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

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 step-by-step 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 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 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
  • n-ary relations
  • Ordered pairs and Cartesian products
  • Reflexive, symmetric and antisymmetric relations
  • Closure properties
Functions and Mappings
  • Single-valued and multi-valued functions
  • Surjective, injective and bijective mappings
  • Transpositions
  • Composition of 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
  • Operations
  • Countable and uncountable sets
  • Decimal representation
  • Dedekind cuts
The Complex Numbers
  • Operations and properties
  • Trigonometric and inverse trigonometric functions
  • Roots; primitive roots of unity
  • Polynomials
Rings and Fields
  • Types of rings; subrings
  • Homomorphisms and isomorphisms
  • Ideals
  • Integral domains
  • Fields; examples of fields
Groups
  • Overview of group theory
  • Types of groups
  • Invariant subgroups
  • Product of subgroups
Linear Spaces and Matrix Theory
Linear Spaces: Overview and Scope
  • What is a linear space?
  • Why are linear spaces important?
  • Background
  • Relationship with numerical linear algebra
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
  • Linear functionals and duality
  • Annihilators
  • Dual of the dual space
  • Dual transformations
Norms and Inner Products
  • Normed linear spaces
  • Orthogonal complements
  • Adjoint transformations
  • Isometry
  • Self-adjoint transformations
Analysis I Real Analysis Fundamentals
Continuous Functions
  • Motivation
  • Definition of continuity
  • Fundamental limit theorems
  • Composite functions
Differential Calculus
  • The derivative of a function
  • The algebra of derivatives
  • Chain rule
  • Extreme values of functions
Fundamental Theorems
  • Intermediate value theorem
  • Extreme value theorem
  • Mean value theorem
  • Rolle’s theorem
  • Taylor’s theorem
Functions of Several Variables
  • Limits and continuity
  • Partial differentiation
  • Composite and implicit functions
  • Higher order derivatives
Further Results in Several Variables
  • Taylor's theorem in several variables 
  • Implicit function theorem
  • Inverse function theorem
  • Jacobians
Other Topics
  • Homogeneous functions
  • Dependent and independent variables
  • Differentials and directional derivatives
  • Jacobians
Sequences
  • Zeno’s paradox
  • What is a sequence?
  • Limit of a sequence of real numbers
  • Bounded, unbounded and monotonic sequences
  • One-step iterative methods
Series
  • Infinite series
  • Telescoping series
  • Geometric series
  • Convergent and non-convergent series
Tests for Convergence
  • Integral tests
  • Comparison test
  • Uniform convergence
  • Root and tests
Other Topics
  • Alternating series
  • Conditional and absolute convergence
  • Dirichlet and Abel tests
  • Rearrangement of series
  • Series and sequences of functions
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

  • Motivation
  • Length, area and volume
  • Concept of measure
  • Lebesgue measure and Borel sets
  • Theorems on measure
  • “Almost everywhere”
  • Measurable Functions
The Lebesgue Integral
  • Definition
  • Geometric interpretation of the Lebesgue integral
  • Lebesgue integral for bounded measurable functions
  • Relationship of Riemann and Lebesgue integrals
Theorems
  • Lebesgue’s dominated convergence
  • Monotone convergence (Beppo Levi)
  • Fatou’s theorem
  • Radon-Nikodym
  • Girsanov

 Elementary Probability
An Introduction to Probability, I
  • Sample space and events
  • Finite probability space
  • Conditional probability
  • Independent events
  • Random variables and random processes
An Introduction to Probability, II
  • Distributions and densities
  • Expected value and variance
  • Law of large numbers
  • Central limit theorem
  • Generating function
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
  • Vectors, matrices and arrays
  • Sets, stack and queues
  • Linked lists
Complexity Analysis
  • Computational and asymptotic complexity
  • Big-O notations
  • Other measures of complexity
  • Potential problems
  • NP-completeness
Recursion
  • Basic concepts
  • Function calls and recursive implementation
  • Tail, nontail and nested recursion
  • Backtracking
Binary Trees
  • Mathematical properties
  • Complete and full binary trees
  • Computed the depth of a binary tree
  • 2-trees
Introduction to Graph Theory
  • Directed and Undirected Graphs
  • Properties of Graphs and Digraphs         
  • Paths and Connectivity 
  • Special Types of Graphs
Graph Structure and Algorithms
  • 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              
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
  • Stochastic processes
  • Stationarity and strict stationarity
  • The autocorrelation function
  • Some common models
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
  • ARCH models
  • Asymmetric GARCH
  • Estimation
  • GARCH for option pricing models
  • GARCH forecasting
Forecasting
  • Univariate procedures
  • Exponential smoothing
  • Box-Jenkins procedure
  • Prediction of stationary procedures

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 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:

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.            Full-time employee Euro 2435
2.            Full-time 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
 


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