination events describe observed data without detailing the underlying regulatory mechanisms, thus limiting the effective and controlled exploitation of stem cells for clinical applications. Our integrated experimental and computational approach allows us to evaluate different mechanisms of stem cell fate regulation, and is useful in the design and development of experiments to improve stem cell culture systems.
ABSTRACT: Spectacular advances in information technology and large-scale computing are producing huge and very high dimensional data sets. These data sets arise naturally in a variety of contexts such as text/web mining, bioinformatics, imaging for diagnostics and surveillance, astronomy and remote sensing. The dimension of these data is in the hundreds or thousands. The traditional clustering algorithms do not work efficiently for data sets in such high dimensional spaces because of the inherent sparsity of data. This is well known as the curse of dimensionality. This dissertation develops a new neural network architecture PART (Projective Adaptive Resonance Theory) and related algorithms based on the neural dynamics to provide a solution to the difficulties in clustering high-dimensional data. The PART architecture is based on the well known ART developed by Carpenter and Grossberg, and a major modification (selective output signaling mechanism) is provided in order to deal with the inherent sparsity of the data points in high dimensional space from many data-mining applications. We provide a rigorous proof of the regular dynamics of the PART model which is a large scale and singularly perturbed system of differential equations coupled with a reset mechanism. Our simulations and comparisons show that the resulting algorithms based on the PART model are effective and efficient in finding projected clusters in high dimensional data sets. In the second part of this dissertation, we propose a provably correct clustering algorithm IMC (Iterative Mean Clustering) and the related mathematical theory. We provide a rigorous proof of the convergence of this algorithm. In particular, in one-cluster case where the data distribution is unimodal, this algorithm converges to the center of the unique cluster starting from an arbitrary initial value. In multi-clusters case where the data distribution is multimodal, this algorithm converges to the center of a cluster that is close to the initial value. Finally, we develop a neural network implementation of the IMC algorithm, called IMC-ART, and introduce a variation of PART algorithm, called PART-A, which combines PART architecture with IMC algorithm.
ABSTRACT: Let f be an irreducible polynomial with integer coefficients. We consider the problem of finding primes p for which the reduction of f modulo p is also irreducible. This problem goes back to Frobenius and is a Chebotarev theorem in disguise.
ABSTRACT: In this talk we will give an introduction to the Metropolis-Hastings Independent Algorithm (MHIA) and we will see how this algorithm can be improved. Our first approach will use a control variate based on the sample generated by the proposal. We will derive the variance of our estimator for fixed sample sizes n and show that, as n tends to infinity, the variance of our estimator is asymptotically better than the one obtained with the MHIA . Our second approach will be based on Jensen's inequality. We will use a Rao-Blackwellisation and we will exploit the lack of symmetry in the MHIA. We will find and upper bound on the improvements that we can obtain by these methods.
ABSTRACT: Success in distinguishing "junk" emails from "good" emails (or one text type from another) depends on many factors, including the inherent separation of the defined text types. Multidimensional Scaling - MDS ( a well-known statitical technique) provides a means of visualizing the separation, and so helps one decide whether or not a given method is worth pursuing for a given type of text. This presentation will discuss work in progress and potential implications of using MDS in linguistic dialect studies.
ABSTRACT: A picture of the complete graph K_n consists of n labelled points in the plane, connected with (n choose 2) lines. I'm going to talk about the algebraic relations that must hold among the slopes of these lines. This sounds like a problem in classical geometry, but it turns out that the tools to attack it come from combinatorics. First, the equations defining a picture can be described using the theory of combinatorial rigidity of graphs. Second, once one knows what these equations are, one can apply another combinatorial idea, the theory of Stanley-Reisner rings, to obtain geometric invariants of the space of all solutions. Finally, various sorts of labelled trees play important roles in describing these invariants combinatorially.
ABSTRACT: Many on-line abstracts and articles have become available to researchers. For example, MEDLINE offers access to more than 12 millions articles publsihed on medical journals. To search, dowload and read these articles can be very time consuming and even impossible. In an attempt to free researchers from this tedious research routine, we propose a simple but effective algorithm that can "read" through all the abstracts of interests and make predictions. We will demostrate this approach by using 15,000 abstracts from MEDLINE. I will also outline other bioinformatics projects that we are working on.
ABSTRACT: The talk is devoted to stochastic dynamical systems
arising in biology and finance, and consists of two parts:
The first part deals with the limit theorems and stability for
biological systems in random media. The second part is devoted to the
study of (B,S)-securities markets with delayed volatility: completeness,
equation for expectation of volatility, option pricing formula.
1) biological systems in random media;
2) stochastic financial systems with delayed volatility.
ABSTRACT: Different approaches have been proposed in literature to simplify neural networks in order to achieve better generalization. One way of achieving this goal is by adding extra term to the standard (sum of squared errors) cost function that penalizes the complexity. The simple version of this approach includes penalizing the sum of the squares of the weights or penalizing the number of nonzero weights. Hinton and Nowlan(1992) proposed a more complicated penalty term in which the distribution of weight values is modeled as a mixture of multiple Gaussian distributions. This regularization term encourages a group of weights to have similar values. The division of weights into a group, the mean weight value for each group, and the spread of values within the group, are all considered as adjustable parameters to be determined as part of learning process.
ABSTRACT: Let a(t) be a polynomial which has positive integer coefficients, a constant term of one, and only real zeros. We show that a(t) appears in the numerator of the Hilbert series of some Cohen-Macaulay ring, and present some evidence in favor of the stronger conjecture that a(t) is the f-polynomial of a simplicial complex. This is joint work with Jason Bell.
ABSTRACT: This talk describes an efficient sampler for the Bayesian estimation of covariance matrices using Markov chain Monte Carlo (MCMC). The methods apply to decomposable covariance selection models with a hyper inverse Wishart (HIW) prior for the covariance matrix. The conjugate properties of the HIW distribution are used to generate from reduced conditional distributions. In particular, the covariance matrix is integrated out of all conditional distributions and is not generated in the MCMC. The resulting sampler is shown to have a much faster convergence rate than existing methods. The computational complexity of one iteration of the MCMC is shown to be similar to existing methods, so the gain in convergence rate is significant. An efficient mixture estimate of the posterior mean of the inverse covariance matrix is given.
ABSTRACT: This seminar will present a study of the applicability of
mathematical modeling in engineering. Models will be built interactively
from scratch to allow the audience to give their input and to alter the
modelling process itself. The idea will be to show the benefits of
modeling in an education environment, as a compliment to theoretical and
experimental studies.
The seminar will specifically look at:
Based on MATLAB, FEMLAB is a modeling tool that solves any arbitrary
nonlinear coupled Partial Differential Equation. It also consists of
specialized application modules for Chemical Engineering/Transport
Phenomena, Structural Mechanics and Electromagnetics. Further information
can be found at www.comsol.com.
Physics can be applied in FEMLAB as model equations in tailored,
ready-to-use forms, or specified freely to suit any arbitrary type of
physical phenomenon (linear, non-linear or time dependent). Several
problems can be combined and coupled in a single model - multiphysics
modeling - meaning that your simulations can encompass all fields of
physics and engineering.
(1) Coupled momentum and heat transfer in flow through a heat
exchanger
(2) Solving fundamental physics on complex geometries, using weak
formulations
(3) Specifying two-phase flow in the general mode of FEMLAB
(4) Coupling 2D and 3D geometries in a structural mechanics/acoustics
multiphysics problem
(5) Command line programming from the MATLAB environment
ABSTRACT: Collocation with piecewise polynomials is a
simple-to-implement and integration-free discretization method for one- or
multi-dimensional Boundary Value Problems (BVPs). Collocation using
splines (i.e. piecewise polynomials with maximum continuity) gives rise to
small linear systems, usually nicely behaved, and with small bandwidth.
Therefore, spline collocation is a reasonable alternative to Galerkin.
However, spline collocation has not yet been extensively used for the
solution of BVPs. It is known that the standard formulation of the method
gives rise to suboptimal approximations with respect to convergence order.
Relatively recently several optimal spline collocation methods have been
derived based on splines of degree 2, 3, 4 and 5. However, the success of
the methods is hindered by the fact that all these methods are derived on
uniform grids.
We will first review the development of optimal spline collocation
methods on uniform partitions. Next, we will describe the extension of
optimal Quadratic and Cubic Spline Collocation (QSC and CSC) methods for
the solution of linear second-order two-point Boundary Value Problems
(BVPs) discretized on non-uniform partitions. To derive the methods, we
use a mapping function between uniform and non-uniform partitions and
develop expansions of the error at the non-uniform collocation points of
some appropriately defined spline interpolants. The existence and
uniqueness of the QSC and CSC approximations are shown, under some
conditions. Optimal global and local orders of convergence of the spline
approximations and derivatives are derived, similar to those of the
respective methods for uniform partitions. The jth derivative of the QSC
approximation, for j>=0, is O(h^{3-j}) globally, and O(h^{4-j}) locally on
certain points. The jth derivative of the CSC approximation, is O(h^{4-j})
globally, for j>=0, and O(h^{5-j}) locally on certain points, for j>0. The
non-uniform partition QSC and CSC methods are integrated with adaptive
grid techniques, and grid size and error estimators. Numerical results on
a variety of problems, including problems with boundary or interior
layers, verify the theoretically expected behaviour of the methods.
ABSTRACT: The purpose of this project is to develop a computer-based system which requires minimum user interaction to measure retinal vessel caliber. Several image processing techniques including directional edge enhancement and edge searching algorithms were incorporated into a method which requires a single observer-selected point inside the vessel to measure its width. Diameters of first, second, third and fourth order retinal veins from red free retinal photographs taken with a Kodak Megaplus 1.4e digital camera mounted on a Topcon TRC-50x fundus camera, were then measured with this semi-automated method and compared to an obserber-driven method. The coefficient of variation of the results with the semi-automated method was between 0.05-8.8% (depending on the vessel diameter), which was much lower than the observer-driven estimates (5.45-29.3%). These numbers compare well with those reported by Newsom et al '92 using a computer-driven method which employed two observer selected edge points. The technique facilitates the analysis of retinal vessel calibers which are reflections of retinal blood flow and retinal vessel pressures. Future directions in using this technique are also discussed.
