Table of Content

Abstract:

The presented project aims at having an insight on one of the most famous, hard but beautiful problems in number theory—Catalan’s Conjecture (This conjecture has become a theorem in 2002). Throughout this project, very advanced techniques and results established are avoided. Most of the results established in this report only require the concepts in elementary number theory, for example: divisibility and congruence. Yet, these techniques can be used delicately to establish a number of particular cases.

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In the paper, we generalize the process of cutting a Mőbius strip and similar strips to a larger extent than Mőbius, Listing, Ball-Coxeter and Fatehi’s papers. We generalize the object from a strip to a “twisted solid torus” (which we abbreviate to tst) and consider the result after cutting it. The Argand diagram, together with the usage of complex numbers, has been used to describe the lines in the cross section of tst. In our derivation, we have used a technique of checking the concurrence of lines defined by parametric equations by applying the concept of pole-polar duality from inversive geometry. Euler’s celebrated formula on graphs has also been employed. Then we study the resultant objects formed from the cutting process and call them “knotted tst”. We then deduce a general formula for the number of different knotted tsts. After that, we consider the links that are formed from the cutting of tsts, which we call “tst links”. General forms of their braid words, Seifert matrices and Alexander polynomials are then deduced. Then we generalize the results further and consider cutting a tst in the form of a non-trivial knot and study the resultant links. Finally, we study the cutting of combinations of more than one tsts in the form of virtual knots, which we call “tst products”, and derive a general formula for the result.

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Many real-world problems can be modeled mathematically as graphs. Some of these graphs are complex because of their large numbers of vertices and edges. To develop applications over any of these graphs, a graph which is less complex but having characteristics similar to the original graph will always be very useful. We propose in this report a new graph reduction method by performing a singular value decomposition on the adjacency matrix of a complex graph. We also propose a notion of loop decomposition which is a generalization of graph triangulation, from which we also derive a measure of graph complexity.

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It is known that a light ray must obey the law of reflection when it is reflected by a plane mirror. In this report, we are going to find out whether a light ray in a regular pentagon$^1$ formed by 5 congruent plane mirrors can go back to the starting position and what the possible emitting angles are. Also, we will investigate the looping of the light trajectory after finite reflection.

First, we make an observation on some special cases. Then, we will consider the general cases and try to classify the looping trajectories. Properties of looping trajectories will be studied. Lastly, another approach, vectors, will be used to investigate this problem.

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Previous articles have discussed about the properties of orthocentric tetrahedrons: nine-point circles on each face cospherical and the 3D Euler line. This paper aims at finding the sufficient and necessary conditions for the nine-point circles to be cospherical in the triangular polyhedrons. First, we discussed the conditions for the nine-point circles to be cospherical in a tetrahedron, in a hexahedron and in an octahedron. Next, we found that the 3Dorthocenter H_c , the center of the 24-point sphere (48-point sphere) N_c and the 3D circumcenter O_c of a tetrahedron (an octahedron), if they exist, must be collinear and the ratio of the distance between them is H_{cNc : NcOc} = 1 : 1. After studying the properties of triangular polyhedrons, we have found that the existence of the 3D orthocenter and the 3D circumcenter is the necessary condition for the nine-point circles to be cospherical.

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In our report, we will manipulate the Fermat’s Equation by allowing one of the exponents to be arbitrary. It turns out that if a prime base is restricted, there are either no solutions or a unique primitive solution, depending on the residue class that the prime belonging to modulo 4.

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The ultimate objective of this paper is to examine the periodicity of the Generalized Fibonacci Sequence (GFS) modulo j with different starting numbers. In this paper, we introduce a brand new method to study the period of the sequence inspired by the hand game ‘Chopsticks’ usually played in primary schools.

We first prove that the period of GFS modulo a prime p other than 5 is either half of the p-th Pisano Period or exactly equal to it in Theorem 16. We then investigate the decomposition from the period of the game modulo j to the least common multiple of the periods of the game modulo the primepower factors of j in Theorem 23. We continue our investigation on the periodicity of GFS modulo p other than 5 and prime powers p^k in Corollary 18-20, Lemma 7 and Theorem 26. Finally, we use Theorem 27 to give a general expression for the period of GFS modulo j in terms of the p_i-th Pisano period, where p_i’s are the prime factors of j.

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Abstract:

Given a regular polygonal paper inscribed in a unit circle, the paper is cut along its radii and each division (consisting of one or more subdivisions) is made into a cone. These cones are allowed to be slanted to obtain a greater capacity. The purpose of this study is to maximize the total capacity of cones made from the paper over all ways of divisions.

The methodology in this report is streamed into two parts – minimax strategy and bounds by inequalities. For triangular paper, the rims of cones are parameterized before their water depths are expressed explicitly. The capacities of cones are maximized over angles of slant. Different ways of division are compared to and out the optimal solution. Probing into general cases, various inequalities are set up analytically and exhaustively to bound the total capacities for comparisons.

To obtain the greatest capacities, cones made from one sub-division should be slanted but those from multiple sub-divisions should be held vertically. For a polygonal paper of six or more sides, it should be divided into two divisions, each comprising two or more sub-divisions with a central angle ratio of 0.648:1.352, approaching the way of division in circular paper.

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