Centre de recherches mathématiques, Annual Report 1995-1996

Part 3

Research Programs

Coherent states and generalizations
Twareque Ali

Twareque Ali's research during the past six years, much of it carried out in collaboration with J.-P. Antoine, Université Catholique de Louvain, Belgium and J.-P. Gazeau, Université Paris-7, has been in the area of square-integrable group representations and their applications to coherent states, quantization and wavelet analysis.

Basically, these collaborators have developed a far-reaching generalization of the concept of square-integrability of a group representation, based on a homogeneous space of the group. This notion unifies all the different approaches to the study of coherent states for locally compact groups found in the literature, the theory of frames in Hilbert spaces and the analysis of signals using wavelet-like transforms. It also make a connection between the method of geometric quantization and quantization using coherent states and Berezin's technique of quantization on Kähler manifolds. As a consequence of their generalized theory of square-integrability, it is possible to derive wavelet-like transforms using almost any locally compact group. These transforms can then be used to analyze functions on the group space or on homogeneous spaces a fact that can be exploited to analyze signals realized as these functions. In quantization problems, their techniques allow for the use of general vector bundles, so that internal symmetries can be accounted for, and furthermore, many of the obstructions inherent in standard geometric quantization are eliminated.

Numerical methods for nonlinear hyperbolic systems
Paul Arminjon

P. Arminjon's main research interest lies in the domain of numerical methods for nonlinear hyperbolic systems, with applications to engineering problems in gas dynamics and electrostatics/electrodynamics. For transonic/supersonic compressible flows, P. Arminjon studies, with his collaborators, A. Dervieux and M.C. Viallon, the design and numerical analysis of high accuracy finite difference, finite element or finite volume methods, and their application to typical flows arising in aerodynamics and aerospace engineering. Recently, they have obtained a family of non-oscillatory 2nd-order accurate schemes based on:

i) a 2-step finite volume Richtmyer-Galerkin scheme with a TVD-controlled artificial viscosity, ii) a TVD-controlled barycentric combination of the Richtmyer-Galerkin and Osher's first order scheme, iii) a 2nd-order version of Osher's scheme using MUSCL-extrapolated, TVD-controlled, cell-interface flux values, and iv) a new finite volume extension, for 2-dimensional conservation equations, of the Nessyahu-Tadmor non-oscillatory 1-dimensional centred difference scheme.

In joint work with M.C. Viallon, they have recently proved the convergence of this latter scheme for a linear conservation equation, and they are presently extending the proof to the nonlinear case.

Nonlinear delayed equations
Jacques Bélair

Nonlinear dynamics gives an interpretation of complex changes in physiological rhythms (as bifurcations) when the values of the control parameters are modified. The theory leads to predictions for the possible behaviours in experimental settings and gives a unified explanation for the various regimes. Bélair's work has concentrated on nonlinear delayed feedback in control and in hormonal and neuromuscular system oscillations, stressing the role of the delay, the multiple feedback loops and the variable delays in the generation of periodic (oscillatory) or irregular behaviours.

Recent work has also applied a technique to detect the onset of single- or multiple-frequency periodic rhythms to 'simple' systems of artificial neural networks (of low dimension), as well as to a prototype of the simplest plausible oscillator in neuromuscular control. This same approach is currently used to design a controlled release drug administration system.

Algebraic transformation groups and algebraic geometry
Abraham Broer

Abraham Broer is interested in connections between algebraic geometry and representation theory. He studies, for example, nilpotent varieties and cotangent bundles of flag manifolds and common generalizations; algebraic properties like normality and rational singularities are established.

He proved a vanishing theorem for the higher cohomology of line bundles on the cotangent bundle of a flag manifold for the purpose of proving that the subregular nilpotent variety is normal. Applications are in rings of differential operators and representation theory of Hecke-algebras.

He recently obtained another vanishing theorem, this time of the Dolbeault cohomology of homogeneous vector bundles on flag manifolds, generalizing Borel-Weil's vanishing result for ordinary sheaf-cohomology. It is expected to have applications in algebraic geometry.

Control and nonsmooth analysis
Francis Clarke

An important question in control addresses the stabilization of a system through a feedback command. In particular the following question is central: given that the system is commandable, is there an associated feedback? This question has important consequences in applications of control systems. When the system is linear, it is a classical result that the commandability implies stabilization through feedback. In the nonlinear case, the question remained long unresolved until recently when F. Clarke, in collaboration with Yu. Ledyaev, E. Sontag and A. Subbotin, answered it positively. A key point in their analysis is a new definition of a system solution whenever the feedback command is a discontinuous function.

