ottawa mathematics conference june 16-18th · we provide some examples to illustrate the concepts....

Dynamical Systems, Geometric Optimiza-tion, Partial Differential Equations, Control Theory, Abstract Algebra, Projectivization, Probability, Number Theory, Numerical Analysis, Differential Geometry, Theoretical Computer Science, Statistical Modeling, To-pology, Computational Biology, Fluid Me-chanics, Combinatorics, Graph Theory, Cryptography, Category Theory, Machine Learning, Graph Theory, Tensor Analysis, Functional Analysis, K-Theory, Operator Theory, Algebraic Topology, Game Theory, Information Theory, Logic, Complex Analy-sis, Group Theory, Commutative Algebra

The 10th Annual

Ottawa Mathematics Conference

June 16-18th Ottawa, Ontario, Canada

Organizing CommitteeJames DowdallChris DelaneyLuiz CordeiroAdèle BourgeoisGaël Giordano

Contents

1 Acknowledgements 3

2 Keynote Speakers 4

3 uOttawa Best Student Paper Award Lecture 7

4 Schedule 8

5 Invited Speakers 9

6 Maps and Directions 19

1 Acknowledgements

James Dowdall PresidentChris Delaney VP Finance

Organizing Committee Luiz Cordeiro VP AcademicAdèle Bourgeois VP InternalGaël Giordano VP ExternalDr. Benoit Dionne University of OttawaDr. Rafal Kulik University of Ottawa

Administrative Support Janick Rainville University of OttawaMayada El Maalouf University of OttawaDiane Demers University of Ottawa

Ottawa Community Christina Amos City of OttawaMary Sayewich City of Ottawa

The 10th Annual Ottawa Mathematics Conference is proudly supported by

The Student Committee of the Canadian Mathematical Society

The FIELDS Institute

Local 2626 of The Canadian Union of Public Employees

The Graduate Student Association of the University of Ottawa.

The Department of Mathematics and Statistics and Faculty of Sci-ence at the University of Ottawa.

2 Keynote Speakers

Dr. Mayer Alvo, University of Ottawa

Friday, June 16th, 9:00-10:00 in Room MRT212

Dr. Mayer Alvo is a professor of statistics at the University of Ottawa. He re-ceived his Ph. D from Columbia University where he worked under HerbertRobins. Dr. Alvo was named a fellow of the Fields Institute in recognitionof his outstanding contributions to the Canadian Mathematical Society andFields activities. Dr. Alvo was the driving force which led to the Universityof Ottawa becoming a Principal Sponsoring University of the Fields Insti-tute in 2002, a sponsorship which directly benefits our department, throughthe funding of postdoctoral fellows and of several research workshops andconferences each year.

The Many Facets of Correlation

The popular Pearson correlation coefficient is used to measure the linear dependence between two randomvariables. Its interpretation in practice depends upon the underlying distribution of the data. In this talk, we revisitthe concept of correlation and define it through the use of the ranks of the data. This avoids the reliance on theunderlying distribution. We then extend the notion of rank correlation to situations where the data is partiallymissing. We do this by introducing the concept of compatibility. In another direction we extend rank correlationto deal with angular data. We provide some examples to illustrate the concepts.

Dr. Kevin Cheung, Carleton University

Sunday, June 18th, 9:00-10:00 in Room MRT212

Kevin Cheung is an Associate Professor at Carleton University in the Schoolof Mathematics and Statistics. After completing his Ph.D. at the Universityof Waterloo in the Department of Combinatorics and Optimization under thesupervision of William H. Cunningham in 2003, he spent two years at theMassachusetts Institute of Technology as an NSERC Postdoctoral Fellowunder the mentorship of Michel X. Goemans. In addition to discrete opti-mization, his current academic interests include designing and developingteaching tools and exploiting technology to help students attain mastery. In2014 Dr. Cheung was honoured with the Capital’s Educator Award in recog-nition of his outstanding achievements as an educator.

How much do you trust computed results?

Mathematical software is used by mathematicians and non-mathematicians alike. Computations are used inpractical applications as well as in establishing theoretical results. An example of the latter is the Four ColourTheorem. When one delegates a large chunk of mathematical derivations to a computer, a natural question arises:How much can one trust the results? Establishing correctness of computations is an active area of research. Thereare two complementary points of view that one can take: one is to develop provably correct software. The other,which is the focus of this talk, is to obtain ways to verify computed answers. A brief overview of some of the ideasinvolved in previous work will be given. A recent effort in verifying mixed-integer linear programming results willbe described. Some future research directions will be discussed.

