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This module entitled MAT 2244 : Multivariable Analysis is designed for level II students following  MPE, MBE, MGE and MsCE combinations at UR-CE. It is an extension of MAT 1242. One real variable analysis.

This module introduces the student to functions of many variables: Domains of definition, levels lines and surfaces, Limits and continuity; Partial derivatives, Schwartz theorem, chain rule; Total differentiation, directional derivative and gradient; Optimization: critical points given a formula, maxima and minima, second derivative test; Multiple integrals, curvilinear and surface integrals and their applications; Gradient, divergence and curl of vector valued functions and Green and Stokes theorems.

Functional analysis is the study of infinite-dimensional vector spaces equipped with extra structure.  Such spaces arise naturally as spaces of functions. As well as being a beautiful subject in its own right,
functional-analysis has numerous applications in other areas of both pure and applied mathematics, including Fourier analysis, the study of the solutions of certain differential equations, stochastic processes,   quantum physics,  ordinary and partial differential equations,
numerical analysis, calculus of variations, approximation theory, integral equations, optimization and approximation theory and much more.  Apart from an introductory chapter, where we review basic concepts used in functional analysis, the module develops the theory of metric spaces, normed spaces, Hilbert spaces, linear operators, and linear functionals.  The module will deal with these topics at a basic level appropriate for undergraduates students in Applied Mathematics.

This module introduces the basic concepts of Partial Differential Equations by concrete Examples. PDEs are used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatistics, electrodynamics, fluid dynamics, electricity, gravitation and quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs.  Partial differential Equations are used in modelling of multidimensional systems.

This course will permit the users to work in industry. Environmental modelling is also tackled using PDEs. Climate change, Chemical reactions, competitions between species, modelling of dynamics of infectious diseases, among others are described using PDEs.

The course will introduce modern methods of statistical inference which use computational methods of analysis. These methods such as permutation tests and bootstrapping are nowadays used in business, finance, and scientific research. The focus is on statistical methods that use computation to replace certain assumptions. Students will learn how to manipulate data, design and perform simple Monte Carlo experiments, and be able to use resampling methods such as the Boot-strap. Although the main focus will be put on understanding the methods, programming language R will be used to implement them.

Expected outcomes

1)      Should be able to write R code, and be able to modify and understand existing R code.

2)      Understand basic data structures and algorithms in statistical applications.

3)      Understand basic numerical methods such as optimization, sampling, etc.

4)      Learn some statistics topics such as bootstrap, linear/logistic regression.

Indicative Content

Unit 1: R Programming Introduction

Unit 2: Introduction: Distributions of random variables; Classical parametric hypothesis testing; p-values.

Unit 3: Nonparametric tests: Permutation tests; Rank tests; Matched pairs.

Unit 4: The bootstrap: The jackknife; The empirical distribution; The nonparametric bootstrap; The parametric bootstrap; Bootstrap confidence intervals; Bootstrapping linear models.

Unit 5: Cross-validation: Leave-one-out cross-validation; Cross-validation for smoothing splines; k-fold cross-validation; Cross-validation for likelihood-based models.

Reference

James E. Gentle, Elements of Computational Statistics.

Efron and Tibshirani, An Introduction to the Bootstrap.

Computational Statistics Handbook with MATLAB®.

G.H. Givens and J.A. Hoeting, “Computational Statistics”, 2nd edition, Wiley, 2012.

Welcome to this course MAT2345:Analytical geometry. This course concerns analytic geometry in 2D and 3D Euclidean spaces. The course is mainly focused on the geometry of point,line and quadrics(curves and surfaces). The classification of the second order curves  and surfaces in 2D and 3D respectively  is done  systematically by means of the techniques borrowed from the linear and multilinear algebra covered in the previous modules with codes MAT1343 and MAT2243.In this module the marriage  of Algebra and Geometry is shown nicely.This marriage allows geometry problems to be solved algebraically  and algebra problems to be solved geometrically.

Aims of the module:

• Define and characterize affine Euclidean spaces
  
• Solve problems related to affine  coordinate transformations
• Appreciate the role of scalar, vector and box products in determining metric properties of various geometric objects in Affine Euclidean spaces
• Determine various positions between point-line and plane
• Classify Second Order Curves in 2D
• Classify Second Order Surfaces in 3D

Facilitators:

Mr Theoneste Hakizimana, Tel:0788592204, e-mail: htheoneste2000@yahoo.fr

Mr Emmanuel Iyamuremye, Tel:0785228276, e-mail:

The aim of this module is to study the basic theory introduce the stochastic processes in discrete and continuous time.  We use mathematical techniques to explore the behaviour of these processes.  We introduce to the students the Markov chain both discrete and, continuous and their application to Queueing models, Martingale and Gaussian Processes, and finally simulation of some stochastic processes using R, or MATLAB.

This module intends to equip students with critical thinking and problem solving skills. It introduces to students on concepts of first and second order both Ordinary and Partial Differential Equations (ODEs & PDEs). In addition it discusses systems of ODEs and Applications of PDEs in other science subjects particular in Physics, Biology, Chemistry and Geography. 

The study of Dynamical Systems explores how systems evolve over time, focusing on the long-term behavior of these systems under different conditions. It provides the mathematical framework to analyze how a system's state changes as a function of time, which can model a wide range of real-world phenomena, from physical and biological processes to economics and engineering applications. Dynamical Systems  is delivered mainly through lectures backed up by tutorial sessions. The lecture includes interactive elements whereby students in groups apply principles to simple problems to ensure their involvement and so gain understanding.  Handouts are used so that students can concentrate on the material of the lecture, but with gaps where students either have to fill in or make separate notes. Problem sheets will be given out to students and after time, the problems will be discussed in class.  The assignment will require the students to undertake some investigation on their own and to develop ideas and apply them.  They will also produce a report for each.