Shape optimization, intrinsic differential geometry and the theory of thin shells
Michel Delfour

The main theme of Michel Delfour's research program is the optimization with respect to the shape or the geometry of a domain on which one or a system of partial differential equations are defined. This is a central problem in optimal design (aeronautics, heat control, image processing, etc.). At the theoretical level it is necessary to introduce appropriate topologies on families of subsets to give a meaning to derivatives and problem formulations. These include those induced by the distance functions or families of functions parametrized by sets and embedded in a functional space. In particular the algebraic distance gives a powerful tool to do differential calculus on submanifolds. This yields a totally intrinsic approach to the theory of thin shells and extends the shape calculus to differential equations defined on these submanifolds.

Biorthogonal wavelets
Gilles Deslauriers

David Donoho has used dyadic interpolation to generate wavelets. There exists a relationship between splines and wavelets. It is then possible to construct compact support wavelets with a given number of vanishing moments that are orthogonal to wavelets obtained from spline functions. Gilles Deslauriers' research project pursues, in this context, a recent idea on the lifting of wavelets.

Estimation in statistics and in actuarial studies
Louis Doray

Louis Doray's research deals mainly with two themes. In general insurance, his interests lies in modelling damages that were incurred but not reported to the insurer (IBNR) using regression models, time series and compound Poisson processes. He studies various parameter estimators, the adjustment of the model and the reserve prediction for IBNR accidents.

In statistics, he is interested in families of discrete laws defined on the nonnegative integers whose probability function can be expressed recursively. Some of these functions do not have a closed form. The estimation of parameters by the maximum likelihood method is then very difficult. However the iterated weighted least square method gives very efficient estimators that are easier to calculate. Moreover a statistic to test the adjustment of the model to data can be easily obtained, as well as its asymptotic distribution. Tests differentiating the various members of the family are being analysed. Doray is also studying the problem of the explicative variables for these discrete law families.

Analysis of curve and surfaces
Serge Dubuc

Serge Dubuc's main goal has been the development of mathematical analysis for the design, construction and perception and for the study of various planar and spatial figures such as curves and surfaces. Two themes are under study. Iterative interpolation: simultaneously with other authors, Dubuc invented a new technique of interpolation, the iterative interpolation (or fractal interpolation) in one or several variables. For the case of one variable, this technique is very close to the theory of wavelets. Dubuc is planning to develop further the multidimensional inter polation on rectangular and triangular lattices. Many irregular surfaces obtained in this context are difficult to study. Analysis of fractal objects: the aim of this theme is a better understanding of the theory of fractional dimensions. One subject of interest is the errors made in calculating the dimension of a regular object. Dubuc hopes to determine the dimension of certain lattices of curves that way.

Stochastic processes in actuarial studies and in finance
Daniel Dufresne

Daniel Dufresne's recent research can be divided into the following three topics: (i)Asiatic options, (ii)properties of gamma laws and (iii)the application of the theory of martingales to the general principles of actuarial evaluation.

  1. An Asiatic option is on the average of the values of a financial title (e.g. a share of IBM). Mathematically, the problem consists in finding the law of the geometric brownian motion average. No exact form is known for this law. Dufresne obtained this year new results concerning a famous identity due to Bourgerol, closely related to the average mentioned above. A simple proof was found for this identity.
  2. New affine properties for gamma and beta laws were found. These results are important in the study of various stochastic difference equations that are used in actuarial studies and finance.
  3. One can show that actuarial gains and losses that are studied in actuarial practice are in fact differences of martingales. This leads to better understanding of the behaviour of the actuarial surplus or deficit in time and can even influence the way actuarial evaluations are done.

Fractional generalization of the KdV equation
Stéphane Durand

Using supersymmetry it is possible to generalize in a non-trivial way the Korteweg-de Vries equation (KdV) to an integrable system of two coupled differential equations (Mathieu). Knowing that the supersymmetry can itself be extended (parasupersymmetry and fractional supersymmetry (Durand, Vinet)), it is natural to look for generalizations to integrable systems of several coupled differential equations. The formalism of fractional superspace introduced by

Durand allows such a generalization in a natural way. This result is reached using the fractional extension of supersymmetry, the hamiltonian structure of the fractional pseudo-classical mechanics and the fractional generalization of superextension of Virasoro algebra (and/or its q-deformations).

