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Journal articles on the topic 'Solution of mathematical problems'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Arcavi, Abraham, and Zippora Resnick. "Generating Problems from Problems and Solutions from Solutions." Mathematics Teacher 102, no. 1 (2008): 10–14. http://dx.doi.org/10.5951/mt.102.1.0010.

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Many mathematical problems can be completely solved analytically. Usually, when we reach a solution, we tend to be satisfied and consider the task finished. In this article, we illustrate the practice of reexamining the result of a full analytic solution, an approach that leads to the search for an alternative geometrical solution and thus enhances our understanding of both solutions.

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Arcavi, Abraham, and Zippora Resnick. "Generating Problems from Problems and Solutions from Solutions." Mathematics Teacher 102, no. 1 (2008): 10–14. http://dx.doi.org/10.5951/mt.102.1.0010.

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Many mathematical problems can be completely solved analytically. Usually, when we reach a solution, we tend to be satisfied and consider the task finished. In this article, we illustrate the practice of reexamining the result of a full analytic solution, an approach that leads to the search for an alternative geometrical solution and thus enhances our understanding of both solutions.

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3

Okino, Takahisa. "New Mathematical Solution for Analyzing Interdiffusion Problems." MATERIALS TRANSACTIONS 52, no. 12 (2011): 2220–27. http://dx.doi.org/10.2320/matertrans.m2011137.

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4

Cezikturk, Ozlem. "Multiple solution problems and mathematical thinking: Wasan geometry example." Academic Perspective Procedia 2, no. 1 (2019): 37–46. http://dx.doi.org/10.33793/acperpro.02.01.10.

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Wasan geometry flourished in Edo period of Japan as arty, multiple solutioned and systematic. Teachers are not open and papers are not evaluated. Multiple solutions are seen in parallel with advanced mathematics thinking. 80 middle school and 37 high school pre service math teachers were given a Wasan problem and grouped according to correct answers, different answers, and originality. 24 different answers (equality of diagonals of a square, symmetry, equality of tangent lines, auxiliary lines and auxiliary shapes, cosines’ theorem, tan (67.5), similarities, types of triangles etc.) were detected. Answers were also coded for composite answers (who include two different rules), or simple one ways, first answers, similar solution ways and 5-6 different solutions.

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Krivulin, N. K., and J. V. Romanovsky. "Solution of mathematical programming problems using tropical optimization methods." Vestnik St. Petersburg University, Mathematics 50, no. 3 (2017): 274–81. http://dx.doi.org/10.3103/s1063454117030104.

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6

Gandhi, K. Raja Rama. "Solution of some Problems in Hungarian Mathematical Competition - IV." Bulletin of Society for Mathematical Services and Standards 5 (March 2013): 46–48. http://dx.doi.org/10.18052/www.scipress.com/bsmass.5.46.

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7

Ursell, F. "Water-wave problems, their mathematical solution and physical interpretation." Journal of Engineering Mathematics 58, no. 1-4 (2006): 7–17. http://dx.doi.org/10.1007/s10665-006-9092-8.

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8

Senchukov, Viktor. "The solution of optimization problems in the economy by overlaying integer lattices: applied aspect." Economics of Development 18, no. 1 (2019): 44–55. http://dx.doi.org/10.21511/ed.18(1).2019.05.

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The results of generalization of scientific approaches to the solution of modern economic optimization tasks have shown the need for a new vision of their solution based on the improvement of existing mathematical tools. It is established that the peculiarities of the practical use of existing mathematical tools for solving economic optimization problems are caused by the problems of enterprise management in the presence of nonlinear processes in the economy, which also require consideration of the corresponding characteristics of nonlinear dynamic processes. The approach to solving the problem of integer (discrete) programming associated with the difficulties that arise when applying precise methods (methods of separation and combinatorial methods) is proposed, namely: a fractional Gomorrhic algorithm – for solving entirely integer problems (by gradual &quot;narrowing&quot; areas of admissible solutions of the problem under consideration); the method of branches and borders - which involves replacing the complete overview of all plans by their partial directional over. Illustrative examples of schemes of geometric programming, fractional-linear programming, nonlinear programming with a non-convex region, fractional-nonlinear programming with a non-convex domain, and research on the optimum model of Cobb-Douglas model are given. The advanced mathematical tools on the basis of the method of overlaying integer grids (OIG), which will solve problems of purely discrete, and not only integer optimization, as an individual case, are presented in the context of solving optimization tasks of an applied nature and are more effective at the expense of reducing the complexity and duration of their solving. It is proved that appropriate analytical support should be used as an economic and mathematical tool at the stage of solving tasks of an economic nature, in particular optimization of the parameters of the processes of organization and preparation of production of new products of the enterprises of the real sector of the economy.