ABSTRACT: Aeroelasticity is concerned with the dynamic interaction
between an aerodynamic flow and an elastic structure, such as aircraft
wings in high speed flight, long span bridges and tall buildings
responding to wind loadings, or airflow through the mouth and lungs.
Classical works on aeroelasticity assume linear models for dynamics,
aerodynamics, and structures. However, structural nonlinearities arise
from worn hinges of control surfaces, loose control linkages, material
behavior and various other sources. An understanding of the nonlinear
behavior of the system is crucial to the efficient and safe design of
aircraft wings and control surfaces. There are distributed and
concentrated structural nonlineairties, while the latter is commonly found
in control mechanisms.
This study focuses on the concentrated structural nonlinearities, which
can be classified roughly into three types: cubic spring, freeplay and
hysteresis nonlinearities. The principle interest for the aeroelastician
is the asymptotic motion behavior (convergence, divergence, limit cycle
oscillation) and the amplitude and frequency of the limit cycle
oscillations. The model under investigation is a two-dimensional airfoil
(aircraft wing section) oscillating in pitch and plunge and exposed to
subsonic aerodynamics. By using the analytical techniques: the center
manifold theory, the principle of normal form, the perturbation method,
and the point transformation method, we accurately predict the nonlinear
aeroelastic response. For the aeroelastic system with structural
nonlinearities, damped, period-one, period-one with harmonics, period-two,
period-two with harmonics, and chaotic motions are detected and the
amplitudes and frequencies of limit cycle oscillations are predicted for
the velocities below the linear flutter speed. Although time-integration
numerical methods have often been used to study the response of the
aeroelastic system with structural nonlinearities, the importance and
necessity of analytical techniques are revealed through a detailed study
of the numerical errors resulting from the Runge-Kutta method.
Refreshments will be served at 1:30p.m. in N620 Ross.
ABSTRACT: The topological sensitivity analysis consists in studying
the variation of a cost function with respect to a modification of the
topology of a domain. It is a basic tool for topological shape of the
domain. The topological sensitivity provides a descent direction for
updating the shape of the domain. The topological sensitivity analysis
provides an asymptotic expansion of a shape function with respect to the
insertion of a small obstacle inside a domain.
We developped such an expansion for the Poisson problem and for Stokes
equations with general shape functions and arbitrary shaped holes. It is
shown that this expansion depends on the shape of the obstacle in the
three dimensional case, whereas it is independent of the shape in the two
dimensional case.Numerical example illustrate the use of the topological
sensitivity in a shape optimization problem.
Refreshments will be served in N620 Ross at 3:30p.m.
You are invited to attend a statistics seminar and dinner hosted by The Applied Biostatistics Association (TABA). Please RSVP for both the seminar and dinner to Tina Haller, TABA Treasurer, at tina@statcon.ca by October 2, 2002.
A cheque for dinner can be made payable to TABA and forwarded to Tina Haller at 6 Matson Dr. RR#2, Bolton ON, L7E 5R8.
Lehana Thabane, Department of Clinical Epidemiology and Biostatistics,
McMaster University, will speak on "
ABSTRACT: The talk is an introduction to Bayesian reasoning in the
analysis of data in health research where the primary goal is to inform
decision making in presence of uncertainty. Statistics methods are mostly
based on the concept of probability as a measure of uncertainty.
Therefore, the talk will start by reviewing the common interpretations of
probability which are key to understanding the various school of thoughts,
namely the Bayesian and classical approaches. I will examine the Bayesian
methods in detail, using a numerical example of clinical trial data on
AZT, to illustrate the implementation of the Bayesian methods. I will put
more emphasis on how to use the Bayesian methods to incorporate evidence
from previous studies in the analysis and how to summarize the results of
the analysis. Comparisons with the classical results will also be made.
Advantages and challenges of the Bayesian methods; and similarities
between the Bayesian and classical approaches will also be highlighted.
Learning Objectives
(1) Gain basic knowledge of Bayesian reasoning
(2) How to use Bayesian methods to analyze health data.
(3) How to report Bayesian analysis results
(4) Advantages of Bayesian methods and similarities with classical
methods
We thank GlaxoSmithKline, Pharmacia and Bayer for their generous support of our programs.
COST: seminar - FREE
ABSTRACT: The best approach to study nonlinear dynamical systems is
combining theoretical (analytical) and computational (numerical/symbolic)
methods. Normal form theory is one of the most useful and important tools
in attacking complex nonlinear problems, and requires both theoretical and
computational methodologies. In this talk, after giving a brief
introduction, I'll concentrate on the normal form computation using
computer algebra systems.
Recently, it has been noticed that the conventional or classical normal
form (CNF) can be further simplified, leading to the simplest normal form
(SNF). Such a further reduction results in a powerful method, which is
particularly useful for analyzing higher dimensional and/or higher order
nonlinear dynamical systems. However, the computation of the SNF is much
more involved than that of the CNF. Thus, the most important step in the
SNF computation is, based on Lie algebra, to develop efficient
computational methods.
By appropriately combining theoretical and computational approaches, we
have developed an efficient, recursive method for deriving the algebraic
equations at each order. Moreover, several specific techniques have been
established for computing the SNF associated with a number of
singularities. Lie algebra is used for theoretical proofs, while
algorithms and Maple programs are developed for explicitly computing the
coefficient of the SNF and the associated nonlinear transformation, which
greatly facilitates applications in solving real problems.
Refreshments will be served in N620 Ross at 12:00p.m.
ABSTRACT: Once more, the peak phenomenon! The peak set of a
permutation \pi in the symmetric group S_n consists of all 1 In our self-contained approach, there is a particular interest in inner
products in P_n, arising from the ordinary multiplication of permutations.
Peak counterparts of several results on the descent algebra D_n will be
presented, based on the fact that P_n turns out to be a left ideal of D_n.
This includes combinatorial and algebraic characterizations of P_n, the
basics of peak Lie idempotents, and a number of observations on the
structure of P_n and some sub-algebras.
Enough information will be provided to transfer these results to the
setting of Stembridge's peak algebra, by duality.
ABSTRACT: Inside a proton-exchange-membrane fuel cell, condensation of water vapor may occur and subsequent removal of liquid drops is of primary interest. In this talk, we will discuss the motion of a liquid drop on a solid surface driven by gas flows. In particular, we will discuss the behaviour of the liquid-gas-solid three-phase contact point and related modelling issues. We will then outline a front-tracking approach for computing the motion of a two-dimensional drop, based on Peskin's immersed boundary method. We will also present a simple analytical approach to handle the singularity at the contact point and to model the dynamical contact angle. Numerical results will be presented.
Please see M.W. Wong for Abstract information.
ABSTRACT: We discuss how to establish differential equations for general orthogonal polynomials and compare this with Sturn-Liouville systems. We also discuss the connection with the Coulomb gas problem concerning the equilibrium position of N charged particles in an external field. Finally the role of discriminants is discussed and we show how to use them to give closed form expression for the energy of the system at equilibrium. q-analogues may be mentioned.
Refreshements will be served at 3:30p.m. in N620 Ross.
ABSTRACT: A mathematical difficulty that arises in the linear, inviscid theory of waves in shear flows is that a singularity is generally present if there is a point where the mean flow velocity is equal to the phase speed of the perturbation. In the critical layer centred upon such a point, incident waves may be absorbed by the mean flow and, when nonlinear effects are included in the governing equations, wave breaking and reflection sometimes occur. There are several atmospheric phenomena which are known to result from wave-mean-flow interactions. Among these is the quasi-biennial oscillation (QBO) which is observed in the tropical stratosphere and is the focus of my investigations at present. In my talk, I shall describe numerical and asymptotic studies of the nonlinear evolution of wave packets at critical layers.
Refreshments will be served in N620 Ross at 3:30p.m.
ABSTRACT: If G and H are graphs, then G \preceq H if there is an
edge-preserving vertex-mapping, or homomorphism, from G to H. The
quasi-order relation \preceq on the class of finite graphs gives rise to
an order relation in a natural way, calledthe colouring order, written C.
The order C has many intriguing properties; for example, Hedrlin proved in
the 1960's that it is universal: every countable order embeds in C as a
suborder. Hedrlin's proof is long and makes heavy use of category theory.
In a recent tour de force, Nesetril discovered a shorter combinatorial
proof of Hedrlin's theorem.
A retract of a graph is an endomorphism f that is idempotent: f^2=f.
Retracts of graphs and other structures have been widely studied in
semigroup theory, going back to Howie's pioneering work on the idempotents
of the full transformation semigroup. The natural order on retracts is
defined by f \le g if fg=gf=f. While the natural order is a familiar tool
in algebraic semigroup theory, it has only recently attracted the
attention of the graph homomorphism community. After giving some
background on the colouring order C and the natural order, we will see how
these two orders are related. In particular, we will prove that the
natural order on the retracts of the infinite random graph embeds C and so
is universal.
ABSTRACT: I will present the contents of a paper under the above title by the eminent Russian mathematician V.I. Arnol'd.
ABSTRACT: Much of the recent literature on option valuation has successfully applied Fourier analysis to determine option prices. However, most of these numerical methods can be both slow and inaccurate in computation. We propose a classical statistical technique--saddlepoint approximations method for fast and accurate computation of European option prices. The method is applicable to pricing European options whose returns processes are developed in a general equilibrium model with stochastic volatility and stochastic interest rates. The model is calibrated for the $S\&P$ 500 index, and we show that the saddlepoint approximations methodology is accurate and easily implementation.
Survey Paper requirement for Master's students.
Reminder: Master's Mathematics students are expected to attend the
talk.
ABSTRACT: The peak algebra $\Pi$ was introduced by J. Stembridge
in his development of enriched $P$-partitions. It is a Hopf subalgebra of
the quasisymmetric functions $\Qsym.$ The Hopf structure has become
important in connecting $\Pi$ to the enumeration of chains in Eulerian
posets.
