Mathematics

Let p −r (n) denote the r -coloured partition function, and σ(n)= ∑ d|n d denote the sum of positive divisors of n . The aim of this note is to prove the following p −r (n)=θ(n)+ ∑ k=1 n−1 r k+1 (k+1)! ∑ α 1 =k n−1 ∑ α 2 =k−1 α 1 −1 ⋯ ∑ α k =1 α k−1 −1 θ(n− α 1 )θ( α 1 − α 2 )⋯θ( α k−1 − α k )θ( α k ) where θ(n)= n −1 σ(n) , and its inverse σ(n)=n ∑ r=1 n (−1 ) r−1 r ( n r ) p −r (n).

We derive a formula for p(n) (the number of partitions of n ) in terms of the partial Bell polynomials using Faà di Bruno's formula and Euler's pentagonal number theorem.

According to the Abel-Ruffini theorem, equations of degree equal to or greater than 5 cannot, in most cases, be solved by radicals. Due of this theorem we will present a formula that solves specific cases of sixth degree equations using Martinellis polynomial as a base. To better understand how this formula works, we will solve a sixth degree equation as an example. We will also see that all sixth degree equations that meet the coefficient criterion have a resolvent of fifth degree that can be splitted into a second degree and a third degree equation. Throughout the paper we will see a demonstration of the ratio of the coefficients of a sixth degree equation that can be solved with the formula that will be presented this paper

In this paper, we introduce the upper and lower approximations on the invers set-valued mapping and the approximations an established on a powerful set valued homomorphism from a ring R1 to power sets of a ring R2. Moreover, the properties of lower and upper approximations of a powerful set valued are studies. In addition, we will give a proof of the theorem of isomorphism over approximations F-rough ring as new result. However, we will prove the kernel of the powerful set-valued homomorphism is a subring of R1. Our result is introduce the first isomorphism theorem of ring as generalized the concept of the set valued mappings.

A research for a general expression of the triplets of integer sided triangles with a 120 ∘ angle, in parallel with the case of a 60 ∘ angle.

The first part of this article develops a variational formulation for relativistic mechanics. The results are established through standard tools of variational analysis and differential geometry. The novelty here is that the main motion manifold has a n+1 dimensional range. It is worth emphasizing in a first approximation we have neglected the self-interaction energy part. In its second part, this article develops some formalism concerning the causal structure in a general space-time manifold. Finally, the last article section presents a result concerning the existence of a generalized solution for the world sheet manifold variational formulation.

We analyze Toponogov's sine theorem for an infinitesimal geodesic triangle ABC on a C^2 regular surface M, which is given in his book [6, Problem 3.7.2] and we provide a generalization of the law of cosines for ABC on M. By replacing in the law of cosines B=\frac{\pi}{2} on M, we derive the generalized theorem of Pythagoras on a surface: AC^2 = AB^2 + BC^2 + f(\angle A,\frac{\pi}{2},AB,BC)o(AC^2) or AC^2 = AB^2 + BC^2 + (\angle A + \angle C-\frac{\pi}{2})^2 where f(\angle A,\angle B,AB,BC) is a rational function w.r. to cosA; cosB, sinA, sinB, AB and BC.

Given a semigroup S generated by its squares equipped with an involutive automorphism σ and a multiplicative function μ:S→C such that μ(xσ(x))=1 for all x∈S , we determine the complex-valued solutions of the following functional equation.

We show geometrically that n − − √ is irrational for n=3,5,7 by adapting Tennenbaum's geometric proof that 2 – √ is irrational. We also show that this method cannot be used to prove the irrationality of n − − √ for a bigger n .

In this paper, we introduce a geometrical summation method that makes the original Riemann series converge over the critical strip. This method gives an analytical function, that coincides with zêta. This point of view allows us to introduce a quantity of interest that seems to give a characterization of the non-trivial zeros of the Riemann zêta function.