# 8D05401 – Mathematics

### Cipher – name of specialty:

8D05401 – “Mathematics”

### Passage of accreditation (year of passage, period of passage):

–

### Training period:

3 years.

### The awarded academic degree:

doctor of philosophy (PhD) in the educational program “8D05401 – Mathematics»

### Sphere of professional activity:

the science; education; Applied Mathematics; production and economy.

### Objects of professional activity:

research organizations, engineering and design bureaus, firms and companies; educational organizations (higher education institutions, etc.); organization management of the relevant profile; organizations of various forms of ownership, using the methods of mathematics in their work.

### Subject of professional activity:

scientific research in areas using mathematical methods and computer technologies; solving various applied problems using mathematical modeling of processes and objects and software; development of effective methods for solving problems of natural science, technology, economics and management; software and information support of scientific, research and management activities; teaching of mathematical disciplines, organization of the educational process in higher educational institutions and other educational institutions.

### Types of professional activity:

research, educational (pedagogical), organizational and managerial, industrial and technological.

### Practical bases:

Institute of Mathematics and Mathematical Modeling of the SC MES RK, Scientific Center of Applied Mathematics and Computer Science of the ARSU named after K. Zhubanova.

- Existence and uniqueness theorem for the solution of the Cauchy problem for an ordinary differential equation of the first order.
- Linear differential equations of the n-th order with constant coefficients.
- Linear differential equations of the n-th order with variable coefficients.
- Dynamic systems and their research on the phase plane.
- Stability of solutions of linear systems of differential equations.
- The Cauchy-Kovalevskaya theorem for linear partial differential equations.
- Symmetric non-negative linear operators. Eigenvalue problems for the operator of the second derivative.
- Problems of Cauchy and Gursa for a general linear hyperbolic equation.
- Equations of mixed type. The Tricomi problem for the Lavrent’ev-Bitsadze equation.
- Generalized solution of the first initial-boundary value problem for an equation of parabolic type.
- The curvature of a curve on surface.
- The normal cross-section of surface. Meunier’s theorem.
- Methods for calculating the main directions and main curvatures at a given point of surface.
- Geodesic lines. Theorem on the existence of geodesic lines on a regular surface.
- Family of lines, envelope.
- A system of linear algebraic equations. The Kronecker-Capelli theorem on the compatibility of a system of linear equations.
- Laplace’s theorem on the expansion of the determinant over several rows or columns.
- Fundamental theorem of algebra of complex numbers.
- Characteristic roots of the linear transformation and eigenvalues.
- Sturm’s theorem on calculating the roots of a polynomial.
- Reduction to the canonical form of the λ (lambda)-matrix.
- A necessary and sufficient condition for the reducibility of a matrix to diagonal form.
- Random variables, basic laws of distribution.
- The probability distribution functions of a random variable.
- Continuously differentiable functions, fundamental theorems about them. Uniform continuity. Cantor’s theorem.
- Limit points, upper and lower limits of the sequence. Cauchy criterion for the existence of a limit of function.
- Properties of a definite integral. Estimates of integrals. Mean value theorems.
- Improper integrals, convergence tests. The principal value of the improper integral.
- Functions of bounded variation, their criterion. The Stieltjes integral, its properties.
- Directional derivative of a function. Gradient. The Hamilton operator, its properties.
- Sufficient conditions for the local extremum of functions of several variables.
- Analytical functions. Cauchy-Riemann conditions. Properties of analytical functions.
- The integral of a function of a complex variable. Cauchy’s theorem. The Cauchy integral formula.
- Absolute and conditional convergence of series. Signs of absolute convergence. Properties of convergent series.
- Uniform convergence of functional sequences and series. Signs of uniform convergence. Properties of uniformly convergent series.
- Laurent series. Isolated singular points of an analytic function.
- Residue of the function relative to the singular point and its calculation.
- Definition and examples of complete metric spaces. Continuous maps of metric spaces.
- Definition and examples of normed spaces. Subspaces. Factor-space.
- Hilbert space. The isomorphism theorem.
- Linear functionals on normed spaces. Conjugate space. Examples.
