We improve the convergence properties of the iterative scheme for solving unconstrained optimisation problems introduced in Petrovic et al. [‘Hybridization of accelerated gradient descent method’, Numer. Algorithms (2017), doi:10.1007/s11075-017-0460-4] by optimising the value of the initial step length parameter in the backtracking line search procedure. We prove the validity of the algorithm and illustrate its advantages by numerical experiments and comparisons.
One of the main tasks in teaching mathematics is to develop students’ constructive thinking. In order to effectively accomplish this task, it is necessary to make a good selection of instructional materials and teaching aids. In order to make good selection and improve the teaching of mathematics, it is, also, necessary to include a statistical analysis of the certain factors’ impact that affect mathematics curriculum. For the purpose of this research, we used the software computational approach ANFIS (adaptive neuro fuzzy inference system) to determine the qualitative impact of several factors on improving students’ ability to create constructive thinking.
Contemporary mathematics teaching is mostly reduced to the application of algebraic formulas and algebraic procedures. The visual-logical approach in solving mathematical tasks is very little represented in teaching mathematics. Such practice should be changed since visualization is of great importance in the process of learning and understanding mathematics as well as in solving mathematical tasks. This paper suggests the possibility of developing students' ability to perceive lawfulness among numbers by introducing figurative numbers in mathematics teaching. Considering the visual presentation of figurate numbers and obvious rules among their members, students find them interesting and easy for understanding. They can also be a very good paradigms for many tasks with numerous arrays. The research carried out in this paper has shown that figurative numbers contribute to a visual-logical approach in solving tasks with numerous arrays and provide long-term storage of numerous data.
In this paper, we are going to demonstrate a method for determining the generating functions of tetrahedral, hexahedral, octahedral, dodecahedral, and icosahedral figurative numbers. The method is based on the differences between the members of the series of the mentioned figurative numbers, as well as on the previously specified generating functions for the sequence ∑n≥0n+1xn and geometric sequence ∑n≥0xn.
In this paper, we deal with the dominating set and the domination number on an icosahedral-hexagonal network. We will consider all cases of successive halving of the edges of triangles that are the sides of icosahedrons and thus obtain icosahedral-hexagonal networks.
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