We will describe the structure of $\Pi,$ showing it to be a free
polynomial algebra, a cofree graded coalgebra, and a free module over
Schur's $Q$-function algebra. These results mirror results on the
structure of $\Qsym$ and its relationship to the symmetric functions. We
introduce a new basis of {\em monomial peak functions} for $\Pi$ which
behaves much like the monomial basis for $\Qsym.$ For example, the
stucture constants relative to this new basis count quasi-shuffles of {\em
peak compositions.}
By duality, our results have implications for the algebra of
chain-enumeration functionals on Eulerian posets. Earlier joint work with
L. Billera and S. van Willigenburg identified the ${\bf cd}$-index as the
dual basis to Stembridge's basis of fundamental peak functions. Here we
find that the monomial basis for $\Pi$ is dual to N. Reading's
Charney-Davis index, which appears to be an Eulerian analog of the flag
$f$-vector.
A knowledge of quasisymmetric functions and flag $f$-vectors is useful
but not required for this talk.
ABSTRACT: We consider M-estimators for partly linear models with possibly dependent observations such as those from a longitudinal study. We approximate the nonparametric function in the model by a regression spline and show that any M-estimation algorithm for the usual linear models can be used to obtain consistent estimates of the semiparametric model and valid large sample inferences on the linear components without any specification of the error distribution and the covariance structure. Included as special cases are the analysis of the conditional mean and median functions for longitudinal data and for certain spatial data. Advantages of this approach from both the theoretical and practical points of view are discussed in the talk.
ABSTRACT: Data clustering is the unsupervised process of classifying patterns into groups, aiming at discovering structure which is hidden in a data set. Applications in various domains often lead to very high-dimensional data. Clustering such high-dimensional data sets is a contemporary challenge. Successful algorithms must avoid the curse of dimensionality but at the same time should be computationally efficient. The dissertation plans to develop a neural network architecture and related algorithms based on the neural dynamics to provide a solution to the challenging high-dimensional clustering problem.
The syllabus for this exam is available for inspection in N519 Ross.
ABSTRACT: A short survey will be provided for the recent development in the theory and applications of reaction diffusion equations with both retarded arguments and non-local spatial interactions. Models arising from structured populations with spatial dispersal will be discussed, and some new approaches and results regarding traveling waves will be reported together with their implication for biological invasion and range expansion.
ABSTRACT: A concept of equation morphism is introduced for every
endofuctor $F$ of a cocomplete category $\Ce$. Equationally defined
classes of $F$--algebras for which free algebras exist are called
varieties. Every variety is proved to be monadic over $\Ce$, and
conversely, every monadic category is equivalent to a variety. And the
Birkhoff Variety Theorem is proved for ``\Set--like'' categories.
By dualizing, we arrive at a concept of coequation such that
covarieties, i.e., coequationally specified classes of coalgebras with
cofree objects, precisely correspond to comonadic categories. Natural
examples of covarieties are presented.
ABSTRACT: Fulton's Universal Schubert polynomials represent general
degeneracy loci for maps of vector bundles with rank conditions coming
from a permutation. The Buch-Fulton Quiver formula expresses this
polynomial as an integer linear combination of products of Schur
polynomials in the differences of the bundles. We present a positive
combinatorial formula for the coefficients. Our formula counts sequences
of semi-standard Young tableaux satisfying certain conditions.
This is joint work with Anders Buch, Andrew Kresch and Harry Tamvakis.
ABSTRACT: Prediction of the nonlinear response of a dynamical
system is a crucial step in many science and engineering applications. For
example, in the study of nonlinear aeroelasticity, understanding the
nonlinear behavioir of aircraft structures will lead to more efficient and
safe design of aircraft wings and control surfaces. Traditionally,
mathematical theory and numerical simulation have been successfully
applied to study the response of nonlinear dynamical systems. In this
approach, a mathematical model is developed and the system parameters must
be known. In some practical applications, only the dynamic response due to
a given excitation is available. The recorded nonlinear response is
usually noisy, nonstationary, and may have high dimensional dynamics.
Consequently, the traditional approach may be difficult to deal with these
practical problems.
In this talk, we propose to analyze the dynamics from data instead of
using mathematical equations and numerical simulations. An expert data
mining system (EDMS) is developed, in which a short term data is taken as
input to EDMS. The output of EDMS provides a prediction of the long term
dynamic behavior and it can also extract important features of the
corresponding nonlinear response. The key modules in the proposed EDMS
include artificial neural networks, nonlinear time series models and
filtering techniques. Applications of the proposed EDMS to simulated data
and real experimental data from nonlinear aeroelastic systems modeling a
two degree of freedom airfoil oscillating in pitch and plunge will be
reported.
ABSTRACT: The theory of quasigroups and loops is a fairly young
discipline which takes its roots from geometry, algebra and combinatorics.
In geometry, it arose from the analysis of web structures; in algebra,
from non-associative products; and in combinatorics, from Latin squares.
Today it has applications in many different parts of mathematics and
physics (algebraic nets, differential geometry, designs theory (Steiner's
systems), coding and encoding, cryptography, graph theory, ...)
Any associative quasigroup is a group and vice versa. Therefore the
identities different from associativity are of the interest. We shall say
a few words about the law of mediality ab . cd = ac . bd and relevant
identities.
Refreshments will be served at 2:30p.m. in N620 Ross.
ABSTRACT: In the first edition of his book "The theory of groups of finite order" (1897) Burnside wrote: " No simple group of odd order is at present known to exist. Also there is no known simple group whose order involves fewer than three different primes." These statements were to lead to two of the most important results of the theory of finite groups. The second of these was proved by Burnside--the "p^a q^b theorem": "Every group of order p^a q^b (p,q primes) is solvable" (while the first one, the "odd-order problem", had to wait for another 60 years). The "p^a q^b theorem" together with Burnside's Conjugacy Class Theorem will be presented with full proofs - assuming the theory of characters.
ABSTRACT: Multilevel models are models specifically geared toward the statistical analysis of data that have a hierarchical or clustered structure. Such data arise routinely in various fields, for instance in educational research, where pupils are nested within schools, family studies with children nested within families, medical research with patients nested within physicians or hospitals, and biological research, for instance the analysis of dental anomalies with teeth nested within different persons' mouths. This paper will explain the theory aspects of the two-level regression model, methodology, development, accuracy of the parameters estimates, when a normal distribution is assumed for the dependent variable. Then an illustration of these will be performed.
ABSTRACT: Hopf algebras are a natural setting for the study of many
combinatorial problems, while quasi-symmetric functions play an important
role as generating functions that encode information about the objects
being studied and as a source of Hopf morphisms that translate problems
from one area to another.
We introduce quasi-symmetric functions, comparing them to the
better-known symmetric functions, and describe three important Hopf
algebras which they form. In this we are aided by an association of
quasi-symmetric functions with partially ordered sets, which allows us to
describe their Hopf algebra structure in terms of operations on partially
ordered sets.
ABSTRACT: When one buys a standard savings bond, it is implicit that the holder may redeem the issue (ie: get their money back) at any time without penalty. Fixed annuities are financial instruments manufactured by U.S.-based insurance companies that are very similar to a typical savings bond (at least in the pre-retirement accumulation phase). However, fixed annuities have liquidity restrictions that prevent their redemption for some period of time. As such, it is expected that a fixed annuity with these restrictions should have a higher guaranteed rate of return than a comparable savings instrument that can be redeemed any time. This talk endeavours to mathematically quantify this yield premium subject to a particular set of liquidity restrictions.
Seminar requirement for Masters students. Reminder: Master's Mathematics students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: Mixed-effects models, allow us to model hierarchical data, and flexibly represent the variability structure that is induced by the clustering of the data. First, a mixed model formulation will be presented, followed by some background and historical information on the best linear unbiased predictors (BLUPs) for random effects. The concept of the bootstrap will then be presented and the usefulness of this technique will be discussed in a multilevel setting. The application of the bootstrap techniques to mixed effects models would benefit researchers when model assumptions do not hold (in situations when we don't have normality), or when the sample size is very small and estimation of fixed and random effects is problematic. Since in multilevel data we have a certain hierarchical structure, the bootstrap should produce samples that mimic that hypothetical distribution from which we obtained our observed multilevel data. Different bootstrap methods (parametric, semi-parametric, nonparametric) in multilevel models will be presented followed by applications and simulations of these techniques.
Survey Paper requirement for Master's students. Reminder: Master's Mathematics students are expected to attend the talk.
ABSTRACT: The $L^p$ norms of Marcinkiewicz integrals with rough kernels of functions $f$ with respect to certain weights are estimated in terms of the $L^p$ norms of the functions $f$ with respect to maximal functions of the weights. Some applications will be described. (This is joint work with Professor Yong Ding of Beijing Normal University and M. W. Wong of York University.)
ABSTRACT: I will discuss stochastic resonance, the coupling of noise and periodic forcing, as it occurs in climate cycles. Specifically, I will highlight how it is the driving force behind the 100 000 yr. glacial cycle, the 1500 yr. North Atlantic thermal/circulation cycle and how it factors into the El-Nino/Southern Oscillation (ENSO).
ABSTRACT: I will be discussing existence of solutions to the martingale problem for an elliptic operator in nondivergence form. One of two main theorems shows that continuity of the coefficents and boundedness of the drift of the operator is a sufficient condition for such existence and the other, utilizing the Girsanov theorem, shows that existence of the solution to the martingale problem for a process with drift follows from such existence for a process without drift, if the operator is uniformly elliptic.
Seminar requirement for Masters students. Master's Mathematics students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: This is the third and the last, but not the least, of a series of talks on heat equations on Riemannian manifolds. We give in this talk an existence and uniqueness theorem for Cauchy problems of heat equations on homogeneous spaces such as $S^2=SO(3)/SO(2)$ with initial data in the space of hyperfunctions.
ABSTRACT: We give a theorem on the existence and uniqueness of solutions for Cauchy problems of Heat equations on compact and connected Lie groups. Since the Cauchy data are in the category of hyperfunctions, our result may be considered to be in the most general form for theorems of this kind. We give as applications several results in harmonic analysis such as the Bochner-Godement type theorem and the Schwartz type kernel theorem.
ABSTRACT: We present two results in this talk. The first is on a relaxation of the growth condition in time for the uniqueness of solutions of the Cauchy problem for the heat equation on a complete Riemannian manifold $M$ with ${\rm{dim}}(M)=n$ and ${\rm{Ric}}(M)\geq -K$ for some $K\geq 0.$ Then we give an integral representation for every positive solution of the heat equation on the manifold. These results extend the corresponding ones obtained by Cheng, Li and Yau in [1] and Li and Yau in [2]. This is joint work with M. W. Wong.