Values omitted by classes of univalent functions
Richard Fournier

Richard Fournier and his collaborator (St. Ruscheweyh) are working at describing explicitly the values omitted by various normalized classes of univalent functions on the unit disk in the complex plane. It seems that these values might be described in simple terms by certain combinations of Taylor coefficients of the functions. Moreover it appears that the omitted values characterize, in a certain sense, various classes of univalent functions, for example the convex ones. This work had led to new inequalities on Taylor coefficients and the modulus of convex conformal transformations. It is hoped that these results can be used to solve some problems on homographic transformations of convex univalent functions.

Critical points of multivalued functionals
Marlène Frigon

The theory of critical points of univalent and continuously differentiable functionals and the multivalent analysis are two important and active topics in mathematics. Marlène Frigon's work is concerned with the development of the theory of critical points for multivalent functionals. This theory will then be applied to partial differential inclusions.

Image processing & target recognition applied to aerial surveillance
Langis Gagnon

Langis Gagnon and two students of Jiri Patera assess and devise new methods in image processing and target recognition for radar and infrared images. The goal is to develop a ship recognition system starting from a set of sensors mounted on an airborne platform. The algorithms studied here use various modern techniques of information processing like mathematical morphology, wavelets and artificial neural networks. Recent accomplishments include: (i) a study of a new method for reducing the speckle noise in images from synthetic aperture radar (SAR) in "strip-map" mode using the wavelet transform and (ii) the target segmentation in a SAR image in "spotlight" mode.

Approximation in complex analysis
Paul Gauthier

Walsh has shown that any continuous function on a curve without double points in the complex plane can be approximated by complex polynomials. Thomas Bagby, Aurel Cornea and Paul Gauthier have shown a similar result using harmonic polynomials. Any continuous function on a curve without double points in Euclidean space can be approximated by harmonic polynomials. We are working on determining whether a similar result holds for functions defined of hypersurfaces.

Applications of wavelet transform and artificial neural networks
Bernard Goulard and Jean-Marc Lina

Bernard Goulard and Jean-Marc Lina are currently in the last year of a three year R&D NSERC collaborative project whose purpose is to extend the capability of Atlantic Nuclear Services monitoring and diagnostic systems through research and development of its Artificial Neural Network (ANN) technology and through the introduction of wavelet transforms in its signal processing parts. First, in collaboration with Y. Bengio (Département d'informatique et de recherche opérationnelle, Univ. de Montréal) and a student, F. Gingras, they are putting the finishing touches to a modular ANN based on a "mixture of experts" to classify various regimes of a reactor. A gaussian modelling has been applied to the occurrence probability of data and to differentiation between typical and atypical data of a class. This "inference machine" has been successfully tested on both simulated and real (reactor) data. Second, mathematical properties of complex wavelets made explicit by J.M. Lina (symmetry minimizing the usual shift variance of real wavelets and complex nature making easier the use of the information carried by the phase to code transient signals) have been illustrated in two papers by J.M. Lina and two students, P. Drouilly and J. Scott. These properties have led them to extend the study of wavelets to the domain of 2-d signals, i.e. image processing (nonlinear regression based on wavelets, multiscale algorithms for digital imaging, multifractal analysis by wavelets). One of them (B.G.), in collaboration with R. Roy (Polytechnique) and a student (A. Qaddouri), is also investigating parallel iterative processes to solve the Boltzmann transport equations which govern neutron distribution in a reactor and their possible extension to other fields.

Symmetries and solutions of nonlinear systems
Michel Grundland

Michel Grundland's research in the last few years has dealt with symmetry-reduction methods and Riemann-invariant methods and their application to equations of nonlinear field theory, condensed matter physics, as well as fluid dynamics. The development of these methods has provided several new tools to study nonlinear phenomena in physics, especially those described by multidimensional systems of partial differential equations (pde) that were not solved by other methods (like inverse scattering). Grundland's research program can be divided into 4 projects:

  1. conditional symmetries for nonlinear pde systems;
  2. comparison of the various methods based on Lie groups used to study pde's;
  3. invariant solutions and partially invariant solutions of Navier-Stokes equations;
  4. multiple Riemann waves for quasilinear pde systems and their relation to symmetry reduction methods.

