Table of Content

Abstract:

In the report, we want to answer the following question: How to deform a curve such that the rate of change of perimeter is minimum while the area and the total kinetic energy are fixed? This means that the perimeter shrinks fastest when \(${dP\over dt}>0$\), increases most slowly when \(${dP \over dt}>0$\) First we work on isosceles triangle as a trial. Then we study smooth simple closed curve and obtain the following results:

• The radial velocity of each point of the curve in polar coordinates. (3.1.6)
• The magnitude of the velocity at each point of the curve along the normal direction is equal to standard score of the curvature at that point (3.2.2).
• Application of the results on Isoperimetric inequality (3.3).
• The velocity for the dual isoperimetric problem (4).

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Abstract:

Huffman’s coding provides a method to generate a weight-balanced tree, but it is not generating progressively. In other words, we cannot have meaningful output if we terminate the algorithm halfway in order to save time. For this purpose, we want to design an alternative algorithm, therefore this paper aims at finding out a sufficient condition of being a weight-balanced tree.

In this paper, we have found out a sufficient condition. Besides, as the solution of building a weight-balanced can be applied to solving other problems, we abstract the problem and discuss it in the manner of graph theory. The applications are also covered.

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Published in 1659, the solutions of Fermat Point problem help people find out the point at which the sum of distances to 3 fixed points in the plane is minimized. In this paper, we are going to further discuss the case when the number of fixed point is more than 3. Also, we would like to find out if there exists a way such that the location of point minimizing the sum of distances to more than three given points can be determined just by compasses and ruler, or approximated by mathematical methods.

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The aim of our project is to investigate the \(3n + 1\) conjecture. It is very hard to give a general path for each natural number to arrive at \(1\). So we investigate its negation i.e. there exists a natural number \(k\) with no path to \(1\). There are two possibilities: either \(k\) takes a path which becomes a cycle to after \(n\) steps, or its path is increasing indefinitely. These two possibilities lead us to study pre-numbers of any odd natural number and the number of peaks of paths. In the project, several interesting results were obtained by studying backward paths, number of peaks and cycles or forward paths.

Firstly, we tried our best to trace back the path by studying all the pre-numbers of any natural number. We successfully showed that every odd number not divisible by \(3\) has infinitely many pre-numbers by finding them explicitly, and hence obtained a beautiful known result: the odd numbers \(A\) and \(4A + 1\) fall into the same number. Moreover, we also found that two third of these pre-numbers has infinitely many pre-numbers. Continuing in this way, we constructed a decreasing path of any length to arrive at any given number. This leads to a beautiful corollary: there are infinitely many distinct decreasing paths of different lengths to \(1\).

Secondly, we studied the peaks of the path of any number and obtained a theorem that for any T-number \(k\) and natural number \(r\), there exists a path to \(k\) of length \(2r\) with exactly \(r\) peaks.

Thirdly, we assumed, on the contrary, that there is a path with the same beginning and ending, and obtained a constraint on both the length and the sum of powers of \(2\).

Fourthly, we found infinitely many pairs which meet before \(1\) and fall into the same path afterward.

Finally, after investigating the possibilities of general \((a, b, c)\)-Conjectures, we concluded that the only possible conjectures are \(3n + b\) Conjectures.

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Abstract:

In this report, our team has explored the mathematical structure of Graeco-Latin squares. Although we give a review of the scope of this field, our focus is on Euler’s Conjecture. According to this conjecture, Graeco-Latin squares of certain orders do not exist. In this report, we disprove this conjecture by demonstrating a means to construct an infinite number of these so-called non-existent squares, following Sade. This branch of mathematics is related to group theory, combinatorics, and transversal design; therefore, we will also provide a brief overview of these topics throughout this report.

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The equidecomposition problem is to divide a shape into pieces, and then use the pieces to form another shape. In this project, we are going to investigate the conditions under which a given shape can be broken down and combined into another specified shape. The classical problem on the equidecomposability of polygons has already been solved by mathematicians. We start by presenting the proof of the classical problem, which is the keystone of this research. Then the problem is generalized to weighted shapes, shape with curves, etc. Some interesting new results are obtained.

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Abstract:

When dividing a cake of circle shape into equal parts, it is quite easy to divide it from the centre. However, if we need to divide it from its edge, how can we accomplish this task accurately?

This report aims to find a method to divide the area of a circle into \(3\) equal parts with two straight lines by Euclidean construction, i.e. the construction with compass and straightedge only. However, we were aware that it is difficult, if not impossible, to find the exact method of construction. Therefore, we try to find some methods to divide the area of circle approximately into three equal parts. In this report, we have three analytic approaches: by Lagrange Interpolating Polynomial, by infinite series of sine function and by method of bisection. Then, we will discuss three methods of construction, which are, inscribing a regular polygon with a large number of sides, inscribing a regular polygon with a small number of sides and bisecting the slope. At last, we will give a comparison of these three methods.

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