[1] S. Y. Cheng, P. Li and S. T. Yau, On the upper estimate of the heat
kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981),
1021-1063.
[2] P. Li and S. T. Yau, On the parabolic kernel of the Schr\"odinger
operator, Acta Math. 156 (1986), 153-201.
ABSTRACT: We will prove the existence of the Perron-Frobenius eigenvalue in semiprimitive matrices and discuss its applications to the classification of Cartan matrices.
ABSTRACT: In this survey paper, we present a data analysis of quality of life study associated with MA.5 clinical trial, which was conducted by the National Cancer Institute of Canada. This trial involved patients with the positive-node breast cancer who were treated by two adjuvant chemotherapies. The objective of this study is to discover statistical evidence regarding whether or not there is a global difference between these two chemotherapies in the aspect of patients' quality of life. Because of patients' dropouts, the collected data is unbalanced with unequal numbers of repeated measures for each patient. This paper discusses three statistical techniques that are suitable for the global comparison under different types of missing patterns. The standard analysis and the growth curve model approach are applied under missing pattern of MAR, and a semi-parametric method approach is employed under informative missing pattern. We find that the three approaches give a consistent conclusion. That it, the two chemotherapies have no statistically significant difference for the benefits of quality of life.
Survey paper requirement for Masters students. Reminder: Master's Mathematics students are expected to attend the talk.
ABSTRACT: I will begin with a brief review of Circulant Matrices, and then show how Circulant Matrices can be used to find the zeros of low degree polynomials. The idea is to construct a Circulant Matrix with a specified characteristic polynomial. The roots of the polynomial thus become the eigenvalues, which are trivially found for circulant matrices.
Seminar requirement for Masters students. Master's Mathematics students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: The proposal discusses motivation for developing a numerical integration method for general nonlinear and non-Gaussian state-space models based on the best quadrature formulas. We review relevant literature concerning especially the Gaussian quadrature approach. We present a general framework in which a rough idea with regard to the thesis research is proposed.
ABSTRACT: In this talk I will discuss the projection estimator of the covariance function of a stochastic process, which can be viewed as a special case of the MINQUE estimator. I will consider the special case of the isotropic stationary stochastic processes, which gives rise to an estimator with particularly simple properties. Finally, I will present some ongoing work - an application to the problem of estimating the volatility of commodity futures curves.
ABSTRACT: We investigate a dynamical portfolio selection problem
when the appreciation rates are not directly available. For this setting,
optimal strategies require estimation of the appreciation rates of stocks
from historical data, and the estimates depend on prior distribution on
parameters. We propose some original filters for the appreciation rates,
in particular, for a hypothesis that takes into account correlation with
trading volume. We evaluate this hypothesis and other hypothesis
numerically by applying it to a generic optimal investment problem for a
real stock market. We discovered that the Gaussian hypothesis is not the
best one and that the performance of a standard Merton's strategy can be
improved via including trading volume into consideration.
ABSTRACT: Circulant matrices are of the form:
A B C D
D A B C
C = C D A B
B C D A
where the elements of each row of C are identical to those of the previous row but are moved one position to the right and wrapped around. They have some properties in common with Permutation matrices and Fourier matrices. This talk will be an introduction to these matrices and the relationship between them.
Seminar requirement for Masters students. Master's Mathematics students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: Existence and uniqueness of both strong and weak solutions to stochastic differential equations will be discussed and a proof of strong existence and uniqueness will be detailed. The connection between these solutions to SDEs and the solution to the local martingale problem will be discussed. More specifically, the weak solution will be reformulated as a solution to the local martingale problem and a proof that the strong solution implies a solution to the local martingale problem will be given.
Seminar requirement for Masters students. Master's Mathematics students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: Recently, credit risk has been attracting a great deal of attention within financial industry. Several models have been proposed to quantify credit risk. This study overviews the three most widely used credit risk models: the credit migration approach(CreditMetrics), the option pricing approach(KMV) and the actuarial approach(CreditRisk+). In addition, detailed analysis is done on simulated data and numerical results discussed to show that how CreditMetrics methodology is used for measuring credit risk at portfolio level.
Seminar requirement for Masters students. Master's Mathematics students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: This SIAM Visiting Lecture features examples of geometry's dominating influence in the automotive manufacturing process. The lecture begins with the design and manufacture of sheetmetal components that motivated advances in mathematical applications for Computer Aided Design. After a discussing the mathematics developed for the geometric aspects of this manufacturing process, the lecture examines an application of the same mathematics to robotics. The next topic relates the previous geometric constructions with the analysis of automotive components for fatigue, stress and strain. The lecture ends with the award winning video tape "Ballet Robotique".
ABSTRACT: Nonlinear dispersive waves in rods have been the subject of many studies. In particular, nonlinear axisymmetric waves that propagate axial-radial deformation in circular cylindrical rods composed of a homogeneous isotropic material have been considered by many authors. Here, we mention in particular three related pieces of work by Wright (1982, 1985) and Coleman and Newman (1990). Wright (1982) seems to have been the first to take into account the full nonlinearity for such a problem. In a sequel paper, Wright (1985) considered traveling waves in a rod composed of an incompressible hyperelastic material. He pointed out the existence of a variety of types of waves. In particular, he conjectured that sharp crested solitary waves can arise. This is one type of solitary waves with a first-order derivative discontinuity at the wave peak. Here, we shall call them solitary shock waves. Coleman and Newman (1990) derived the one-dimensional rod equation for a general incompressible hyperelastic material. Their work was concerned with smooth solutions only. Here, we shall resolve Wright's conjecture by showing that solitary shock waves can indeed arise in a Mooney-Rivlin rod. The explicit solution expressions for these waves and the physical existence conditions are also obtained.
References:
Coleman, B.D. and Newman, D.C. 1990 On waves in slender elastic rods.
Arch. Rational Mech. Anal. 109, 39-61.
Wright, T. 1982 Nonlinear waves in rods. In Proc. IUTAM Symp. on
Finite Elasticity (ed. D.E. Clarkson and R.T. Shields). The Hague:
Martinus Nijhoff.
Wright, T. 1985 Nonlinear waves in rods: results for incompressible
elastic material. Stud. Appl. Math. 72, 149-160.
ABSTRACT: In this talk some inverse problems of reconstruction the boundary, initial condition or lower odder term for the boundary value problem of heat equation are investigated. The radial basis functions are used to solve corresponding direct problems. Some numerical results show the efficiency of this method.
ABSTRACT: In this talk we will first consider Ising models on non-Euclidean lattices (including hyperbolic lattices). We will show that the models exhibit two phase transitions. This contrasts with the Euclidean case where the models have only one phase transition. We will also review results for percolation on these lattices. If time permits, we will discuss some new results for self-avoiding walks on hyperbolic lattices. Part of the results is a joint work with N. Madras.
ABSTRACT: After a quick overview of Loday's new "types of algebras", I will concentrate on the most recently defined "dendriform trialgebras" (there will be more scary names than this, but we will only be doing elementary algebra throughout). The free such trialgebra is based on the space of all rooted trees (Loday and Ronco). I will then describe several examples of trialgebras, all with a combinatorial flavor, and a general construction, which links trialgebras to one of Rota's favorite notions: Baxter operators. The punch line of the talk will involve quasi-symmetric functions, for a change.
Luiz Marcio Cysneiros, Univeristy of Toronto, will speak on "RE for the Health Care Domain" from 11:00a.m. to 12:00p.m. in N638 Ross.
ABSTRACT: There are several different approaches to elicit requirements, each one has its strengths and weaknesses. However, in many different domains these methods and techniques not always can be taken to the word. Health care domain is one of these domains. It is a complex domain with many subtleties, together with many social, political and legal issues that have to be taken into account. Moreover, non-functional aspects like safety, security and privacy are crucial to be dealt with. This work brings some of the lessons learned in more than six years working in several hospitals and laboratories. It will present many elicitation techniques that had to be adapted in order to comply with the constraints imposed by several peculiarities intrinsic to this domain. It also points out some pitfalls that must be avoided during the requirements engineering process, regardless the approach one uses.
ABSTRACT: The talk is devoted to the description of discrete, Brownian and fractional (B,S,X)-securities markets and some pricing formula with respect to these models. By (B,S,X)-securities markets we mean (B,S)-securities markets in random environment X. Firstly, we give some survey of results connected with pricing formula for classical discrete and Brownian (B,S)-securities markets, and for fractional (B,S)-securities markets. Secondly, we consider some results concerning stochastic models for interest rates, including one-factor models, some two-factor models and three-factor models, and pricing formula for them. Also, we consider stochastic models for interest rates with jumps. And finally, some pricing formula for discrete, Brownian and fractional (B,S,X)-securities markets are considered.
ABSTRACT: In this talk we introduce the notion of verbal and marginal subgroups of a given group, with respect to a variety of groups. Also the concept of varietal Schur-multiplier of a group will be defined and then we discuss some of their properties.
ABSTRACT: The goal of the talk is to introduce some recent progress by the speaker and his students/collaborators in the area of data analysis, in order to seek feedback from the statistics group.We first give a short introduction to the connection between the global dynamics and congitive tasks such as associative memory and pattern recognition of neural networks. We then focus on the clustering problem for data sets in high dimensional feature spaces, and we describe a new neural network architecture, recently designed in collaboration with Y. Cao, and report some numerical simulations as well as comparisons with other clustering algorithms. The talk shall keep the technical details to minimal.
ABSTRACT: We describe a method of lines (MOL) technique for the
simulation of taxis-diffusion-reaction (TDR) systems. These time-dependent
PDE systems arise when modelling the spatio-temporal evolution of a
population of organisms which migrate in direct response to e.g.
concentration differences of a diffusible chemical in their surrounding
(chemotaxis). Examples include pattern formation and different processes
in cancer development. The effect of taxis is modelled by a nonlinear
advection term in the TDR system (the taxis term).
The MOL-ODE is obtained by replacing the spatial derivatives in the TDR
system by finite volume approximations. These respect the conservation of
mass property of the TDR system, and are constructed such that the MOL-ODE
has a nonnegative analytic solution (positivity). The latter property is
natural (because densities/concentrations are modelled).