- Linear operators, their continuity, compactness.
- Inverse operator, invertibility.
- Measurable functions and their properties. Almost everywhere convergence. Convergence in measure.
- Definition of the Lebesgue integral on a set of finite measure. Limit transition under the sign of the Lebesgue integral.
- Implicit functions. Existence, continuity, differentiability of implicit functions.
- Operator norm. Functional norm.
- Spectrum of an operator. Resolvent.
- Power series in the real and complex domain. The radius of convergence. Properties of power series.
- Fourier series. Sufficient conditions for the representability of a function by a Fourier series.
- Construction of a fundamental solution to a homogeneous differential equation with constant coefficients of the n-th order.
- Using the Euler method, construct a solution to a linear system of differential equations with constant coefficients.
- Integration of a linear system of differential equations with constant coefficients by the method of variation of arbitrary constants.
- Construct the solution of the differential equation by the method of undefined coefficients.
- Construct a solution to an inhomogeneous system by reducing a system of n linear equations to one equation of the n-th order.
- Integration of differential equations using power series.
- Matrix method of integration of linear systems of differential equations.
- Continuous dependence of the solution of a normal system of differential equations on the initial data and parameters.
- Using the phase plane method, construct a phase portrait of an autonomous system of the second order.
- Investigation of stability by the method of Lyapunov functions.
- Solve the Cauchy problem for the two-dimensional wave equation by the descent method.
- Construct the solution of the Cauchy and Gursa problems for an equation of hyperbolic type by the Riemann method.
- Solve the initial-boundary value problem for the parabolic equation by the method of separation of variables.
- Construct the Green’s function of the initial-boundary value problem for an equation of parabolic type.
- Using the continuation method, construct a solution to the boundary value problem for the diffusion / heat conduction equation on the semi axis.
- Apply the Riemann method to find a solution to the Cauchy problem of the telegraph equation.
- Construct a solution to the Dirichlet problem for the Poisson equation by Green’s method.
- Construct the solution of the Neumann problem for the Poisson equation by Green’s method.
- Using the potential theory method, solve the first boundary value problem for the Laplace equation in a half-space.
- Using the method of energy integrals, construct a solution to the mixed problem for an equation of hyperbolic type.
- The asymptotic lines of surface. Properties of asymptotic lines.
- The first and second quadratic forms of the surface of rotation.
- Surfaces of constant curvature.
- The contact of curves.
- Equation of a line on plane. Parametric representation of the line.
- Equation of a line in different coordinate systems.
- Two types of tasks related to the analytical representation of the line.
- The evolute of a plane curve.
- Applications of the Taylor (Maclaurin) formula with various forms of residual terms.
- The method of indeterminate Lagrange multipliers studies of functions on a conditional extremum.
- Inequalities for sums and integrals (Jung, Helder, Minkowski).
- Reducing a multiple integral to integrals by individual variables.
- Calculation of integrals (proper and improper) that depend on the parameter.
- Application of line integrals in vector analysis. Basic differential operations of vector analysis in curvilinear coordinates.
- Theorems on residues and their application to the calculation of contour integrals.
- Analytical continuation of the function. The Uniqueness theorem.
- The principle of compressive mappings and its applications.
- Compactness in metric spaces.Arcel’s theorem.
- The nested sphere theorem. Baer’s theorem. Completion of space.
- Convex sets and convex functionals. The Hahn-Banach theorem.
- Decomposition of square-summable functions in a series by orthogonal systems.
- Fourier transform, properties and applications.
- Self-adjoint operators in a Hilbert space and their properties.
- Recovering a function by its derivative. Absolutely continuous functions, their properties.
- Bounded linear operators. Equivalence of the concepts of linear continuous and linear bounded operators.
- Differential operators. Integral operators in spaces of functions.
- Solve systems of linear equations by method of sequential elimination of unknowns (or by the Gauss method)
- Determining the common roots of two polynomials by the Euclidean algorithm.
- Reducibility of matrices to the canonical form.
- Reducibility of matrices to Jordan normal form.
