نشریه ریاضی و جامعه
سال ششم شماره 2 (تابستان 1400)

The fractal dimension of Koch and Cesàro curves has different values. These two curves are not differentiable and inte- grable in the sense of ordinary calculus. This paper shows how fractal calculus can be applied to these two curves. To do this, we need to obtain the corresponding staircase function. Calculating the path integral on these two fractal curves is the starting point for formulating and extracting Lagrange equations. Fractal integration and fractal derivation on these two fractal paths are performed analytically, and the results are plotted in a MATLAB environment. In addition, we present an application by finding light path equations in this fractal space using the generalized Fermat’s principle in fractal calculus.

The solution of fractional-order differential problems requires in the majority of cases the use of some computational ap- proach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results.

Let G be a finite group and let Irr(G) be the set of all irreducible characters of G. We say that an element g in G is a vanishing element if there exists some character χ ∈ Irr(G) such that χ(g) = 0. We can easily show that the set of vanishing elements of G is the union of some conjugacy classes. In this paper, we obtain the set of vanishing elements of Frobenius groups whose kernel is nilpotent of class 2, and then, we will try to find a suitable lower bound for the number of vanishing conjugacy classes of these groups. Moreover, we will classify Frobenius groups containing at most six vanishing conjugacy classes.

Many readers of mathematical texts are familiar with the application of mathematics in other sciences. For some of them, it is surprising to hear about the applications of the other sciences in mathematics. In this article, we intend to study this subject. We survey the history of sciences, in order to get an acquaintance with the ideas and methods of the great mathematicians for solving some of the olden mathematical problems with the aid of other scientific laws. Then, with the same approach, we provide the solu- tions of some mathematical problems.

In this article, we state some sufficient conditions for an operator to be subspace-chaotic. The properties of the direct sum of operators are interesting topics in operator theory. In this paper, we show that the direct sum of the two subspace-chaotic operators is subspace-chaotic with respect to at least two subspaces, and the same is true for the subspace-transitive and subspace-hypercyclic operators. We also make interesting examples of these operators with the help of the stated theorems. In addition, we prove that if the direct sum of two operators is subspace-chaotic or subspace-transitive, then each of those two operators is subspace-chaotic or subspace-transitive, respectively. We also generalize these results to the direct sum of a finite number of these operators.

Let C be a code over a finite field. The r-th Generalized Hamming weight(GHW), denoted by dr (C), is defined the minimum of support sizes of r- dimensional subcodes of C.GHW is important for its application in several fields such as Cryptog- raphy, Electronic Engineering, Computer and telecommunication Engineering. In this work, we will investigate this conception and its application.