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Intervalles Maths Coursework

VARIABLES ALÉATOIRES RÉELLES (3/3)

Nous terminons cette semaine le Cours 3 avec, d'une part, un résultat important permettant de calculer la loi d'une variable aléatoire et, d'autre part, des inégalités très utiles.

Graded: QCM de la semaine

  1. Video: Séance 6 : CALCUL DE LOIS
  2. Video: Séance 7 (1/2) : INÉGALITÉS FONDAMENTALES
  3. Video: Séance 7 (2/2) : INÉGALITÉS FONDAMENTALES
  4. Reading: Loi lognormale, homographie gaussienne (*)
  5. Reading: Calculs de lois (**)
  6. Reading: Entropies probabilistes (***)
  7. Reading: Normes Lp et inégalités (***)
  8. Reading: Concentration de la loi de Poisson
  9. Video: Loi lognormale, homographie gaussienne
  10. Video: Calculs de lois
  11. Video: Entropies probabilistes
  12. Video: Normes Lp et inégalités
  13. Video: Concentration de la loi de Poisson

Not all of the listed courses are given each year. The course is offered in the language in which it is described.

Course codes in parentheses are for Carleton University. A 3-unit course at the University of Ottawa is equivalent to a 0.5-unit course at Carleton University.

MAT 5105 Discrete Applied Mathematics I: Gra. Theory (3 units)

Paths and cycles, trees, connectivity, Euler tours and Hamilton cycles, edge colouring, independent sets and cliques, vertex colouring, planar graphs, directed graphs. Selected topics from one or more of the following areas: algebraic graph theory, topological theory, random graphs. This course is equivalent to MATH 5818 at Carleton University.

MAT 5106 Combinatorial Optimization (3 units)

Network flow theory and related material. Topics will include shortest paths, minimum spanning trees, maximum flows, minimum cost flows. Optimal matching in bipartite graphs. This course is equivalent to MATH 5808 at Carleton University.

MAT 5107 Discrete Applied Mathematics II: Combinatorial Enumeration (3 units)

Ordinary and exponential generating functions; product formulas; permutations; partitions; rooted trees; cycle index; WZ method. Lagrange Inversions; singularity analysis of generating functions and asymptotics. Selected topics from one or more of the following areas: random graphs, random combinatorial structures, hypergeometric functions. This course is equivalent to MATH 5819 at Carleton University.

MAT 5121 Introduction to Hilbert Space (3 units)

This course is equivalent to MATH 5009 at Carleton University.

MAT 5122 Banach Algebras (3 units)

This course is equivalent to MATH 5003 at Carleton University.

MAT 5125 Real Analysis I (3 units)

General measure and integral, Lebesgue measure and integration on R, Fubini's theorem, Lebesgue-Radon-Nikodym theorem, absolute continuity and differentiation, Lp-Spaces. Selected topics such as Daniell-Stone theory. This course is equivalent to MATH 5007 at Carleton University.

MAT 5126 Real Analysis II (3 units)

Banach and Hilbert spaces, bounded linear operators, dual spaces. Topics selected from: weak- and weak-topologies, Alaoglu's theorem, compact operators, differential calculus in Banach spaces, Riesz representation theorems. This course is equivalent to MATH 5008 at Carleton University.

MAT 5127 Complex Analysis (3 units)

This course is equivalent to MATH 5005 at Carleton University.

MAT 5131 Ordinary Differential Equations I (3 units)

This course is equivalent to MATH 5405 at Carleton University.

MAT 5133 Partial Differential Equations I (3 units)

First-order equations, characteristics method, classification of second-order equations, separation of variables, Green's functions. Lp and Soboloev spaces, distributions, variational formulation and weak solutions, Lax-Milgram theorem, Galerkin approximation. Parabolic PDes. Wave equations, hyperbolic systems, nonlinear PDes, reaction diffusion equations, infinite-dimensional dynamical systems, regularity. This course is equivalent to MATH 5406 at Carleton University.

