Exploring methods for finding solutions to polynomial equations

textThere are many methods for solving polynomial equations. Dating back to the Greek and Babylonian mathematicians, these methods have been explored throughout the centuries. The introduction of the Cartesian Coordinate Plane by Rene Descartes greatly enhanced the understanding of what the solutions actually represent. The invention of the graphing calculator has been a tremendous aid in the teaching of solutions of polynomial equations. Students are able to visualize what these solutions represent graphically. This report explores these methods and their uses.Mathematic

A locus construction in the hyperbolic plane for elliptic curves with cross-ratio on the unit circle

This project demonstrates how an elliptic curve f defined by invariance under two involutions can be represented by the locus of circumcenters of isosceles triangles in the hyperbolic plane, using inversive model

A Reorganization in the Continuity of Subject Matter in Mathematics

This thesis considers a reorganization in the order of arrangement of certain topics in elementary and undergraduate mathematics; i.e. , arithmetic, algebra, plane geometry, solid geometry, trigonometry, analytic geometry, and calculus. Two terms important in the discussion are reorganization, the process of changing the relative position of topics or proofs in mathematics to an earlier or later place in the development of subject matter, and continuity, the logical order of topics arranged according to the need of one to explain the other. The purpose of the thesis is two-fold; First, to show what arrangement of topics may be desirable; and, Second, to justify the proposed changes by showing that such a reorganization will make it possible to give a simpler and more complete presentation of mathematics without affecting the logical sequence of topics. The discussion reviews the recent changes in elementary mathematics during the past forty years. These changes, in general, may be thought of as either of a general character indicating a trend or of a special character indicating a rearrangement in the order of particular topics. The general arrangement of the thesis is somewhat as follows. It is observed that propositions in elementary mathematics have been proved by methods of analytic geometry and calculus. Proofs of certain propositions in plane geometry are possible by coordinate methods. When they are presented in algebra, these proofs are not only simple but provide further understanding of topics in algebra, such as graphs, ratio and proportion, and the operations of algebra. Proofs of certain propositions, or formulas, from elementary mathematics are possible by means of integration. Such proofs by calculus are too difficult to be presented in algebra. These proofs should be postponed to calculus where the simple method of integration justifies the omission of any earlier type of proof of these propositions in elementary mathematics. In the conclusion of this discussion a rearrangement of topics in elementary mathematics (seventh year mathematics, eighth year mathematics, first year algebra, second course in algebra, and plane geometry) with special attention to the continuity of subject matter is given. Such a rearrangement, of necessity, implies changes in the order of some of the topics in later mathematics

College Algebra Slide Decks

This collection of slide decks is designed to be used in concert with the following Open Eduational Resources: - OpenStax College Algebra 2e Open Texbook - Department of Mathematics Video Playlist - Homework questions from myOpenMath Concepts include are: A study of equations, graphs, and inequalities for linear, quadratic, polynomial, rational, logarithmic, exponential, and absolute value functions. Transformations on graphs, complex numbers, circles, systems of inequalities, and systems of equations including matrices.https://scholars.fhsu.edu/all_oer/1005/thumbnail.jp

Hyperbolicity of direct products of graphs

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G(1) x G(2) is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).This work was supported in part by four grants from Ministerio de Economía y Competititvidad (MTM2012-30719, MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain