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In this work, we study a new implicit method to compute the solutions of ill-posed linear operator equations of the first kind under the setting of compact operators. The regularization theory can be used to demonstrate the stability and convergence of this scheme. Furthermore, we obtain convergence results and effective stopping criteria according to Morozov's discrepancy principle. Numerical performances are calculated to show the validity of our implicit method and demonstrate its applicability to deblurring problems.

The system of linear acoustic equations is hyperbolic. It describes the process of the acoustic wave propagation in deformable media. An important property of the schemes used for the numerical solution is their high approximation order. This property allows one to simulate the perturbation propagation process over sufficiently large distances. Another important property is monotonicity of the schemes used, which prevents the appearance of non-physical solution oscillations. In this paper, we present linear quasi-monotone and hybrid grid-characteristic schemes for a linear transport equation and a one-dimensional acoustic system. They are constructed by a method of analysis in the space of unknown coefficients proposed by A.S. Kholodov and a grid-characteristic monotonicity criterion. Wide spatial stencils with five to seven nodes of the computational grid are considered. Reflection of a longitudinal wave with a sharp front from the interface between media with different parameters is used to compare the numerical solutions.

In this study, we introduce multi-step collocation methods (MSCM) for solving the Volterra integral equation (VIE) of the auto-convolution type such that without increasing the computational cost, the order of convergence of the proposed one-step collocation methods will be increased. A convergence analysis of the MSCM is investigated using the Peano theorems for interpolation and, finally, two numerical examples are introduced to clarify the significant advantage of the MSCM.

The demands of accuracy in measurements and engineering models today render the condition number of problems larger. While a corresponding increase in the precision of floating point numbers ensured a stable computing, the uncertainty in convergence when using residue as a stopping criterion has increased. We present an analysis of the uncertainty in convergence when using relative residue as a stopping criterion for iterative solution of linear systems, and the resulting over/under computation for a given tolerance in error. This shows that error estimation is significant for an efficient or accurate solution even when the condition number of the matrix is not large. An Ο(1) error estimator for iterations of the CG algorithm was proposed more than two decades ago. Recently, an Ο(κ²) error estimator was described for the GMRES algorithm which allows for non-symmetric linear systems as well, where κ is the iteration number. We suggest a minor modification in this GMRES error estimation for increased stability. In this work, we also propose an Ο(n) error estimator for A-norm and l₂-norm of the error vector in Bi-CG algorithm. The robust performance of these estimates as a stopping criterion results in increased savings and accuracy in computation, as condition number and size of problems increase.

The relation between complex matrices H and A, given by the equality H A = ĀH is called the pseudo-commutation. The set S_H of all A that pseudo-commute with a nonsingular n × n matrix H is called the pseudo-commutation class defined by H . Every class S_H is a subspace of the space M_n(C) interpreted as a real vector space of dimension 2n². Under the assumption dim_R S_H = n², we find a necessary and sufficient condition for the possibility to decomplexify all the matrices in S_H by one and the same similarity transformation.

In the article, formulas for exact calculation of the approximation error of multiple Itô stochastic integrals based on their orthogonal expansion are obtained. As an example, stochastic Itô integrals with multiplicities 2-4 are considered, which are used in the numerical methods for solving stochastic differential equations with orders of strong convergence 1-2.

The paper discusses the question of why intervals, which are the main object of Interval Analysis, have exactly the form that we know well and habitually use, and not some other. In particular, we investigate why traditional intervals are closed, i.e. contain their endpoints, and also what is wrong with an empty interval. A second question considered in the work is how expedient it is to expand the set of traditional intervals by some other objects. We show that improper («reversed») intervals and the arithmetic of such intervals (the Kaucher complete interval arithmetic) are very useful from many different points of view.

The paper presents an analytical review of the multivolume floristic synopsis, “Conspectus Florae Asiae Mediae” (1968-1993), edited by the eminent Russian botanist, Professor Aleksei Ivanovich Vvedensky (1898-1972). Сheck-lists for new species as well as for new nomenclature combinations published in these books are compiled. Overall from 1968 to 1993, 10 volumes of synopsis of the Middle Asian flora were published with 8102 species, 1151 genera and 125 families. According to the materials published in the additional 11^th volume of the synopsis, the total number of vascular plants in Middle Asian flora is 9347 species.

The article is dedicated to the 90th anniversary of the birth of the famous Russian florist, algologist Tatiana Andreevna Safonova. Her work has made a great contribution to the study of the diversity, morphology and distribution of freshwater algae.

On February 15, 2023, Olga Dmitrievna Nikiforova, a well-known taxonomist of Siberian vascular plants, Doctor of Biological Sciences, a real ascetic of science and a person of great soul, passed away. All scientific activities of O.D. Nikiforova is associated with the Central Siberian Botanical Garden of the Siberian Branch of the Russian Academy of Sciences. For the multi-volume edition “Flora of Siberia” she studied the morphology, geography and ecology of more than 200 species from the families Poaceae, Fabaceae, Brassicaceae, Boraginaceae. She is the author of 114 scientific publications. In the study of the flora of Siberia and neighboring states, she described 24 new species and subspecies, 50 new sections, subsections and series in the system of genera Vicia, Mertensia, Myosotis.