MAT 5134 Topics in Differential Equations (3 units)

This course is equivalent to MATH 5407 at Carleton University.

MAT 5141 Algebra I (3 units)

Groups, Sylow subgroups, finitely generated abelian groups. Rings, field of fractions, principal ideal domains, modules. Polynomial algebra, Euclidean algorithm, unique factorization. This course is equivalent to MATH 5107 at Carleton University.

MAT 5142 Algebra II (3 units)

Field theory, algebraic and transcendental extensions, finite fields, Galois groups. Modules over principal ideal domains, decomposition of a linear transformation, Jordan normal form. This course is equivalent to MATH 5109 at Carleton University.

MAT 5143 Lie Algebras (3 units)

This course is equivalent to MATH 5104 at Carleton University.

MAT 5144 Commutative Algebra (3 units)

Prime spectrum of a commutative ring (as a topological space); localization of rings and modules; tensor product of modules and algebras; Hilbert's Nullstellensatz and consequences for finitely generated algebras; Krull dimension of a ring; integral dependence, going-up, going-down; Noether Normalization Lemma and dimension theory for finitely generated algebras over a field; noetherian rings and Hilbert Basis Theorem; introduction to affine algebraic varieties and their morphisms. This course is equivalent to MATH 5001 at Carleton University.

MAT 5145 Group Theory (3 units)

This course is equivalent to MATH 5106 at Carleton University.

MAT 5146 Rings and Modules (3 units)

This course is equivalent to MATH 5103 at Carleton University.

MAT 5147 Homological Algebra and Category Theory (3 units)

This course is equivalent to MATH 5108 at Carleton University.

MAT 5148 Groups Representations and Applications (3 units)

This course is equivalent to MATH 5102 at Carleton University.

MAT 5149 Algebraic Geometry (3 units)

Brief overview of commutative algebra, Hilbert's Nullstellensatz, algebraic sets, and Zariski topology. Affine and projective varieties over algebraically closed fields. Regular functions and rational maps. Additional topics chosen from: the relation of varieties over complex numbers to complex analytic manifolds, genus, divisors, line bundles, Riemann-Roch Theorem, Bézout's Theorem. This course is equivalent to MATH 5002 at Carleton University.

MAT 5150 Topics in Geometry (3 units)

This course is equivalent to MATH 5201 at Carleton University.

MAT 5151 Topology I (3 units)

Topological spaces, product and identification topologies, countability and separation axioms, compactness, connectedness, homotopy, fundamental group, net and filter convergence. This course is equivalent to MATH 5205 at Carleton University.

MAT 5152 Topology II (3 units)

Covering spaces, homology via the Eilenberg-Steenrod axioms, applications, construction of a homology functor. This course is equivalent to MATH 5206 at Carleton University.

MAT 5155 Differentiable Manifolds (3 units)

This course is equivalent to MATH 5208 at Carleton University.

MAT 5158 Lie Groups (3 units)

This course is equivalent to MATH 6104 at Carleton University.

MAT 5160 Mathematical Cryptography (3 units)

Analysis of cryptographic methods used in authentication and data protection, with particular attention to the underlying mathematics, e.g. Algebraic Geometry, Number Theory, and Finite Fields. Advanced topics on Public-Key Cryptography: RSA and integer factorization, Diffie-Hellman, discrete logarithms, elliptic curves. Topics in current research. This course is equivalent to MATH 5300 at Carleton University.

MAT 5161 Mathematical Logic (3 units)

A basic graduate course in mathematical logic. Propositional and Predicate logic, Proof theory, Gentzen's Cut-Elimination, Completeness, Compactness, Henkin models, model theory, arithmetic and undecidability. Special Topics (time permitting) depending on interests of instructor and audience. This course is equivalent to MATH 5301 at Carleton University.