Dr. Damien Roy, University of Ottawa

Saturday, June 17th, 9:00-10:00 in Room MRT212

Dr. Damien Roy was born in Verdun, Quebec and is currently an associateprofessor at the University of Ottawa. He obtained his B.Sc. from the Uni-versity of Montreal, his M.Sc. from McGill University, and his Ph.D. fromLaval University. He was a post-doctoral student at Paris VI, Laval and at theCentre Interuniversitaire en Calcul Mathématique Algébrique before joiningthe University of Ottawa. Dr. Roy is a distinguished professor of NumberTheory at the University of Ottawa where he has been named Researcher ofthe Year. He has made substantial, original and lasting contributions, includ-ing progress on conjectures that have stood for decades. His research hastaken on an incredible momentum, with stunning breakthroughs on a varietyof long-standing problems in just the past few years.

On the values of the exponential function

We know that e, π and eπ are transcendental numbers, meaning that they are not the roots of nonzero poly-nomials with integer coefficients, but what about e+π? In this talk, we present results and conjectures about the(algebraic independence of the) values of the exponential function and discuss recent advances on the topic.

3 uOttawa Best Student Paper Award Lecture

Kevin Church, University of Waterloo

Saturday, June 17th, 2:30-3:30 in Room MRT212

Kevin received his MSc from the University of Ottawa where he workedunder Dr. Robert Smith? In 2015-2016 Kevin was named the recipientof University of Ottawa’s Outstanding Student Paper award for his paper"Comparing malaria surveillance with periodic spraying in the presence ofinsecticide-resistant mosquitoes: should we spray regularly or based on hu-man infections?". This award distinguishes an exceptional paper publishedby a student of the Department of Mathematics and Statistics during theirstudies. The paper is extremely well written having high caliber in terms ofboth mathematical research and mathematical writing. Kevin is now a Ph.D Candidate at the University of Waterloo where he is working under thesupervision of Dr. Xinzhi Liu and Dr. Jun Liu.

Control of Malaria in the Presence of Insecticide-Resistant Mosquitoes

Surveillance data is a hugely important tool in the design and implementation of public health policy andstrategies. In regions of the world where malaria is endemic, there is a need to understand how such data caninform malaria control policies. Indeed, the World Health Organization has identified effective surveillance as acritical component in malaria elimination and has called for stronger surveillance systems to track and preventoutbreaks in endemic regions.

Vector control methods – that is, methods that control the mosquito population responsible for spreading thedisease to humans – have been highly effective at reducing prevalence in endemic areas. However, there is evidenceof resistance to the more insecticide-based control measures currently in use. In this presentation, we will introduceand analyze a compartmental model of malaria transmission that features an impulsive insecticide control as wellas a mutant strain of mosquito that is resistant to the insecticide. The control is applied at times that are determinedby passive surveillance data.

While the model is eight-dimensional, the dynamics pertaining to the insecticide control and the resistant strainof mosquitoes can be understood by considering a two-dimensional submodel. This submodel, while seeminglysimple, can exhibit a wealth of interesting dynamical phenomena. We present a brief introduction to bifurcationtheory of impulsive systems and demonstrate how the theory can be used to determine safe control guidelines tolimit the growth of the mutant strain.

4 Schedule

Friday,June 16th

Saturday,June 17th

Sunday,June 18th

9:30-10:00 Coffee and Registration Coffee and Registration Coffee and Registration

10:00-11:00Dr. Mayer Alvo

The Many Facets ofCorrelation

Dr. Damien RoyOn the values of theexponential function

Dr. Kevin CheungHow much do you trust

computed results?11:00-11:15 Break Break Break

11:15-11:35Becem Saidani

Multivariate stabledistributions and stable CLT

Jason BramburgerSpiral wave solutions tolambda-omega lattice

dynamical systems

Clemonell Lord BaronatBilayi-Biakana

Tail empirical processes forstochastic volatility models

11:40-12:00

Xin Yang LuCentroidal Voronoi

tessellations and Gersho’sconjecture in 3D

Omid MakhmaliDifferential Geometric

Aspects of Causal Structures

Aram DermenjianThe facial weak order and its

lattice quotients

12:05-12:25Keith O’Neil

Smoothness in CodifferentialCategories

Gustavo Felisberto ValenteBratteli diagrams andDynamical Systems

Yousef AlqawasmehPersistence conditions and

traveling periodic waves forJuveniles-Adults model in

heterogeneous12:30-2:15 LUNCH LUNCH LUNCH

2:15-2:35Melkior Ornik

An Overview of ReachControl Theory

2015 University of OttawaBest Student Paper Award

Lecture:

