Mathematics

The goal of the work is to take on and study one of the fundamental tasks studying Diophantine n-gons (the author of the paper considers an integral n-gon is Diophantine as far as determination of combinatorial properties of each of them requires solution of a certain Diophantine equation (equation sets)).

In this paper we establish a new formula for the arithmetic functions that verify f(n)= ∑ d|n g(d) where g is also an arithmetic function. We prove the following identity, ∀n∈ N ∗ , f(n)= ∑ k=1 n μ( k (n,k) ) φ(k) φ( k (n,k) ) ∑ l=1 ⌊ n k ⌋ g(kl) kl where φ and μ are respectively Euler's and Mobius' functions and (.,.) is the GCD. First, we will compare this expression with other known expressions for arithmetic functions and pinpoint its advantages. Then, we will prove the identity using exponential sums' proprieties. Finally we will present some applications with well known functions such as d and σ which are respectively the number of divisors function and the sum of divisors function.

In this paper, we de?ne a new type curve as V-Mannheim curve, V Mannheim partner curve and generating curve of Mannheim curve. We give characterization of these curve. In addition, we study a relation between Mannheim curve and spherical curve. Eventually, with Salkowski method, we give an example of the Mannheim curve.

We define two categories, the category FG of fuzzy subgroups, and the category FC of F -inverse covers of inverse monoids, and prove that FG fully embeds into FC . This shows that, at least from a categorical viewpoint, fuzzy subgroups belong to the standard mathematics as much as they do to the fuzzy one.

The aim of this paper is to provide a new class of series identities in the form of four general results. The results are established with the help of generalizatons of the classical Kummer's summation theorem obtained earlier by Rakha and Rathie. Results obtained earlier by Srivastava, Bailey and Rathie et al. follow special cases of our main findings.

In this paper, we define a new type Bertrand curve and this curves are said V-Bertrand curve, f-Bertrand curve and a-Bertrand curve. In addition, we give charectarization of the V-Bertrand curve and we define a Bertrand surface.

We outline a simple proof of PC without surgeries using the homogeneous flow introduced in [O].

In the world of mathematical analysis, many counterintuitive answers arise from the manipulation of seemingly unrelated concepts, ideas, or functions. For example, Euler showed that e iπ +1=0 , whereas Gauss proved that the area underneath y= e − x 2 spanning the whole real axis is π − − √ . In this paper, we will determine the closed-form solution of the improper integral I n = ∫ ∞ 0 lnx x n +1 dx, ∀n∈R, with n>1. Determining closed-form solutions of improper integrals have real implications not only in easing the solving of similar, yet more difficult integrals, but also in speeding up numerical approximations of the answer by making them more efficient. Following our calculations, we derived the formula I n = ∫ ∞ 0 lnx x n +1 dx=− π 2 n 2 cot π n csc π n =− d dn [Γ(1− 1 n )Γ( 1 n )].

A combination a+ib where i 2 =−1 and a,b are real vectors is called a bivector. Gibbs developed a theory of bivectors, in which he associated an ellipse with each bivector. He obtained results relating pairs of conjugate semi-diameters and in particular considered the implications of the scalar product of two bivectors being zero. This paper is an attempt to develop a similar formulation for hyperbolas by the use of jay-vectors - a jay-vector is a linear combination a+jb of real vectors a and b , where j 2 =+1 but j is not a real number, so j≠±1 . The implications of the vanishing of the scalar product of two jay-vectors is also considered. We show how to generate a triple of conjugate semi-diameters of an ellipsoid from any orthonormal triad. We also see how to generate in a similar manner a triple of conjugate semi-diameters of a hyperboloid and its conjugate hyperboloid. The role of complex rotations (complex orthogonal matrices) is discussed briefly. Application is made to second order elliptic and hyperbolic partial differential equations. Keywords: Split complex numbers, Hyperbolic numbers, Coquaternions, Conjugate semi-diameters, Hyperboloids and ellipsoids, Complex rotations, PDEs MSC (2010) 35J05, 35L10, 74J05

An identity is proved connecting two finite sums of inverse tangents. This identity is discretized version of Jacobi's imaginary transformation for the modular angle from the theory of elliptic functions. Some other related identities are discussed.