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Mačys, Juvencijus, and Jurgis Sušinskas. "On Lithuanian mathematical olympiads." Lietuvos matematikos rinkinys, no. 59 (December 20, 2018): 54–60. http://dx.doi.org/10.15388/lmr.b.2018.8.

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Some problems of Lithuanian school mathematical olympiad 2018 and Vilnius University mathematicalolympiad 2018 are considered. Different methods of solution are compared. Some usefulcommon advices for solving mathematical problems are given.

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10

TREFETHEN, LLOYD N. "SERIES SOLUTION OF LAPLACE PROBLEMS." ANZIAM Journal 60, no. 1 (2018): 1–26. http://dx.doi.org/10.1017/s1446181118000093.

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At the ANZIAM conference in Hobart in February 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components.

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11

Сальков and Nikolay Sal'kov. "Graph-analytic Solution of Some Special Problems of Quadratic Programming." Geometry & Graphics 2, no. 1 (2014): 3–8. http://dx.doi.org/10.12737/3842.

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Quadratic programming problems are one of special cases of mathematical programming problems. Mathematical programming problems solution is of great importance, because these problems are those of optimizing of solution related to presented issues from multitude of possible ones. The mathematical programming problems are linear, nonlinear, dynamic and others. It is suggested to consider a graph-analytic solution of quadratic programming’s special problems, which, taken together, constitute the quadratic programming problems for two and three variables. A total of eight special problems have been considered.

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12

Bulatov, V. P., and T. I. Belykh. "Numerical solution methods for multiextremal problems connected with inverse problems in mathematical programming." Russian Mathematics 51, no. 6 (2007): 11–17. http://dx.doi.org/10.3103/s1066369x07060023.

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13

Wu, Min-Thai, Tzung-Pei Hong, and Chung-Nan Lee. "Using the ACS Approach to Solve Continuous Mathematical Problems in Engineering." Mathematical Problems in Engineering 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/142194.

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Ant colony system (ACS) has been widely applied for solving discrete domain problems in recent years. In particular, they are efficient and effective in finding nearly optimal solutions to discrete search spaces. Because of the restriction of ant-based algorithms, when the solution space of a problem to be solved is continuous, it is not so appropriate to use the original ACS to solve it. However, engineering mathematics in the real applications are always applied in the continuous domain. This paper thus proposes an extended ACS approach based on binary-coding to provide a standard process for solving problems with continuous variables. It first encodes solution space for continuous domain into a discrete binary-coding space (searching map), and a modified ACS can be applied to find the solution. Each selected edge in a complete path represents a part of a candidate solution. Different from the previous ant-based algorithms for continuous domain, the proposed binary coding ACS (BCACS) could retain the original operators and keep the benefits and characteristics of the traditional ACS. Besides, the proposed approach is easy to implement and could be applied in different kinds of problems in addition to mathematical problems. Several constrained functions are also evaluated to demonstrate the performance of the proposed algorithm.

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14

Rajić, Dušan. "Mathematical-physical model of solving inventive problems." FME Transactions 49, no. 3 (2021): 726–33. http://dx.doi.org/10.5937/fme2103726r.

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The spatial-temporal LT-contradiction matrix is an inventology tool that enables exact calculations of certain parameters in an engineering system through mathematical-physical modeling. It objectifies the decision-making process and creates the preconditions to finding an adequate resource (X-element) with a higher probability, and thus to reach a higher degree of ideality solution (HDIS) of an inventive problem as well. Any engineering system that generates an inventive problem can be described using the LT-contradiction matrix. By crossing the appropriate parameters in the LT-contradiction matrix, with the help of the differential geometry of the tensor, a qualitative-quantitative analysis and calculation of relevant degree all contradictions that exist in the inventive problem can be performed. After that, the path to finding the physical characteristics of the X-element in the mathematical-physical model is facilitated, i.e. finding a real resource that will enable a HDIS of the inventive problem in an engineering system.

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15

Bellomo, N., P. LeTallec, and B. Perthame. "Nonlinear Boltzmann Equation Solutions and Applications to Fluid Dynamics." Applied Mechanics Reviews 48, no. 12 (1995): 777–94. http://dx.doi.org/10.1115/1.3005093.