Classical and quantum integrable systems
John Harnad

During the past year, John Harnad's main research interest were all related to the modern theory of integrable systems. The topics studied were:

  1. isospectral deformations and classical R-matrices,
  2. isomonodromic deformations and applications and
  3. quantum integrable systems.

A recent work, in collaboration with A.R.Its, carries on the study of dual isomonodromic deformations but also initiates a new program relating the latter to computation of correlation functions in integrable quantum and statistical models and the spectral distributions of random matrices, in which a special class of Fredholm integral operators arise, whose Fredholm determinants are the correlation functions in question. These are computed through the Riemann-Hilbert problem "dressing method," adapted to the case ofisomonodromic deformations, leading to integral representations of importance in the calculation of asymptotics of such correlation functions. A key result derived in this work is the fact that the "dual" isomonodromic representations, deduced generally from the R-matrix structure, follow in this context from the invariance of the Fredholm determinant under Fourier transform of the integral kernel.

Geometry and physics
Jacques Hurtubise

Jacques Hurtubise' research work deals with geometrical and topological aspects of objects originating from mathematical physics. His projects are divided into two rather disjoint topics.

The first one studies the relationship between the solution spaces of several field equations of mathematical physics like those of the sigma model or Yang-Mills equations, and the functional spaces in which they lie. The questions are mostly topological in nature, like the proof of topological stability theorems. These theorems have been extended this year to the most general case known today. The solution spaces are here characterized as minima or critical sets of an action functional, and the techniques used in the proofs involve also analytic subtleties from the calculus of variations.

The second one addresses the algebro-geometric properties of completely integrable mechanical systems. An invariant has been recently introduced that allows for a measurement of the complexity of a large number of these mechanical systems; whenever this complexity is minimal, the system possesses very natural coordinate systems that seem to be related to its quantization.

Geometry of partial differential equations / quasi-exactly solvable systems
Niky Kamran

Niky Kamran's research deals with the properties of partial differential equations (pde) whose nature is essentially geometric. Some of the most important questions arising for pde's can be studied in a precise way in combining classical analytic techniques and powerful tools from differential geometry, differential topology and the theory of representation. Kamran's recent work has addressed global existence of variational principles, geometric integrability questions of hyperbolic equations, existence of conservation laws and singularity formation in solutions. Kamran has also contributed to the development of rigorous foundations for the theory of quasi-exactly solvable potentials in quantum mechanics using original cohomological methods together with fundamental theorems of the classical theory of invariants.

Boundary conditions for finite statistical models
Robert Langlands and Yvan Saint-Aubin

Robert Langlands and Yvan Saint-Aubin are trying to better understand the relationship between boundary conditions and partition functions of simple statistical models. Their aim is to obtain a description sufficiently precise to be able to obtain the partition function on a given (finite) lattice starting from the partition functions of two complementary sublattices. A first step has been accomplished by Langlands for the free boson theory: he constructed an application from the space of boundary conditions into the Hilbert space describing the model. An effort is now being made to construct a similar application for the Ising model. These problems are part of a larger program to define finite statistical models (i.e. with a finite number of degrees of freedom) that have a renormalization transformation and a non-trivial fixed point under this application. Such a family exists for percolation and the critical exponents for the simplest model in this family, calculated numerically, are very close to the critical exponents obtained through physi cal arguments.

Symmetry classes of polyominoes
Pierre Leroux

Polyominoes are important combinatorial structures for mathematical physics. They appear naturally in polymer models and the study of percolation. Recent work of the Bordeaux and Australian schools have given an enumeration with respect to area, perimeter and other finer parameters, for many classes of polyominoes having minimal convex properties. In a geometrical or combinatorial context, it is natural to consider convex polyominoes up to a symmetry or a rotation, i.e. as objects free to move in space. Pierre Leroux is currently working at enumerating them. This uses the study of orbits under the action of the dihedral group on convex polyominoes and, due to the Burnside lemma, the enumeration of various symmetry classes of polyominoes. Many of these classes are intimately related to certain classical families of discrete models in statistical mechanics. For example, the class of convex polyominoes with a diagonal symmetry is related to that of directed and convex polyominoes (or animals) with compact diagonal source.

