Course Content (Syllabus)
Analysis
1. Real and complex numbers.
2. Sequences and series of numbers.
3. Functions of one real variable: continuity, differentiability, Taylor formula, Riemann integral.
4. Sequences and series of functions: pointwise and uniform convergence; differentiability and integrability term by term.
5. Power series, elementary functions.
6. Improper Riemann integral, functions defined by integrals (Euler integrals).
7. Solution of ordinary differential equations
8. Multivariate functions. Fubini-Tonelli theorem. Theorems of Green, Stokes, Gauss.
9. Lebesgue integral. Monotone and dominated convergence theorem.
Algebra and Geometry
1. General notions about some algebraic structures: groups, rings, fields.
2. General properties about polynomials with real and complex coefficients.
3. Finite dimensional vector spaces over real and complex numbers: base and dimension.
4. Linear transformations and matrices; eigenvalues, eigenvectors, diagonal form and applications.
5. Quadratic forms. Plane and and solid analytical geometry: lines, planes, conics, quadrics.
Number Theory
1. Divisibility, congruences modn.
2. Theorems of Fermat, Euler, Wilson.
3. Quadratic residues. Multiplicative structure of reduced residues modn.
Probability and Combinatorics
1. Random walks on the plane and space.
2. Geometric probability.
3. Generating functions.
Course Bibliography (Eudoxus)
1. Problems in Real Analysis: Advanced Calcuclus on the Real Axis, by T.-L. Radulescu, V. Radulescu, T. Andreescu. Springer, 2009.
2. Putnam and Beyond, by R. Gelca, T. Andreescu. Second edition, Springer 2017.
3. Essential Linear Algebra with Applications: A Problem Solving Approach, by T. Andreescu. Springer 2014.