Distance Learning  Mathematics Foundations course  (code DLMFC)
The goal of this distance learning course is to introduce and elaborate the most important mathematical concepts, structures and methods that are used and needed in engineering, computational finance and science. The course is also suitable for university graduates who wish to improve their mathematical skills.
Subjects Covered
We discuss the following major categories:

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
The details of each category are described below. The approach in this course is to take a stepbystep approach by motivating each topic, first in simple terms in combination with clear examples and then progressing to more advanced concepts and examples. The emphasis is on practical issues rather than only theorem proving. We close each module with a set of exercises that we recommend that the student complete and check with the mentor if they are correct.
Course Benefits
This selfcontained course discusses the most important numerical methods are used in reallife 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 endofterm 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 Originator
This course was originated, developed and is supported by Daniel J. Duffy. He has a BA degree in Mathematics as well as MSc and PhD in Numerical Analysis (the numerical solution of partial differential equations). He has many years industrial and business experience and is author of several books on numerical methods, C++ and applications to engineering and computational finance.
Course Contents
Algebra I Fundamental Algebra
Sets

The concept of sets

The role of set theory in mathematics

Set properties and operations

Power set; de Morgan’s laws
Set Relations

Binary and equivalence relations

Ordering in sets

Commutative and associative operations

Isomorphisms and permutations
Relations

Partial and total ordering

nary relations

Ordered pairs and Cartesian products

Reflexive, symmetric and antisymmetric relations

Closure properties
Functions and Mappings
Algebra II Algebraic Systems
Natural Numbers and Integers

Peano postulates

Operations

Mathematical induction

Order relations
Properties of Integers

Divisors; primes

Greatest common divisor

Congruences

Residue classes
Rational and Real Numbers
The Complex Numbers

Operations and properties

Trigonometric and inverse trigonometric functions

Roots; primitive roots of unity

Polynomials
Rings and Fields
Groups

Overview of group theory

Types of groups

Invariant subgroups

Product of subgroups
Linear Spaces and Matrix Theory
Linear Spaces: Overview and Scope
Linear Spaces: Properties

Linear dependence

Bases, components and dimension

Subspaces

Sums and intersections of subspaces

Morphisms of linear spaces
Linear Transformations

Linear forms and linear operators

Rank and nullity

Sums, scalar product, inverse and composition of linear transformations

Invariant subspaces and projections

Eigenvalues and eigenvectors
Coordinate Transformations

Transformation to a new basis

Consecutive transformations

Transformation of the Matrix of a linear operator

Tensors
Dual Vector Spaces
Norms and Inner Products
Analysis I Real Analysis Fundamentals
Continuous Functions
Differential Calculus

The derivative of a function

The algebra of derivatives

Chain rule

Extreme values of functions
Fundamental Theorems
Functions of Several Variables
Further Results in Several Variables
Other Topics
Sequences

Zeno’s paradox

What is a sequence?

Limit of a sequence of real numbers

Bounded, unbounded and monotonic sequences

Onestep iterative methods
Series
Tests for Convergence

Integral tests

Comparison test

Uniform convergence

Root and tests
Other Topics
Analysis II Integration and Function Spaces
Fundamentals of Integral Calculus

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
Integrals of more general Functions

Upper and lower integrals

Monotonic and piecewise monotonic functions

Bounded monotonic functions

Integration of polynomials
Relation between Integration and Differentiation

Derivative of an indefinite integral

Second fundamental theorem of calculus

Integration by substitution

Integration by parts
Measure Theory
The Lebesgue Integral

Definition

Geometric interpretation of the Lebesgue integral

Lebesgue integral for bounded measurable functions

Relationship of Riemann and Lebesgue integrals
Theorems
Elementary Probability
An Introduction to Probability, I
An Introduction to Probability, II
Linear and Nonlinear Data Structures
Overview

Abstract data types and algorithms

Taxonomy of data structures

Mathematical tools for algorithm analysis

Linear and nonlinear data types

Design strategies
Review of Fundamental Data Structures
Complexity Analysis
Recursion
Binary Trees
Introduction to Graph Theory
Graph Structure and Algorithms

Graph data structures and operations on graphs

Minimum spanning tree (MST) problems

Depthfirst and Breadthfirst searches in graphs

Shortest path problems

Connected components
Fundamentals of Time Series

Introduction

What are time series?

Objectives of time series analysis

Approaches to time series analysis

Examples and application areas
Fundamental Techniques
Time Series Analysis

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
GARCH Models
Forecasting
Prerequisites
We assume that the student has reached a certain level of mathematical sophistication in order to follow and understand this course. For example, some knowledge of integral and differential calculus of one variable is certainly a prerequisite. This is sufficient in order to follow this course. For those students who feel that they do not have the necessary background please do not hesitate to contact me dduffy AT datasim.nl.
Who should attend?
This focused and practical course introduces and discusses in reasonable detail all the major topics in algebra, analysis and their applications to prepare the student for entry program to university Master degree courses, computational finance, engineering and science. In particular, in our opinion the course is suitable for the following groups:

Professionals working in business who wish to refresh their mathematical knowledge or learn new mathematical skills using a handson 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:
Discrete Mathematics, S. Lipschutz and M. Lipson Schaum
Modern Algebra, F. Ayres Schaum
Real Variables, M. R. Spiegel Schaum
Duration, price, date, locations and registration
Course duration: 
Distance learning. You study in your own pace. Under normal circumstances, this should take you between 1 and 1.5 years to complete. 
Dates and location: 
(click on dates to print registration form) 
Date(s) 
Location 
Price 
Language 
Any time 
Distance Learning 
See below 
English 
Click here to register.
Attention
Course Resources
The optimal way to learn in our opinion is by executing the following steps. This discussion pertains to studying and learning the contents of a single module:
1. Listen to the audio show and use the printed PowerPoint slides as printed backup
2. Read the relevant material in the provided book(s)
3. Do the exercises; compile and run the programs
4. If you are having problems, go back to one of more of steps 1, 2, 3
5. If step 4 has been unsuccessful then post your problem on the Datasim forum
6. Go to next module
Prices and when to start etc.
You can start the course any time and you receive lifelong access to the resources. You decide the pace that is most appropriate for you. The price per student category is (all prices exclude VAT if applicable (no VAT paid if you live outside the EU)):
1. Fulltime employee Euro 2435
2. Fulltime student at a recognised university Euro 1495
3. Between jobs? contact dduffy AT datasim.nl
4. For groups of employees in a company, contact info AT datasim.nl