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This paper provides a review of the mathematical results on the solution of the nonlinear Boltzmann equation. The survey deals both with analytical and computational aspects: Mathematical formulation of problems, initial and/or boundary value problems, a survey of the qualitative analysis of solutions, and computational treatment of fluid dynamical problems. A discussion of some problems deserving further study concludes this work.

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Krivulin, Nikolai K., and Joseph V. Romanovsky. "Solution of mathematical programming problems using methods of tropical optimization." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 4(62), no. 3 (2017): 448–58. http://dx.doi.org/10.21638/11701/spbu01.2017.307.

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17

Donets, G. A., and Bin Zhan. "Statement and solution of some problems on a mathematical safe." Cybernetics and Systems Analysis 42, no. 3 (2006): 311–19. http://dx.doi.org/10.1007/s10559-006-0067-6.

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18

Preziosi, L., G. Teppati, and N. Bellomo. "Modeling and solution of stochastic inverse problems in mathematical physics." Mathematical and Computer Modelling 16, no. 5 (1992): 37–51. http://dx.doi.org/10.1016/0895-7177(92)90118-5.

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19

Andreani, Roberto, and José Mario Martı´nez. "On the solution of mathematical programming problems with equilibrium constraints." Mathematical Methods of Operations Research 54, no. 3 (2001): 345–58. http://dx.doi.org/10.1007/s001860100158.

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20

Shcherbakova, E. E., and S. Yu Knyazev. "Numerical solution of mathematical physics problems by the collocation method." IOP Conference Series: Materials Science and Engineering 1029 (January 19, 2021): 012037. http://dx.doi.org/10.1088/1757-899x/1029/1/012037.

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21

Muhtarom, Muhtarom, Ali Shodiqin, and Novita Astriani. "Exploring senior high school student's abilities in mathematical problem posing." JRAMathEdu (Journal of Research and Advances in Mathematics Education) 5, no. 1 (2020): 69–79. http://dx.doi.org/10.23917/jramathedu.v5i1.9818.

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The ability of problem posing is very essential for the student’s. However, there are still many students who don't realize the importance of these abilities. This research describes the senior high school student’s ability in mathematical problem posing, especially in the material system of linear equations in three variables. Research data were collected from 7 student’s using written test and interview techniques. The validity of the data used triangulation methods by comparing the results of written tests and interviews. Data were coded, simplified, presented, and triangulated for the credibility and conclusion drawing. The results show that there were still very few students who have all three classifications of problem posing abilities, namely pre-solution posing, within-solution posing, and post-solution posing. Students who have the ability of pre-solution posing can ask questions based on the data provided and can arrange problem solving. Students who have the ability of within-solution posing can write what is given and asked of the problem, raise supporting questions which is relevant to the problem and arrange solutions to the supporting questions and problems that are given correctly. Students who have the ability of post-solution posing can raise similar mathematics problem after solving the problem. Students can also arrange solutions to problems that have been made. Teacher needs to practice pre-solution posing, within-solution posing, and post-solution posing to the students.

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22

Vermolen, F. J., C. Vuik, E., and S. Van der Zwaag. "Review on some Stefan Problems for Particle Dissolution in Solid Metallic Alloys." Nonlinear Analysis: Modelling and Control 10, no. 3 (2005): 257–92. http://dx.doi.org/10.15388/na.2005.10.3.15124.

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This paper is a review of a suite of mathematical models of increasing complexity on particle dissolution in metallic alloys. This work deals with models for multi-component particle dissolution in multi-component alloys, where various chemical species diffuse simultaneously, and a two-dimensional model incorporating interfacial reactions as in the model of Nolfi [1]. The work is mathematically rigorous where asymptotic solutions and solution bounds are derived but is also of a practical nature as particle dissolution kinetics is modelled for industrially relevant conditions.

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23

Hosseinpour, Sahereh, Mir Mohammad Reza Alavi Milani, and Hüseyin Pehlivan. "A Step-by-Step Solution Methodology for Mathematical Expressions." Symmetry 10, no. 7 (2018): 285. http://dx.doi.org/10.3390/sym10070285.

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In this paper, we propose a methodology for the step-by-step solution of problems, which can be incorporated into a computer algebra system. Our main aim is to show all the intermediate evaluation steps of mathematical expressions from the start to the end of the solution. The first stage of the methodology covers the development of a formal grammar that describes the syntax and semantics of mathematical expressions. Using a compiler generation tool, the second stage produces a parser from the grammar description. The parser is used to convert a particular mathematical expression into an Abstract Syntax Tree (AST), which is evaluated in the third stage by traversing al its nodes. After every evaluation of some nodes, which corresponds to an intermediate solution step of the related expression, the resulting AST is transformed into the corresponding mathematical expression and then displayed. Many other algebra-related issues such as simplification, factorization, distribution and substitution can be covered by the solution methodology. We currently focuses on the solutions of various problems associated with the subject of derivative, equations, single variable polynomials, and operations on functions. However, it can easily be extended to cover the other subjects of general mathematics.

