Notations, sets and logical symbols. Euclidean spaces, metric and norm .
Sets on the real line, upper and lower bounds ofthe numeric sets. Numeric
sequences and their basic properties. Bounded and unbounded sequences.
Converging sequences and their main properties. Monotone sequences. Rudin, Ch. 1, 2
Lecture 2
Definition of a function. Basic classification of functions Limit of a
function. Comparison of functions. Infinitesimal functions. Notions of o(f(x))
and O(f(x)). Continuity of functions. Arithmetic operations with continuous
functions. Discontinuous functions, piecewise continuous functions. Main properties
of continuous functions. The sign of a continuous function. Theorem about the intermediate
values. Boundedness of a continuous function. Theorem about the upper and the lower bound
of a continuous function. Uniform continuity. Composite function. Inverse function. Rudin, Ch. 4, Simon and Blume, 2.1-2.2.
Lecture 3
Definition of a derivative. Derivatives of sums, differences, products,
and ratios of functions. Some examples. Derivative of an inverse
function. Derivative of a composite function. Higher order derivatives.
Properties of differentiable functions (Mean value theorems). The Taylor's theorem.
Using the Taylor's formula for the computation of limits: the L'Hospital's rule and its
generalization. Rudin, Ch. 5, Simon and Blume, 2.4-2.7, 3.1-3.4.
A handout with some useful formulas
Lecture 4
The Riemann's integral. Definition and existence.
Integrability of continuous and some discontinuous functions.
Main properties of integrals. Evaluation of integrals.
The Newton-Leibnitz's formula. Change of variables in integrals. Rudin, Ch. 6, Simon and Blume, A4.
Lecture 5
Linear (vector) spaces.
Bases, change of a basis.
Matrices. Basic operations with matrices. Inverse matrix.
Determinants. Hoffman, Kunze, Ch. 1, 2, 4, Simon and Blume, 10, 11, 8, 9, 26.
Lecture 6
Systems of linear equations.
Linear subspaces. Linear and affine transformations.
Eigenspaces. Matrix diagonalization and the notion of the Jordan form.
Quadratic forms. Hoffman, Kunze, Ch. 3, 7, 10, Simon and Blume, 27, 23.1, 23.3, 23.4.
Lecture 7
Properties of functions in multi-dimensional spaces. Partial derivaties
and gradients. Smooth mappings. Higher order partial derivaties.
Convex and concave functions. Homothetic functions.
The Taylor's formula for multiple dimensions. The hessian. Quadratic form approximation.
The Jacobi's matrix. Implicit function theorem for smooth mappings.
Differentiation of functions defined by implicit relationships. Rudin, Ch. 9, Simon and Blume, 12, 13 14, 15.
Lecture 8
The notion of the Jordan measure.
The Riemann's integral in multiple-dimensional spaces
Change of variables. Change of the order of integration.
Parametric integrals. Generalization of the Leibnitz's rule. Rudin, Ch. 10.
Lecture 9
Unconstrained optimization. Necessary and sufficient conditions.
Optimization with equality constraints.
Examples. Mas-Collel, Whinston, Green, M.J , Simon and Blume, 17 . Solution recipes for static optimization with constraints
Lecture 10
Optimizaion with inequality and equality constraints.
Elements of convex programming.
The Kuhn-Tucker theorem, the Slater's conditions.
Duality. The solution of the convex
programming problem as a saddle point of the Lagrangian.
Envelope theorem. Mas-Collel, Whinston, Green, M.K , Simon and Blume, 18, 19, 21.5 .
Lecture 11
Ordinary differential equations. First order differential equation.
Solution of differential equations in some simple cases.
Linear differential equations.
Linear dynamic systems: general solutions.
The notion of the phase space. Equilibria of dynamic systems.
Linearization of non-linear dynamic system.
The analysis of a 2 dimensional case: the cases of real
and complex eigenvalues as well as some degenerate cases.
Solution of linear equations with constant coefficients.
Lecture 12
Linear operators in Banach spaces. Contraction mappings.
Fixed points of contraction mappings.
Existence and uniqueness of the solution of a Cauchy problem
for a first order ordinary differential equation.
Functionals. Differentiation of functionals. The Lagrange variation.
The general problem of the calculus of variation.
Necessary conditions for the problem with constraints.
Examples. Mas-Collel, Whinston, Green, M.H, M.I.
Lecture 13
The optimal control problem. The notion of an optimally
controlled process. Pontryagin's maximum principle in
the Lagrange's form. Necesary conditions for an extremum.
Examples of some optimal control problems. Phase diagrams.
Optimal control problem in the Hamilton's form.
The role of constraints on the state variables.
Optimal control in discrete time. The Bellman equation.
Lecture 15
Introduction to computational techniques using Matlab and Stata
