Mathematics and Applications

Mathematics and Applications

Admission requirements are available here.  

MATHEMATICAL ANALYSIS

1. Subsequence and partial limit. Existence of upper and lower limits of the sequence. Bolzano-Weierstrass lemma
2. Cauchy's congruence principle for sequences
3. Intermediate values ​​of a continuous function on a segment. Boundedness.
4. Uniform continuity, Cantor's theorem
5. Lagrange's theorem, the formula for finite increments
6. Definite integral. the necessary and sufficient condition of existence, classes of integrable functions. Newton-Leibniz formula 
7. Differentiability for multivariable functions and convergence of series with positive terms
8. Functional properties of the sum of a functional series
9. Power series, Cauchy-Hadamard formula
10. Taylor series
11. Functions of bounded variation, necessary and sufficient condition
12. Complex differentiation. Cauchy-Riemann conditions
13. Derivation of Cauchy's integral formula
14. Power series expansions of analytic functions
15. Remainder, the main theorem of the theory of remainders
16. Continuity of the Lebesgue measure 
17. Definition, correctness of the Lebesgue integral
18. Passage of the limit under the Lebesgue integral sign
19. Absolute continuous functions.

LITERATURE

[M1] Մուսոյան Վ. Խ., Մաթեմատիկական անալիզ, I մաս, 2009
[M2] Մուսոյան Վ. Խ., Մաթեմատիկական անալիզ, II մաս, 2012
[F1] Фихтенгольц Г. М., Курс дифференциального и интегрального исчисления, I, 2009
[F2] Фихтенгольц Г. М., Курс дифференциального и интегрального исчисления, II, 2009
[F3] Фихтенгольц Г. М., Курс дифференциального и интегрального исчисления, III, 2009
[1] Маркушевич А. И., Краткий курс теории аналитических функции, М., 1978.
[2] Шабат Б. В., Введение в комплексный анализ, М., 1969.
[3] Колмогоров А. Н., Фомин С. В., Элементы теории функций и функционального анализа, М., 2001

DIFFERENTIAL EQUATIONS AND FUNCTIONAL ANALYSIS

1. Cauchy problem for ordinary differential equations, solution existence and uniqueness theorems (without proof)
2. Solving linear homogeneous differential equations with constant coefficients ([1], page 73). Proof for simple roots
3. Solving linear normal inhomogeneous equations by the method of variation of constants
4. Lyapunov's theorem on the stability of the equilibrium position (without proof)
5. Contraction mapping principle
6. Hilbert space, orthonormal systems, extremal property
7. Linear operator, limit, continuity, norm
8. Theorem of Banach-Steinhaus, consequences, (mark of congruence)

LITERATURE

[1] Ղազարյան Հ.Գ., Հովհաննիսյան Ա.Հ., Հարությունյան Տ.Ն., Կարապետյան Գ.Ա., Սովորական դիֆերենցիալ հավասարումներ, Երևան, 2002։

[2] Колмогоров А. Н., Фомин С. В., Элементы теории функций и функционального анализа, «Наука», Москва 1976 г.

[3] Люстерник Л. А., Соболев В. И., «Элементы функционального анализа», Москва

ALGEBRA AND GEOMETRY

1. The determinant of the product of matrices
2. Invertible matrix. The formula for calculating the inverse matrix. Cramer's formulas
3. Greatest common divisor of polynomials. Euclid's algorithm
4. Polynomial analysis of the product of irreducible polynomials
5. Rank of a system of vectors in a linear space. Basis and dimension of a linear space
6. Criterion for isomorphism of finite-dimensional linear spaces. Dimension of sum of subspaces and direct sum
7. The relation between the kernel and image of a linear map
8. Gram-Schmidt orthogonalization algorithm. The isomorphism criterion of Euclidean spaces
9. Dot, vector and mixed products of vectors. Equations of lines and planes in space. The distance of the point from the line
10. Ellipse, hyperbola and parabola
11. The general equation of the plane. The distance of the point from the plane

LITERATURE

[1] Մովսիսյան Յու.Մ., Բարձրագույն հանրահաշիվ և թվերի տեսություն, 2023:

[2] Александров П.Г., Курс аналитической геометрии, М..

[3] Փիլիպոսյան Վ.Ա., Օհնիկյան Հ.Հ., Վերլուծական երկրաչափության խնդրագիրք Երևան, 2012, մաս I, II.

[4] Աթաբեկյան Վ. Հանրահաշվի ներածություն 2005:

 

PROBABILITY THEORY AND MATHEMATICAL STATISTICS

1. Axioms of probability
2. Conditional probability, independence of events. Full Probability and Bayes Formulas
3. Independent experiments. Bernoulli's formula. A random variable. Distribution function and its properties. Independence of random variables (no proof)
4. The mathematical expectation and variance of a random variable
5. The law of large numbers. Markov and Chebyshev theorems
6. Central limit theorem for independent and uniformly distributed random variables
7. Sampling distribution function. Glivenko's theorem
8. The Rao-Kramer inequality and efficient estimates
9. Methods of moments and maximum likelihood
10. Confidence intervals for parameters of normal and Bernoulli distributions

LITERATURE

[1] Համբարձումյան Գ.Հ., Հավանականությունների տեսություն, Լույս, 1977:

[2] Боровков А.А.,Теория вероятностей, М., 2006.

[3] Ширяев А.Н., Вероятность, М., Наука, 2011.

[4] Հարությունյան Ե. և ուրիշներ, Հավանականություն և կիրառական վիճակագրություն, Եր. 2000.

[5] Боровков А.А. , Математическая статистика. Оценка параметров. Проверка гипотез, М. 1984.