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24

Ben-Chaim, David, Yechiel Shalitin, and Moshe Stupel. "Historical mathematical problems suitable for classroom activities." Mathematical Gazette 103, no. 556 (2019): 12–19. http://dx.doi.org/10.1017/mag.2019.2.

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The human mind, by nature, is curious and enjoys dealing, both independently and competitively, with intellectual challenges. Throughout time, mathematical tasks, riddles, and puzzles have offered such challenges. In today's modern era, marketing companies and the media even offer prizes and rewards for the successful solution of puzzles, expanding the audience exposed to the various challenges. In addition, the styles, subjects and depths of the puzzles have become diversified. Experience gained in solving puzzles aids in developing reasoning abilities and deepening thought.

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25

Mashunin. "Mathematical Apparatus of Optimal Decision-Making Based on Vector Optimization." Applied System Innovation 2, no. 4 (2019): 32. http://dx.doi.org/10.3390/asi2040032.

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We present a problem of “acceptance of an optimal solution” as a mathematical model in the form of a vector problem of mathematical programming. For the solution of such a class of problems, we show the theory of vector optimization as a mathematical apparatus of acceptance of optimal solutions. Methods of solution of vector problems are directed to problem solving with equivalent criteria and with the given priority of a criterion. Following our research, the analysis and problem definition of decision making under the conditions of certainty and uncertainty are presented. We show the transformation of a mathematical model under the conditions of uncertainty into a model under the conditions of certainty. We present problems of acceptance of an optimal solution under the conditions of uncertainty with data that are represented by up to four parameters, and also show geometrical interpretation of results of the decision. Each numerical example includes input data (requirement specification) for modeling, transformation of a mathematical model under the conditions of uncertainty into a model under the conditions of certainty, making optimal decisions with equivalent criteria (solving a numerical model), and, making an optimal decision with a given priority criterion.

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Evirgen, Fırat, and Mehmet Yavuz. "An Alternative Approach for Nonlinear Optimization Problem with Caputo - Fabrizio Derivative." ITM Web of Conferences 22 (2018): 01009. http://dx.doi.org/10.1051/itmconf/20182201009.

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In this study, a fractional mathematical model with steepest descent direction is proposed to find optimal solutions for a class of nonlinear programming problem. In this sense, Caputo-Fabrizio derivative is adapted to the mathematical model. To demonstrate the solution trajectory of the mathematical model, we use the multistage variational iteration method (MVIM). Numerical simulations and comparisons on some test problems show that the mathematical model generated using Caputo-Fabrizio fractional derivative is both feasible and efficient to find optimal solutions for a certain class of equality constrained optimization problems.

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NOUY, A., and C. SOIZE. "Random field representations for stochastic elliptic boundary value problems and statistical inverse problems." European Journal of Applied Mathematics 25, no. 3 (2014): 339–73. http://dx.doi.org/10.1017/s0956792514000072.

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This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of non-Gaussian positive-definite matrix-valued random fields, adapted to the statistical inverse problems in high stochastic dimension for their experimental identification, is introduced and its properties are analysed. A minimal parameterisation of discretised random fields belonging to this general class is proposed. Using this parameterisation of the general class, a complete identification procedure is proposed. New results of the mathematical and numerical analyses of the parameterised stochastic elliptic boundary value problem are presented. The numerical solution of this parametric stochastic problem provides an explicit approximation of the application that maps the parameterised general class of random fields to the corresponding set of random solutions. This approximation can be used during the identification procedure in order to avoid the solution of multiple forward stochastic problems. Since the proposed general class of random fields possibly contains random fields which are not uniformly bounded, a particular mathematical analysis is developed and dedicated approximation methods are introduced. In order to obtain an algorithm for constructing the approximation of a very high-dimensional map, complexity reduction methods are introduced and are based on the use of sparse or low-rank approximation methods that exploit the tensor structure of the solution which results from the parameterisation of the general class of random fields.

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Lubyshev, Fedor V., and Mahmut E. Fairuzov. "On an iterative process for the grid conjugation problem with iterations on the boundary of the solution discontinuity." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 21, no. 3 (2019): 329–42. http://dx.doi.org/10.15507/2079-6900.21.201903.329-342.

