Physical Sciences and Mathematics Commons^™

A Comparison Of Recent Results On The Unicity Conjecture Of The Markoff Equation, Brandon John Metz

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this thesis we discuss the positive integer solutions to the equation known as the Markoff equation

x² + y² + z² = 3xyz.

Each solution to the equation is a permutation of a triple (x,y,z) with [mathematical equation refer to PDF] which is called a Markoff triple and each integer of the triple is referred to as a Markoff number.

In 1913, Frobenius conjectured that given an ordered Markoff triple (x,y,z), then both x and y are uniquely determined by z. In other words, if both (x₁,y₁,z) and (x₂,y_{2 …}

Generalized Markoff Equations, Euclid Trees, And Chebyshev Polynomials, Donald Joseph Mcginn

UNLV Theses, Dissertations, Professional Papers, and Capstones

The Markoff equation is x^2+y^2+z^2 = 3xyz, and all of the positive integer solutions

of this equation occur on one tree generated from (1, 1, 1), which is called the

Markoff tree. In this paper, we consider trees of solutions to equations of the form

x^2 + y^2 + z^2 = xyz + A. We say a tree of solutions satisfies the unicity condition

if the maximum element of an ordered triple in the tree uniquely determines the

other two. The unicity conjecture says that the Markoff tree satisifies the unicity

condition. In this paper, we show that there exists …

Time-Dependent Random Effect Poisson Random Field Model For Polymorphism Within And Between Two Related Species, Shilei Zhou

UNLV Theses, Dissertations, Professional Papers, and Capstones

Molecular evolution is partially driven by mutation, selection, random genetic drift, or combination of the three factors. To quantify the magnitude of these genetic forces, a previously developed time-dependent fixed effect Poisson random field model provides powerful likelihood and Bayesian estimates of mutation rate, selection coefficient, and species divergence time. The assumption of the fixed effect model that selection intensity is constant within a genetic locus but varies across genes is obviously biologically unrealistic, but it serves the original purpose of making statistical inference about selection and divergence between two related species they are individually at mutation-selection-drift inequilibrium. By relaxing …

The Zeta Function Of Generalized Markoff Equations Over Finite Fields, Juan Mariscal

UNLV Theses, Dissertations, Professional Papers, and Capstones

The purpose of this paper is to derive the Hasse-Weil zeta function of a special class of Algebraic varieties based on a generalization of the Markoff equation. We count the number of solutions to generalized Markoff equations over finite fields first by using the group structure of the set of automorphisms that generate solutions and in other cases by applying a slicing method from the two-dimensional cases. This enables us to derive a generating function for the number of solutions over the degree k extensions of a fixed finite field giving us the local zeta function. We then create an …