The MOL-ODE is stiff and of large dimension. We develop integration
schemes which treat the discretization of taxis and diffusion/reaction
differently (splitting). This is achieved through operator
(Strang-)splitting. To solve the resulting non-stiff subproblem we employ
strong-stability preserving (SSP) Runge-Kutta methods and for the stiff
subproblems a linearly implicit W-method with approximate matrix
factorization is applied. Optimal SSP Runge-Kutta methods with number of
stages larger than their order are used because of their favourable
positivity preserving properties.
Numerical experiments confirm the broad applicability of the splitting
schemes for the solution of TDR systems and show the effect of using SSP
Runge-Kutta methods with many stages.
ABSTRACT: I will be proving the following: For every separable metrizable space X, we have that the small inductive dimension of X = the large inductive dimension of X = the covering dimension of X.
Seminar requirement for Masters students
Reminder: Master's Mathematics students are expected to attend the
talks of other students. Documented evidence at 6 such talks is expected.
Attendance sheets can be picked up from N519 Ross.
ABSTRACT: In Utility Theory the concept of representable linear order is fundamental. Recall that a chain $(L,\prec)$ is {\em representable} if it can be order-embedded into ${\mathbb R}\,$. We classify representable lexicographic products of chains in terms of some characteristics of their factors. Then, we extend the notion of representability as follows: define a chain $(L,\prec)$ to be $\alpha$-{\em representable} ($\alpha$ being any ordinal) if it can be order-embedded into the lexicographic power ${\mathbb R}_{lex}^\alpha$. The least ordinal $\alpha$ such that $(L,\prec)$ is $\alpha$-representable is its {\em representability number} repr$(L)\,$. We will prove that repr$({\mathbb R}_{lex}^{\,\alpha}) = \alpha\,$ for all ordinals $\alpha\,$. We will also state a conjecture where we classify chains into those which are $\alpha$-representable for some $\alpha < \omega_1$, and those which show some kind of uncountable behavior (long, Aronszajn, etc.).
ABSTRACT: In this talk we will review the definition of crystal bases of an irreducible module of a symmetrizable Kac-Moody Lie algebra and show how they can be used to answer questions about the representation theory of this algebra. We will talk about describing these bases in terms of Young Tableaux.
ABSTRACT: In genetic linkage analysis, families are examined to
find patterns of genetic marker transmissions that coincide with patterns
of disease. This analysis can identify chromosomal regions likely to
harbour genes that increase susceptibility to disease. However, for
complex diseases, it is likely that multiple genes act on different
symptoms or comorbid conditions to increase risk of disease. Although
clinical data on symptoms is normally available, it is not clear what is
the best way to use such information.
I will describe an extension to "model-free" linkage analysis methods
that uses multivariate clinical data. This method adaptively finds the
individual characteristics that are associated with the strongest evidence
for linkage, through the use of classification and regression trees
(CART). Bootstrapping can be used to stabilize cutpoint selection, and
cross-validation optimizes the size of the regression tree. The methods
will be illustrated on a data set of 68 families ascertained to have at
least two cases of asthma. Due to the adaptive nature of the algorithm,
results must be interpreted cautiously and validated in independent
data.
ABSTRACT: I will discuss aspects of the number-theoretic work of Lagrange, Gauss, Dirichlet, and Riemann.
ABSTRACT: A problem that arises from cryptography is to study the primality of the number of points on an elliptic curve over a finite field. We shall show how sieve methods give some information on this problem. We will begin by describing the cryptographic motivation and background.
ABSTRACT: After a brief tour of the development of the Nullstellensatz through David Hilbert in the 1890s and Emmy Noether in the 1920s, we follow up on a fairly recent idea by Bill Lawvere and exhibit the categorical essence of the Nullstellensatz via Birkhoff's Subdirect Representation Theorem for general algebras. Hence, we shall prove this theorem for quite general categories without requiring any particular expertise by the audience.
ABSTRACT: Dynamic geometry is the exploration of geometric
relationships by observing geometric configurations in motion. Although,
in most classrooms, these configurations are constructed by the students
themselves, sketches pre-constructed by the teacher or downloaded from a
website can also be used. There is a continuing discussion in the
educational technology community about whether it is better to give
students powerful general-purpose programming and construction tools or to
have them interact with pre-constructed, interactive models. Some strongly
support student constructions because they believe that students develop a
deeper understanding of the object by explicitly connecting the parts.
Others believe that pre-constructed models are valuable as learning tools
because ability to recognise connections between geometric objects is a
necessary stage before students can effectively carry out many
constructions.
In an attempt to inform this debate my research investigated the
benefits and limitations of using JavaSketches--web-based, interactive,
dynamic geometry sketches--with senior high school students in geometric
activities related to proof. In this seminar I will present some of my
findings, and demonstrate the features of pre-constructed sketches that
helped students focus attention on mathematically meaningful details. By
reflecting on my results in relation to the extensive research on
Geometer's Sketchpad and Cabri, I will attempt to characterise situations
in which pre-constructed sketches can play a purposeful role in the
geometry program.
ABSTRACT: In 1989, Jones used spin models to construct link
invariants and braid group representations. In particular, the Jones
polynomial can be obtained from the simplest spin model called the Potts
model. In 1995, Bannai and Bannai constructed four-weight spin models,
which are generalization of spin models that give invariants for oriented
links.
In an important paper by Jaeger, he discovered the first connection
between spin models and Bose-Mesner algebras. In 1995, Jaeger and Nomura
showed that there is a Bose-Mesner algebra attached to every spin model.
A similar result for four-weight spin models also holds. Bose-Mesner
algebras, which are equivalent to association schemes, have connections to
a vast number of combinatorial objects such as designs, codes and distance
regular graphs. Hence Jaeger's result draws the attention of
combinatorialists to spin models.
Let $\schur$ denote the Schur product of two matrices, that is, $(A
\schur B)_{ij} = A_{ij} B_{ij}$. Given a matrix $C$ in $M_n({\Bbb C})$, we
define endomorphisms $X_C$ and $\Delta_C$ on $M_n({\Bbb C})$ by
\begin{equation*} X_C(M) = CM, \quad \Delta_C(M) = C \schur M.
\end{equation*} A pair of matrices $(A,B)$ from $M_n({\Bbb C})$ is called
a Jones pair if $X_A$ and $\Delta_B$ are invertible and they give a
representation of braid groups using Jones' construction. (We save the
details for the talk.)
Jones pairs provide a generalization of both spin models and
four-weight spin models. Moreover, Jones pairs also have Bose-Mesner
algebras attached to them. In this talk, we give an introduction of Jones
pairs and discuss the Bose-Mesner algebras associated with spin models,
four-weight spin models and Jones pairs. In particular, we describe a
family of five Bose-Mesner algebras attached to each Jones pair.
This is joint work with Chris Godsil and Akihiro Munemasa.
ABSTRACT: A motivation for the use of entropy to express the outcomes of a probabilistic experiment, then will give a proof of the Uniqueness Theorem for entropy.
Seminar requirement for Masters students. Reminder: Master's Mathematics students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: The higher Bruhat orders B(n,d) were introduced by Manin and Schechtman as generalizations of weak Bruhat order: B(n,1) is weak Bruhat order on the symmetric group S_n, and B(n,0) is the Boolean lattice Q_n. The higher Stasheff-Tamari posets are defined as partial orders on the set of triangulations of a cyclic polytope. There are reformulations of these families of posets which make the connections between them more obvious: a convex-geometric reformulation of the higher Bruhat orders due to Kapranov and Voevodsky, and a combinatorial reformulation of the Stasheff-Tamari posets due to myself. I will discuss various maps between higher Stasheff-Tamari posets and higher Bruhat orders which specialize to familar maps between S_n and Q_{n-1}, and from S_n to planar binary trees.
ABSTRACT: Let Q be a n x n matrix over a field of K. Consider
the K-algebra
-1 -1
P(Q) = K[x ,x , ...,x , x ] with x x = q x x . Basic properties,
including
1 1 n n i j ij j i
the simplicity, of P(Q) will be discussed.
Seminar requirement for Masters students. Reminder: Master's mathematics students are expected to attend the talks of other students. Documented evidence at 6 such talks is expected. Attendance sheets can be picked up from N519 Ross.
ABSTRACT: Consider a continuous natural exponential family on the line for which we test whether the parameter is equal to a fixed value or not. For this we can use either the two sided UMPU test or the generalized likelihood ratio test. For a given level of acceptance the results can coincide. This is what happens for a Gaussian family with known variance, for a gamma family and -although this is more difficult to check- for an inverse Gaussian family. We prove that, conversely, if the tests coincide for any level of acceptance, then the family is of one of the three types above. The proof is achieved by the cancellation of a determinant of order 6, leading to a differental equation. Computations have needed Mathematica..
ABSTRACT: I will discuss the connection between the Banach-Tarski Parado and measure Theory. I will sketch a proof of the Banach-Tarski Paradox which states the following: If A and B are bounded subsets of R3 with non-void interiors, then A and B are equivalent by finite decomposition.
Seminar requirement for Masters students.
Reminder: Master's Mathematics students are expected to attend the
talks of other students. Documented evidence at 6 such talks is expected.
Attendance sheets can be picked up from N519 Ross.
ABSTRACT: An introduction to derivative securities (specifically, options) will be given before motivating the 2-D analog to the Black-Scholes equation which is used to price derivative securities. The resulting linear convection-diffusion type equation will then be transformed into a PDE with a nonlinear source term in order to implement the penalty method used to handle the early exercise feature of American options. The nonlinear PDE is then solved via a finite element method.
Seminar requirement for Masters students.
Reminder: Master's Mathematics students are expected to attend the
talks of other students. Documented evidence at 6 such talks is expected.
Attendance sheets can be picked up from N519 Ross.
ABSTRACT: Proofs of Wedderburn's Little theorem and Jacobson's Commutativity theorem will be presented.
Seminar requirement for Masters students.
Reminder: Master's Mathematics students are expected to attend the
talks of other students. Documented evidence at 6 such talks is expected.
Attendance sheets can be picked up from N519 Ross.