Analysis of population genetic models
Sabin Lessard

Sabin Lessard's research interests include a wide variety of population genetic models and the concomitant evolutionary dynamics. His ultimate goals are: a) to explain the maintenance of variability in biological populations, b) to develop mathematical and statistical techniques to analyse population genetic structures, c) to deduce general evolutionary principles, and d) to study populations with complex interactions between individuals.

q-special functions / The Efimov effect
Jean LeTourneux

Most special functions of mathematical physics admit q-analogs, namely deformations involving a parameter q. Just as Lie algebras provide a unifying framework for discussing special functions, q-deformations of these algebras provide a unifying framework for discussing q-special functions. In collaboration with Luc Vinet (CRM) and Roberto Floreanini (Trieste), Jean LeTourneux carries out a systematic investigation of the quantum algebraic interpretation of the q-special polynomials encompassed in the scheme of Askey-Wilson polynomials.

According to the Efimov effect, a three-body system has an infinite number of bound states when it involves two-body interactions that marginally bind the two-body system. Formal proofs of this effect are too complex to provide any physical intuition. Simpler proofs, given for special cases within the framework of the Born-Oppenheimer approximation, break down as soon as one goes beyond the lowest order approximation. With Bertrand Giraud (Saclay) and Yukap Hahn (Univ. of Connecticut), Jean LeTourneux investigates a certain number of questions raised by this situation.

Optimal methods of statistical modelling
Brenda MacGibbon

Brenda MacGibbon's main research interest is the optimal estimation of constrained parameters and its application to both parametric and non-parametric models of real life problems. She is particularly interested in the use of tools from harmonic analysis such as Fourier and wavelet analysis in functional estimation, for example:

  1. often there is a need to consider discrete data as in binary 0-1 signals or in the count data observed in many problems in positron emission tomography;
  2. in random censorship-truncation models, in cancer trials, only incomplete data is available;
  3. with econometric data and many biometric applications, prediction is important;
  4. how may these techniques best be used for regression smoothing of binary data in small area estimation in complex sample surveys?

Numerical studies of wave perturbation interaction and shear flows in aerodynamics and geophysics
Sherwin Allan Maslowe

Chonghui Liu is currently simulating the transition toward turbulence of a limit layer. The important case is when the flow is subject to an adverse pressure gradient. This is the case, for example, in the neighbourhood of a flow stripping point around an incident wing profile. A spectral element method is used in order to combine the advantages of finite element methods (adaptability of the geometry) and the fast convergence of a spectral method. L. Campbell, a master's student, is currently studying the interaction of a forced wave packet of the Rossby type with the zonal shear flow in the framework of the beta plane often used in meteorology. This study will use asymptotic methods as well as finite differences. Its goal is to generalize the preceding studies by considering a wave packet instead of a normal mode. The recent analysis of the critical layer for a wave packet of Maslowe, Benney and Mahoney (1994) will play an important role here.

Nonseparable wavelets in 2-d
Michel Mayrand

Wavelets are used in signal processing in several ways (filtering, noise removal, compression, etc.). For 2-d signals (e.g. in imaging), tensor products of 1-d wavelets (separable wavelets) have been used until recently. However the most efficient wavelets in one dimension are those that exhibit symmetry properties. Since 2-d separable wavelets are not symmetric under the trivial transformations like rotation, it is natural to introduce new families of nonseparable wavelets. Michel Mayrand is currently working on defining new nonseparable wavelets and classifying their symmetry properties. As in the 1-d case, it will be useful to parametrize the 2-d wavelets according to their properties of orthogonality, of symmetry and continuity of their derivatives.

Non-crystallographic root systems
Jiri Patera

Jiri Patera's recent research has been devoted to two areas: (i)non-crystallographic root systems and (ii)deformations of semisimple Lie algebras and their representations. In collaboration with R.V.Moody (Alberta), he has laid down the mathematical foundations of non-crystallographic root systems, stressing their relationship with quasicrystals. Their consequences should be visible in publications over the next few years. A considerable effort was devoted to preparation and running of two events (jointly with R.V. Moody): a NATO Advanced Study Institute entitled "Mathematics of Aperiodic Long Range Order" and a semester program around the same subject (both held at the Fields Institute). He has also pursued the study and exploitation of simultaneous deformations of semisimple Lie algebras and their representations. The main tool here is the approach which he has invented recently, requiring that a fixed grading be preserved during the deformation.

