Table of Content

Abstract:

The Trapezoidal Peg Problem, as one of the generalizations of the famous Square Peg Problem, asks when a prescribed trapezoid can be inscribed in a given Jordan curve. We investigated a possible approach towards the problem by first weakening the similarity condition, in which we have shown that for any trapezoid, some classes of convex curves can actually inscribe, up to two kinds of weaker forms of similarity, infinitely many trapezoids. Our main theorem further analyzed the properties of one of these infinite family of trapezoids, and showed that any given trapezoid can be uniquely inscribed in any strictly convex \(C_1\) curve, which we named `oval’, up to only translation and a kind of transformation, which we called `stretching’, but without rotation, and the resulting trapezoid moves continuously when the given trapezoid rotates.
Through this, we consequently obtained a necessary and sufficient condition for an oval to inscribe an arbitrary trapezoid up to similarity, which could be taken as an answer to the problem among ovals. Some other variations are also discussed.

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In this paper, we propound an efficacious method to derive the p-adic valuation of the Catalan number by analyzing the properties of the coefficients in the base p-expansion of n. We unearth a new connection between those coefficients and the p-adic valuation of the Catalan number. In fact, we have discovered that the highest power of p dividing the Catalan number is relevant to the number of digits greater than or equal to half of \(p + 1\), the nature of distribution of digits equal to half of \(p + 1\) and the frequency of carries when 1 is added to n. Meanwhile, we remark that the method we apply is more natural than the current way used by Alter and Kubota which is quite artificial.
Applications, examples of our new formula and details about Catalan numbers are also included in this paper.

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This essay will analyze a function that is a generalization of the Gauss sum. The function happens to be closely related to the cycle index of the symmetric group, which will also be analyzed. Some properties of the Gauss sum will be generalized. A number-theoretic inequality is also obtained.

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The central problem we are investigating is based on a problem from the 2018 Singapore International Mathematics Challenge. It is about a mathematical model of the probabilities that the people on a footbridge from two sides meet. In our paper, we generalize the contest problem in various cases. We develop a Markov model then formulate a transition matrix to solve the generalized problem. Also, we deffine an expansion rule of the transition matrices to reduce the time complexity to compute. Furthermore, we propose a new topic on the expected number of collisions. We tackle the problem by performing Jordan decomposition. Lastly, we optimize the method of finding eigenvalues by observing the recursive relationships in transition matrices.

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Residue theorem has been frequently used to tackle certain complicated deffinite integrals. However, it is never applied for indeffinite integrals. Therefore, in this report, residue theorem and a some small tricks are applied to find antiderivatives. The are mainly three interesting results:

1. Antiderivatives can be found without integration: antiderivatives can be represented by residues, while calculation of residues requires no knowledge of integration. For residues at poles, only differentiation is needed.
For residues at essential singularities, Taylor series manipulation is required; still, it is just differentiation with algebra work. This allows fast computation of antiderivatives of rational functions, especially those with only simple poles, providing an alternative to partial fraction decomposition. This is also applicable for other functions. Moreover, this result implies that integration is not only the reverse of differentiation, integration is indeed equivalent to differentiation.

2. A universal functional form of antiderivative can be obtained: antiderivatives obtained by this method has a functional form that converges wherever it should converge. The functional form has the largest possible region of convergence on the complex plane.

3. As a weak tool for analytic continuation: since the universal functional form of antiderivative is obtained, differentiating yields a universal functional form of the integrand. If one knows the behaviour of f in the vicinity of every singularity of f, one can analytically continue f to its largest possible domain by the method presented in this report.

Residue theorem is the central tool to be used throughout this report. Certain simple inequalities such as triangle inequality and estimation lemma are occasionally applied

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This paper aims to investigate the integral solutions of the Mordell’s Equation \(y^2 = x^3 + k\) for a particular class of integers $k$. We employ some classical approaches, i.e. factorization in number fields and quadratic reciprocity. When \(k = p^2\) for certain primes \(p\), we can determine the set of solutions. Two other classes of integers \(k\) are also solved in this paper.

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The classical triangle centres, namely centroid, circumcentre, incentre, excentre, orthocentre and Monge point, will be generalized to tetrahedra in a unified approach as points of concurrence of special lines. Our line characterization approach will also enable us to create new tetrahedron centres lying on the Euler lines, which will be a family with nice geometry including Monge point and twelve-point centre.

Two tetrahedron centres generalizing orthocentre of triangles from new perspectives will be constructed through introducing antimedial tetrahedra, tangential tetrahedra and a new kind of orthic tetrahedra. The first one, defined as the circumcentre of the antimedial tetrahedron of a tetrahedron, will be proved to lie on the Euler line. The second one, defined as the incentre or a suitable excentre of the new orthic tetrahedron of a tetrahedron, will be discovered to be collinear with its circumcentre and twenty-fifth Kimberling centreX25. Surprisingly, these two differently motivated geometric generalizations turn out to have analogous algebraic representations.
A clear definition of tetrahedron centres, as a generalization of triangle centres to tetrahedra, will be coined to set up a framework for studying analogies between geometries of triangles and tetrahedra. Fundamental properties of tetrahedron centres will be studied.

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