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We find sufficient conditions for the solvability and construction of the generalized Green’s operator for linear Noether boundary value problem for linear differential-algebraic system.

We find sufficient conditions for the solvability and construction of the generalized Green’s operator for linear Noether boundary value problem for degenerate linear differential-algebraic system with interface conditions.

A new statement of Integral Geometry problem where the image of function in each point is taken as an integral with respect to measure which depends on the point is suggested. Such Measure System is named Measure Induction. It is shown that an inversion formula exists for class of measures having a unit atom in corresponding
point and limited on whole space. Previously obtained results for average on systems of measurement dissections and for weight average on graphs are generalized.

The concept of an operator is an almost algebraic with respect to two-sided ideal of the algebra of linear operators in some finite-dimensional linear spaces, it extended to the case when the ideal is left. We prove a theorem on the following equation particular solution $\sum\limits^{n, m}_{i=0, j=0} a_{ij} A^i B^j u = f$, where $A$ and $B$ is a linear operator, $f$ is an element of a linear space. The result is applied to the differential-difference equations.

We consider integration Clein–Gordon and Dirac equations in Bianchi IX cosmology model. Using the noncommutative integration method we found the new exact solutions for Taub universe.

Noncommutative integration method for Bianchi IX model is based on the use of the special infinite-dimensional holomorphic representation of the rotation group, which is based on the nondegenerate orbit adjoint representation, and complex polarization of degenerate covector. The matrix elements of the representation of form a complete and orthogonal set and allow you to use the generalized Fourier transform. Casimir operator for rotation group under this transformation becomes constant. And the symmetry operators generated by the Killing vector fields in the linear differential operators of the first order from one dependent variable. Thus, the relativistic wave equation on the rotation group allow non-commutative reduction to ordinary differential equations. In contrast to the well-known method of separation of variables, noncommutative integration method takes into account the non-Abelian algebra of symmetry operators and provides solutions that carry information about the non-commutative symmetry of the task. Such solutions can be useful for measuring the vacuum quantum effects and the calculation of the Green’s functions by the splitting-point method.

The work for the Taub model compared the solutions obtained with the known, which are obtained by separation of variables. It is shown that the non-commutative solutions are expressed in terms of elementary functions, while the known solutions are defined by the Wigner function. And commutative reduced by the Klein–Gordon equation for Taub model coincides with the equation, reduced by separation of variables. A commutative reduced by the Dirac equation is equivalent to the reduced equation obtained by separation of variables.

Small practical value of many numerical methods for solving single-ended systems of linear equations with ill-conditioned matrices due to the fact that these methods in the practice behave quite differently than in the case of precise calculations. Historically, sustainability is not enough attention was given, unlike in numerical algebra ‘medium-sized’, and emphasis is given to solving the problems of maximal order in data capabilities of the computer, including the expense of some loss of accuracy. Therefore, the main objects of study is the most appropriate storage of information contained in the sparse matrix; maintaining the highest degree of rarefaction at all stages of the computational process. Thus, the development of efficient numerical methods for solving unstable systems refers to the actual problems of computational mathematics.

In this paper, the approach to the construction of numerically stable direct multiplier methods for solving systems of linear equations, taking into account sparseness of matrices, presented in packaged form. The advantage of the approach consists in minimization of filling the main lines of the multipliers without compromising accuracy of the results and changes in the position of the next processed row of the matrix are made that allows you to use static data storage formats. The storage format of sparse matrices has been studied and the advantage of this format consists in possibility of parallel execution any matrix operations without unboxing, which significantly reduces the execution time and memory footprint.

Direct multiplier methods for solving systems of linear equations are best suited for solving problems of large size on a computer — sparse matrix systems allow you to get multipliers, the main row of which is also sparse, and the operation of multiplication of a vector-row of the multiplier according to the complexity proportional to the number of nonzero elements of this multiplier.

As a direct continuation of this work is proposed in the basis for constructing a direct multiplier algorithm of linear programming to put a modification of the direct multiplier algorithm for solving systems of linear equations based on integration of technique of linear programming for methods to select the host item. Direct multiplicative methods of linear programming are best suited for the construction of a direct multiplicative algorithm set the direction of descent Newton methods in unconstrained optimization by integrating one of the existing design techniques significantly positive definite matrix of the second derivatives.

Multiplicative methods for sparse matrices are best suited to reduce the complexity of operations solving systems of linear equations performed on each iteration of the simplex method. The matrix of constraints in these problems of sparsely populated nonzero elements, which allows to obtain the multipliers, the main columns which are also sparse, and the operation of multiplication of a vector by a multiplier according to the complexity proportional to the number of nonzero elements of this multiplier. In addition, the transition to the adjacent basis multiplier representation quite easily corrected. To improve the efficiency of such methods requires a decrease in occupancy multiplicative representation of the nonzero elements. However, at each iteration of the algorithm to the sequence of multipliers added another. As the complexity of multiplication grows and linearly depends on the length of the sequence. So you want to run from time to time the recalculation of inverse matrix, getting it from the unit. Overall, however, the problem is not solved. In addition, the set of multipliers is a sequence of structures, and the size of this sequence is inconvenient is large and not precisely known. Multiplicative methods do not take into account the factors of the high degree of sparseness of the original matrices and constraints of equality, require the determination of initial basic feasible solution of the problem and, consequently, do not allow to reduce the dimensionality of a linear programming problem and the regular procedure of compression — dimensionality reduction of multipliers and exceptions of the nonzero elements from all the main columns of multipliers obtained in previous iterations. Thus, the development of numerical methods for the solution of linear programming problems, which allows to overcome or substantially reduce the shortcomings of the schemes implementation of the simplex method, refers to the current problems of computational mathematics.

