3. Different forms of the line equation. Mutual positions of two lines in the space.

4. Mutual positions of a line and a plane in the space. Angles between lines, a line

and a plane in the space.

5. Linear mappings in finite-dimensional spaces. Matrix form of the linear mapping.

6. Eigenvectors and eigenvalues of linear mappings and their properties.

7. Real and complex Euclidean spaces. Cauchy-Bunyakovsky inequality.

8. The adjoint operator, the definition and its properties.

9. Definition of the orthogonal (unitary) operator. Definition of the orthogonal

(unitary) matrix, its properties.

10. Quadratic forms (QF), types of QFs. Sylvester’s law of inertia.

Discrete Mathematics

1. Sets. Operations on sets (unions, intersections, complements, difference,

symmetric difference).

2. Correspondences, their properties (completely defined correspondence,

surjection, injection, functionality).

3. Relations. Its properties (reflexivity, anti-reflexivity, symmetry, anti-symmetry,

transitivity, connection).

4. Boolean functions, their representations. Theorem on the number of different

Boolean functions.

5. Boolean formulas. Solving problems on sets theory by means of Boolean

functions.

6. Theorem on disjunctive decomposition in the set of variables. Perfect disjunctive

normal form and disjunctive normal form. Theorem on conjunctive decomposition in

the set of variables. Perfect conjunctive normal form and conjunctive normal form.

Zhegalkin polynomial.

7. Closureness and completeness of Boolean functions. Post classes.

8. Normal form minimization.

9. Finite state machines. Mealy machine. Moore machine.

10. Turing machine. Encoding of Turing machine.

11. Basic notions of combinatorics. Combinations, types of combinations. Newton

binomial.

12. Graphs. Basic notions of graph theory. Isomorphism, connection. Trees, their

properties. Rooted trees, estimation of their number. Geometric realization of graphs.

Differential Equations

1. Differential equations of the first order. Theorem on the existence and uniqueness

of the solution to the Cauchy problem. Equations with separable variables.

2. Homogeneous differential equations, equations reducing to them.

3. Linear equations, Bernoulli equations.

4. Total differential equations. Integrating multiplier.

5. Equations of high orders. Equations of the type

y( n)= f ( x ); y''= f ( x , y

' ) ; y'

'= f ( y ; y

' ); y'

''= f ( y ; y

' ; y

'' )

6. Linear inhomogeneous differential equations. Method of variation of parameters.

7. Method of undetermined coefficients for linear inhomogeneous equations with

inhomogeneity of special types.

8. Systems of differential equations. Reduction of the system to the differential

equation of the corresponding order.

9. Method of integrable combinations.

10. Matrix method for solving systems of differential equations.

Numerical methods

1. Causality of numerical problem.

2. Causality of matrices, systems of linear algebraic equations.

3. Solving of the systems of linear algebraic equations by Gauss’ method. LU-

decomposition.

4. Orthogonal methods for solving systems of linear algebraic equations. Theorem

on the QR- decomposition.

5. Simple iteration method for solving systems of linear algebraic equations.

6. Canonical forms for iteration methods for solving systems of linear algebraic

equations (stationary, non-stationary, explicit and implicit methods, convergence

conditions).

7. Convergence of Newton’s method. Solving of the nonlinear equations.

8. Lagrange interpolating polynomial.

9. General quadratic formulas for the numerical evaluations of integrals.

10. Numerical differentiation, basic formulas.

11. Causality of numerical differentiation.

12. Runge-Kutta methods.

13. Method of finite differences.

14. Notions of well-posed and not well-posed problems.

Probability Theory

1. Random variable: definition. Distribution function, its properties.

2. Discrete random variable, distributive law of random variables.

3. Continuous random variables. Probability density function, its properties.

4. Basic numerical characteristics of random variables. Examples.

5. Random vectors. Distribution function for random vectors, its properties.

6. Discrete random vectors, distributive law of discrete random vectors.

7. Continuous random vectors. Probability density function for random vector, its

properties.

8. Basic numerical characteristics of random vectors. Correlation matrix and its

properties.

9. Correlation coefficient, its properties, probability meaning.

10. Independence of random variables. Conditions of independence.

11. Uncorrelatedness of random variables, its connection with independence.

Example.

12. The law of large numbers. The law of large numbers for independent identically

distributed random variables.

13. Central limit theorem for independent identically distributed random variables.

Programming languages and translation methods

1. Standard data types. Representation of integers in computer memory. Two’s

complement. Integer data types (with an example of a particular algorithmic

language).

2. Real numbers. Representation of real numbers in binary notation. Storage of

floating-point numbers in computer memory: sign, fraction, exponent. Floating-point

data types (with an example of a particular algorithmic language).

3. Program structure (with an example of a particular algorithmic language).

Internal and external procedures (functions). Elements of modular programming.

Working with program libraries (modules).

4. The basic elements of structural programming: jump statements, loops,

conditional statements, selection statements (with an example of a particular

algorithmic language).

5. Array as a data structure. Storage of an array in computer memory. Declaration

and initialization of an array. Access to array elements. Description of the

implementation of the standard algorithm, choose from: insertion and deletion of

array elements; linear search in an array; search for the maximum element in an

array.

6. Storage of symbols in computer memory. Character encodings. ASCII code.

String as a character array. Representation of strings (with an example of a particular

algorithmic language).

7. Input and output of strings. String operations: access to elements, concatenation.

Concatenation properties. Comparison of strings. Search for a substring in a string.

8. Binary and text files. Basic procedures for working with files: opening / closing,

reading / writing, moving the pointer, analyzing data exhaustion (with an example of

a particular algorithmic language).

9. Static and dynamic memory allocation. Management of dynamically allocated

memory in high-level programming languages. Pointers.

10. Definition of dynamic data structures. Examples and methods of their

implementations. Lists. Examples of the usage.

Programming technologies

1. Object-oriented programming. Basic concepts. Description of classes. Class

modifiers. Access modifiers for class elements.

2. Description of class fields. Field modifiers. Description of class methods. Method

modifiers. Features of methods, allowable actions in methods. Class constructors.

3. Inheritance, its aspects and varieties. Multiple inheritance. Abstract and final

classes. Polymorphism (with examples).

4. Classification of computing system architectures. Basic architectures of parallel

computers.

5. Parallel programming models. Basic properties of parallel algorithms.

6. Main stages of developing a parallel program. Decomposition. Types of

decompositions. Communication. Types of communications.

7. Characteristics of the MPI library. Basic functions of MPI (minimal set). MPI

library. Organization of data receiving / sending between separate processes.

8. OpenMP technology. Strategy of the approach. Definition of a parallel section.

Distribution of calculations between threads. Synchronization.

9. Definition of a distributed application, reasons for their creation. Principles of

constructing distributed systems. Basic paradigms of distributed programming.

10. Models of distributed systems. Models of distributed system architectures. Thin

clients.

Databases and expert systems

1. Basic concepts of data integration and management (databases, database

management systems (DBMS)). Modeling of data integration and processing

(infologic model, hierarchical model, network and relational models, models of

distributed processing).

2. Database designing. Basic stages of designing (domain modelling, data structures,

storage structures).

3. Concept of a knowledge-based system. Basic functional modules. Forms of

knowledge representation in expert systems and organization of logic inference.

4. Data operations. Set operations on relations. Operations of relational algebra.

Give examples of each operation.

5. Physical organization of databases. Internal database model. Storage structures.