Luiz CordeiroHiperlinear Groups and

Invariant States

2:40-3:00James Dowdall

Centre Manifold Theory andWave Propogation

Kevin Church

Evans BoadiSymmetric Polynomials,

Combinatorics andMathematical Physics

3:05-3:25Kaveh Mousavand

Application of τ-tiltingtheory in preprojectivization

Control of malaria in thepresence of

insecticide-resistantmosquitoes

Kris ChambersThe Isotropy Group for theTopos of Continuous G-Sets

3:30-3:45 Break Break Break

3:45-4:05

Amir El-AooitiCircuit Complexity of Solving

Constraint SatisfactionProblems with Few

Subpowers

Priyabrata SenapatiEffective Multilevel Monte

Carlo Methods for StochasticBiochemical Kinetics

Carleton University

4:10-4:30Chris Delaney

Quantum Calculus andq-Differential Categories

Adèle BourgeoisIrreducible Representations

of the p-adic IntegersStudent Lecture Series

4:35-4:55Damien Rioux Lavoie

A penalty method applied tothe fractional heat equation

Noe Aubin-CadotIntroduction to Atiyah-Floer

conjectureConcluding Remarks

5 Invited Speakers

The 2017 Ottawa Mathematics ConferenceOttawa, Ontario, CanadaJune 16-18, 2017

Spiral wave solutions to lambda-omega lattice dynamical systems

Jason Bramburger, University of Ottawa

In this talk we inspect an infinite system of coupled lambda-omega differential equations indexed by thetwo-dimensional integer lattice and show there exists a spiral wave solution. It is demonstrated that the infinite-dimensionality of the system poses unique problems in that traditional techniques from systems of finitely manydifferential equations cannot be directly applied. In particular, the existence proof requires extensive results fromBanach space theory and a “Hard" Implicit Function Theorem. Some brief remarks on the history and relevanceof lambda-omega systems are also provided to properly frame the results.


Centroidal Voronoi tessellations and Gersho’s conjecture in 3D

Xin Yang Lu, McGill University

Centroidal Voronoi Tessellations (CVT) are tessellations using Voronoi regions of their centroids. CVTs areuseful in data compression, optimal quadrature, optimal quantization, clustering, and optimal mesh generation.Many patterns seen in nature are closely approximated by a CVT. Examples of this include the Giant’s Causeway,the cells of the cornea, and the breeding pits of the male tilapia.

This is closely related to Gersho’s conjecture, which states that there exists an asymptotically optimal CVTwhose Voronoi regions are all rescaled copies of the same polytope. Straightforward in 1D, and proven in 2D,Gersho’s conjecture is still open for higher dimensions. One of the main difficulties is that Gersho’s conjecture is astrongly nonlocal, infinite dimensional minimization problem (even in 3D). In this talk we will present some recentresults which reduce Gersho’s conjecture to a local, finite dimensional problem in 3D. Joint work with RustumChoksi.


An Overview of Reach Control Theory

Melkior Ornik, University of Toronto

The behaviour of controlled mechanical systems is commonly interpreted in terms of ordinary differential equa-tions, where the equation guiding the system trajectories in some way depends on the control input applied on thesystem. A classical problem in the area is that of controllability, i.e., of determining whether there exists a controlinput which drives the system trajectory from one point in the state space to another. Standard methods in controlapply only to Euclidean state spaces, and do not consider the existence of physical obstacles or other constraintsaffecting the system behaviour. Reach control theory is an approach to solving the problem of controllability on aconstrained state space. The method it uses is to triangulate the state space into simplices, and then design a simplecontroller on each of the simplices to drive the system trajectory from the starting point to a desired ending point.Reach control admits a substantial mathematical structure, with a top layer based on graph theory, computationalgeometry, and formal logic, and a lower layer driven by dynamical equations. In this talk, I will give an overviewof the current state of the art in reach control theory, and outline the theoretical challenges it is still facing. Finally,I will illustrate the method and use of reach control by presenting an application to the adaptive cruise controlfeature in vehicles. Along with an appealing real-world use, this application motivates a number of new theoreticalproblems in reach control.


