Table of Content

Abstract:

The question investigated in this essay is: Given the dimensions of a cylindrical container and the volume of the water contained in it, which position would give the minimum wet contact area? In part 1, we will discuss about which position of the container, horizontal or vertical, will give a smaller wet contact surface area when the volume of water varies. In part 2, we will still discuss the wet contact area in the cylinder, but considering the volume of water as a constant and allowing the cylinder to incline with a variable angle \(\alpha \). We will try to find out the value of \(\alpha \) such that the total wet surface area is minimum.

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Abstract:

It is possible to prove the Prime Number Theorem(PNT) by elementary methods. A. Selberg sketched his original elementary proof in a paper in 1949. This article is an attempt to complete the proof of the PNT by following the ideas in Selberg’s paper.

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In this project, we study Poincar\(‘{e} \) disk model of hyperbolic geometry and compare it with Euclidean geometry we have learnt in school. We investigate some basic properties of the model and derive some theorems comparable to those in Euclidean geometry.
The main objective of our work is to construct four common (non-Euclidean) tangents to two circles with Euclidean compass and Euclidean straightedge, as well as two other construction problems, in Poincar\(‘{e} \) disk model. With non-Euclidean transformations, we can transform a point to anywhere inside the Poincar\(‘{e} \) disk, with lengths and angles preserved. So we first focus on performing the transformation by compass and straightedge, and then solve the problems with a centre of the circle placed at the centre of the disk. Finally we can transform the picture back to the given position by the inverse function.

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We want to prove that in \(mathbb{R}^2\), the greatest density of unit circle packing is equal to \(\frac{\pi}{2\sqrt{3}}\).
Elementary techniques were mainly used in the following proof(s). Dissection method was used to form a house circumscribing the circle. After all cases were considered, the regular hexagonal house was found with the smallest area.
Hence, the circle inside this regular hexagonal house is of the highest density, which is \(\frac{\pi}{2\sqrt{3}}\).

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Abstract:

The aim of this project is to study the problem of Threshold Scheme. After reading the book “In Code: A Mathematical Journey”[1] written by Sarah Flannery, we found an interesting problem. When 11 persons keep a secret and any 6 of them are allowed to open it, we need 462 locks and each person needs 252 keys. As the author gives the answer without explanation, we are interested in Threshold Scheme. We call this problem the Key Distribution scheme “KD Scheme”. Afterwards, we think about how the Chinese Remainder Theorem (CRT) works to implement the Threshold Scheme. We call it “CRT Scheme”. As CRT requires some pairwisely prime (prp) numbers, we create algorithms to generate prp numbers. The results of this report include:

1. Solving the KD scheme completely.
2. Constructing the CRT Scheme.
3. Comparing the CRT Scheme with the KD scheme.
4. Constructing a set of prp numbers by the method of \(\{M\pm1\}\cup\{M+p_i\}\).
5. Constructing a set of prp numbers by the “Sieve of PRP Numbers”.
6. Making a conjecture that the maximum number of prp numbers within the range \([s, s+k]\) is approximately equal to \(\pi(k)\).

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In this project, we try to find out the patterns of Sudoku and the best strategy of a game in game theory.

For Sudoku part, we added three symmetries and used permutation to deal with the problem. We found that there are only 16 possible permutation patterns. Some properties of the combinations of these 16 patterns are also included in the report.

The game in game theory is game about 3 players choose a number that is greater than or equal to 3 respectively. The player who chooses the smallest and not repeated number is the winner. We successfully found the best strategy.

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Abstract:

First, we define three kinds of measures of angles in order to handle rolling and rotation.

In Part One, we give the relations among them:
\(d \theta = d \varphi – d\phi\)

And the accumulated exterior angles \(\varphi\) and \(\phi\) are highly depended on the geometric features of the objects concerned ( i.e. curvatures of the boundaries of objects):
\(dD = R(\varphi)d\varphi\) and  \( dD = -r(\phi)d\phi.\)

In Part Two, we further study the length of the trajectory formed by certain point on the rolling object. The result is:
\(dT_p = L_p(\phi)|d\theta|.\)

Also we will give some applications of these results under the circumstance that the curvatures are uniform (The curvatures can be positive, zero and negative).

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With the aim of finding new alternatives to resolve large Fibonacci or Lucas numbers, we have immersed ourselves in these two sequences to find that there are other fascinating phenomena about them. We have, in the first part of the report, successfully discovered four new methods to resolve large Fibonacci and Lucas numbers. From the very beginning, we have decided to adopt the normal investigation approach: observe, hypothesize, and then prove. The first two methods were discovered. Then we move on and try to explore these sequences in the two dimensional world. From the tables and triangles thereby created, we have discovered various surprising patterns which then help us generate the third and fourth formula to resolve large Fibonacci and Lucas numbers. In the second part of the report, we have focused on sequences in two dimensions and discovered many amazing properties about them.

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Abstract:

In a common personal computer environment, 3D objects are projected on the screen 2-dimensionally, and the project result is based on: (1) The relative position and angle of the object and the observer; (2) Observing method, for example, alpha channel or image distortion. Among most of the 3D objects, human models usually are very high regarded; indeed, creating motive human model is the technique which the game designers want to master, as human are so curious about themselves, and the related mathematics and the technique of computer efficiency are treated as a challenge. Although this is not a fresh topic, obviously, it is still very specialized, we want to express it effectively and simply by mathematics.
Though the research of the construction, appearance design, and the limbs motions of human model, we hope we can extend to other 3D objects, and let the program designer to design a popular 3D model designing tools by using the result of the research.

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