ABSTRACT: A number of combinatorial objects--labeled trees, allowable pairs of input-output permutations for priority queues, factorizations of an n-cycle into transpositions, and parking functions--are all enumerated by the same formula: (n+1)^(n-1). A series of related bijections have been constructed between two or more of these. Here we introduce and prove a direct bijection between priority queue allowable pairs and labeled trees that has an additional property not present in previous direct bijections: our bijection maps increasing sequences in the output permutation of the priority queue allowable pair to leaves in the tree. This gives us a full understanding of the underlying tree structure of priority queue allowable pairs. For instance, we could use this understanding to construct the analogue of a Prufer code for allowable pairs. This is joint work with A. Yong.
ABSTRACT: We obtain a selection criterion based on the residual log-likelihood, RIC, for regression models including classical regression models, Box-Cox transformation models, weighted regression models, and regression models with ARMA errors. We show that RIC is a consistent criterion, and for all four models simulation studies indicate that RIC provides better model order choices than AIC, AICC, FPE, R_{adj}^2 and C_p, except when the sample size is small and the signal-to-noise ratio is weak, and when none of the criteria perform well. Monte Carlo results also show that RIC is superior to the consistent criterion BIC when the signal-to-noise ratio is not weak, and it is comparable to BIC when the signal-to-noise ratio is weak and the sample size is large.
ABSTRACT: We briefly review the modern history of immersion-theoretic topology, beginning with the seminal work of S. Smale in the U.S.A. and finishing with the work of the Leningrad school of topology in Russia, especially the work of M. Gromov. We discuss also the interesting role that jet spaces of maps played in the formulation of results for the Leningrad school. The presentation is non-technical with emphasis on historical developments. We conclude with excerpts from a video, due to W. Thurston, which illustrates Smale's theorem on "turning a sphere inside out".
ABSTRACT: The theory of stochastic differential equations in
Hilbert and Banach spaces is applied extensively in stochastic modelling
to describe random phenomena studied in science and engineering.
Consequently, existence and uniqueness of the solutions and their
measurability are crucial.
Consider the following stopped semimonotone nonlinear integral
equation:
for $t\leq \tau(\omega)$, in a real separable Hilbert space $H$ where
$\tau$ is a stopping time, $U(t,s)$ is an $H$-valued contraction-type
evolution operator, and $f$ is a semimonotone nonlinear function in $H$.
We show that the above integral equation has a unique measurable strong
solution $u$. We also prove known results under weaker assumptions.\[
u(t,\omega) = U(t,0) u_0(\omega)
+ \int_0^t U(t,s) f(s,\omega,u(s,\omega)) ds
+ V(t,\omega)
\]
ABSTRACT: Given a sequence of points which lie over a
differentiable 2-manifold M embedded in R^3, we propose a method which
allows the construction of approximating or interpolating curves which
respect intrinsic geometry of the manifold. In particular we desire exact
representation of geodesic arcs and of a class of spiral-like curves which
orthogonally project to geodesic arcs on the manifold. In the particular
case when M is a sphere applications exist in the domain of geological and
geographical mapping, for instance the creation of topographical contour
lines or isotherms, and in the field of video production where it is
desirable to have smooth camera trajectories interpolating fixed camera
positions.
For a differentiable Riemannian manifold M and a point x\in M, it is
well known that there is a neighborhood V of x \in M such that for every
y\in V, there exists a geodesic in M connecting x with y. In the case that
M is compact , the Hopf-Rinow theorem tells us that the later statement is
true for every y\in M. This is the starting point. In the case that the
points to be approximated or interpolated lie in the manifold, natural
generalizations of the de Boor, de Casteljau, and Aitken algorithms are
presented for the construction of analogs of B'ezier, B-spline, or
Lagrange interpolatory curves which lie in the manifold M. The algorithms
are constructed from the corresponding Euclidean algorithms by
substituting line segments with geodesic arcs. For instance, the de
Casteljau algorithm, as we know, constructs, for given functions of a
given stage, a line segment which joins values of the functions evaluated
at a parameter t. The value of the function of the next stage is taken to
be that point which has a relative distance of $t$ along the constructed
line segment. To generalize this to the setting of a differentiable
manifold, one simply substitutes euclidean line segments with geodesic
arcs. For the case in which the points to be approximated or interpolated
lie above the manifold, the Euclidean algorithms are similarly modified by
substituting line segments with arcs of the spiral-like curves mentioned
earlier.
One would like to know that the standard repertoire of CAGD techniques
for manipulating curves also have natural generalizations to the manifold
context. Early results indicate that this is not the case. Many standard
techniques depend explicitly on the affine and algebraic structures of
Euclidean three space for which there is no apparent counterpart in the
manifold context. In particular there seems to be no succinct
representation of the manifold curves by basis functions which has the
functionality of the familiar representations of the B'ezier, B-spline,
and Lagrange curves. Thus, there are many aspects of the theory of
manifold curves that remain open questions.
Although curves constructed lack the versatility of their Euclidean
cousins, they do however possess the important characteristic that they
respect aspects of intrinsic geometry of the manifold. In particular,
should control/interpolation points lie on a geodesic, then so also does
the constructed curve. Constructed curves also satisfy a convex hull
property in which the description of the Euclidean convex hull is slightly
altered to provide a notion of geodesic convex hull.
In order to effectively construct the proposed curves, an efficient
algorithm for the calculation of geodesics is presented and applied to the
construction of these curves on well-known 2-manifolds. In the case of a
sphere, examples are presented which demonstrate the construction of
curves which lie over the sphere.
The methods introduced have been touched on by Shoemake. He uses
properties of quaternion arithmetic to describe curves on the quaternion
sphere. Leversly and Ragozin using different methods speak of Lagrange
interpolation in a differentiable manifold. Walker describes the
techniques in the special case when the manifold M is a sphere.
ABSTRACT: Adaptive designs have often been proposed as a way
sequentially to assign more patients to better treatments, based on
outcomes of previous treatments in clinical trials.
In this talk, we discuss the optimal adaptive designs for two-arm
clinical trial based on the following optimality criterion: for a fixed
power of a test and fixed signifant level, minimizes the expected number
of treatment failures.
An optimal sequential design is proposed and is compared to some other
adaptive designs.
ABSTRACT: In teaching and thinking about least-squares regression
we tend to use the geometry of ‘subject space' where points are variables
and the axes represent subjects.
However, many important concepts for applications, including paradoxes
like Simpson's paradox, are perhaps more easily understood in ordinary
‘variable space' where points are observations and axes, variables.
The simple geometry of the ellipse seems to be the key to visualizing
regression concepts in variable space.
This talk will explore ways in the connections between data ellipsoids
and confidence ellipsoids can be used to provide clearer insights into
properties of regression that are frequently misunderstood.
ABSTRACT: Consider an incompressible fluid in which a thin elastic membrane is immersed. The motion of the membrane is then coupled with the motion of the fluid and it is necessary to solve the incompressible Navier-Stokes equations with a singular forcing term along the membrane. This must be coupled with boundary conditions requiring that the elastic and fluid forces balance across membrane, and that the membrane moves with the local fluid velocity. Peskin's immersed boundary method is a popular approach to solving such problems, especially in biofluid dynamics applications. Rather than attempting to follow the deforming geometry with a moving grid, the fluid dynamics equations are solved on a uniform Cartesian grid, with a forcing term given by spreading the singular force to the grid using discrete delta functions of finite width. This method is robust but typically only first order accurate due to this spreading. I will discuss an approach to achieving better accuracy by using the elastic force to impose jump conditions directly on the pressure in the process of solving the elliptic equation that arises in a projection method for the Navier-Stokes equations. Membranes with mass can also be handled, in which case there is an additional inertial force that must be included in the force balance.
Professor LeVeque will give a two-week short course on numerical conservation laws at The Fields Insitute.
ABSTRACT: We define the Hilbert function of a finite set of points in projective n-space. We then state the result of Geramita, Maroscia and Roberts which characterizes the Hilbert functions of points in terms of Macaulay's O-sequences. For every possible Hilbert function, they construct a set of points having that Hilbert function. Their constructions, called k-configurations, have several properties of interest in themselves. For example, their Hilbert function, minimal free resolution and the degrees of each point are easily determined. We generalize k-configurations to constructions which preserve these properties. Furthermore, since O-sequences are defined only in terms of the binomial coefficients, we are able to use a generalized Pascal's triangle to generalize Macaulay's O-sequences. We use this generalization to characterize the Hilbert functions of our genereralized k-configurations.
ABSTRACT: Data Mining or Knowledge Discovery in Databases (KDD) is a non-trivial process of identifying valid, useful and understandable patterns in large Databases. Clustering large disk-resident data sets is one of the fundamental issues in Data Mining. In this talk, I will give an overview of classical clustering algorithms. Then I will introduce advanced clustering algorithms suitable for large disk-resident data sets proposed by computer scientists. Frameworks proposed by Statisticians will be discussed as well. I will also present our clustering approaches: clustering by shrinking and clustering by orthogonal regression. Clustering by Shrinking (CLUES) is a novel approach for clustering based on local gravitational fields. Simulation studies and data analysis have demonstrated high accuracy and fast convergence. The idea of clustering by orthogonal regression is based on the idea of fitting several hyperplanes to the data space to capture the existing trends. No assignment of response variable is required in this algorithm.
Refreshments will be served in N620 Ross at 10:00a.m.
ABSTRACT: Pro-p groups interact with many areas of mathematics. They arise naturally as Galois groups, open subgroups of algebraic and Lie groups over local fields and groups of automorphisms of trees. They are used to study number theory, infinite groups, finite p-groups, and more. In addition, many tools are used to study pro-$p$ groups such as $p$-groups, Lie theory, Galois theory, and even model theory. I will present a short overview of the theory of pro-$p$ groups and in particular "small" pro-$p$ groups. Then I will show how the notion of Hausdorff dimensions can be used to study the subgroup structure of pro-p groups. This will be demonstrated in the particular example of the Nottingham group. The main idea is to use Lie theoretic methods. The main theorems I am going to present are joint with Klopsch, Shalev, and Zelmanov. This talk is self-contained and there is no need to know what a pro-p group is or what Hausdorff dimension is.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: In this paper, we introduce a new model for locating
competitive facilities, which captures key aspects of competitive
facility location models such as market expansion and market
cannibalization, while retaining computational solvability that allows for
realistic-size applications. The model presents a direct generalization
for the class of traditional spatial interaction models.