Estimation of the precision matrix
François Perron

François Perron's research interests are related to decision theory and multidimensional analysis. His results are concerned with finding minimax estimators for the estimation of the average vector and the covariance matrix for a multinormal population. The idea underlying the finding of minimax estimators is the following. In general, an estimator never gives the exact value of the parameter it is supposed to estimate. There is always an error associated with the estimator precision and this error varies with respect to the value taken by the parameter to be estimated. For a given estimator, one can find the largest estimation error by varying the parameter on its domain. The minimax estimator is the one that gives the smallest maximal error. Krishnamoorthy and Gupta have tried to show, without success, that a certain estimator of the precision matrix was indeed minimax. In fact, they noticed through simulations that the result was plausible and formulated a conjecture that they believed to be true. In the paper On a Conjecture of Krishnamoorthy and Gupta, Perron has shown that the conjecture as stated is false even though the estimator of the precision matrix is indeed minimax. Perron's future projects will be related to Bayesian analysis and Monte Carlo simu lation methods.

Clones and their applications
Ivo Rosenberg

Ivo Rosenberg has carried on the study of clone lattices (in universal algebra and multivalent logics) mostly on finite universes. He has studied maximal subclones of clones of isotone operations with respect to a bounded order, and clones that are not finitely generated. The clone network can be partitioned in a countable set of intervals called monoidal. For a universe of three elements, Fearnley and Rosenberg have undertaken a classification of these intervals by their size (1, finite, countable or with the cardinality of the continuum). Rosenberg showed the natural correspondence between hyperalgebras on an universe A and the subclones of the clone of isotone operations on the universe of nonempty subsets of A ordered by inclusion. He has started the classification of maximal subclones of this clone for finite A. In this correspondence, hypergroups become particular semigroups and Rosenberg is currently applying the results of the theory of semigroups to hypergroups. With Hikita he has also worked on a general criterion for completeness of uniform delay operations.

Qualitative study and bifurcations in differential equations
Christiane Rousseau

One of the long-term goals of Christiane Rousseau's research program is the completion of the proof for the existence part of Hilbert's 16th problem for quadratic systems, i.e. to show that there exists a uniform bound for the number of limit cycles in a quadratic system. This project, initiated in 1991 with F. Dumortier and R. Roussarie, is progressing steadily. An important step made recently by Rousseau and H. Zoladek by exploiting simultaneously Khovanskii and Bautin's techniques for the centres and Roussarie's techniques for blowing up of families, allows one to hope for a complete solution in the coming three to five years.

All the techniques introduced here have an intrinsic interest going far beyond their application to the above problem. With Roussarie, Rousseau has applied some of them to the study of certain homoclinic loops in 3-dimensional space and their Ph.D. student, L.S. Guimond, is making further progress in that direction.

Another aspect of Rousseau's research project will be devoted to algebro-geometric methods applied to the study of polynomial vector fields. She is working on the problem of the centre (in collaboration with D. Schlomiuk) and on the geometric characterization of isochrone vector fields (with P. Mardeiç and L. Moser-Jauslin).

This study of polynomial vector fields has a direct impact on still another aspect: the study of singularities of vector fields of higher codimension (typically larger than or equal to 3). The bifurcations of these singularities are organizing centres of bifurcation diagrams occurring in many applied models.

Algebraic theory of discrete structures
Gert Sabidussi

Gert Sabidussi's research interests lie in the algebraic theory of discrete structures, in particular graphs, his two main research axes being two algebraic structures associated with graphs: graph symmetries as expressed by the automorphism group, and invariants of certain groups of linear transformations induced by the graph. Under the first heading (symmetries), his research deals with algebraic properties of several classes of highly symmetric graphs that have their origin in theoretical computer science where they model the interconnection networks in parallel computing. These models exist in large number, giving rise to an abundance of different algorithms for a given task. The algebraic theory aims at reducing this profusion by laying a theoretical basis for the design of general algorithms applicable to all interconnection networks with a sufficiently rich symmetry structure. Under the second title (invariants), his research is less oriented towards applications and addresses mainly the relationships between chromatic properties of graphs and the existence of certain types of invariants.

Biomathematics and sociolinguistics
David Sankoff

In biomathematics, David Sankoff works on algorithms for the analysis of DNA sequences and he has, within the context of the human genome project, extended this discipline to the development of methods for studying genome evolution resulting from the process of chromosomal rearrangement. This has resulted in the development of algorithms (in collaboration with John Kececioglu and Gopalakrishnan Sundaram) for sorting permutations using a small set of operations: reversals, transpositions, translocations. Sankoff and Vincent Ferretti study syntenic sets of genes in collaboration with Joseph Nadeau, a geneticist at Case Western Reserve, and several mathematics and statistics students. In phylogeny, Sankoff and Ferretti have developed a method of nonlinear phylogenetic invariants.