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An iterative process for the grid problem of conjugation with iterations on the boundary of the discontinuity of the solution is considered. Similar grid problem arises in difference approximation of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions. The study of iterative processes for the states of such problems is of independent interest for theory and practice. The paper shows that the numerical solution of boundary problems of this type can be efficiently implemented using iterations on the inner boundary of the grid solution discontinuity in combination with other iterative methods for nonlinearities separately in each of the grid subregions. It can be noted that problems for states of controlled processes described by equations of mathematical physics with discontinuous coefficients and solutions arise in mathematical modeling and optimization of heat transfer, diffusion, filtration, elasticity theory, etc. The proposed iterative process reduces the solution of the initial grid boundary problem for a state with a discontinuous solution to a solution of two special boundary problems in two grid subdomains at every fixed iteration. The convergence of the iteration process in the Sobolev grid norms to the unique solution of the grid problem for each initial approximation is proved.

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Alderremy, A. A., Hassan Khan, Rasool Shah, Shaban Aly, and Dumitru Baleanu. "The Analytical Analysis of Time-Fractional Fornberg–Whitham Equations." Mathematics 8, no. 6 (2020): 987. http://dx.doi.org/10.3390/math8060987.

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This article is dealing with the analytical solution of Fornberg–Whitham equations in fractional view of Caputo operator. The effective method among the analytical techniques, natural transform decomposition method, is implemented to handle the solutions of the proposed problems. The approximate analytical solutions of nonlinear numerical problems are determined to confirm the validity of the suggested technique. The solution of the fractional-order problems are investigated for the suggested mathematical models. The solutions-graphs are then plotted to understand the effectiveness of fractional-order mathematical modeling over integer-order modeling. It is observed that the derived solutions have a closed resemblance with the actual solutions. Moreover, using fractional-order modeling various dynamics can be analyzed which can provide sophisticated information about physical phenomena. The simple and straight-forward procedure of the suggested technique is the preferable point and thus can be used to solve other nonlinear fractional problems.

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30

Lovianova, Iryna V., Dmytro Ye Bobyliev, and Aleksandr D. Uchitel. "Cloud calculations within the optional course Optimization Problems for 10th–11th graders." Освітній вимір 53, no. 1 (2019): 95–110. http://dx.doi.org/10.31812/educdim.v53i1.3835.

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The article deals with the problem of introducing cloud calculations into 10th–11th graders’ training to solve optimization problems in the context of the STEM-education concept. After analyzing existing programmes of optional courses on optimization problems, the programme of the optional course Optimization Problems has been developed and substantiated implying solution of problems by the cloud environment CoCalc. It is a routine calculating operation and not a mathematical model that is accentuated in the programme. It allows considering more problems which are close to reality without adapting the material while training 10th–11th graders. Besides, the mathematical apparatus of the course which is partially known to students as the knowledge acquired from such mathematics sections as the theory of probability, mathematical statistics, mathematical analysis and linear algebra is enough to master the suggested course. The developed course deals with a whole class of problems of conventional optimization which vary greatly. They can be associated with designing devices and technological processes, distributing limited resources and planning business functioning as well as with everyday problems of people. Devices, processes and situations to which a model of optimization problem is applied are called optimization problems. Optimization methods enable optimal solutions for mathematical models. The developed course is noted for building mathematical models and defining a method to be applied to finding an efficient solution.

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31

Ulpah, Maria. "Characteristics of Students’ Intuitive Thinking in Solving Mathematical Problems." International Conference of Moslem Society 3 (April 12, 2019): 48–57. http://dx.doi.org/10.24090/icms.2019.2327.

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Intuition is one of important thing in the process of solving mathematical problems. It works as cognitive mediation. In this understanding, intuition can be made as a bridge to students' understanding so that it can be accessed in linking imagined objects with the desired alternative solutions. In other words, students can determine what strategies or steps should be taken to get a problem solution, especially contextual problems that have completion steps that cannot be accessed directly. Intuitive thinking often occurs in mathematical problem solving. This was also seen in the mathematical students of IAIN Purwokerto. Based on the teaching experience so far, it was found that many students gave spontaneous answers without analyzing first. So, the researcher studied how characteristics of students’ intuitive thinking are. This research used qualitative with descriptive-exploratory type of research and used test to identify the characteristics of students’ intuitive thinking in solving mathematical problems. Results showed that students’ characteristics consisted of extrapolative, implicitly, persistently, coercively, and the power of synthesis.

