On a Logarithmic Weighted Power Distribution: Theory, Modelling and Applications

Engineers, economists, hydrologists, social scientists, and behavioural scientists often deal with data belonging to the unit interval. One of the most common approaches for modeling purposes is the use of unit distributions, beginning with the classical power distribution. A simple way to improve its applicability is proposed by the transmuted scheme. We propose an alternative in this article by slightly modifying this scheme with a logarithmic weighted function, thus creating the log-weighted power distribution. It can also be thought of as a variant of the log-Lindley distribution, and some other derived unit distributions. We investigate its statistical and functional capabilities, and discuss how it distinguishes between power and transmuted power distributions. Among the functions derived from the log-weighted distribution are the cumulative distribution, probability density, hazard rate, and quantile functions. When appropriate, a shape analysis of them is performed to increase the exibility of the proposed modelling. Various properties are investigated, including stochastic ordering (first order), generalized logarithmic moments, incomplete moments, Rényi entropy, order statistics, reliability measures, and a list of new distributions derived from the main one are offered. Subsequently, the estimation of the model parameters is discussed through the maximum likelihood procedure. Then, the proposed distribution is tested on a few data sets to show in what concrete statistical scenarios it may outperform the transmuted power distribution.

A Possible use of the Cubit Rod in Ancient Egypt to Measure and Draw Lengths Based on Fractions

We will discuss about a possible method of using the cubit rod by the architects and the surveyors of Ancient Egypt to measure and draw lengths, comparing it with the other interpretations present in Literature. Instead of the modern decimal notation, which sees the use of comma to represent a number or a measure, at that time there was a wide use of fractions in calculations. The current work proposes that, through the cubit rod and its partitions of the finger into fractions, it could be possible to obtain very accurate measurements.

Extension Property of Automorphism of Modules

In this paper we generalize Schupp’s result for groups to modules. For an injective module, every automorphism satisfies the extension property. We characterize the automorphisms of a module M satisfies the extension property.

An Evaluation of Music’s Adherence to Benford’s Law Throughout History

Western Music history can be divided into six major categories: Medieval, Renaissance, Baroque, Classical, Romantic, and Post-War. We analyzed a large collection of music from each time period and discovered a clear mathematical connection. Within each time period, we found that the note frequencies measured in hertz (Hz) and note durations are all Benford distributed. We also found that as music progressed through time, note lengths adhered closer and closer to the Benford distribution with the exception of the Post-War time period.

On Some Applications of Kakutani’s Fixed Point Theorem in Game Theory

In this paper, we discuss some major applications of Kakutani’s fixed point theorem in game theory. In the course of research work we mostly use the idea of mathematical set, functions, topological properties and Brouwer’s fixed point theorem to make the Kakutani’s fixed point theorem more conspicuous. In the key point of idea, we include how this theory can play the effective role to highlight new fixed point results and their applications in different fields of game theory.

Schur-Convexity of a Class of Elementary Symmetric Composite Functions

In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we provide much simpler proofs of the Schur-convexity, Schur-geometric convexity on and Schur-harmonic convexity on for a composite function of the elementary symmetric functions.

On the Sirs Epidemic Model with Free Boundaries

We investigate an epidemic non-linear reaction-diffusion system with two free boundaries. A free boundary is introduced to describe the expanding front of the infectious environment. A priori estimates of the required functions are established, which are necessary for the correctness and global solvability of the problem. We get sufficient conditions for the spread or disappearance of the disease. It has been proven that with a base reproductive number the disease disappears in the long term if the initial values and the initial area are sufficiently small.

Hom-Hyporeductive Triple Algebras

Hom-hyporeductive triple algebras are defined as a twisted generalization of hyporeductive triple algebras. Hom-hyporeductive triple algebras generalize right Hom-Lie-Yamaguti and right Hom-Bol algebras as the same way as hyporeductive triple algebras generalize right Lie-Yamaguti and right Bol algebras. It is shown that the category of Hom-hyporeductive triple algebras is closed under the process of taking nth derived binary-ternary Hom-algebras and by self-morphisms of binary-ternary algebras. Some examples of Hom-hyporeductive triple algebras are given.

Correlation between No-Slip and Slip Boundary Conditions Associated with A Two-Dimensional Navier-Stokes Flows in a Plane Diffuser

We examine the viscous effects of slip boundary conditions for the model describing two-dimensional Navier-Stokes flows in a plane diffuser. It is shown that the velocity profile is related to a half period shifted Weierstrass function with two parameters. This allows to approximate the explicit solution by a Taylor series expansion with two new micro- parameters, that can be measured in physical experiments. It is shown that the assumption for no-slip boundary conditions is stable in the sense that a small perturbation of the boundary values result in a small perturbation in the solutions.

Fixed Point Theorems on Generalized Rectangular Metric Spaces

In this paper, we introduce the notion of generalized rectangular metric spaces which extends rectangular metric spaces introduced by Branciari. Analogues of the some well-known fixed point theorems are proved in this space. With an example, it is shown that a generalized rectangular metric space is neither a G-metric space nor a rectangular metric space. Our results generalize many known results in fixed point theory.