Application of τ-tilting theory in preprojectivization

Kaveh Mousavand, Université du Québec à Montréal (UQAM)

Preprojective algebras were firstly introduced by Gelfand and Ponomarev, in 1979 via a combinatorial description,and a few years later by Baer, Geigle and Lenzing (1987) through a homological approach. In 1998, M. Ringlelshowed that these seemingly different constructions give rise to the same algebra, which led into applications invarious areas, including representation theory of associative algebras, algebraic geometry, combinatorics, latticetheory, Coxeter groups and cluster theory, with growing linkage between other subjects.

On the other hand, in 2013, Adachi, Iyama and Reiten introduced the notion of tau-tilting theory, as a naturalgeneralization of the well-known tilting theory, in which the Auslander-Reiten translation plays a crucial role. Thishas provided an efficient tool in the study of module category of associative algebras and related areas.

In this talk, we look at the connections between τ-tilting theory and preprojective algebras and will show howa good knowledge of Auslander-Reiten translation in the module category of a given algebra paves the way forpreprojectivization of non-hereditary algebras. We use gentle algebras to depict our direction, as this class ofalgebras embodies the right framework for this approach.


Differential Geometric Aspects of Causal Structures

Omid Makhmali, McGill University

In this talk I will define causal structures as a field of tangentially non-degenerate projective hypersurfaces inthe projectivized tangent bundle of a manifold. As a result, causal structures are a generalization of conformalpseudo-Riemannian structures. The local equivalence problem of causal structures on manifolds of dimension atleast four is solved using Cartan’s method of equivalence, which allows one to geometrically interpret the essentiallocal invariants of such geometries as a natural generalization of the sectional Weyl curvature and the Fubini cubicforms of the fibers. After giving examples, in dimension four, a generalization of the notion of self-duality forindefinite conformal structures to causal structures with ruled null cones will be given. This is carried out byextending Penrose’s original realization about self-duality to causal structures, i.e., the existence of a 3-parameterfamily of surfaces whose tangent planes at each point rule the null cone.


Tail Empirical Processes for Stochastic Volatility Models

Clemonell Lord Baronat Bilayi-Biakana, University of Ottawa

We consider the tail empirical process (TEP) related to a distribution with a regularly varying tail. This is animportant tool used in nonparametric estimation of extremal quantities, like the Hill estimator of the index ofregular variation, or various risk measures. In this talk, we consider a long memory stochastic volatility model ofinterest in finance. We first start by investigating some probabilistic properties of this model. We establish centraland non-central limit theorems for the TEP and apply these results to investigate the asymptotic behaviour of theaforementioned extremal quantities. Our theoretical results are illustrated by simulation studies. This talk is basedon joint work with Rafal Kulik and Gail Ivanoff.


Introduction to Atiyah-Floer conjecture

Noé Aubin-Cadot, Université de Montréal

In 1987, M. F. Atiyah conjectured an eventual isomorphism between instanton Floer homology and LagrangianFloer homology. After a short/superficial reminder about gauge theory and symplectic topology, I’ll state theconjecture and summarize various attempts that happened since.


Circuit Complexity of Solving Constraint Satisfaction Problems with FewSubpowers

Amir El-Aooiti, Ryerson University

Although combinatorial problems represented in the form of Constraint Satisfaction Problems (CSPs) are gener-ally known to be NP-complete, placing restrictions on the constraint template can yield tractable subclasses. Bystudying the operations in the polymorphism of algebras of the constraint languages, we can construct algorithmswhich solve our CSP in polynomial time. It was shown by (Bulatov and Dalmau, 2006) that constraint languageswith polymorphisms containing Mal’tsev operations are tractable, and it was shown later by (Idziak et al, 2007)that constraint languages with polymorphisms containing k-edge operations (equivalent to having Few subpowers)for integers k > 1 (a general form of the Mal’tsev operation for arities greater than 3) are also tractable. We seekto improve on this result by presenting an algorithm which can solve ’k-edge’ constraints more efficiently withcircuit complexity in NC. By doing this, we can potentially expand on the Dichotomy Conjecture presented by(Feder and Vardi, 1999) by hypothesizing that tractable CSP subclasses are either P-complete or in NC.

