Distance Learning - Mathematics Foundations course - (code DL-MFC)

•  Algebra I Fundamental Algebra
• Algebra II Algebraic Systems
• Linear Spaces and Matrix Theory
• Analysis I Real Analysis Fundamentals
• Analysis II Integration and Function Spaces
• Elementary Probability
• Linear and Nonlinear Data Structures
• Fundamentals of Time Series

This self-contained course discusses the most important numerical methods are used in real-life applications. The major benefits for the student are:

• A full treatment of essential mathematics that are needed in many applications.
• Very competitively priced.
• Practical, relevant, and up to date. Lifelong access to the online resources.
• Extensive exercises and end-of-term project (leading to a certificate).
• Interaction with, and feedback from mentor (using email or on the forum).
• This is one of the few courses dealing with this range of topics in this way.
• WYSIWYG (all topics in this outline are discussed in detail).
• Student exercises, project and certificate on successful completion of course.

Course Contents

• The concept of sets
• The role of set theory in mathematics
• Set properties and operations
• Power set; de Morgan’s laws

• Binary and equivalence relations
• Ordering in sets
• Commutative and associative operations
• Isomorphisms and permutations

• Partial and total ordering
• n-ary relations
• Ordered pairs and Cartesian products
• Reflexive, symmetric and antisymmetric relations
• Closure properties

• Single-valued and multi-valued functions
• Surjective, injective and bijective mappings
• Transpositions
• Composition of mappings

• Peano postulates
• Operations
• Mathematical induction
• Order relations

• Divisors; primes
• Greatest common divisor
• Congruences
• Residue classes

• Operations
• Countable and uncountable sets
• Decimal representation
• Dedekind cuts

• Operations and properties
• Trigonometric and inverse trigonometric functions
• Roots; primitive roots of unity
• Polynomials

• Types of rings; subrings
• Homomorphisms and isomorphisms
• Ideals
• Integral domains
• Fields; examples of fields

• Overview of group theory
• Types of groups
• Invariant subgroups
• Product of subgroups

• What is a linear space?
• Why are linear spaces important?
• Background
• Relationship with numerical linear algebra

• Linear dependence
• Bases, components and dimension
• Subspaces
• Sums and intersections of subspaces
• Morphisms of linear spaces

• Linear forms and linear operators
• Rank and nullity
• Sums, scalar product, inverse and composition of linear transformations
• Invariant subspaces and projections
• Eigenvalues and eigenvectors

• Transformation to a new basis
• Consecutive transformations
• Transformation of the Matrix of a linear operator
• Tensors

• Linear functionals and duality
• Annihilators
• Dual of the dual space
• Dual transformations

• Normed linear spaces
• Orthogonal complements
• Adjoint transformations
• Isometry
• Self-adjoint transformations

• Motivation
• Definition of continuity
• Fundamental limit theorems
• Composite functions

• The derivative of a function
• The algebra of derivatives
• Chain rule
• Extreme values of functions

• Intermediate value theorem
• Extreme value theorem
• Mean value theorem
• Rolle’s theorem
• Taylor’s theorem

• Limits and continuity
• Partial differentiation
• Composite and implicit functions
• Higher order derivatives

• Taylor's theorem in several variables 
• Implicit function theorem
• Inverse function theorem
• Jacobians

• Homogeneous functions
• Dependent and independent variables
• Differentials and directional derivatives
• Jacobians

• Zeno’s paradox
• What is a sequence?
• Limit of a sequence of real numbers
• Bounded, unbounded and monotonic sequences
• One-step iterative methods

• Infinite series
• Telescoping series
• Geometric series
• Convergent and non-convergent series

• Integral tests
• Comparison test
• Uniform convergence
• Root and tests

• Alternating series
• Conditional and absolute convergence
• Dirichlet and Abel tests
• Rearrangement of series
• Series and sequences of functions

• Concept of area as a set function
• Partitions and step functions
• Sum and product of step functions
• Definition of the integral for a step function
• Integral for more general functions

• Upper and lower integrals
• Monotonic and piecewise monotonic functions
• Bounded monotonic functions
• Integration of polynomials

• Derivative of an indefinite integral
• Second fundamental theorem of calculus
• Integration by substitution
• Integration by parts

Measure Theory

• Motivation
• Length, area and volume
• Concept of measure
• Lebesgue measure and Borel sets
• Theorems on measure
• “Almost everywhere”
• Measurable Functions

• Definition
• Geometric interpretation of the Lebesgue integral
• Lebesgue integral for bounded measurable functions
• Relationship of Riemann and Lebesgue integrals

• Lebesgue’s dominated convergence
• Monotone convergence (Beppo Levi)
• Fatou’s theorem
• Radon-Nikodym
• Girsanov

• Sample space and events
• Finite probability space
• Conditional probability
• Independent events
• Random variables and random processes

• Distributions and densities
• Expected value and variance
• Law of large numbers
• Central limit theorem
• Generating function

• Abstract data types and algorithms
• Taxonomy of data structures
• Mathematical tools for algorithm analysis
• Linear and nonlinear data types
• Design strategies

• Vectors, matrices and arrays
• Sets, stack and queues
• Linked lists

• Computational and asymptotic complexity
• Big-O notations
• Other measures of complexity
• Potential problems
• NP-completeness

• Basic concepts
• Function calls and recursive implementation
• Tail, nontail and nested recursion
• Backtracking

• Mathematical properties
• Complete and full binary trees
• Computed the depth of a binary tree
• 2-trees

• Directed and Undirected Graphs
• Properties of Graphs and Digraphs         
• Paths and Connectivity 
• Special Types of Graphs

• Graph data structures and operations on graphs              
• Minimum spanning tree (MST) problems             
• Depth-first and Breadth-first searches in graphs
• Shortest path problems               
• Connected components              

•  Introduction
• What are time series?
• Objectives of time series analysis
• Approaches to time series analysis
• Examples and application areas

• Stochastic processes
• Stationarity and strict stationarity
• The autocorrelation function
• Some common models

• Time plots
• Trends
• Seasonality
• Randomness in data

Stationary ARMA Processes

• Moving Average (MA(q)) processes
• Autoregressive (AR(p)) processes
• ARMA(p,q) processes
• Autocovariance function of an ARMA(p,q) process
• ARIMA(p,d,q) processes

• ARCH models
• Asymmetric GARCH
• Estimation
• GARCH for option pricing models
• GARCH forecasting

• Univariate procedures
• Exponential smoothing
• Box-Jenkins procedure
• Prediction of stationary procedures

Prerequisites

Who should attend?

• Professionals working in business who wish to refresh their mathematical knowledge or learn new mathematical skills using a hands-on approach.
• For university graduates who need to upgrade their mathematical skills for acceptance into MSc and MFE degree programs.
• For software developers who wish to learn a number of mathematical methods underlying Computer Science.

Course Form

This is a distance learning course and you can do it in your own time. Ideally, you should strive to finish the course in one year after commencement of the course. You will be given 12 sets of exercises, each exercise being based on one section in the course. As unique feature, we provide C++ code to help you understand the algorithms and numerics even better.

The books provided with this course are:

Duration, price, date, locations and registration