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32

Konovalov, Dmitriy, Anatoly Vershinin, Konstantin Zingerman, and Vladimir Levin. "The Implementation of Spectral Element Method in a CAE System for the Solution of Elasticity Problems on Hybrid Curvilinear Meshes." Modelling and Simulation in Engineering 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/1797561.

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Modern high-performance computing systems allow us to explore and implement new technologies and mathematical modeling algorithms into industrial software systems of engineering analysis. For a long time the finite element method (FEM) was considered as the basic approach to mathematical simulation of elasticity theory problems; it provided the problems solution within an engineering error. However, modern high-tech equipment allows us to implement design solutions with a high enough accuracy, which requires more sophisticated approaches within the mathematical simulation of elasticity problems in industrial packages of engineering analysis. One of such approaches is the spectral element method (SEM). The implementation of SEM in a CAE system for the solution of elasticity problems is considered. An important feature of the proposed variant of SEM implementation is a support of hybrid curvilinear meshes. The main advantages of SEM over the FEM are discussed. The shape functions for different classes of spectral elements are written. Some results of computations are given for model problems that have analytical solutions. The results show the better accuracy of SEM in comparison with FEM for the same meshes.

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Denton, Brian H., A. M. Yaglom, and I. M. Yaglom. "Challenging Mathematical Problems with Elementary Solutions." Mathematical Gazette 72, no. 462 (1988): 337. http://dx.doi.org/10.2307/3619973.

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34

Pollard, W. F. "Challenging Mathematical Problems with BASIC Solutions." Mathematical Gazette 73, no. 465 (1989): 253. http://dx.doi.org/10.2307/3618475.

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35

Khare, Avinash. "Mathematical Physics:With Applications, Problems & Solutions." Current Science 115, no. 5 (2018): 987. http://dx.doi.org/10.18520/cs/v115/i5/987-987.

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36

Zillober, Ch. "On the solution of large engineering design problems via mathematical programming." PAMM 1, no. 1 (2002): 480. http://dx.doi.org/10.1002/1617-7061(200203)1:1<480::aid-pamm480>3.0.co;2-7.

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37

Gurin, Artem L., Irina S. Grashchenko, and Lidia V. Savchenko. "PARAMETRIC METHOD OF SOLVING PROBLEMS OF MATHEMATICAL SAFE ON GRAPHS." Journal of Automation and Information sciences 2 (March 1, 2021): 5–10. http://dx.doi.org/10.34229/0572-2691-2021-2-1.

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We consider one method of solving the problem of mathematical safe on certain graphs called parametric. Its gist consist in denoting some variables, corresponding to graph vertices, by certain parameters. Other unknown variables are expressed through these parameters. Then unknown variables chosen in special way are compared and the mentioned parameters are found by solving additional system of equations for these parameters. Dimension of this system is equal to the number of parameters. Solution to the problem i.e. all unknown variables of the original system, are found by solving additional system of equations. In the paper this method is described on specially chosen examples. The method is demonstrated by solving the mathematical safe problem on the graphs of «chain», «ladder» and «window» types that showed its efficiency. Besides special attention is paid to special cases when solution does not exist. This occurs in the cases when the weighed sum of system equations is not divisable without remainder to its modulo. In such cases, to find solution the initial state of the vector b is corrected in such a way that the weighted sum of equations satisfies the above mentioned condition. Then solution of the problem is performed according to the general method scheme.

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38

Hald, O. H., and J. R. McLaughlin. "Solution of inverse nodal problems." Inverse Problems 5, no. 3 (1989): 307–47. http://dx.doi.org/10.1088/0266-5611/5/3/008.

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39

Judice, J. J., and G. Mitra. "Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods." Journal of Optimization Theory and Applications 57, no. 1 (1988): 123–49. http://dx.doi.org/10.1007/bf00939332.

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40

Polyakov, Sergey, and Viktoriia Podryga. "Multiscale Multilevel Approach to Solution of Nanotechnology Problems." EPJ Web of Conferences 173 (2018): 01010. http://dx.doi.org/10.1051/epjconf/201817301010.