MAT 5162 Mathematical Foundations of Computer Science (3 units)

Foundations of functional languages, lambda calculi (typed, polymorphically typed, untyped), Curry-Howard Isomorphism, proofs-as-programs, normalization and rewriting theory, operational semantics, type assignment, introduction to denotational semantics of programs, fixed-point programming. Topics chosen from: denotational semantics for lambda calculi, models of programming languages, complexity theory and logic of computation, models of concurrent and distributed systems, etc. This course is equivalent to MATH 6807 at Carleton University.

MAT 5163 Analytic Number Theory (3 units)

This course is equivalent to MATH 5305 at Carleton University.

MAT 5164 Algebraic Number Theory (3 units)

This course is equivalent to MATH 5306 at Carleton University.

MAT 5165 Theory of Automata (3 units)

This course is equivalent to MATH 5605 at Carleton University.

MAT 5167 Formal Language and Syntax Analysis (3 units)

This course is equivalent to MATH/COMP 5807 at Carleton University.

MAT 5168 Homology Theory (3 units)

This course is equivalent to MATH 5202 at Carleton University.

MAT 5169 Foundations of Geometry (3 units)

This course is equivalent to MATH 5207 at Carleton University.

MAT 5170 Probability Theory I (3 units)

Probability spaces, random variables, expected values as integrals, joint distributions, independence and product measures, cumulative distribution functions and extensions of probability measures, Borel- Cantelli lemmas, convergence concepts, independent identically distributed sequences of random variables. This course is equivalent to STAT 5708 at Carleton University.

MAT 5171 Probability Theory II (3 units)

Laws of large numbers, characteristic functions, central limit theorem, conditional probabilities and expectation, basic properties and convergence theorems for martingales, introduction to Brownian motion. This course is equivalent to MATH 5709 at Carleton University.

MAT 5172 Topics in Stochastic Processes (3 units)

This course is equivalent to STAT 5508 at Carleton University.

MAT 5173 Stochastic Analysis (3 units)

Brownian motion, continuous martingales and stochastic integration. This course is equivalent to STAT 5604 at Carleton University.

MAT 5174 Network Performance (3 units)

The course will focus on advanced techniques in performance evaluation of large complex networks. Topics may include classical queueing theory and simulation analysis; models of packet networks; loss and delay systems; blocking probabilities. This course is equivalent to STAT 5704 at Carleton University.

MAT 5175 Robust Statistical Inference (3 units)

This course is equivalent to STAT 5506 at Carleton University.

MAT 5176 Advanced Statistical Inference (3 units)

Pure significance tests; uniformly most powerful unbiased and invariant tests; asymptotic comparison of tests; confidence intervals; large sample theory of likelihood ratio and chi-square tests; likelihood inference; Bayesian inference. Topics such as empirical Bayes inference, fiducial and structural inference, resampling methods. This course is equivalent to STAT 5507 at Carleton University.

MAT 5177 Multivariate Normal Theory (3 units)

This course is equivalent to STAT 5500 at Carleton University.

MAT 5180 Numerical Analysis for Differential Equations (3 units)

Floating pointing arithmetic; numerical solution of ordinary differential equations; finite difference methods for partial differential equations; stability, consistency and convergence: von Neumann analysis, Courant-Friedrichs-Lewy condition, Lax theorem; finite element methods: boundary value problems and elliptic partial differential equations; spectral and Pseudo-spectral methods. This course is equivalent to MATH 5806 at Carleton University.

MAT 5181 Data Mining I (3 units)

Visualization and knowledge discovery in massive datasets; unsupervised learning: clustering algorithms; dimension reduction; supervised learning: pattern recognition, smoothing techniques, classification. Computer software will be used. This course is equivalent to STAT 5703 at Carleton University.

MAT 5182 Modern Applied and Computational Statistics (3 units)

Resampling and computer intensive methods: bootstrap, jackknife with applications to bias estimation, variance estimation, confidence intervals, and regression analysis. Smoothing methods in curve estimation; statistical classification and pattern recognition: error counting methods, optimal classifiers, bootstrap estimates of the bias of the misclassification error. This course is equivalent to STAT 5702 at Carleton University.

