Mathematics

We describe the explicit form of a left greatest common divisor and a least common multiple of solutions of a solvable linear matrix equation over a commutative elementary divisor domain. We prove that these left greatest common divisor and least common multiple are also solutions of the same equation.

The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines R(s)=α . An approach based on a two-dimensional principal component analysis matching the dimensionality of the complex plane, which is built on a parametric ellipsoidal shape, has been undertaken to achieve this result. This lower bound, which is expressed as ∀s∈C s.t. R(s)∈P , |η(s)|≥|1− 2 √ 2 α | , where η is the Dirichlet eta function, has implications for the Riemann hypothesis as |η(s)|>0 for any s∈C s.t. R(s)∈P , where P is a partition spanning one half of the critical strip, on either sides of the critical line R(s)=1/2 depending upon a variable.

This research work aims to explore the distortions in distance in equidistant cylindrical projection. The horizontal bending that occurs in the projection process can be assessed by performing a geometric analysis using Tissot's indicatrices. In addition, the concept of the spherical coordinates, alongside with trigonometrical identities, can be used to illustrate the route from a point to another as a curve. With a combination of the knowledge extracted from the examination of the projection using those two theories, this research aims to fully unravel the degree of distortion in distance in equidistant cylindrical projections.

In this paper we introduce a new multiple-parameters generalization of the Wright function arose from an eigenvalues problem concerning an hyper-Laguerre-type operator involving Caputo derivatives. We show that by giving particular values at the parameters including in the function, it leads right to well known special functions (the classical Wright function, the α -Mittag-Leffler function, the Tricomi function etc...). In addition, we investigate a nonlinear fractional differential equation admitting the new generalization Wright function as solution, and in particular isochronous solutions.

The understanding of probability can be difficult for a few young scientists. Consequently, this new mathematical symbol, related to binomial coefficients and simplicial polytopic numbers, could be helpful to science education. Moreover, one can obtain kinds of remarkable identities and generalize them to a sort of "Newton's binomial theorem". Finally, this symbol could be perhaps useful to other scientific subjects as well, such as computer science.

The Collatz conjecture is an unsolved problem in mathematics which introduced by Lothar Collatz in 1937. Although the prize for the proof of this problem is 1 million dollar, nobody has succeeded in proving this conjecture. However in this article, we will discuss the results of the author's research and come closer to the proof of this conjecture.

The first part of the paper proves that a subset of the general set of Ermakov-Pinney equations can be obtained by differentiation of a first-order non-linear differential equation. The second part of the paper proves that, similarly, the equation for the amplitude function for the parametrix of the scalar wave equation can be obtained by covariant differentiation of a first-order non-linear equation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary 1-forms (psi,rho). The 1-form psi satisfies the divergenceless condition div(psi)=0, whereas the 1-form rho fulfills the non-linear equation div(rho)+rho**2=0. The auxiliary 1-forms (psi,rho) are evaluated explicitly in Kasner space-time, and hence also amplitude and phase function in the parametrix are obtained. Thus, the novel method developed in this paper can be used with profit in physical applications.

In the current note, we present a new, short proof of the famous AM-GM-HM inequality using only induction and basic calculus.

In this paper, we propose a new fractional derivative, which is based on a Caputo-type derivative with a smooth kernel. We show that the proposed fractional derivative reduces to the classical derivative and has a smoothing effect which is compatible with ℓ 1 regularization. Moreover, it satisfies some classical properties.

A newly-generalized problem from a problem initially thought for the Mathematical Olympiad and the methods to solve it.