In sociolinguistics, David Sankoff directs a programme whose goal is a rigorous statistical methodology for the analysis of syntactic variation and phonology in spoken language, based on computerized transcriptions of corpora of free speech. With David Rand, he developed and distributed a software package (GoldVarb) for linguistic data analysis. His empirical interests include bilingual syntax, specifically methods for distinguishing alternating borrowing codes, and the study of particles of speech.

Local and global studies of analytic vector fields
Dana Schlomiuk

Using interdisciplinary methods, Dana Schlomiuk is constructing various tools adapted to the global analysis of polynomial dynamical systems in the plane. Joining concepts from algebraic geometry and the theory of bifurcations, methods are being constructed that allow a better understanding of the global geometry of systems and leading to a better organization in the bifurcations arising in families of dynamical systems. Dana Schlomiuk's work, done alone or in collaboration with J. Pal or Y. Dupuis, gives a good description of the global dynamics of certain classes of quadratic nonlinear systems. The methods developed are well suited for the problem of algebraic integrability of systems and further research is being pursued in that direction. Other aspects of the project cover the study of centre singularities (some work is in progress with L. Farell) as well as the gluing and resummation of local first integrals around a singularity or global ones.

Multi-sensor data fusion
Elisa Shahbazian

Elisa Shahbazian is responsible for conception, prioritization, and coordination of all R&D activities at Lockheed Martin Electronic Systems Canada (LMESC). LMESC is a leader in the integration and management of complex programs and systems. These systems require applications of Image Analysis and Data Fusion technologies for enhancing their decision aid capabilities by: (a) integrating information from multiple dissimilar sources to derive maximum information about the phenomenon being observed (Level 1 of Data Fusion or Multi-Sensor Data Fusion); (b) to evaluate/make inferences about the meaning of this phenomenon (Level 2 and 3 of Data Fusion or Situation and Threat Assessment); and (c) to propose actions that should be taken in the evaluated situations (Level 4 of Data Fusion and Resource Management).

Elisa Shahbazian leads a team of 10 scientists specializing in various areas of Data Fusion and Image Analysis and a team of engineers who build the high performance computer infrastructure necessary to demonstrate the enhanced decision aid capabilities for the complex programs and systems of interest to LMESC.

Elisa Shahbazian's current research activities fall within the field of Multi-Sensor Data Fusion (MSDF) and analyses/selection of MSDF techniques and architectures for integration into existing systems, where data management is performed using conventional techniques.

MSDF is one of the key future technologies whose applications can range from the military to the commercial, from computer vision and medical diagnostics to smart structures and image recognition for space satellites, and to surveillance, search and rescue.

As a technology, MSDF is actually the integration and application of many traditional disciplines and new areas of engineering to achieve the fusion of data.

These areas include communication and decision theory, epistemology and uncertainty management, estimation theory, digital signal processing, computer science and artificial intelligence. Methods for representing and processing data (signals) coming from dissimilar sensors are adapted from each of these disciplines to perform data fusion.

Nonsmooth analysis: theory and applications
Ronald Stern

The field of nonsmooth analysis, pioneered by F.H. Clarke in the 1970's, provides a "calculus" for functions which are nondifferentiable and possibly not even continuous, and which are therefore not amenable to treatment by standard (i.e. smooth) methods. On the geometric side there have been many important applications of this theory in recent years, notably in optimization, control, and general dynamical systems (invariance theory and existence of equilibria). Ron Stern, in collaboration with F.H. Clarke, Yu. S. Ledyaev, P.R. Wolenski, and J.J. Ye, has been contributing in these areas in recent years. At present, a general problem Stern is working on is the construction of control feedback laws in certain control problems, using the tools of nonsmooth analysis.

Quantum integrable systems
John Toth

John Toth's principal area of research is the semiclassical spectral asymptotics of quantum integrable Hamiltonians. In particular, his work focuses on pointwise bounds for joint eigenfunctions, in terms of fractional powers of the semiclassical parameter. He is also working on analogous bounds for Toeplitz eigenfunctions on compact, CR manifolds.

The relevant techniques involve microlocal, Carleman-type estimates for the F.B.I. transforms of eigenfunctions.

Data fusion
Pierre Valin

Pierre Valin's main research interest is multi-sensor data fusion.

