Abstract:In this paper, we study the existence of non-torsion solutions of a homogeneous linear system over a commutative ring. More precisely, we determine the minimal positive integer n such that any homogeneous systems of m equations with n variables over a given ring R gives a non-torsion solution, i.e. a solution x =( x1, x2, . . . , xn) such that at least one coordinate xi is not a zero-divisor. We proved that over Noetherian rings, a non-trivial lower bound to the minimal number can be guaranteed via the use of primary decomposition. We also consider the number of generators of ideals in R and the localisations of R. For some classes of Noetherian rings, such as principal ideal rings and reduced rings, we show that such minimal number exists.
Abstract:We study the problem of finding the maximum number of maximal chains in a given size-k subset of a square poset [n] × [n]. This was proposed by Johnson, Leader, and Russell but not yet solved. Kittipassorn had given a conjectural solution to the problem. We verify Kittipassorn’s conjecture for 0 ≤ k ≤ 3n − 2 and solve a variant problem for the case 3n − 1 ≤ k ≤ 4n − 4, which also supports the conjecture. For general k, we find that the optimal configuration is given by a 1-Lipschitz function. We also generalize the problem to rectangle posets and give a solution to one particular poset.
Abstract:A new and practical test for determining the solvability of the general Pell’s equation \(x^2 - Dy^2 = n\) will be offered through proving one necessary condition and one sufficient condition for this renowned quadratic Diophantine equation to be solvable in integers. The test involves only prime factorization and checking of certain simple quadratic residuosity relations. While the necessary condition will be comparatively more straightforward, the sufficient condition in a form of conditional converse of the former will require algebraic number theory tools to formulate and analyze. To prove this sufficient condition, the solvability in question will be transformed into the question of principality of certain well-designed ideal class in a real quadratic field Q(√D) of class number two.
Abstract:The Basel problem is about finding the sum of the reciprocals of all perfect squares. This problem is first posed by Pietro Mengoli in 1650 and was solved by Leonhard Euler in 1734. Euler proved that the sum of the series is \(π^2/6\). In this report, inspired by an idea suggested by the YouTube channel 3blue1brown in 2018, we attempt to give a new proof to the Basel problem. After that, we discuss some possible generalizations of the Basel problem, by finding the sum of reciprocals of squares and cubes of the form an + b. Furthermore, we discuss how the sum of reciprocals of integral powers of an + b can be computed, and the relation between ζ(3) and the results we have achieved.
Abstract:This is an investigation on the ring of integer-valued polynomials on the Gaussian integers and the ring of integer-valued continuous function on rational integers, inspired by the results from integer-valued polynomials on the rational integers. Polynomials in the first ring map Gaussian integers to Gaussian integer values while functions in the second ring map rational integers to rational integers. This investigation explores their properties as rings, following a chain of class inclusions, which includes the most commonly known domains. The properties of rings of polynomials over algebraic integers, continuously differentiable functions on rational integers and continuous functions on Gaussian integers are also discussed.
Abstract:A lot of work has been put into solving special cases of the famous Square Peg Problem, which focuses on the two dimensional space. The aim of this project is to investigate the generalization of the Triangle Peg Problem into manifolds of higher dimensions.
Abstract:The Buffon-Laplace needle problem is a variation of the well-known Buffon’s needle problem, which asks what the probability of a needle, after being dropped to a rectangular grid, intersects, or touches the grid is. In this paper, we aim to solve some generalizations of this problem. We generalized the problem by dropping regular polygons to the grid instead of dropping a needle. We solved this generalization of the problem, by first using the rotational and reflectional symmetry of the regular polygons, then splitting the number of sides of the polygon into 4 cases, then solving each case. We also generalized the problem by dropping arbitrary 2D shapes. We found a general formula and an algorithmic solution to the problem. Apart from generalizations to the problem, we also considered some variations of the problem, like dropping right regular polygon prisms in a 3D space, with the grid being planes in each axis. We used a similar method to solve this problem and provided a formula. We also considered dropping a needle into a n-dimensional space. However, we failed to get a closed form for the formula.