In this paper, the approach to the construction of numerically stable direct multiplier methods for solving problems in linear programming, taking into account sparseness of matrices, presented in packaged form. The advantage of the approach is to reduce dimensionality and minimize filling of the main rows of multipliers without compromising accuracy of the results and changes in the position of the next processed row of the matrix are made that allows you to use static data storage formats.

As a direct continuation of this work is the basis for constructing a direct multiplicative algorithm set the direction of descent in the Newton methods for unconstrained optimization is proposed to put a modification of the direct multiplier method, linear programming by integrating one of the existing design techniques significantly positive definite matrix of the second derivatives.

In the article a theory of the implicit iterative line-by-line recurrence method for solving the systems of finite-difference equations which arise as a result of approximation of the two-dimensional elliptic differential equations on a regular grid is stated. On the one hand, the high effectiveness of the method has confirmed in practice. Some complex test problems, as well as several problems of fluid flow and heat transfer of a viscous incompressible liquid, have solved with its use. On the other hand, the theoretical provisions that explain the high convergence rate of the method and its stability are not yet presented in the literature. This fact is the reason for the present investigation. In the paper, the procedure of equivalent and approximate transformations of the initial system of linear algebraic equations (SLAE) is described in detail. The transformations are presented in a matrix-vector form, as well as in the form of the computational formulas of the method. The key points of the transformations are illustrated by schemes of changing of the difference stencils that correspond to the transformed equations. The canonical form of the method is the goal of the transformation procedure. The correctness of the method follows from the canonical form in the case of the solution convergence. The estimation of norms of the matrix operators is carried out on the basis of analysis of structures and element sets of the corresponding matrices. As a result, the convergence of the method is proved for arbitrary initial vectors of the solution of the problem.

The norm of the transition matrix operator is estimated in the special case of weak restrictions on a desired solution. It is shown, that the value of this norm decreases proportionally to the second power (or third degree, it depends on the version of the method) of the grid step of the problem solution area in the case of transition matrix order increases. The necessary condition of the method stability is obtained by means of simple estimates of the vector of an approximate solution. Also, the estimate in order of magnitude of the optimum iterative compensation parameter is given. Theoretical conclusions are illustrated by using the solutions of the test problems. It is shown, that the number of the iterations required to achieve a given accuracy of the solution decreases if a grid size of the solution area increases. It is also demonstrated that if the weak restrictions on solution are violated in the choice of the initial approximation of the solution, then the rate of convergence of the method decreases essentially in full accordance with the deduced theoretical results.

In this paper, we consider conditional gradient methods for optimizing strongly convex functions. These are methods that use a linear minimization oracle, which, for a given vector $p \in \mathbb{R}^n$, computes the solution of the subproblem

\[ \text{Argmin}_{x\in X}{\langle p,\,x \rangle}. \]There are a variety of conditional gradient methods that have a linear convergence rate in a strongly convex case. However, in all these methods, the dimension of the problem is included in the rate of convergence, which in modern applications can be very large. In this paper, we prove that in the strongly convex case, the convergence rate of the conditional gradient methods in the best case depends on the dimension of the problem $ n $ as $ \widetilde {\Omega} \left(\!\sqrt {n}\right) $. Thus, the conditional gradient methods may turn out to be ineffective for solving strongly convex optimization problems of large dimensions.

Also, the application of conditional gradient methods to minimization problems of a quadratic form is considered. The effectiveness of the Frank – Wolfe method for solving the quadratic optimization problem in the convex case on a simplex (PageRank) has already been proved. This work shows that the use of conditional gradient methods to solve the minimization problem of a quadratic form in a strongly convex case is ineffective due to the presence of dimension in the convergence rate of these methods. Therefore, the Shrinking Conditional Gradient method is considered. Its difference from the conditional gradient methods is that it uses a modified linear minimization oracle. It's an oracle, which, for a given vector $p \in \mathbb{R}^n$, computes the solution of the subproblem \[ \text{Argmin}\{\langle p, \,x \rangle\colon x\in X, \;\|x-x_0^{}\| \leqslant R \}. \] The convergence rate of such an algorithm does not depend on dimension. Using the Shrinking Conditional Gradient method the complexity (the total number of arithmetic operations) of solving the minimization problem of quadratic form on a $ \infty $-ball is obtained. The resulting evaluation of the method is comparable to the complexity of the gradient method.

The work is devoted to modeling the processes of disassembling complex products in CADsystems. The ability to dismantle a product in a given sequence is formed at the early design stages, and is implemented at the end of the life cycle. Therefore, modern CAD-systems should have tools for assessing the complexity of dismantling parts and assembly units of a product. A hypergraph model of the mechanical structure of the product is proposed. It is shown that the mathematical description of coherent and sequential disassembly operations is the normal cutting of the edge of the hypergraph. A theorem on the properties of normal cuts is proved. This theorem allows us to organize a simple recursive procedure for generating all cuts of the hypergraph. The set of all cuts is represented as an AND/OR-tree. The tree contains information about plans for disassembling the product and its parts. Mathematical descriptions of various types of disassembly processes are proposed: complete, incomplete, linear, nonlinear. It is shown that the decisive graph of the AND/OR-tree is a model of disassembling the product and all its components obtained in the process of dismantling. An important characteristic of the complexity of dismantling parts is considered — the depth of nesting. A method of effective calculation of the estimate from below has been developed for this characteristic.