A detailed analysis of the brachistochrone problem

If A and B are two points in the plane, with B lower and to the right of A, then we may consider the trajectory of an object travelling from A to B under the influence of gravity. The search for the trajectory minimising the time taken by the object gives rise to a mathematical optimisation problem involving an indefinite integral. Although the solution of this problem is known, a full detailed handling of the problem does not seem to be available. The aim of this article is to provide such a detailed study

On the image of Euler's totient function

In this article we study certain properties of the image of Euler's totient function; we also consider the structure of the preimage of certain elements of the image of this function

Some remarks on Euler's totient function

The image of Euler's totient function is composed of the number 1 and even numbers. However, many even numbers are not in the image. We consider the problem of finding those even numbers which are in the image and those which are not. If an even number is in the image, then its preimage can have at most half its elements odd. However, it may contain only even numbers. We consider the structure of the preimage of certain numbers in the image of the totient function.Comment: This article is a revision and extension of my previous article 'On the image of Euler's totient function' arXiv:0910.222

On powers of polynomials

3 pagesIn this paper we show that, under certain conditions, when the coefficients of a power of a polynomial over a field lie in a subfield, then the coefficients of the polynomial itself lie in the subfield

On Valuation Rings

Valuation rings occur in a wide variety of situations. In this paper we bring together many basic results concerning this important class of rings. In particular, we consider the relation between valuation rings and Dedekind domains

Hilbertian fields and Hilbert's irreducibility theorem

Hilbert's irreducibility theorem plays an important role in inverse Galois theory. In this article we introduce Hilbertian fields and present a clear detailed proof of Hilbert's irreducibility theorem in the context of these fields

Terrapin technologies manned Mars mission proposal

A Manned Mars Mission (M3) design study is proposed. The purpose of M3 is to transport 10 personnel and a habitat with all required support systems and supplies from low Earth orbit (LEO) to the surface of Mars and, after an eight-man surface expedition of 3 months, to return the personnel safely to LEO. The proposed hardware design is based on systems and components of demonstrated high capability and reliability. The mission design builds on past mission experience, but incorporates innovative design approaches to achieve mission priorities. Those priorities, in decreasing order of importance, are safety, reliability, minimum personnel transfer time, minimum weight, and minimum cost. The design demonstrates the feasibility and flexibility of a Waverider transfer module

Water management and irrigation for bulb onion (Allium cepa L.) growth and development in the Papua New Guinea Highlands

Water is crucial for nutrient intake, transportation, temperature regulation, and photosynthesis in bulb onion (Allium cepa L.) growth and development. Water scarcity, caused by climate variability and particularly during prolonged dry periods, has proved to be an obstacle to cultivating bulb onions in the Papua New Guinea (PNG) Highlands. Farmers have a limited grasp of the interdependence of soil, water, and plants. They have traditionally depended on precipitation, and water from streams and rivers, to irrigate their bulb onion crops. The main method for providing irrigation support is manual irrigation, with agricultural organisations assisting with basic irrigation technologies. Despite farmers and organisations efforts to improve irrigation practices, the prolonged dry season still raises labour demands for bulb onion farms. Farmers’ lack of knowledge about irrigation technology and soil water conservation contributes to this issue. Moreover, the lack of irrigation suppliers worsens the issue of limited soil water in onion farms. This paper gives an overview of onion production in PNG, focusing on irrigation practices and constraints in the Highlands region. It aims to stress irrigation’s importance in crop growth and explore PNG’s irrigation methods and soil moisture conservation practices. There is a need for a sustainable irrigation and soil water conservation system that is easy to use and incorporates crop water requirements, while offering farmers an improvement in crop profitability. This system is vital for watering onions and conserving soil moisture, promoting their growth and yield. In this context, the use of mulch and irrigation systems can preserve and enhance soil moisture during prolonged dry phases