The prominent mathematician Leopold Kronecker (1823 – 1891) is often relegated to footnotes and mainly remembered for his strict philosophical position on the foundation of mathematics. He held that only the natural numbers are intuitive, thus the only basis for all mathematical objects. In fact, Kronecker developed a complete school of thought on mathematical foundations and wrote many significant algebraic works, but his enigmatic writing style led to his historical marginalization. In 1887, Kronecker published an extended version of his paper, “On the Concept of Number," translated into English in 2010 for the first time by Edward T. Dean, who confirms that Kronecker is “notoriously difficult to read." In his paper, Kronecker proves that a socalled “algebraic number," meaning any root of a polynomial with integer coefficients, can be isolated from the other roots of that polynomial, as Dean says, “using solely talk of natural numbers." To ease the reader’s comprehension of Kronecker’s prose, here we explicate in detail the argument contained in that paper.
Serre showed that, for a discrete valuation field, the group of linear fractional transformations acts on an infinite regular tree with vertex degree determined by the residue degree of the field. Since the p-adics and the polynomials over the finite field of order p act on isomorphic trees, we may ask whether pairs of actions from these two groups are ever conjugate as tree automorphisms. We analyze permutations induced on finite vertex sets, and show a permutation classification result for actions by these linear fractional transformation groups. We prove that actions by specific subgroups of these groups are conjugate only in specific special cases.
This paper defines the Lebesgue measure preserving Thompson monoid, denoted by G, which is modeled on the Thompson group F except that the elements of G preserve the Lebesgue measure and can be non-invertible. The paper shows that any element of the monoid G is the composition of a finite number of basic elements of the monoid G and the generators of the Thompson group F. However, unlike the Thompson group F, the monoid G is not finitely generated. The paper then defines equivalence classes of the monoid G, use them to construct a monoid H that is finitely generated, and shows that the union of the elements of the monoid H is a set of equivalence classes, the union of which is G.
Let f(x) be a random integral polynomial of degree d ≥ 2 with coefficients uniformly and independently drawn from [−N,N ]. It is well known that the probability that f(x) is irreducible over the integers with Galois group Sd tends to 1 as N →∞. However, finding more precise estimations for these probabilities is still an active area of research. In this paper, we survey the classic work on this problem as well as a recent method introduced by Rivin. Additionally, we discuss the precision of Rivin’s argument for special classes of polynomials and end by investigating a toy case of cubic trinomials.
Entropy is a single value that captures the complexity of a group action on a metric space. We are interested in the entropies of a family of ideal pants groups ΓT, represented by projective reflection matrices depending on a real parameter T > 0. These groups act on convex sets ΩΓT which form a metric space with the Hilbert metric. It is known that entropy of ΓT takes values in the interval (1 2 ,1 ] ; however, it has not been proven whether 1 2 is the sharp lower bound. Using Python programming, we generate approximations of tilings of the convex set in the projective plane and estimate the entropies of these groups with respect to the Hilbert metric. We prove a theorem that, along with the images and data produced by our code, suggests that the lower bound is indeed sharp. This theorem regards the degeneration of the Hilbert metric on the convex set ΩΓT .
A graph is considered to be totally colored when one color is assigned to each vertex and to each edge so that no adjacent or incident vertices or edges bear the same color. The total chromatic number of a graph is the least number of colors required to totally color a graph. This paper focuses on k-regular graphs, whose symmetry and regularity allow for a closer look at general total coloring strategies. Such graphs include the previously defined Möbius ladder, which has a total chromatic number of 5, as well as the newly defined bird’s nest, which is shown to have a total chromatic number of 4. Furthermore, a total 4-coloring of the Petersen graph is examined and the total (k +1)colorings of k-regular graphs is discussed. More specifically, it is proposed that any (k +1)-coloring of any k-regular graph is inherently equitable for all 3 ≤ k ≤ 5 given a bound on the order of the graph. That is to say that every color is used no more than one time more than any other color when totally coloring the graph.
Many properties are known about analytic functions, however the class of harmonic functions which are the sum of an analytic function and the conjugate of an analytic function is less understood. We wish to find conditions such that linear combinations of univalent harmonic functions are univalent. We focus on functions whose image is convex in one direction i.e. each line segment in that direction between points in the image is contained in the image. M. Dorff proved sufficient conditions such that the linear combination of univalent harmonic functions will be univalent on the unit disk. The conditions are: the mappings must be locally univalent, their images must be convex in the imaginary direction and they must satisfy a normalization which states that the right and left extremes of the image are the image of 1 and -1 respectively. In this paper we generalize this existing theorem. The conditions of this theorem are geometric, and we would like to maintain this feature in the generalization. We show that the image may be convex in any direction and that any points on the boundary of the domain, which no longer must be the unit disk, can be the points that are mapped to the extrema, which now must be in the direction perpendicular to the direction of convexity.
Error-correcting codes (ECC), found in coding theory, use methods to handle possible errors that may arise from electronic noise, to a scratch of a CD in a way where they are detected and corrected. Recently, ECC have gone beyond their traditional use. ECC can be used in applications from performing magic tricks to detecting and repairing mutations in DNA sequencing. This paper investigates an application of the Hamming Code, a type of ECC, in the form of a magic trick which uses Andy Liu’s description of the Hamming Code through set theory and a known card trick. Finally, connections between this new card trick and the properties of the Hamming Code are explained.
Artin’s Primitive Root Conjecture represents one of many famous problems in elementary number theory that has resisted complete solution thus far. Significant progress was made in 1967, when Christopher Hooley published a conditional proof of the conjecture under the assumption of a certain case of the Generalised Riemann Hypothesis. In this survey we present a description of the conjecture and the underlying algebraic theory, and provide a detailed account of Hooley’s proof which is intended to be accessible to those with only undergraduate level knowledge. We also discuss a result concerning the qx +1 problem, whose proof requires similar techniques to those used by Hooley.
We introduce a relationship between the concavity of a C2 function and the area bounded by its graph and secant line. We utilize this relationship to develop a method of numerical integration. We then bound the error of the approximation, and compare to known methods, finding an improvement in error bound over methods of comparable computational complexity.
In this paper, we find patterns and count the number of distinct generalised Fibonacci sequences under modular arithmetic. We will start with the repetition of the normal Fibonacci sequence modulo an integer m ≥ 2 and make connections to its dependency on the prime factorisation of m. We will then extend the complexity of the problem into generalised Fibonacci sequences with different starting values. Finally we will present some interesting observations that are still open problems.
Map colorings refer to assigning colors to different regions of a map. In particular, a typical application is to assign colors so that no two adjacent regions are the same color. Map colorings are easily converted to graph coloring problems: regions correspond to vertices and edges between two vertices exist for adjacent regions. We extend these notions to 4x4 Sudoku puzzles, known as Shidoku puzzles, and standard 9x9 Sudoku puzzles by demanding unique entries in rows, columns, and regions. Motivated by our study of ring and field theory, we expand upon the standard division algorithm to study Gröbner bases in multivariate polynomial rings. We utilize Gröbner bases of an ideal of a multivariate polynomial ring over a finite field to solve coloring, Shidoku, and Sudoku problems. In the last section, we note Gröbner bases are also well-suited to hypergraph coloring problems.
The digraphs of commutative rings under modular arithmetic reveal intriguing cycle patterns, many of which have yet to be explained. To help illuminate these patterns, we establish a set of new theorems. Rings with relatively prime moduli a and b are used to predict cycles in the digraph of the ring with modulus ab. Rings that use Pythagorean primes as their modulus are shown to always have a cycle in common. Rings with perfect square moduli have cycles that relate to their square root.