MAT 5185 Asymptotic Methods of Applied Mathematics (3 units)

Asymptotic series: properties, matching, application to linear and nonlinear differential equations. Asymptotic expansion of integrals: elementary methods, methods of Laplace, Stationary Phase and Steepest Descent, Watson's Lemma, Riemann-Lebesgue Lemma. Perturbation methods: regular and singular perturbation for differential equations, multiple scale analysis, boundary layer theory, WKB theory. This course is equivalent to MATH 5408 at Carleton University.

MAT 5187 Topics in Applied Mathematics (3 units)

This course is equivalent to MATH 5403 at Carleton University.

MAT 5190 Mathematical Statistics I (3 units)

Statistical decision theory; likelihood functions; sufficiency; factorization theorem; exponential families; UMVU estimators; Fisher's information; Cramer-Rao lower bound; maximum likelihood and moment estimation; invariant and robust point estimation; asymptotic properties; Bayesian point estimation. This course is equivalent to STAT 5600 at Carleton University.

MAT 5191 Mathematical Statistics II (3 units)

Confidence intervals and pivotals; Bayesian intervals; optimal tests and Neyman-Pearson theory; likelihood ratio and score tests; significance tests; goodness-of-fit tests; large sample theory and applications to maximum likelihood and robust estimation. This course is equivalent to STAT 5501 at Carleton University.

MAT 5192 Sampling Theory and Methods (3 units)

Unequal probability sampling with and without replacement; unified theory of standard errors; prediction approach; ratio and regression estimation; stratification and optimal designs; multistage cluster sampling; double sampling; domains of study; post-stratification; non-response; measurement errors. Related topics. This course is equivalent to STAT 5502 at Carleton University.

MAT 5193 Linear Models (3 units)

Theory of non-full-rank linear models: estimable functions, best linear unbiased estimators, hypothesis testing, confidence regions; multi-way classification; analysis of covariance; variance component models: maximum likelihood estimation, MINQUE, ANOVA methods. Miscellaneous topics. This course is equivalent to STAT 5503 at Carleton University.

MAT 5194 Stochastic Processes and Times Series Analysis (3 units)

This course is equivalent to STAT 5504 at Carleton University.

MAT 5195 Design of Experiments (3 units)

Overview of linear model theory; orthogonality; randomized block and split plot designs; Latin square designs; randomization theory; incomplete block designs; factorial experiments; confounding and fractional replication; response surface methodology. Miscellaneous topics. This course is equivalent to STAT 5505 at Carleton University.

MAT 5196 Multivariate Analysis (3 units)

This course is equivalent to STAT 5509 at Carleton University.

MAT 5197 Stochastic Optimization (3 units)

Topics chosen from stochastic dynamic programming, Markov decision processes, search theory, optimal stopping. This course is equivalent to STAT 5601 at Carleton University.

MAT 5198 Stochastic Models (3 units)

Markov systems, stochastic networks, queuing networks, spatial processes, approximation methods in stochastic processes and queuing theory. Applications to the modelling and analysis of computer-communications systems and other distributed networks. This course is equivalent to MATH 5701 at Carleton University.

MAT 5301 Topics in Combinatorial Mathematics (3 units)

This course is equivalent to MATH 5609 at Carleton University.

MAT 5303 Linear Optimization (3 units)

This course is equivalent to MATH 5801 at Carleton University.

MAT 5304 Nonlinear Optimization (3 units)

This course is equivalent to MATH 5803 at Carleton University.

MAT 5307 Topics in Operations Research (3 units)

This course is equivalent to MATH 5804 at Carleton University.

MAT 5308 Topics in Algorithm Design (3 units)

This course is equivalent to MATH 5805 at Carleton University.

MAT 5309 Harmonic Analysis on Groups (3 units)

This course is equivalent to MATH 6002 at Carleton University.

