Volume 8 (2024)

Abstract:We study analytical solutions of a bi-dimensional low-mass gaseous disc slowly rotating around a central mass and submitted to small radial periodic perturbations. Hydrodynamics equations are solved for the equilibrium and perturbed configurations. A wave-like equation for the gas-perturbed specific mass is deduced and solved analytically for several cases of exponents of the power law distributions of the unperturbed specific mass and sound speed. It is found that, first, the gas perturbed specific mass displays exponentially spaced maxima, corresponding to zeros of the radial perturbed velocity; second, the distance ratio of successive maxima of the perturbed specific mass is a constant depending on disc characteristics and, following the model, also on the perturbation’s frequency; and, third, inward and outward gas flows are induced from zones of minima toward zones of maxima of perturbed specific mass, leading eventually to the possible formation of gaseous annular structures in the disc. The results presented may be applied in various astrophysical contexts to slowly rotating thin gaseous discs of negligible relative mass, submitted to small radial periodic perturbations.

Abstract:This paper aims to present Hermite-Hadamard type inequalities for a new class of functions, which will be denoted by \(Q_m^{h,g}(F;I)\) an and called class of quasi \(F-(h,g;m)\)-convex functions defined on interval \(I\). Many well known classes of functions can be recaptured from this new quasi convexity in particular cases. Also, several publish results are obtained along with new kinds of inequalities.

Abstract:We provide a semi-local convergence analysis of a seventh order four step method for solving nonlinear problems. Using majorizing sequences and under conditions on the first derivative, we provide sufficient convergence criteria, error bounds on the distances involved and uniqueness. Earlier convergence results have used the eighth derivative not on this method to show convergence. Hence, limiting its applicability.

Abstract:Squares of odd index Fibonacci polynomials are used to define a new function \(\Phi\left(10^{n}\right)\) to approximate the number \(\pi\left(10^{n}\right)\) of primes less than \(10^{n}\). Multiple of 4 index Fibonacci polynomials are further used to define another new function \(\Psi\left(10^{n}\right)\) to approximate the number \(\Delta\left(\pi\left(10^{n}\right)\right)\) of primes having \(n\) digits and compared to a third function \(\Psi’\left(10^{n}\right)\) defined as the difference of the first function \(\Phi\left(10^{n}\right)\) based on odd index Fibonacci polynomials. These three functions provide better approximations of \(\pi\left(10^{n}\right)\) than those based on the classical \(\left(\frac{x}{log\left(x\right)}\right)\), Gauss’ approximation \(Li\left(x\right)\), and the Riemann \(R\left(x\right)\) functions.

Abstract:We show that Euler’s relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations to be found among the sums of two consecutive cubes and the sum of two other cubes, in the form of two families. Their recursive and parametric equations are calculated.

Abstract:This study introduces theorems concerning matrix products, which delineate the transformations of sequences or series into other sequences or series, ensuring either the preservation of limits or the guarantee of convergence. Previous literature has explored the properties of matrices facilitating transformations between sequences, series, and their combinations, with detailed insights available in references [1,2,3].

Abstract:The concept of weak UP-algebras (shortly wUP-algebra) is an extension of the notion of UP-algebras introduced in 2021 by Iampan and Romano. In this report, an effective extension of a (weak) UP-algebra to a wUP-algebra is created. In addition to the previous one, the concept of atoms in wUP-algebras is introduced and their important properties are registered. Finally, the concept of wUP-filters in wUP-algebras was introduced and its connections with other substructures in wUP-algebras were analyzed.

Abstract:In normed spaces, Birkhoff orthogonality and isosceles orthogonality can be used to characterize space structures, and many scholars have introduced geometric constants to quantitatively describe the relationship between these two types of orthogonality. This paper introduces a new orthogonal relationship – Skew orthogonality – and proposes a new geometric constant to measure the “distance” of difference between skew orthogonality and Birkhoff orthogonality in normed spaces. In the end, we provide some examples of specific spaces.