- Reduction of the Cauchy problem for a linear differential equation to the Volterra integral equation and its solvability.
- Invariance of a linear differential equation with respect to any transformation of the independent variable and with respect to a linear transformation of the desired function.
- Efficiency of application of the method of successive approximations (Picard’s method) in the research of the problem of existence and uniqueness of the initial problem for some differential equations.
- The structure of the fundamental system of solutions of a homogeneous linear system with constant coefficients and the influence on the structure of elementary divisors of the matrix of coefficients of the system.
- Analysis of the behavior of second-order dynamical systems on the phase plane.
- The connection between the autonomous system and the corresponding system in a symmetric form.
- Criterion of stability in the first approximation.
- Oscillatory character of solutions of linear homogeneous equations of the second order.
- Boundary value problems for an ordinary differential equation of the second order and their physical content.
- The Cauchy problem for a linear partial differential equation of the first order.
- Well-posed of problem statement of mathematical physics. Examples of ill-posed boundary value problems.
- Construction of a system of eigenfunctions, completeness of orthogonal systems of functions in various functional spaces.
- Reducibility of the Sturm-Liouville problem to an integral equation.
- Uniqueness and stability of the solution to the first boundary value problem for an equation of parabolic type.
- Construction of eigenvalues and eigenfunctions of the Laplace operator in a circle.
- Apply potential theory to reduce boundary value problems to integral equations: The Dirichlet problem for the Laplace equation.
- Apply potential theory to reduce boundary value problems to integral equations: The Neumann problem for the Laplace equation.
- Using the Tricomi method, prove the uniqueness of the solution of the T-problem for the Lavrent’ev-Bitsadze equation.
- Application of difference methods for solving problems of mathematical physics: Solution of a mixed problem for the diffusion equation by the method of finite differences.
- Application of difference methods for solving problems of mathematical physics: Solution of the Dirichlet problem for Poisson’s equation in a rectangle by the method of finite differences.
- Semi-geodesic coordinate systems.
- Basic equations of the theory of surfaces.
- Investigation of the shape of second-order surfaces by their canonical equations.
- Average curvature. Minimal surfaces.
- Full curvature. Surfaces of constant negative curvature.
- Theorems on implicit functions and their applications.
- Relationship between Volterra integral equations and linear differential equations.
- Application of contraction mapping principle to systems of linear algebraic equations.
- Application of contraction mapping principle in the theory of differential equations.
- Application of the method of finding a fixed point of mapping a metric space into itself for constructing solutions to nonlinear ordinary differential equations.
- Application of contraction mapping principle to integral equations.
- Application of the Fourier transform to the solution of differential equations.
- Basic integral formulas of analysis and their applications. Green’s Formulas.
- Generalized functions. Fundamental solutions of linear differential operators with constant coefficients.
- Applications of power series theory.
- Gradient method for finding extremums of strongly convex functions.
- Harmonic functions and their properties. Application of harmonic functions in mathematical physics.
- Application of Fourier series in solving boundary value problems of mathematical physics.
- Solving variational problems with fixed ends. Particular cases of the Euler equation.
- Conformal mappings and examples of their application.
- Applications of the matrix rank calculation method in solving vector algebra problems.
- Comparative analysis of methods for calculating the rank of a matrix.
- Comparative analysis of the Euclid algorithm and the Gorner method.
- Application of the basic theorem of the algebra of complex numbers in mathematical analysis and algebra.
- Finding the parameters of the sample equation of the straight line of the root-mean-square regression from ungrouped data.
- Finding the parameters of a sample equation of a straight regression line from grouped data.
- Method for calculating the sample correlation coefficient.
- Testing the hypothesis of the normal distribution of the general population. Pearson’s criterion of agreement
- Sampling Spearman’s rank correlation coefficient and testing the hypothesis of its significance.
- The integral of a random function and its characteristics.

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