Data fusion coming from dissimilar sensors allows for an optimal synergy leading to better tracking and an identification of the target that is faster and safer. Various algorithms for sensors of the Canadian frigate are presently under consideration for the Canadian surveillance airplane Aurora (CP-140). Among the new challenges are:

  1. the presence of imaging sensors (infrared and synthetic aperture) that call for algorithms for shape recognition,
  2. the presence of a higher rate of false contacts due to waves and
  3. a usually higher density of targets.

Distinguishing and identifying several boats of various shapes in close formation on an agitated sea presents more difficulties than tracking well-separated airplanes in a blue sky.

Statistical and quantum field methods in solid state physics
Carolyne Van Vliet

Carolyne Van Vliet's recent research interests have been concentrated on the following two subjects: (i) development of kinetic equations for transport properties in solid-state systems, together with detailed computations of the electrical conductance in such systems and (ii)electrical current fluctuations (called "electrical noise") in small systems and devices.

All kinetic equations in statistical mechanics are based on two patterns. First, following Liouville and von Neumann, one considers all interactions and externally applied fields on the many-body level. While this requires a "rather complete" Hamiltonian for the whole system, the advantage is that the equations on this level are linear so that standard functional analysis applies. Second, in many cases, one can consider the behaviour of quasi-particles, such as "dressed electrons" which are virtually independent, a process started by Boltzmann. One particle energies are easily formulated but the ensuing kinetic equations are quadratic or higher order in the collision terms. New quantum versions of this approach were successfully obtained by Van Vliet and Vassilopoulos.

Short noise and thermal noise, and their combined effect in mesoscopic quasi-one-dimensional conductors, was investigated by Van Vliet and Sreenivasan. This research is on the very edge of current developments in ultra-small electronic conductors and devices. The first low frequency result was given by Landauer and Martin. We extended and improved an approximation first used by Kuhn and Reggiani, employing quantum-field methods, obtaining both equilibrium results and non-equilibrium results under applied fields valid up to a terahertz (10 12 hz). Full agreement with low frequency formulas of Landauer and Martin is obtained.

Quantum physics and combinatorics
Luc Vinet

The main objectives of Luc Vinet's research projects are: (i)to develop the appropriate theoretical tools for solving important models of quantum many-body physics; (ii)to advance the theory of symmetric functions. Last year, in collaboration with his Ph.D. student Luc Lapointe, Luc Vinet made a major step towards obtaining an algebraic solution of the Calogero-Sutherland model, and in so doing proved longstanding conjectures on some of the most important symmetric polynomials in algebraic combinatorics. With Roberto Floreanini (Trieste) and Jean LeTourneux, Luc Vinet has pursued his systematic investigation of the quantum algebraic interpretation of q-special functions. He has also undertaken a study of difference equations form the symmetry point of view.

Lie symmetries of difference equations
Pavel Winternitz

Lie groups as symmetry groups of differential equations provide powerful tools for solving such equations, especially when combined with singularity theory and other attributes of modern integrability theory. Pavel Winternitz, together with Decio Levi (University of Rome III) and Luc Vinet, is developing a formalism that should be equally useful for treating difference equations. Two different approaches are being considered simultaneously. One applies to differential difference equations, involving both continuous and discrete variables. Transformations involving the continuous variables are treated via Lie algebras, the discrete ones are treated globally. In the second approach all variables are continuous, but their increments are discrete, i.e. differences figure instead of derivatives. The symmetry group is then constructed via "discrete prolongation" techniques, adapted from the usual Lie techniques used for differential equations. In order to recover all Lie point symmetries of a differential equation in the continuous limit, it turns out to be necessary to consider a much larger class of symmetries in the discrete case. They act simultaneously on the entire lattice, not just at one point.

Assessing the quality of bootstrap samples and estimates
Yannis Yatracos

Yannis Yatracos' research deals with the problem of estimation intrinsically associated with the bootstrap that gives motivation for the method to be used in evaluating the quality of bootstrap samples and estimates. For a large class of models it has been shown that, as the dimension d of the model increases, the quality of the sample obtained by resampling decreases compared with that of the original sample, and it is less probable that the bootstrap estimate will be close to the target. In particular, the quality of the sample obtained for the case of a uniform law is comparable to that of a sample from a model of infinite dimension. Finally measures are introduced to determine the efficiency of estimators obtained by resampling and the compatibility of different models with the resampling.

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