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The paper is devoted to a multiscale multilevel approach for the solution of nanotechnology problems on supercomputer systems. The approach uses the combination of continuum mechanics models and the Newton dynamics for individual particles. This combination includes three scale levels: macroscopic, mesoscopic and microscopic. For gas–metal technical systems the following models are used. The quasihydrodynamic system of equations is used as a mathematical model at the macrolevel for gas and solid states. The system of Newton equations is used as a mathematical model at the mesoand microlevels; it is written for nanoparticles of the medium and larger particles moving in the medium. The numerical implementation of the approach is based on the method of splitting into physical processes. The quasihydrodynamic equations are solved by the finite volume method on grids of different types. The Newton equations of motion are solved by Verlet integration in each cell of the grid independently or in groups of connected cells. In the framework of the general methodology, four classes of algorithms and methods of their parallelization are provided. The parallelization uses the principles of geometric parallelism and the efficient partitioning of the computational domain. A special dynamic algorithm is used for load balancing the solvers. The testing of the developed approach was made by the example of the nitrogen outflow from a balloon with high pressure to a vacuum chamber through a micronozzle and a microchannel. The obtained results confirm the high efficiency of the developed methodology.

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Stanojević, Bogdana, Milan Stanojević, and Sorin Nădăban. "Reinstatement of the Extension Principle in Approaching Mathematical Programming with Fuzzy Numbers." Mathematics 9, no. 11 (2021): 1272. http://dx.doi.org/10.3390/math9111272.

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Optimization problems in the fuzzy environment are widely studied in the literature. We restrict our attention to mathematical programming problems with coefficients and/or decision variables expressed by fuzzy numbers. Since the review of the recent literature on mathematical programming in the fuzzy environment shows that the extension principle is widely present through the fuzzy arithmetic but much less involved in the foundations of the solution concepts, we believe that efforts to rehabilitate the idea of following the extension principle when deriving relevant fuzzy descriptions to optimal solutions are highly needed. This paper identifies the current position and role of the extension principle in solving mathematical programming problems that involve fuzzy numbers in their models, highlighting the indispensability of the extension principle in approaching this class of problems. After presenting the basic ideas in fuzzy optimization, underlying the advantages and disadvantages of different solution approaches, we review the main methodologies yielding solutions that elude the extension principle, and then compare them to those that follow it. We also suggest research directions focusing on using the extension principle in all stages of the optimization process.

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Rustamov, G. A. "Some Feasibility Problems in the Exact Solution of Сontrol Exercises". Mekhatronika, Avtomatizatsiya, Upravlenie 21, № 10 (2020): 555–65. http://dx.doi.org/10.17587/mau.21.555-565.

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There is studied the mathematical foundations of the synthesis methodology in the engineering interpretation of a number of popular feedback control systems and the reasons for the impracticability of the results due to the appearance in the synthesis equations of pure differentiation operators and sources of various types of roughness violation. The global reason for the increasingly accelerated divergence of control theory from practice is associated with the impact on creative thinking such as mutation, incompatibility, randomness, fuzziness, asymmetry which underlies the evolution of synergetic systems. Both the "methodological crisis" and a number of seemingly insignificant engineering inconsistencies lead to a decrease in the planned efficiency of the developed control systems. There is a tendency to solve this practical problem through its excessive mathematization. As a result, there is nonsense — "the more mathematics, the worse", which leads to a "mathematical labyrinth", to exit from which the mathematical apparatus becomes more and more complicated until the creation of a new theory. It is shown that the use of even "correct" mathematical relations, which are the basis of the synthesis method, often leads to a violation of feasibility and rudeness. It is cited that the neglect of important poorly formalized technical indicators and the conditions of rudeness (robustness) when setting the problem does not allow us to obtain a constructive solution and is one of the main reasons for the discrepancy between theoretical results and practice. A number of popular directions of the classical theory of feedback control are considered: an inverse approach-compensation method, which forms the basis for constructing astatic, invariant, robust and other compensation systems; synthesis methods for systems with a finite settling time; assessment and control methods based on the concept of " inverse dynamics problems"; high gain limit systems. Violation of various types of feasibility and rudeness is demonstrated by specific examples tested on Matlab / Simulink. Computer research has made it possible to draw a number of positive conclusions that have important applied value.

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Orlov, Victor, and Oleg Kovalchuk. "Mathematical problems of reliability assurance the building constructions." E3S Web of Conferences 97 (2019): 03031. http://dx.doi.org/10.1051/e3sconf/20199703031.

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The paper deals with a mathematical model of console type based on the nonlinear differential equation having a mobile feature of the General solution (or a mobile singular point). The presence of mobile singular points indicates affiliation of this type of equations to the class of intractable in the general case in of quadratures. This fact, taking into account the interpretation of mobile singular point as the coordinate of structural failure, actualizes the development of an analytical approximate method for solving nonlinear differential equations. Taking into account these features for of structural analysis increases the authenticity of results and reliability of construction.