This module provides a basic understanding of the theory and practice of  research. It describes  both the qualitative and quantitative methods in scientific research. It differentiates the types of research (strategic, applied and fundamental)  and their respective associated designs.  The module starts with the philosophical meaning of research , then objectives of research, motivation of research, research approaches and significance of research. Finally all the process of research starting from Problem identification and formulation to interpretation of results and report are described.

This module covers topics in time series analysis and some statistical techniques on forecasting, such as time series regression, decomposition methods, exponential smoothing and Box-Jenkins forecasting methodology.

 The main aim of the module is to equip students with various forecasting techniques and knowledge on modern statistical methods for analyzing time series data.

This module will  enable students understand the fundamental ideas of group representation theory and apply them in solving physical problems. The module consists of Algebraic structure, Groups and their representations, and finally finite groups.

Module Facilitators: Dr. Alphonse Uworwabayeho (auworwabayeho@ur.ac.rw)

                                    Mr. Emmanuel Iyamuremye (eiyamuremye@aims.ac.tz).

The module investigates the following concepts of :Recall about Fourier series ,Special improper integrals, special Ordinary  differential equations and their different applications in the fields of mathematics and physics, concepts of Orthogonal polynomials : Classification of Equations of Mathematical Physics and their applications, related boundary values problems (IVP and BVP) in n-2D.

The module aims to equip students with the knowledge and confidence to help them to develop skills, attitudes and values to successfully implement the mathematics CBC at secondary schools. It equips students with different methods of data analysis, basic concepts of sampling distribution, notion of parameters estimate, confidence intervals and statistical hypothesis tests. It allows to make analysis and predictions (“inferences”) from data.

This module intends to equip students with critical thinking and problem solving skills. It introduces to students on concepts of first and second order both Ordinary and Partial Differential Equations (ODEs & PDEs). In addition it discusses systems of ODEs and Applications of PDEs particular in Physics. 

Welcome to this Course MAT3244: Introduction to classical Differential Geometry.This course employs the principles of multivariable Calculus , both differential  and integral as well as Multilinear algebra, Analytic geometry and Differential equations to solve geometry problems related to curves and surfaces in Affine Euclidean spaces.This course is divided into two main parts. The first part  deals with  the differential geometry of smooth  curves in 2D and 3D  Affine Euclidean space  and the  second part is focused on the differential geometry of regular surfaces in 3D Affine Euclidean space.

Aims of the module:

• Define and Present a regular  curve in 2D and 3D space
• Find the length, curvature and torsion of a curve
• Establish Frenet-Serret equations for a curve
• Give the geometric interpretation of curvature and torsion
• Define and give examples of regular surfaces
• Determine the first and Second fundamental forms of a surface
• Explain the intrinsic geometry of a surface using first quadratic form
• Explain the extrinsic geometry of a surface using second quadratic form
• Calculate and give geometric interpretations of various curvatures on a surface
• Determine some class es of lines on surface

Facilitators:

1. Mr Theoneste Hakizimana , Tel :07888592204,e-mail: htheoneste2000@yahoo.fr

2.Mr Emmanuel Hagaburimana,Tel: 0788779986,e-mail: hagabura@gmail.com

Markov Chains, Queueing, Martingales :

The Poisson process, the compound Poisson process, discrete time Markov chains: classification of states, stationary distributions, time reversibility. Continuous time Markov chains. Markov queueing systems (M/M/c/K), Markovian queueing systems (M/Er/1, Er/M/1), Markov networks, M/G/1 queueing systems, Pollaczek-Khinchin transform equation. Discrete time martingales: Conditional expectation, martingale convergence theorems, Doob’s inequality, optional stopping Theorems. Birkhoff’s ergodic theorem.

Time Series :

Basic forecasting Tools (Time pots and time series patterns, Seasonal plots,  Scatter plots,  Auto correlation, Prediction intervals, Least Square estimation),  Time series models (Auto regressive (AR) models, Moving Average models (MA), Auto Regressive Moving Average (ARMA), Auto Regressive Integrated Moving Average (ARIMA), Exponential Smoothing), Box-Jenkins methodology for ARIMA models, Assumptions in Box-Jenkins fitting models, Forecasting using ARIMA models,  Introduction to Non –Linear time series model.

This module focuses on analysis of problems in biology by applying the techniques of mathematical modelling mainly using differential equations along with numerical solution techniques.  The module presents models of population dynamics, epidemics, biochemical reaction networks and molecular networks (metabolic reactions and gene regulation).

Actuarial mathematics is a field of financial mathematics which focus on risks measurements particularly insurance industry. Students undertaking this module are thus trained to apply financial mathematics in insurance industry. It is thus reasonable in the contents of the course to have some reviews of financial mathematics, mortality concepts and mortality tables, some annuities computation, life insurance, premium computations  and reinsurance. 

At the end of the module, the learner should be able to explain confidently basic concepts used in financial mathematics, to simulate the prices of financial products (fixed income products and derivatives) using available IT software (R, Python). The contents of this module to include arbitrage theory, pricing derivatives, martingales and martingale representations, differentiation in stochastic environments, the Wiener process, Levy processes and rare events in financial markets, Integration in stochastic environments and Ito's formula and its usage in financial mathematics.