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**Main achievements by CC FEB RAS in 2014 **

A new criterion is obtained for the validity of the weighted inequality for the Hardy integral operator with Oinarov’s kernel. All the earlier known criteria of this type had a discrete form. This new result is obtained with a discretization method which is carried out using a sequence constructed on the weight function and allows to get the final result in the integral form.

The three-dimensional problems of diffraction of acoustic waves in unbounded domains with local inclusions have been investigated. Boundary integral equations with one unknown function are applied for the numerical solution of these problems. Their use is preferable from a computational point of view since the equations’ requirements on computer resources is minimal. The results of further computational experiments demonstrate the effectiveness of this approach to the solution.

The existence and the uniqueness of the R-generalized solution is established for boundary value problems with inconsistent degeneracy of input data. There is no need for the coefficient of the function u(x) in the differential equation to be sufficiently large. This has allowed to investigate the problem for the existence and uniqueness of the solution, and to build high-performance numerical methods for boundary value problems with corner singularity, particularly in electrodynamics, hydrodynamics and elasticity theory.

An improved mathematical model of hemodynamics of large blood vessels is built with respect to the data of peripheral arterial pulsation or photoplethysmogram (PPG). The use of this model in medical practice allows to forecast the change in the parameters of the cardiovascular system and to predict outcomes of treatment to taking into account the individual characteristics of patients. Fig. 10 shows the predicted PPG through two cardiac cycles. Actual and predicted PPG shown in Fig. 1 differ slightly, this speaks in favor of the use of short-term predicted data PPG in the model.

Fig. 1. PPG signal: Measured - solid (blue) line, and predicted - dashed (red) line, the abscissa - time (s), the ordinate - PPG signal (mV). **Main achievements by CC FEB RAS in 2013 **

An integral criterion is found for the validity of a certain inequality of Hardy type. Boundedness criteria are obtained for non-linear integral operators of iterative type.

Properties of the Hardy-Steklov integral operator are studied in Lebesgue spaces on the semi-axis. New types of boundedness criteria for the Hardy-Steklov operators are obtained with applications to fractional inequalities of Sobolev theorems’ type. Two-sided estimates for Schatten norms have been found for this class of integral transforms.

A theory of numerical analysis methods is built for boundary value problems with a singularity and with the speed of convergence, independent of the size (magnitude) of the singularity caused by the presence of incoming angles on the boundary and by internal features of the solution. On the basis of the theory the weighted finite element method is developed for solving problems of elasticity theory and electrodynamics. The method is many times more accurate for finding solution and a lot more simple in the formation of the rigidity angles then all known foreign analogues. **Main achievements by CC FEB RAS in 2012**

Boundedness criterion is obtained for supremal integral Hardy operator in weighted Lorentz spaces.

The technique of numerical solutions is developed for spatial boundary value problems of the theory of fluctuations in integral formulation with spectrum points. This contributes to finding approximate solutions for the problems for all values of wave numbers, including the spectrum points in combination with violated conditions of correct solvability and equivalence of differential and integral formulation of the initial problems.

A high-performance technology together with an information system is designed to work with very large sets of data which are adaptable to solving a wide range of multidisciplinary scientific problems in various fields of knowledge.

For boundary value problems with strong singularity and a Dirichlet integral diverging from the solution, on the basis of the introduction of the definition of R-generalized solution, a weight rib finite elements method with a high rate of convergence of approximate solutions to the exact solution non-dependent on the type and order of singularity is established.

An extension of the theory of technical shells is constructed for geometrically nonlinear problems of the dynamics of underground pipelines. A method for analyzing the motion of curved metal pipes is substantiated, the method is based on the geometry of pipes and allows to reduce the number of independent variables.