First, we perform a logspace reduction of our CSP to an equivalent CSP for directed graphs with binaryconstraints as done by (Bulin et al, 2014). The algebra formed by taking the polymorphism of the edge relationsof the digraph will only have strictly simple subalgebras of type 2, 3, and 4 in the sense of the Tame CongruenceTheory. Having set the stage for our main result, we prove that the solution set of the digraph CSP contains asubalgebra which itself is a subdirect product of a set of algebras which are subalgebras of the algebra formedby taking the polymorphism of the digraph edge relations. With this property, we can then arrange the logicalformulas describing the CSP into a binary tree where each leaf represents a constraint in the CSP. As we travel upto the root of the binary tree, we take the conjunction of the constraint formulas yielding partial solutions at everystep until we are left with a set of solutions at the root of the tree which satisfy all the constraints.


Effective Multilevel Monte Carlo Methods for Stochastic Biochemical Kinetics

Priyabrata Senapati, Ryerson University

Stochastic mathematical models are critical for an accurate representation of biochemical processes in a single cell.The effect of random fluctuations at the molecular level may be significant when some species have low populationnumbers. While exact stochastic simulation methods exist, they are prohibitively expensive on most systems ofinterest. The high computational cost of such numerical simulations for complex stochastic models motivated thesearch for more effective strategies. Often, the expected value of some function of the final time solution of thestochastic model is of interest. Then, the approach employing multi-level Monte Carlo methods is more efficientthan traditional techniques. In this talk, we present multi-level Monte Carlo (MLMC) schemes for a reliable andeffective simulation of stochastic models of biochemical kinetics. The advantages of these MLMC strategies areillustrated on several biochemical models arising in applications.


The Facial Weak Order in Finite Coxeter Groups

Aram Dermenjian, Université du Québec à Montréal (UQAM)

We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of thepermutahedron of W. We call order the facial weak order. We first provide two alternative characterizations of thisposet : a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial,that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizationsare then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Björnerfor the classical weak order.


Bratteli diagrams and Dimension Groups

Gustavo Felisberto Valente, University of Ottawa

Bratteli diagrams are graphs with special properties that can be translated into Operator Algebras and DynamicalSystems settings. We use these diagrams to visualize a direct sequence of finite-dimensional algebras and helpdescribe properties of direct limits of such algebras. In this talk I’ll give an overview of these concepts making anintroductory approach to the topics.


A penalty method applied to the fractional heat equation

Damien Rioux Lavoie, Université de Montréal

Since Abel?s success in solving the tautochrone curve problem two centuries ago, fractional differential equations(FDE) have found a great deal of applications in the domain of physics and engineering.

Sadly, due to the fact that the fractional order differential operators are non-local, it is not easy to find analyticalsolutions to most of the important FDE. Therefore, it is necessary to develop efficient numerical tools to solve theseequations.

In this talk, we will present an active penalty method coupled with a high-order Fourier spectral method tosolve the fractional heat equation with non-trivial boundaries and obstacles.


Multivariate stable distributions and stable CLT

Becem Saidani, University of Ottawa

The stable distribution is a natural generalization of the normal distribution with which it shares the property ofbeing stable under addition, a property that is needed in actuarial sciences for continuously compounded returns.Furthermore, stable distributions are capable of capturing excess kurtosis shown by historical stock returns. Thegoal of this report is to give a characterization of stable distributions using Levy-Khinchine and spectral represen-tations and to prove the Stable Central Limit Theorem.


Smoothness in Codifferential Categories

Keith O’Neill, University of Ottawa

In noncommutative geometry, it is useful to invoke notions of smoothness which are compatible with ostensiblypathological spaces; in this way one may apply the tools of differential geometry to spaces for which even topo-logical structure is elusive. In this talk we explore a class of conjectures around which we may orient our variedconceptions of smoothness. Two crucial examples are highlighted.


