Kinds Of Mathematical Problems Research Articles

Thinking development and aesthetic education of students in the process of teaching mathematics by example solutions for one problem

The article discusses one of the leading goals of teaching mathematics - the development of students’ thinking by means of mathematics, it is noted that this requires the creation of special situations, the resolution of which leads to the development of thinking. The possibility of developing logical and figurative thinking of students of secondary schools in the process of aesthetic education using the subject of mathematics is investigated. The beauty of problem solving allows you to increase interest in the study of mathematics. The development of intellectual and creative capabilities, their imagination, as well as broadening the horizons in the application of methodological programs in teaching and evaluating students’ knowledge and skills. The present work is presented on the example of solving one problem by several mathematical methods. Using a mathematical model to solve various kinds of problems with the greatest number of solutions contributes to aesthetic education, allowing you to develop a culture and logical thinking, to form in students the optimal analysis of the task in solving it. The work is presented in an experiment on organizing aesthetic education by knowing the beauty of problem solving, which will fully reveal the criteria for applying the individual approach to learning through the development of thinking in solving various kinds of mathematical problems. The organization of the intensification of the process of mastering knowledge and applying aesthetic beauty is shown by the example of solving problems in mathematics of grades 7-9. We carried out a number of experimental works on the development of thinking and aesthetic education, relying on the activity-based approaches of educational and developmental teaching of mathematics.

Performance of New Hope and CRYSTALS-Dilithium Postquantum Schemes in the Transport Layer Security Protocol

In recent years, there has been a notable amount of research on developing cryptographic schemes that are secure against both quantum and classical computers. In 2016, the National Institute of Standards and Technology (NIST) initiated a process to solicit, evaluate, and standardize one or more quantum-resistant public key cryptographic schemes. This process originated because quantum computers can exploit quantum mechanical phenomena and solve mathematical problems that are difficult or intractable for classical computers. This kind of mathematical problem is the basis of secure public key cryptography. As a consequence, in a near future quantum computers will be able to break many of the public key schemes currently in use. However, the challenge is especially acute for devices with different architectures. They might not be well equipped to run the new standards and interoperate with existing communication protocols and networks. In this work, we analyze the performance of postquantum schemes in the transport layer security (TLS) protocol considering x86 as the server architecture and x86/ARM architectures as clients. All of them lack cloud computing or virtualized environments. Our analysis considers integrating the implementation of two cryptographic schemes that were successful in the second round of the postquantum standardization process, namely, Dilithium and New Hope. The performance of postquantum schemes in the TLS protocol is statistically analyzed in x86 and ARM architectures, giving the relationships, the effects and the survival of the analysis.

Solving Multi-Objective Linear Fractional Stochastic Transportation Problems Involving Normal Distribution using Simulation-Based Genetic Algorithm

In real-life situations, we human beings faced with multi-objective problems that are conflicting and non-commensurable with each other. Especially, when goods are transported from source to locations with a goal to keep exact relationships between a few parameters, those parameters of such problems might also arise in the form of fractions which are linear in nature such as; actual transportation fee/total transportation cost, delivery fee/desired path, total return/total investment, etc. Due to the uncertainty of nature, such a relationship is not deterministic. Mathematically such kinds of mathematical problems are characterized as a multi-objective linear fractional stochastic transportation problem. However, it is difficult to handle such types of mathematical problems. It can't be solved directly using mathematical programming approaches. In this paper, a solution procedure is proposed for the above problem using a stochastic Genetic Algorithm based simulation. The parameters in the constraint of the above problem follow a normal distribution. The probabilistic constraints are handled by stochastic simulation-based GA for the solution procedure of the proposed problem. The feasibility of probability constraints is checked by the stochastic programming through the Genetic Algorithm approach, without finding the equivalent deterministic model. The feasibility is maintained all-over the problem. The stochastic simulation-based Genetic Algorithm is considered to generate non-dominated solutions for the given problem. Then, a numerical case study is provided to illustrate the method.

Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms

This paper is focused on solving different hard optimization problems that arise in the field of insurance and, more specifically, in reinsurance problems. In this area, the complexity of the models and assumptions considered in the definition of the reinsurance rules and conditions produces hard black-box optimization problems (problems in which the objective function does not have an algebraic expression, but it is the output of a system (usually a computer program)), which must be solved in order to obtain the optimal output of the reinsurance. The application of traditional optimization approaches is not possible in this kind of mathematical problem, so new computational paradigms must be applied to solve these problems. In this paper, we show the performance of two evolutionary and swarm intelligence techniques (evolutionary programming and particle swarm optimization). We provide an analysis in three black-box optimization problems in reinsurance, where the proposed approaches exhibit an excellent behavior, finding the optimal solution within a fraction of the computational cost used by inspection or enumeration methods.

Creating Games from Mathematical Problems

The emphasis on problem solving and problem solving courses has become a standard staple of most teacher preparation programs in mathematics. While opinions differ on whether problem solving should be integrated throughout the preparatory courses or teachers should also have a dedicated course for it, everybody agrees that problem solving skills are very important and should be improved. Mathematical games provide excellent opportunities for students at all levels to be engaged in constructing strategies, formulating hypotheses, asking new questions, and posing new problems. Here we present some ideas on how one can create games from different kinds of mathematical problems, and in turn how these games can be used to improve every step of mathematical problem solving as well as to discover the original problems behind the games.

Optimization Algorithm for Solving a Kind of Mathematical Problems

In this paper, we develop an algorithm to globally solve a kind of mathematical problem. Firstly, by utilizing equivalent problem and linear relaxation method, a linear relaxation programming of original problem is established. Secondly, by using branch and bound technique, a determined global optimization algorithm is proposed for solving equivalent problem. Finally, the convergence of the proposed algorithm is proven and numerical examples showed that the presented algorithm is feasible to solve the kind of mathematical problems.

Elastic and elasto-plastic analysis of submarine pipelines as unilateral contact problems

A submarine pipeline is conceived as a slender beam, subjected to vertical loads and undergoing “small” deflections, contrasted without friction by a supporting rigid profile. The relevant contact (unilateral constraint) problem is formulated in approximate, algebraic terms on the basis of discrete models, allowing for the presence of a significant given tension force. The flexural behaviour is first assumed as linearly elastic and the analysis is formulated as a linear complementarity problem (LCP) (or to a variational inequality). Subsequently, nonlinear, elasto-plastic flexural behaviour is considered. This is described by a piecewise linear moment-curvature relation, and the analysis is shown to lead to the same kinds of mathematical problems as in the elastic case. In both cases a pair of dual extremum principles is pointed out, leading to a pair of equivalent dual quadratic programming problems (QPP). Among available algorithms for the numerical solution, systematic overrelaxation is found to be convenient and efficient for problems arising in engineering situations. Computational experience of practical relevance is discussed.