** **** Main achievements of CC FEB RAS in 2010**

Regularization technique for numerical analysis methods based on setting locally adaptive weight basis in the vicinity of the points of singularities has been developed for boundary value problems with consistent and inconsistent degeneration of initial data and with strong singularity of solution in which the Dirichlet integral diverges from the solutions or the generalized solution has a weak regularity. Based on it, the weight edge finite element method for calculating systems of the Mawell's equations with strong singularity has been created and successfully tested.

We have formulated the problem of stability of sandy channel bottom of rectangular form relative to two-dimensional perturbations over the space. The channel stability problem has been solved taking into account the refined formula of sediment discharge. The correction is related to the account of influence of free surface perturbations on drawn sediment transportation. Using the refined formula of sediment allowed to obtain analytical expressions for the velocity of the bottom perturbation and the wave length for maximum fast-growing bottom disturbances for the small Froude numbers.

By potential method the three-dimensional stationary problem of diffraction of elastic waves has been reduced to a system of singular integral equations of the surface. The approximate solution is found by approximating the system of integral equations by a system of linear algebraic equations. Herewith, the partition of identity on the surface of inclusion and a new method of averaging integral operators with weak and strong singularities in the nucleus, coherent with the order of sampling. This eliminates the need for pre-triangulation of the surface and allows to find the numerical solution without using regularization algorithms.

** Main achievements of CC FEB RAS in 2009**

It was shown that in the case of a bounded integral operator acting from the space of functions summable with degree 0<p<1 on a continuous (non-atomic) measure in the Lebesgue space with a countably finite measure, the operator is zero. Criteria for the implementation of the Hardy inequality with three countably finite measures on the real axis for 0<p<1 were obtained. This result complements the known results for the Hardy operator for sums and absolutely continuous measures.

We obtained new necessary and sufficient conditions of Lp-Lq-boundedness of the Hardy-Steklov operator with two increasing on (0, ∞) boundary functions of integration a (x) and b (x) for 1<p≤q<∞ и 0<q<p<∞, p>1. The applications of the results to the weight characterization of the corresponding Lp-Lq-inequality for the geometric Steklov operator and other problems from adjacent areas.

The general theory of differential properties and methods of numerical analysis of boundary value problems with strong singularity of the solution, in which the Dirichlet integral diverges from the solution or a generalized solution has a weak regularity, was built and systematized. The developed theory allows us to find a solution with high accuracy for problems of electrodynamics, hydrodynamics and the theory of elasticity with singularities caused by the presence of the cuts (cracks) and the corner points on the boundary.

The evolutionary development problem of the cross-section of basic trapezoidal channel with different physical and mechanical and grain-size properties of bottom material was developed. The numerical method and solution algorithm were suggested. It was shown that at characteristic transitional period of dominant formative discharge the profile of the bottom surface acquires a shape approximated by power-law, which agrees well with the field experimental data.

The theoretical questions of obtaining solutions of three-dimensional boundary problems of the dynamics of highly viscous homogeneous and inhomogeneous incompressible media were considered. The investigation of the conditions of existence and uniqueness of solutions, the analysis of their differential properties and the continuous dependence on initial data was carried out. The construct solutions that can serve as a basis for constructing efficient algorithms for numerical calculations, as well as for estimating approximations were presented. The theoretical justification of the proposed method is illustrated by the calculations of model examples.

** Main achievements of CC FEB RAS in 2008**

The satisfaction criterion for the Hardy inequality in the Lebesgue spaces with arbitrary countably finite measures for the case 0<p<1. was obtained. It was shown that the only bounded integral operator acting from the space of functions summable with degree 0<p<1 with respect to continuous measure, into the Lebesgue space with a countably finite measure, is the zero operator.

New criteria were obtained for satisfaction of weighted Hardy inequality with variable limits of integration for all 0<p,q<∞. As an application we solved the problem of characterization of the corresponding inequality for the operator of the geometric mean.

** Main achievements of CC FEB RAS in 2007**

*Fig. 1. Integration of the Corporate Network of FEB RAS in educational research networks in Russia and global research and education networks.*