Mathematics

Two kinds of invariance for geometrical objects under transformations are involved in this paper. With respect to these kinds, we obtained novel invariants for almost geodesic mappings of the third type of a non-symmetric affine connection space in this paper. Our results are presented in two sections. In the Section 3, we obtained the invariants for the equitorsion almost geodesic mappings which do not have the property of reciprocity.

We propose and prove a couple of formulas and infinite series involving the floor and the ceiling functions. Formula relating to the difference of floor and ceiling functions is obtained using aforementioned formulas. Partial summations of floor and ceiling of qth roots of natural numbers are equated as simple formulas. Particular cases of the series are taken into consideration and it is proven that both the cases relate to the Riemann-Zeta function. Poles for the both series are mentioned and it is shown that even if both series individually fail to converge at the pole, their difference is convergent at the same. It is shown that our formulas reduce to the Gauss formula and the series reduce to the Riemann-Zeta for a particular value. Further some special cases and scope for future work are discussed.

In this work, we studied various properties of arithmetic function φ ~ , where φ ~ (n)=|{m∈N|1≤m≤n,(m,n)=1,m is not a prime}|.

There are four division algebras over R , namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions. It does not make sense to ask, for example, whether the equation x 2 +1=0 is solvable, without specifying the field in which we want the solutions to be lie. The equation x 2 +1=0 has no solutions in R , which is to say, there are no real numbers satisfying this equation. On the other hand, there are complex numbers which do satisfy this equation in the field C of all complex numbers. How about if we extend the same idea to other two normed division algebras quaternions and octonions. Liping Huang and Wasin So derive explicit formulas for computing the roots of quaternionic quadratic equations. We extend their work to octonionic case and solve monic left octonionic quadratic equation of the form x 2 +bx+c=0 , where a,b are octonions in general.[ We called this form of quadratic equation as left octonion quadratic equation because we can consider x 2 +xb+c=0 as a different case due to non-commutativity of octonions]. Finally, we represent the left spectrum of 2?2 octonionic matrix as a set of solutions to a corresponding octonionic quadratic equation, which is an application of deriving explicit formulas for computing the roots of octonionic quadratic equations.

The first Zagreb index of a graph G is the sum of squares of the vertex degrees in a graph and the second Zagreb index of G is the sum of products of degrees of adjacent vertices in G . The imbalance of an edge in G is the numerical difference of degrees of its end vertices and the irregularity of G is the sum of imbalances of all its edges. In this paper, we extend the concepts of these topological indices for signed graphs and discuss the corresponding results on signed graphs.

The Collatz Conjecture can be stated as: using the reduced Collatz function C(n)=(3n+1)/ 2 x where 2 x is the largest power of 2 that divides 3n+1 , any odd integer n will eventually reach 1 in j iterations such that C j (n)=1 . In this paper we use reduced Collatz function and reverse reduced Collatz function. We present odd numbers as sum of fractions, which we call `fractional sum notation' and its generalized form `intermediate fractional sum notation', which we use to present a formula to obtain numbers with greater Collatz sequence lengths. We give a formula to obtain numbers with sequence length 2. We show that if trajectory of n is looping and there is an odd number m such that C j (m)=1 , n must be in form 3 j ×2k+1,k∈ N 0 where C j (n)=n . We use Intermediate fractional sum notation to show a simpler proof that there are no loops with length 2 other than trivial cycle looping twice. We then work with reverse reduced Collatz function, and present a modified version of it which enables us to determine the result in modulo 6. We present a procedure to generate a Collatz graph using reverse reduced Collatz functions.

In the present article, we first examine the conception of C*-algebra-valued controlled Fc-metric type spaces as a generalization of F-cone metric spaces over banach algebra. Further, we prove some fixed point theorem with different contractive conditions in the framework of C*-algebra-valued controlled Fc-metric type spaces. Secondly, we furnish an example by means of the acquired result.

In this paper we obtain a generalization of some integral inequalities related to Chebyshev`s functional by using a generalized Katugampola fractional integral.

In this paper, we introduce isophote curves on surfaces in Galilean 3-space. Apart from the general concept of isophotes, we split our studies into two cases to get the axis d of isophote curves lying on a surface such that d is an isotropic or a non isotropic vector. We also give the method to compute isophote curves of surfaces of revolution. Subsequently, we show the relationship between isophote curves and slant(general) helices on surfaces of revolution obtained by revolving a curve by Euclidean rotations. Finally, we give an example to compute isophote curves on isotropic surfaces of revolution.

The main purpose of the present article is to give some new Hilbert's sum type inequalities, which in special cases yield the classical Hilbert's inequalities. Our results provide some new estimates to these types of inequalities.