Abstract:

For operators A, it is sometimes possible to define  e^At  as an operator in and of itself provided it meets certain regularity conditions. Like e^λx for ODEs, this operator is useful for solving PDEs involving the operator A. We call the set of  e^At  a semigroup generated by A. In this paper, we discuss the properties of semigroups generated by the fractional integral, an operator appearing in PDEs in increasingly many fields, over Bochner-Lebesgue spaces.

FULL Article

Abstract:

A lot of effort has been devoted into solving the famous Square Packing Problem, which investigates the minimum side length of a square container that can pack n unit squares. This involves the search for the most optimal packing for squares. The aim of this research is to investigate an opposite idea to the original problem. We delve into the least optimal packing of squares, i.e. finding the minimum side length of a square container that can contain all configurations of n unit squares.
By considering the idea of a rotating container, we have successfully found the solution to the case of two squares. At the end, by studying the classification of intersections between the configuration and the container, as well as harnessing analytical methods, we have found the exact solution to the general case of n squares.
 

FULL Article

Abstract:

This study explores Egyptian fractions, focusing on parametrization to construct a unified approach to open problems in this field. The paper introduces a symmetric parametrization for Egyptian fraction equations, demonstrating its effectiveness through three applications. It also investigates conjectures related to the shortest length of Egyptian expansion and the Generalized Erdös-Straus conjecture, and explores connections with semiperfect numbers. The research leverages Geometry to transform Egyptian equations into a parametrized system, offering a novel perspective on tackling open problems with and within the field of Egyptian fractions.

FULL Article

Abstract:

The critical group of a graph is defined as the torsion subgroup of the co-kernel of the Laplacian matrix of the graph. In this paper, we investigated the critical groups of two classes of unitary circulant graphs, which are Cayley graphs on the group of integers modulo n, with connecting set being the set of units modulo n. The explicit group structure of such graphs when n is product of two distinct primes and when n is a prime power, are computed using Ramanujan Sums. Furthermore, we investigated the critical groups of circulant graphs with fixed connecting sets, and expressed one of the components of the group as the greatest common divisor of real and imaginary parts of Chebyshev Polynomials.

FULL Article

Abstract:

This paper rigorously explores the expansion of fractions in the unorthodox number system with a rational negative base -Nb/Db , building on the work of Lucia Rossi and Jörg M. Thuswaldner on multiple number representations in such a base. Our objective is to establish a finite number of recurring expansions, using our novel theories and algorithms. We introduce definitions and conditions for four types of expansions, and present two distinct proofs for the Complete Residue System Theorem, our first main theorem. Our Second Main Theorem outlines the bounds of terminating and recurring expansions in any number system, providing a method to compute all expansions for any fraction m/n . These findings provide a thorough examination of fraction representations in the negative rational base system, enhancing understanding of its intricate characteristics.

FULL Article

Abstract:

In this paper, we conduct an analysis of the problem concerning the mean shadow cast by rotating objects. The original problem was introduced by Cauchy in 1832. He proposed solutions for the 2-D and 3-D scenarios in 1842 and 1850 respectively. In the original problem, the shadow was formed by orthogonal projection. In 2022, the problem was revisited under the 3-D scenario of a light source with finite distance above the rotation center. Instead of 3-D scenario, we focus on the 2-D case and generalize the problem by placing the light source arbitrarily. We derive explicit formulae of the mean shadow. With these formulae, we provide a numerical method to compute the mean shadow, which surpasses the conventional simulation.

FULL Article

Abstract:

In this paper, we have generalised the orthocentre of a triangle as the isogonal conjugate of the circumcentre of a simplex. Along this generalisation, we have also carried two intriguing properties of the orthocentre of a triangle over to higher dimensions, which says that the isogonal conjugate of the circumcentre of a simplex is either the incentre or an excentre of its pedal simplex, and is also the radical centre of the facetal circumhyperspheres of the simplex.
  To this end, we have extended the scope of isogonal conjugation with respect to simplices to non-interior points through developing new algebraic and geometric characterisations for it. We have also obtained a higher-dimensional analogue of a curious property of isogonal conjugates with respect to triangles, which says that when both a point and its isogonal conjugate with respect to a simplex are projected onto the facets, the projections formed are co-hyperspherical.
 

FULL Article