The new model is formulated as an integer program with a ''nasty''
non-linear objective function. We develop a novel approach to solve
problem by using the TLP linear approximation that yields an efficient
piece-wise linear approximation within a specified relative error bound.
We develop a tight worse case error bound for the greedy heuristic, which
is somewhat unexpected in view of the non-linearity of the underlying
model.
We also develop efficient computational approaches - both exact (Branch
and Bound) and approximate ( optimal with controllable error bound ),
allowing for solution of fairly large-scale models.
Refreshments will be served in N620 Ross at 10:00a.m.
Olga Vechtomova will be speaking on "Using word collocation to improve performance of information retrieval systems" from 3:30p.m. to 4:30p.m. in Vari Hall 3000.
ABSTRACT: The user's task of finding useful information from large
databases of unstructured text is difficult. Current information retrieval
systems return large numbers of documents, usually only a small proportion
of which are relevant to the user's need. One of the causes of poor search
performance is the quality of the user's search formulation.
In this seminar I will present techniques I have developed for automatic
enhancement of query formulations. The developed algorithms analyse
long-span co-occurrence patterns of query terms in the documents and add
significantly associated words (collocates) to the query formulation. The
aim is to discover semantically related words, frequently used together in
the same context, and add them to queries in order to retrieve more relevant
documents.
I will describe the developed techniques, their integration with an
information retrieval system, performance evaluation and comparison with
related approaches.
Refreshments will be served in N620 Ross at 4:30p.m.
ABSTRACT: The Wishart distribution is the distribution of the
estimate of the covariance parameter in the multivariate Gaussian model.
In many statistical problems, the test statistic is a fairly complicated
function of this estimate and it is impossible to find its exact
distribution. This distribution therefore has to be approximated using the
moments and inverse moments of the Wishart distribution.
The Wishart distribution can be defined generally on symmetric cones.
We donote by K the orthogonal subgroup of the group of automorphisms of
the cone. We will show how, in this general case, all K-invariant moments
for the Wishart and its inverse can be found using spherical polynomials.
In the particular case of the cone of Hermitian matrices, all moments
(not necessarily K-invariant) can be obtained using the structure of the
symmetric group.
ABSTRACT: The generalized Chinese restaurant process is a sequential random seating rule which can be thought of as a method for inducing a random partition of the integers. Such processes in fact can be used effectively as priors in Bayesian non and semiparametric problems. A careful study of the resulting posteriors under this rich class of priors points to several new and versatile computational algorithms, including the iid GWCR (generalized weighted Chinese restaurant) algorithm as well as other Monte Carlo methods, some Gibbs, and some iid in nature. Examples applied to mixture models, model selection and semiparametric mixture models will be discussed. The method also applies to partial likelihoods such as multiplicative intensity models used in event history analysis, including proportional hazards and spatial marked point process models. Extensions to dependent nonhomogenous Poisson processes can also be handled.
Refreshments will be served at 10a.m. in N620 Ross.
Jianguo Lu, University of Toronto, will speak on "Reengineering of Database Applications to EJB based Architecture" from 2:00p.m. to 3:00p.m. in N638 Ross.
ABSTRACT: The advent and widespread use of Enterprise JavaBean (EJB) technology not only demands more reengineering support for legacy database applications, but also changes the reengineering practice. Initiated from our experience of reengineering database applications to EJB based architecture, this talk addresses two challenges in the mapping between database queries and EJBs. The first is to map a SQL statement to the equivalent EJB client code when the enterprise beans exist. The second is to generate enterprise beans from the set of legacy SQL statements when the EJB architecture does not exist in the first place. We propose the EJB-SQL mediator to solve the first problem, and a view selection algorithm to solve the second one.
Refreshments will be in N620 Ross at 3:00p.m.
ABSTRACT: Maria G. Agnesi was the first woman to publish a mathematical book, in 1748. Two years later she was offered a lectureship in calculus at the University of Bologna. In this paper I argue that Agnesi was able to establish herself as a legitimate scholar and as an authoritative member of the Italian intelligentsia because of particular cultural and religious conditions. I relate her success, and the salient aspects of her mathematical style, to more general features of eighteenth-century Catholic didactics. In other contexts, like England and France, the practice and teaching of advanced mathematics were being otherwise effectively gendered.
ABSTRACT: The Schur functions form a fundamental basis for the
symmetric functions and the coefficients in the expansion of the homogeneous
basis in the Schur basis count column strict tableaux. This is easily shown
using a combinatorial rule for adding a row to the partition indexing the
homogeneous functions due to Pieri (1860-1913).
The operation of adding a column to the indexing partition of a
homogeneous function is a new recurrence which we refer to as the 'ribbon
rule' because the terms may be explained combinatorially as attaching
ribbons to partitions. This operation generalizes by refining the formula
and it leads to several fascinating open questions.
Refreshments will be served in N620 Ross at 2:30p.m.
ABSTRACT: A numerical model for a planar dc discharge with a semi-transparent anode positioned between two symmetrical cathodes was developed. The kinetic and Poisson's equations were solved iteratively by means of the direct Monte Carlo method. The approach employed the basic physical quantum and semi-classical collision models and may be applied for different elements constituent the gas. The behaviour of the electron spectra dependent on the distance from the cathode and on transparency of the anode were studied numerically. The spectra and the magnitude of differential fluxes may be taken into account in plasma diagnostic methods in plasma assisted chemical vapour deposition, in reactive ion etching and in gas laser applications.
ABSTRACT: In this expository talk, we discuss attempts of understanding the structure of compact topological spaces in combinatorial terms. Starting from the more than century old Cantor-Bendixson analysis of dispersed subsets of the reals and the Stone duality linking compact spaces with Boolean algebras, we describe recent results and methods which connect as diverse areas as topology, Boolean algebras, logic and functional analysis.
Refreshments will be served in N620 Ross at 3:00p.m.
ABSTRACT: In this talk, we present an exact solution approach for mixed integer programming. The approach integrates three bodies of research; namely, Lagrangean relaxation, interior point methods and branch-and-bound. The integration is novel as it is one of the first attempts to efficiently use interior-point methods within branch-and-bound. Traditionally, simplex-based methods are used instead. We describe the solution approach in an application to the capacitated arborescence problem (CAP). The CAP is a difficult graph theory problem defined on a directed graph where all the nodes of the graph except the root have a demand quantity to be delivered from the root. The CAP determines a minimum cost directed spanning tree with a specified number of branches and such that the total demand on each branch does not exceed a certain capacity limit. It has several graph theoretic problems as special cases such as the minimum arborescence problem and the minimum spanning tree problem; and has applications in routing and network design. We decompose the problem using Lagrangean relaxation by relaxing the capacity requirements. We compute the Lagrangean bound using a new interior-point cut and column generation algorithm. Primal and dual interior point methods are used to compute the Lagrangean multipliers after cut and column generation, respectively. The Lagrangean bound is embedded within branch-and-bound to find optimal solutions. A warm start strategy is used to recompute the Lagrangean bound after branching; and dominance tests are used to enhance the performance of the overall branch-and-bound method. We present promising numerical results for the CAP and extend the method to the case of the asymmetric vehicle routing problem.
Refreshments will be served at 2:30p.m. in N620 Ross.
ABSTRACT: While linear algebra is primarily concerned with solving linear equations, linear programming is related to solving both linear equations and inequalities. A linear programming (LP) problem is one of maximizing (or minimizing) a linear function of finite number of real variables, subject to a finite number of linear equations and inequalities. In this talk, I will present some of most beautiful results of LP theory including LP duality. Certain applications of LP will be looked at. We will also talk about two most popular LP algorithms. The simplex method is the classical LP algorithm which promises efficiency in practice. Yet the interior point method, which receives much attention only in the last 15 years or so, is the only LP algorithm which is efficient both in practice and in theory. It is also one of the most active research fields in optimization. I will present some basic results from these areas. Towards the end of the talk, I will present some results of computational complexity of solving LPs.
Group Meeting will be in N620 Ross at 4:00p.m.
Refreshments will be served in the Coxeter Library at 4:00p.m.
ABSTRACT: The theory of continuous-time generalized AR(1) processes is developed for modelling non-normal time series with equally or unequally-spaced time observations, which may be count data or positive-valued data. Such a process is a Markov process represented as the sum of a dependent term (involving extended thinning operation) and an innovation term. The stationary distribution of a continuous-time generalized AR(1) process can have support on non-negative integers or positive reals; common distributions such as Poisson, negative binomial and Gamma are included. In this talk, we will introduce the continuous-time generalized AR(1) process and its properties, as well as the characterization of its stationary distribution. The modelling procedure will be illustrated by two real cases with count data and positive-valued data respectively.
Refreshments will be served in N620 Ross at 3:30p.m.
ABSTRACT: I will discuss a number of classical properties of the Schur function basis revealing the importance of combinatorics in symmetric function theory. I will then consider a filtration of the symmetric function space, and introduce new symmetric functions appearing, from their combinatorial properties, to be the Schur functions of the subspaces associated to this filtration. I will finish by discussing the connection between these new symmetric functions and Macdonald polynomials.
ABSTRACT: Semidefinite relaxations arising from combinatorial
optimization are often too large to be handled by the classical interior
point algorithms. Reformulation of this class of problems to eigenvalue
optimization is well known.
An efficient technique in nondifferentiable optimization is the analytic
center cutting plane method (ACCPM). In practice, when applying the ACCPM to
minimize a maximum eigenvalue function, the oracle returns a semidefinite
cut if the maximum eigenvalue has multiplicity more than one. We apply the
ACCPM algorithm to the eigenvalue optimization arising from the
combinatorial problems. Extension of the ingredients of ACCPM to the cone of
positive semidefinite matrices is discussed. We present numerical results
for the max-cut problem.
Refreshments will be served in N620 at 10:30a.m.
ABSTRACT: This talk is about means to construct simple associative algebras, and to use these associative algebras to obtain simple Lie algebras.
Refreshments will be served at 3:30p.m. in N620 Ross.
ABSTRACT: Consider P(N), the power-set of the natural numbers, as a Boolean algebra. A subset I of P(N) is an ideal if it is closed under taking subsets and finite unions of its elements.