Quantum Calculus and q-Differential Categories

Chris Delaney, University of Ottawa

Differential categories, designed a decade ago to categorify notions at the intersection of linear logic and differen-tial calculus, have since found a number of applications to categorical geometry, such as in the study of smoothnessand tangent structures. Category theory is also being fruitfully applied in the quantum realm, such as to quantuminformation theory and TQFTs. Quantum calculus is a quantized version of the classical discipline with connec-tions to Lie algebras and non-commutative geometry, and thus a categorification of its structures in analogy withthe classical one might provide similar insights. In this talk, we’ll review the basics of quantum calculus, andpropose a generalization using variants of the usual categorical differentiation and integration rules, along with akey equalizer.


Irreducible Representations of the p-adic Integers

Adèle Bourgeois, University of Ottawa

Representation theory is a branch in mathematics that allows the study of abstract algebraic structures, such asgroups. It has many applications in other mathematical fields, which include Fourier analysis, geometry and num-ber theory. To understand the representations of a given group, it suffices to study what we call its irreduciblerepresentations. In this talk, we will introduce the group of p-adic integers and determine the irreducible represen-tations of this group.


Hiperlinear groups and invariant states.

Luiz Cordeiro, University of Ottawa

Sofic and hyperlinear groups are generalizations of amenable and residually finite groups which have been heavilystudied due to the existing connections to Operator Algebras, Symbolic Dynamics and other areas. I will motivatethe study of these classes of groups, and explain a new characterization of hyperlinearity (and the analogue forsoficity) in terms of amenable almost-representations recently obtained.


Centre Manifold Theory and Wave Propogation

James Dowdall, University of Ottawa

The growing flow of people and goods around the globe has allowed new, non-native species to establish andspread in already fragile ecosystems. The introduction of invasive species can have a detrimental impact on alreadyestablished species, making it imperative that we understand the mechanisms that facilitate or prevent invasion.Since the pioneering work of Fisher and Skellam, reaction-diffusion models have been used to study the spatialdynamics of invading species such as plants, insects, epidemic agents, etc. Most of the previous work in thisdirection has assumed a homogeneous environment. However, natural environments are becoming increasinglyfragmented due to natural causes and human intervention. The mosaic of habitats can either hinder or facilitateinvasion. The object of this talk is to clearly articulate the current theories behind wave-blocking and show howthey can provide insight into the complex ecological problems we face in modernity.


Symmetric Polynomials, Combinatorics and Mathematical Physics.

Evans Boadi, Brock University

Abstract Symmetric polynomials have interesting connection with Mathematical Physics. The Jack polynomialsare connected to the Hamiltonian of the quantum n body systems. This work present the link between Jack poly-nomials and the CMS operator. The Kerov?s map translates the Jack functions into super Jack polynomials. Thesuper symmetric version of the Jack polynomials turn out to the eigenfunctions of the deformed CMS operators.

6 Maps and Directions

The Ottawa Math Conference is held every year in Ottawa, the capital of Canada. Graduate students, postdoc-toral researchers, and undergraduate students from across the country are invited to attend the conference. Theconference will be held at the University of Ottawa, which is easily reachable by air and ground transportation.

The easiest way to get from either the Ottawa International Airport or the Via Rail Station to the Universityof Ottawa is by using one of the local taxi services. A fare to campus should be approximately $30-$40 from theairport, or $10-$15 from the train station.

Alternatively, the City of Ottawa?s public transportation (OC Transpo) runs a bus from the airport to theUniversity of Ottawa (#97) and two from the Via Rail station to the University (#92,#96).

In the past the organizing committee for the OMC has worked to provide financial support for travel. If youwould like to be considered for financial support please include this when registering for the conference.

This year the conference is held in the Morisset Library of the University of Ottawa. All conference activitiesare limited to rooms 212 and 211. A map of the University of Ottawa is provided on the following page.

��

�

�

��

��

�

��

��

��

��

��

��

�

��

��

��

��

��

��

�

�

��

��

��

��

�

�

��

��

��

��

�

� ��

��

��

��

��

��

�

��

�

��

��

�

��

��

�

��

��

��

� ��

��

��

��

��

��

��

�

��

�

!"#

!##

!#$

!#%

!"$

!"%

!$$

!$%

#& #' "&

#" #(

��

��

��

��

��

��

��

��

��

��

��

��

��

�

�

��

��

��

��

��

��

��

��

��)�*+,

�+)�-.

�/-�0

�1-+��

��,�-��2,+3�-�+��+-�

�1,�4��+�

5�2��26

��)12-�26

��,+�-

��

�!�

�"�!��!