MAT 5312 Topics in Topology (3 units)

This course is equivalent to MATH 6201 at Carleton University.

MAT 5313 Topics in Probability and Statistics (3 units)

This course is equivalent to MATH 6507 at Carleton University.

MAT 5314 Topics in Probability and Statistics (3 units)

This course is equivalent to MATH 6508 at Carleton University.

MAT 5315 Advanced Design of Surveys (3 units)

MAT 5317 Analysis of Categorical Data (3 units)

Analysis of one-way and two-way tables of nominal date; multi-dimensional contingency tables, log-linear models; tests of symmetry, marginal homogeneity in square tables; incomplete tables; tables with ordered categories; fixed margins, logistic models with binary response; measures of association and agreement; biological applications. This course is equivalent to STAT 5602 at Carleton University.

MAT 5318 Reliability and Survival Analysis (3 units)

This course is equivalent to STAT 5603 at Carleton University.

MAT 5319 Topics in Probability and Statistics (3 units)

This course is equivalent to MATH 6507 at Carleton University.

MAT 5324 Games Theory (3 units)

This course is equivalent to MATH 5607 at Carleton University.

MAT 5325 Topics in Information and Systems Science (3 units)

This course is equivalent to MATH 5802 at Carleton University.

MAT 5326 Topics in Analysis (3 units)

This course is equivalent to MATH 6008 at Carleton University.

MAT 5327 Topics in Algebra (3 units)

This course is equivalent to MATH 6101 at Carleton University.

MAT 5328 Topics in Analysis (3 units)

This course is equivalent to MATH 6008 at Carleton University.

MAT 5329 Topics in Analysis (3 units)

This course is equivalent to MATH 6009 at Carleton University.

MAT 5330 Topics in Algebra (3 units)

This course is equivalent to MATH 6102 at Carleton University.

MAT 5331 Topics in Algebra (3 units)

This course is equivalent to MATH 6103 at Carleton University.

MAT 5341 Quantum Computing (3 units)

Space of quantum bits; entanglement. Observables in quantum mechanics. Density matrix and Schmidt decomposition. Quantum cryptography. Classical and quantum logic gates. Quantum Fourier transform. Shor's quantum algorithm for factorization of integers. This course is equivalent to MATH 5821 at Carleton University.

MAT 5343 Mathematical Aspects of Wavelets and Digital Signal Processing (3 units)

Lossless compression methods. Discrete Fourier transform and Fourier-based compression methods. JPEG and MPEG. Wavelet analysis. Digital filters and discrete wavelet transform. Daubechies wavelets. Wavelet compression. This course is equivalent to MATH 5822 at Carleton University.

MAT 5361 Topics in Mathematical Logic (3 units)

This course is equivalent to MATH 6806 at Carleton University.

MAT 5375 Mathematical Statistics (3 units)

Limit theorems; sampling distributions; parametric estimation; concepts of sufficiency and efficiency; Neyman-Pearson paradigm, likelihood ratio tests; parametric and non-parametric methods for two-sample comparisons; notions of experimental design, categorical data analysis, the general linear model, decision theory and Bayesian inference. This course is equivalent to STAT 5610 at Carleton University.

MAT 5505 Mathématiques discrètes appliquées I : Théorie des graphes (3 crédits)

Chemins et cycles, arbres, connexité, parcours eulériens et cycles hamiltoniens, coloration des arêtes, ensembles indépendants et cliques, coloration des sommets, graphes planaires, graphes orientés. Sujets choisis parmi les thèmes suivants : théorie algébrique des graphes, théorie topologique des graphes, graphes aléatoires. Ce cours est équivalent à MATH 5818 à la Carleton University.

MAT 5506 Optimisation combinatoire (3 crédits)

Théorie des flots et thèmes voisins. On traitera parmi d'autres les sujets suivants : chemins minimaux, arbres générateurs de coût minimal, flots de coût maximal, flots de coût minimal. Couplage optimal dans les graphes bipartis. Ce cours est équivalent à MATH 5808 à la Carleton University.