Question: How does a change of the ideal I affect the change of its quotient algebra P(N) /I?
I will survey what is known about this question, by considering it in three different contexts.
Refreshments will be served in N620 Ross at 4:00p.m.
ABSTRACT: A new approach of estimating parameters in multivariate models is introduced. A fitting function will be used. The idea is to estimate parameters so that the fitting function equals or will be close to its expected value. The function will be decomposed into two parts. From one part, which will be independent of the mean parameters, the dispersion matrix is estimated: This estimator is inserted in the second part which then yields the estimators of the mean parameters. The Growth Curve model will illustrate the approach.
ABSTRACT: The lecture concerns the analysis of a mathematical model
which incorporates both malaria epidemics and human population genetics of
the sickle-cell gene. By looking at crucial gene dynamics factors such as
the fitness of the sickle-cell gene, the fitness cost and the selection
strength, the impact of malaria epidemics on the maintenance of the
sickle-cell gene in a population with malaria being prevalent will be mainly
discussed.
As the dynamics of the coupled model can be separated into two
time-scales with a faster time-scale for the epidemics and a slower
time-scale for the change in gene frequencies, the geometric theory of
singular perturbations and dynamical systems techniques are the main tools
for our analysis. Some numerical simulations of the model will also be
presented in the lecture.
Refreshments will be served in N620 at 10:30am.
Eric S. Wheeler will give a talk on "Software and Why it is Hard to do Well" from 3:30p.m. to 4:30p.m. in N638 Ross.
... and that leads into a discussion of some of the things I have done in industry with software compaines and also into ITEC and related matters.
Refreshments will be served in N620 Ross at 3:00p.m.
ABSTRACT: The study of o-minimal expansions of the reals has witnessed a remarkable progress since 1991, when Wilkie proved the o-minimality of $(R, exp)$ (the expansion of the reals by the exponential function). This was a major breakthrough and has motivated further research on this topic. Since the definable sets in an o-minimal expansion share many of the nice geometric properties of semi-algebraicsets, the quest for analogies between the semi-algebraic case and the o-minimal case has guided many aspects of this research. In this talk, I will focus mainly on these aspects, presenting some results of my monograph "Ordered Exponential Fields", The Fields Institute Monograph Series, vol. 12, AMS Publications). In particular, I will discuss construction methods for non-archimedean exponential fields. The basic tool for this is valuation theory, and I will explain how one can develop an exponential analogue for all important notions and methods of the theory of real places and real closed fields. I will show how to use this abstract machinery to give concrete constructions of models with interesting algebraic and model theoretic properties.
Refreshments will be served in N620 Ross at 4:00p.m.
ABSTRACT: The Good-Lagrange formula for the inversion of multivariated formal power series is a central tool in the field of enumeration of tree-like structures. In this talk, we will present an elegant bijective proof of this formula and some application to the random generation of some families of tree-like structures.
Joint work with Michel Bousquet, Gilbert Labelle and Pierre Leroux, LaCIM.
ABSTRACT: We derive approximations to the first three moments of the conditional distribution of the deviance statistic, for testing the goodness of fit of generalized linear models with non-canonical link, by using an estimating equations approach, for data that are extensive but sparse. A supplementary estimating equation is proposed from which the modified deviance statistic is obtained. An application of a modified deviance statistic is shown to binomial and Poisson data. We also conduct a performance study of the modified Pearson statistic derived by Farrington and the modified deviance statistic derived in this paper, in terms of size and power, through a small scale simulation experiment. Both statistics are shown to perform well in terms of size. The deviace statistic, however, shows an advantage of power. Two examples are given.
Refreshments will be served in N620 Ross at 10:00a.m.
ABSTRACT: In this talk, I shall present the nonlinear stability of the front traveling waves to a time-delayed reaction-diffusion equation. For a front wave with a big speed, it is proved to be asymptotically stable, if the initial perturbation decays fast in an exponential spatial rate. Numerical simulations are carried out too. The weighted energy method plays a key role in the proof.
Refreshments will be served at 2:00p.m.
ABSTRACT 1: The need for mass information is evident in all aspects
of everyday life. To help students gain insight and interest in database
technology and how it can help them in their research and problem solving,
this seminar will focus on 4 aspects of Database technology and Information
systems development.
1. The relationship between Database technology and data processing
2. Method and Procedure of design and development the computer data
processing and data management systems using databases
3. The selection of data management systems
4. Project regarding Web-cooperation Research Center in China in which
people from universities that is located around the world work on Internet
and sharing data, resources and results of design.
ABSTRACT 2: The architecture distributed Oracle database system This
seminar hopes to help familiarize the audience with the operation,
management and maintenance of Oracle systems.
Contents of Introduction:
1. Oracle software architecture and composition
2. Oracle INSTANCE which includes:
3. Computing model supported by Oracle
4. Installation of Oracle product
5. Highlights of Oracle feature
POSTPONED
ABSTRACT: We briefly review the modern history of immersion-theoretic
topology, beginning with the seminal work of S. Smale in the U.S.A. and
finishing with the work of the Leningrad school of topology in Russia,
especially the work of M. Gromov. We discuss also the interesting role that
jet spaces of maps played in the formulation of results for the Leningrad
school. The presentation is non-technical with emphasis on historical
developments. We conclude with excerpts from a video, due to W. Thurston,
which illustrates Smale's theorem on "turning a sphere inside out".
ABSTRACT: Scaling the proposal distribution of a multi-dimensional random walk Metropolis algorithm in order to maximize the efficiency of the algorithm has been studied for some collections of target distributions. In particular, Roberts, Gelman and Gilks (1997) proved a weak convergence result as the dimension n, of a sequence of target densities converges to infinity, if the target density has a symmetric product form and is ‘smooth' enough. In this talk, I will discuss some background and some recent work on extending this kind of results to algorithms with other target distributions, such as continuous densities with jumps.
ABSTRACT: We derive a lattice model for a single species population
living in infinite patches in one dimension space connecting locally for
diffusion, and with age structure and a fixed maturation period. Under the
assumption that the death and diffusion rates of the matured population are
age independent, we show that the dynamics of the matured population is
governed by a lattice delay differential equation with global interaction.
We study the well-posedness of the initial value problem and we obtain the
existence of monotone traveling waves with wave speeds $c>c_*$. We also show
that the minimal wave speed $c_*$ gives the asymptotic speed of propagation,
and depends on the maturation period and the diffusion rate of matured
population monotonically.
This is joint work with Huaxiong Huang and Jianhong Wu.
ABSTRACT: Most clustering algorithms do not work efficiently for
data sets in high dimensional spaces. Due to the inherent sparsity of
data points, it is not feasible to find interesting clusters in the
original full space of all dimensions, but pruning off dimensions in
advance, as most feature selection procedures do, may lead to
significant loss of information and thus render the clustering results
unreliable.
In a recent project with Jianhong Wu, we propose a new neural network
architecture Projective Adaptive Resonance Theory (PART) in order to
provide a solution to this feasibility-reliability dilemma in clustering
data sets in high dimensional spaces. The architecture is based on the
well known ART developed by Carpenter and Grossberg, and a major
modification (selective output signaling) is provided in order to deal
with the inherent sparsity in the full space of the data points from
many data-mining applications. Unlike PROCLUS (proposed by Aggarwal et
al. in 1999) and many other clustering algorithms, PART algorithm do not
require the number of clusters as input parameter, and in fact, PART
algorithm will find the number of clusters. Our simulations on high
dimensional synthetic data show that PART algorithm, with a wide range
of input parameters, enables us to find the correct number of clusters,
the correct centers of the clusters and the sufficiently large subsets
of dimensions where clusters are formed, so that we are able to fully
reproduce the original input clusters after a reassignment procedure.
We will also show that PART algorithm is based on rigorous
mathematical analysis of the dynamics for PART neural network model (a
large scale system of differential equations with singular
perturbation), and that in some ideal situations which arise in many
applications, PART does reproduce the original input cluster
structures.
Refreshments will be served in N620 Ross at 2:30p.m.
ABSTRACT: Last September we have discussed: Catalan number that classically enumerate Dyck paths, and investigate the quotient ring $R_n$ of the ring of polynomials $\Q[x_1,x_2,\ldots,x_n]$ over the the ideal generated by non-constant quasi-symmetric polynomials. We expected the dimension of $R_n$ to be the $n$th Catalan number. Now we can prove it all, and say even more...
ABSTRACT: As is well known, Leibniz held that matter is infinitely divided, but refused to countenance infinite number on the ground that it contradicts the part-whole axiom. This position has often been criticized as inconsistent: Leibniz would have done better, his critics allege, to have embraced actually infinite number, as would Bolzano and Cantor in the nineteenth century. Here I defend Leibniz's conception of the infinite as actual but syncategorematic. I show that this is compatible with the infinite division of matter as he conceived it, whereas Cantor's transfinite is not. I show how Leibniz could give a consistent, recursive construal of infinite aggregates that would not commit him to infinite cardinals; and how he might reasonably have rejected Cantor's w, as well as the latter's famous "diagonal argument".
ABSTRACT: We explain what is a pre-Lie algebra, where this notion comes from and try to show why it may be of interest. We will give examples of combinatorial and algebraic constructions of pre-Lie algebras.
ABSTRACT: Recent third order approximate inference procedures have
evolved either from the saddlepoint method (Daniels 1954, 1987;
Barndorff-Nielsen & Cox 1979; Lugannani & Rice 1980) or from the direct
analysis and Taylor series expansion of log density functions
(Barndorff-Nielsen 1986; Fraser 1990; Fraser & Reid 1995). Tail
probabilities at a particular parameter value can be obtained by these
methods; however, it is generally restricted to the canonical parameter of
the exponential model or to the location parameter of the transformation
model. For more general models, Barndorff-Nielsen (1986) proposed a method
which depends on the existence of an ancillary statistic.
A more general approach to obtain third order approximate inference for
any scalar parameter will be presented and will emphasize on applying the
method to various areas.
ABSTRACT: Let n be a positive integer, w a group word. Consider the class of all groups G satisfying the identity w^n=1 and having the verbal subgroup w(G) locally finite. We show that in many cases this is a variety.
All are Welcome!