#��!�

#�$�!��%

�!��&'��&��

��

�71)4�1,

819: �2-91��,+3�-�+�;��,�-�

��<*�,

9

(�)��

�-11:�

�7/=�-��7��-�

(*+��!%!,�*

��&+��&

8��

8�

��

� ��

��

�#��-�

#(

�#�

#��

.�.

/�0

#�

�#0

1�

2��

(�3

��4

��

�-�

�(

-��

(��#

��

�1,��1-:�1>,�-+.?�

� � � � � � ��

��+5�&65��$�5�% #��6��+5�&

7 +�&&�!�6867 +�&&�!�&

7��&�+)�-.��(&��,+3�-�+�/��,+3�-�+�;

��!��686��,5�2��26��5�@$%��12+� ��2-

9$!%�6 �6)�&��!�6��%:��*A�-��9711*�1A��,�?�)�,��)�-�+��$$��2-+�-��

�;��686��)��)�1*1,�*��;��!(!��12+� ��2-

#; �$��686#� �$��2+,.1,��"$!��);�7

�$��$�&686�$��$��,.-1,��#&��-+� �2-+�

/�$�%�;686/�$�%�*

9 �$��!�686� �$��!��)12-�26��@!"$�8��, 8�9B2��2��+�-

9�� &6&�5;��&686�� 6�� &��?�,��!!$��/-�47+, ��-+1,

�$��$�&6 �6%�6&��;686(��%�<6�$��$�&�2+,.1,��"$!��);�7

�$��$�&6&!$��%�&686�!$��%6�$��$�&��3+**1,�.��9+�,9��19+�*��19+�*��9+�,9��2+*.+,?��5��!C&��,+3�-�+�/��,+3�-�+�;

�/?�,.��?�,.

��+1,,�)�,��3+�+��2-��+�+�1-��-:+,?��1�

�+��-+<2�-+9��.��4�-)+��;�D��+�4*�;��-:+,?

��3��72��*�

��-�,�41

�+<*+1�7EB2��+<-�-;

��6+� �41+,��.F�)<�-B2�)�,��6+��+9: 24��1+,�

�,A1-)��+1,

�/*/471,��.��912-��)�-?�,9;��*�471,�

�

=

#

-

-�!!>&

1�&��&1�&��!�&

1�&��&1�&��!�&

��

��

��

��:!

��:!

��:!

��:!

��:!

��:!

��:!

��:!

��:!

��:!

��:!

��:!

��

��,��

�5!��&6�!+5%�?

�!+5%�?�6&5!��:

#��!

�� "!%�@��

��

#��

�1)4*�6��/�+.�,�+�*

�7��

-�!&$��$�&��

�!%!��%6-*

��+.�,�+�*�1)4*�6

��

��

��

��

��

��

��

��

��

��

�

%C%&(C %"%" %( %' '&

!C&(!#(!$

G"

!&C

!&"

!!C!!(!!'!C&

!C'

(&#

%"!!#!&G

!GC

$$&$"'$"(

!#"

�70�

�70�

�70�

�70�

�70�

�70�

�70�

��@�$�6 �&6��$�� &6 "��,�

�;&��@;6'6��&��@�

��,�6/��6��@�$�&

1�&��&1�&��!�&

1�&��&1�&��!�&

�

��

��3��

�72��*�

��3��

�72��*�

��-�,�41

��-�,�41

��-�,�41

��6��&

��

��

� -��

��

#�

#��

��

�-�

#��

�-3

#�6A��B

��

!�-�9;9*��,.�-?-�.2��,A17 +�&&�!��<�-��%$��2-+�-��

C��#��9;9*��-�.2��?�,��!!$��/-�47+, ��-+1,

��

��"$!��);�7H*��+��)42�

��

#��

��@�$��7 +�&&�!�

��

��-�**�

#�6A��B

��/��

��

��

C

C

C

C

C

C

C C

�%�$�6 �6%"��@��&��;.��@��&��*6�D��

��+5�&5��$�5�%#��

��+5�&

�� "!%

��

��6��&��3��

�72��*�

��3��

�72��*�

��-�,�41

��

�

��

�

�

�

�

��

�-�,

�+�>

�;

��

��

��

C��

�1,��-29�+1,

��

��,.

�-�1,

�1,��-29�+1,

��

��

ottawa mathematics conference june 16-18th · we provide some examples to illustrate the concepts....

Documents