Linear Systems and Matrices

Workshop: inverse matrices and solution of linear equations in 3 variables

Quadratic functions and equations

Polynomials and the fundamental theorem of algebra

Introduction to complex numbers. Complex numbers and their operations on the complex plane.

Advanced matrix theory and linear equations

Workshop: inverse matrices and solution of linear equations in 3 or more variables

Advanced methods for solving quadratic equations

Polynomials and advanced solutions to polynomial equations

Solving equations with complex numbers. Complex numbers in polynomial identities and equations

Sequences and series

Arithmetic and geometric sequences and series

Applications in physics and engineering

Introduction to data processing

Seminar: problems solving using sequences and series

Advanced theory of sequences and series

Solving complex problems using sequences and series

Workshop: solutions of nonlinear equations using geometric sequences

Modern problems of data analysis

Seminar: solving complex problems in physics and technology using sequences and series

Introduction to probability and statistics

Finding probabilities of dependent and independent events

Mean, median and Mode of data sets

Conditional probability, variance, normal distribution. Use the rules of probability to compute probabilities of compound events.

Workshop: finding probabilities and applications in finance and banking

Probability and statistics

Basic principals of statistics. Gaussian and Gibbs statistics. Central tendency, standard deviation, combinations and permutations, probability, sampling, and various distributions

Bayesian inference and applications

Random processes underlying statistical experiments. Inferences and conclusions from sample surveys, experiments and observational studies.

Workshop: Prediction theory and applications in the financial sector. Correlation matrix, reasoning and decision making

Classification of geometric figures

Classification of angles

Geometric measures of 2-D figures

Circles

3-D figures

Exercises

Trigonometric and circular functions and their relation

Advanced operations with trigonometric functions

Advanced methods for solving trigonometric equations. Verification of fundamental identities and analytic trigonometry

Graphs of complex trigonometric functions

Workshop: Solving complex real world problems using trigonometric functions

Introduction to analytical geometry

Functions and equations in analytical geometry. Linear functions, slope and intercepts.

Elements of functional analysis

Continuity, limits and derivatives

Workshop: Analytical geometry and functional analysis

Elements of advanced analytical geometry

Analytical geometry and elements of functional analysis

Basic calculus concepts including limits, derivatives, continuity

Calculus of infinitesimals. Taylor series

Workshop: approximation, limits and derivatives

The derivative and the tangent line problem

Basic rules for differentiation. Higher- order derivatives

The chain rule and implicit differentiation

Extrema in the open and closed intervals. Second- order derivatives. Function behavior using first-order and second-order derivatives

Workshop: application of calculus in different areas of science and technology

Advanced rules of differentiation

Differentiation of complex composite functions

Derivatives of higher order and their use in functional analysis

Integration. Substitution rules. Integration by parts. The triangle theorem in integration. The use of geometry for integration of complex functions. Heaviside function and Dirac function.

Workshop: modern methods for differentiation and integration of complex functions