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44

Kojecký, Tomáš. "Iterative solution of eigenvalue problems for normal operators." Applications of Mathematics 35, no. 2 (1990): 158–61. http://dx.doi.org/10.21136/am.1990.104397.

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45

Kané, Ladji, Daouda Diawara, Lassina Diabaté, Moussa Konaté, Souleymane Kané, and Hawa Bado. "A Mathematical Model for Solving the Linear Programming Problems Involving Trapezoidal Fuzzy Numbers via Interval Linear Programming Problems." Journal of Mathematics 2021 (April 16, 2021): 1–17. http://dx.doi.org/10.1155/2021/5564598.

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We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

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Vabishchevich, P. N. "Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids." Computational Methods in Applied Mathematics 5, no. 3 (2005): 294–330. http://dx.doi.org/10.2478/cmam-2005-0015.

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Abstract Mathematical physics problems are often formulated by means of the vector analysis differential operators: divergence, gradient and rotor. For approximate solutions of such problems it is natural to use the corresponding operator statements for the grid problems, i.e., to use the so-called VAGO (Vector Analys Grid Operators) method. In this paper, we discuss the possibilities of such an approach in using gen- eral irregular grids. The vector analysis di®erence operators are constructed using the Delaunay triangulation and the Voronoi diagrams. The truncation error and the consistency property of the di®erence operators constructed on two types of grids are investigated. Construction and analysis of the di®erence schemes of the VAGO method for applied problems are illustrated by the examples of stationary and non-stationary convection-diffusion problems. The other examples concerned the solution of the non- stationary vector problems described by the second-order equations or the systems of first-order equations.

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Chowdhury, Atiqur, Saleh Tanveer, and Xueying Wang. "Nonlinear two-point boundary value problems: applications to a cholera epidemic model." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2234 (2020): 20190673. http://dx.doi.org/10.1098/rspa.2019.0673.

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This paper is concerned primarily with constructive mathematical analysis of a general system of nonlinear two-point boundary value problem when an empirically constructed candidate for an approximate solution ( quasi-solution ) satisfies verifiable conditions. A local analysis in a neighbour- hood of a quasi-solution assures the existence and uniqueness of solutions and, at the same time, provides error bounds for approximate solutions. Applying this method to a cholera epidemic model, we obtain an analytical approximation of the steady-state solution with rigorous error bounds that also displays dependence on a parameter. In connection with this epidemic model, we also analyse the basic reproduction number, an important threshold quantity in the epidemiology context. Through a complex analytic approach, we determine the principal eigenvalue to be real and positive in a range of parameter values.

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Engel, Bill, and Diane Schmidt. "The Galactic Spaceship Tour Challenge." Mathematics Teacher 97, no. 5 (2004): 314–19. http://dx.doi.org/10.5951/mt.97.5.0314.

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Many students believe that the solution to most mathematics problems is fairly direct, involves only one step, and provides one final answer. In most real-life situations, however, mathematical problem solving involves several steps to obtain a complete solution. One cannot design a bridge with a simple mathematical computation. In addition, most problem-solving situations in the real world require teams of people. The National Council of Teachers of Mathematics (NCTM 2000) has promoted the use of complex problems that involve multistep solutions. Many standardized tests present problems that require expanded or extended responses. Because a simple solution with one answer might not be sufficient, rubrics are often developed to address the extent to which a problem has been solved. The mathematics assessment section of the Florida Comprehensive Achievement Test (FCAT) is one such example (Florida Department of Education 2002).

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Engel, Bill, and Diane Schmidt. "The Galactic Spaceship Tour Challenge." Mathematics Teacher 97, no. 5 (2004): 314–19. http://dx.doi.org/10.5951/mt.97.5.0314.

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Many students believe that the solution to most mathematics problems is fairly direct, involves only one step, and provides one final answer. In most real-life situations, however, mathematical problem solving involves several steps to obtain a complete solution. One cannot design a bridge with a simple mathematical computation. In addition, most problem-solving situations in the real world require teams of people. The National Council of Teachers of Mathematics (NCTM 2000) has promoted the use of complex problems that involve multistep solutions. Many standardized tests present problems that require expanded or extended responses. Because a simple solution with one answer might not be sufficient, rubrics are often developed to address the extent to which a problem has been solved. The mathematics assessment section of the Florida Comprehensive Achievement Test (FCAT) is one such example (Florida Department of Education 2002).

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Morhun, S. "Improving the mathematical models applied for the solution of solid assembly constructions thermoelasticity problem." Journal of Mechanical Engineering 20, no. 2 (2017): 42–46. http://dx.doi.org/10.15407/pmach2017.02.042.

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