MAT 5507 Mathématiques discrètes appliquées II : Énumération combinatoire (3 crédits)

Fonctions génératrices ordinaires et exponentielles; formules de produit; permutations; partitions; arborescences; indice de cycle; méthode WZ. Inversion de Lagrange; analyse des singularités des fonctions génératrices et leur comportement asymptotique. Sujets choisis parmi les thèmes suivants : graphes aléatoires, structures combinatoires aléatoires, fonctions hypergéométriques. Ce cours est équivalent à MATH 5819 à la Carleton University.

MAT 5521 Introduction aux espaces hilbertiens (3 crédits)

Ce cours est équivalent à MATH 5009 à la Carleton University.

MAT 5522 Algèbres de banach (3 crédits)

Ce cours est équivalent à MATH 5003 à la Carleton University.

MAT 5525 Analyse réelle I (3 crédits)

Mesure et intégration, mesure de Lebesgue et intégration sur R, théorème de Fubini, théorème de Lebesgue-Radon-Nikodym, continuité absolue et dérivation, espaces Lp. Chapitres choisis comme par exemple la théorie de Stone-Daniell. Ce cours est équivalent à MATH 5007 à la Carleton University.

MAT 5526 Analyse réelle II (3 crédits)

Espaces de Banach et de Hilbert, opérateurs linéaires bornés, espaces duals. Chapitres choisis parmi les suivants : topologies faibles, théorème d'Alaoglu, opérateurs compacts, calcul différentiel dans les espaces de Banach, théorèmes de représentation de Riesz. Ce cours est équivalent à MATH 5008 à la Carleton University.

MAT 5527 Analyse complexe (3 crédits)

Ce cours est équivalent à MATH 5005 à la Carleton University.

MAT 5531 Équations différentielles ordinaires I (3 crédits)

Ce cours est équivalent à MATH 5405 à la Carleton University.

MAT 5533 Équations aux dérivées partielles I (3 crédits)

Ce cours est équivalent à MATH 5406 à la Carleton University.

MAT 5534 Équations différentielles : Chapitres choisis (3 crédits)

Ce cours est équivalent à MATH 5407 à la Carleton University.

MAT 5541 Algèbre I (3 crédits)

Groupes, sous-groupes de Sylow, groupes abéliens de type fini. Anneaux, corps des fractions, anneaux principaux, modules. Algèbre de polynômes, algorithme d'Euclide, unicité de la décomposition. Ce cours est équivalent à MATH 5107 à la Carleton University.

MAT 5542 Algèbre II (3 crédits)

Théorie des corps, extensions algébriques et transcendantes, corps finis, groupes de Galois. Modules sur un anneau principal, décomposition d'une application linéaire, forme normale de Jordan. Ce cours est équivalent à MATH 5109 à la Carleton University.

MAT 5543 Algèbre de lie (3 crédits)

Ce cours est équivalent à MATH 5104 à la Carleton University.

MAT 5544 Algèbre commutative (3 crédits)

Spectre premier d'un anneau commutatif (comme espace topologique); localisation des anneaux et des modules; produit tensoriel des modules et algèbres; théorème des zéros de Hilbert et conséquences pour les algèbres de type fini sur un corps; dimension de Krull d'un anneau; dépendance intégrale, théorèmes de « going-up » et « going-down »; lemme de normalisation de Noether et théorie de la dimension dans les algèbres de type fini sur un corps; anneaux noethériens et théorème « de la base » de Hilbert; introduction aux variétés algébriques affines et à leurs morphismes.

MAT 5545 Théorie des groupes (3 crédits)

Ce cours est équivalent à MATH 5106 à la Carleton University.

MAT 5546 Anneaux et modules (3 crédits)

Ce cours est équivalent à MATH 5103 à la Carleton University.

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