Analysis of the stability of the computational algorithm to a change in the geometric parameters of cylindrical shell structures

This study deals with testing sustainability of a computational algorithm to a change in geometric parameters of cylindrical shell structures. A change in geometry implies the replacement of one type of a cylindrical shell (elliptic, hyperbolic, parabolic) with another so that the quantitative change (the difference in elevations) in the area under consideration is minimal. On the one hand, this test allows to assessing the correctness of the algorithm itself and is relevant for algorithms that use both numerical methods and symbolic calculations. On the other hand, it allows to evaluating the possibility of simplifying calculations by approximating a complex surface with a simpler one both in understanding the surface definition itself and in expressing its basic characteristics such as Lame coefficients and main curvatures. A mathematical model of deformations of shell structures based on the hypotheses of Timoshenko (Mindlin - Reisner) are used in the work. The model takes into account transverse shifts, geometric nonlinearity and orthotropy of the material, and its written in the form of a functional of the total potential strain energy. The calculation algorithm is built on the basis of the Ritz method to reduce the variational problem of the minimum functional to the solution of a system of nonlinear algebraic equations, and on the method of continuing the solution with the best parameter for its solution. All calculations were carried out in dimensionless parameters. Three types of cylindrical panels are calculated, and critical loads of buckling and deflection fields at subcritical and supercritical moments are obtained. It is shown that for the considered class of problems the previously proposed mathematical model and computational algorithm are resistant to changes in the geometry of the structure.

Analysis of the possibility of using modern packages of computer algebra in the synthesis of crypto-primitives

The paper deals with systems of computer algebra - software for symbolic calculations, which allows to conduct the entire cycle of development of a mathematical model. The paper presents the results of the analysis of systems of computer algebra with specialized purpose Magma, evaluation of the possibility of its use for modulation of processes in symmetric and asymmetric cryptographic systems, as well as recommendations for their further improvement. Magma functionality is also analyzed for evaluation of possibility to model and study promising candidates for the post-quantum standard of electronic signature algorithms, asymmetric encryption and key encapsulation, including algorithms based on cryptographic transformations in the lattice-based, the use of hash trees, mathematical codes that are undergoing research during the NIST PQC competition, as well as the draft standard "Vershina 1".

Teaching the Notion Conic Section with Computer

The article discusses the problem of introducing and constructing mathematical concepts using a computer. The Wolfram Mathematica 12 symbolic calculation system is used at each stage of the complex spiral process to form the notion of conic section and the related concepts of focus, directrix and eccentricity. The nature of these notions implies the use of appropriate animations, 3D graphics and symbolic calculations. Our vision of the process of formation of mathematical concepts is presented. The notions ellipse, parabola and hyperbola are defined as the intersection of a conical surface with a plane not containing the vertex of the conical surface. The conical section is represented as a geometric location of points on the plane for which the ratio of the distance to the focus to the distance to the directrix is a constant value. The lines of hyperbola and ellipse are determined by their foci. The equivalence of different definitions for conical sections is commented.

Decomposition of abstract linear operators in Banach spaces

In this paper we investigate a class of integro-differential equations οn a Banach space with nonlocal and initial boundary conditions in terms of an abstract operator equation B1 x=Ax-S0 F(A x)-G0 Φ(Ax)=f ,x∈D(B1) (1) where A, A are linear abstract operators, S0, G0 are vectors and Φ, F the functional vectors. The operator B1 under study has a decomposition of the form B1=B0 B with B and B0 being different abstract linear operators of special forms. Methods. Extensions of operators on Banach spaces are used. The Decomposition Method proposed here is essentially different from other Decomposition Methods in the relevant literature. Results. Our main research result is the existence and uniqueness of solution of and its representation in the closed form. The necessary and sufficient conditions for the correctness of the operator are intermediate, secondary results. Overall, a direct method analytically solving problem (1) is proposed, in an algorithmic procedure that is reproducible in any program of symbolic calculations. The stages of the solution method are illustrated by three examples. Computer algebra system Mathematica is employed to demonstrate the solution outcomes.

Possibility of using CMS Maple to study laws of distribution of random variables

The use of CMS Maple for students' practical and independent work is described. The study of random variable distribution laws is actual. Statistical calculations without computer are difficult and require many functional and quintiles tables of standard distributions. This does not contribute to feeling the element of novelty in the material being studied, to be able to arbitrarily change the conditions of tasks, etc., it takes a lot of time in solving applied production problems, which is inappropriate Thus to determine and research random variable distribution laws both in practical applications and in studying we must use special mathematical packages. The most extended of them are Mathcad, MatLab, Mathematica, Maple. Specialized statistical packages (SAS, SPSS, STATISTIKA, STATGRAPHICS) are not relevant to study. Their use for studying requires very high education level in mathematical statistics. Most of the existing math packages allow users to operate at random variables, including the Computer Mathematics System (CMS) Maple. Thus, the purpose of this article is a description of the studying possibilities of the random variables distribution laws with CMS Maple and the application of the acquired skills to the independent work of students. The Maple Statistics Library has a large set of commands for analyzing data, computing various numerical characteristics of random variables, graphing their distribution laws, and for statistical data processing. Thanks to a powerful set of statistical tools, the possibility of symbolic calculations and data processing of CMS Maple, wide possibilities of graphical interpretation of the results obtained not only in a static but also in a dynamic form, it is advisable to use it when studying the topic "Distribution Laws of Random Variables" in students' practical and independent work to use their acquired skills in solving applied problems of science and technology.

Understanding Process Dynamics and Control

Presenting a fresh look at process control, this new text demonstrates state-space approach shown in parallel with the traditional approach to explain the strategies used in industry today. Modern time-domain and traditional transform-domain methods are integrated throughout and explain the advantages and limitations of each approach; the fundamental theoretical concepts and methods of process control are applied to practical problems. To ensure understanding of the mathematical calculations involved, MATLAB® is included for numeric calculations and MAPLE for symbolic calculations, with the math behind every method carefully explained so that students develop a clear understanding of how and why the software tools work. Written for a one-semester course with optional advanced-level material, features include solved examples, cases that include a number of chemical reactor examples, chapter summaries, key terms, and concepts, as well as over 240 end-of-chapter problems, focused computational exercises and solutions for instructors.