Gabriel B. Costa

Polynomial solutions of certain classes of ordinary differential equations

Several classes of ordinary differential equations which have polynomial solutions are studied. In particular, generalizations of the Hermite, Laguerre, Legendre, and Chebyshev equations are given for which such solutions exist. These solutions turn out to be generalizations of well‐known polynomials and enjoy similar properties.

Solving second-order differential equations with variable coefficients

A method is developed in which an analytical solution is obtained for certain classes of second-order differential equations with variable coefficients. By the use of transformations and by repeated iterated integration, a desired solution is obtained. This alternative method represents a different way to acquire a solution from classic power series techniques and other approaches. It is, at times, more involved than traditional methods.

Visualization of two dimensional to three dimensional transformations – exploration through technology

A small class of functions is described that easily lend themselves to two-dimensional and three-dimensional visualizations at the basic calculus level. The intended audience is those educators involved in the instruction of elementary calculus. This note is an educational piece that begins with the question: ‘What happens if a function defined on a short interval in the first quadrant is rotated about an axis? This question is answered visually using a graphical software package Mathematica®.

Families of separable partial differential equations

In this paper the authors investigate classes of partial differential equations (mostly in two independent variables, of order at most 4) which can be solved by the well‐known technique of separation of variables. In particular, several classical ordinary differential equations, some of which have been generalized (by means of certain parameters), are used to assist in the construction of such partial differential equations. The process gives rise to (potentially) many families of equations which appear to be difficult to solve. The article not only illustrates the technique used in the determination of solutions to such equations, but also contrasts the effect of the nature of solutions as certain parameters effect them. In addition, the paper indicates how other classes of solvable equations might be obtained; equations with more than two variables and of any order. Only analytical methods are discussed; however it would not be difficult to use computing machinery as the number of independent variables ...

9 – Probability and Markov Chains

Abstract In this chapter, the concept of probability is introduced. Both independent and disjoint events are covered, as are basic properties of probability. Bernoulli events and counting (combinatorics) are also discussed. Markov chains used in modeling are also presented. Applications are given as well.

6 – Eigenvalues and Eigenvectors

1 Determinants Recall that determinant is a function from square matrices to real numbers, det : M n×n −→ R, satisfying the following properties: • det I n = 1 • det(E ij A) = − det A • det(E i (c) A) = c det A • det(E ij (k) A) = det A In addition, it provably satisfies the following properties: • det A ̸ = 0 iff A is nonsingular. In other words, if there is linear dependence among the rows or columns, then det A = 0 • det(AB) = det A · det B • det(A T) = det A There are two ways for computing det A: • Using definition, that is, by applying elementary row/column operations to transform A to a lower-or upper-triangular form while recording how the determinat is changed in each step; • Using cofactors (along any row or column).

The keystone combination: team teaching in a sabermetrics course

ABSTRACT The Society for American Baseball Research (SABR), formed in 1971, has given rise to the term “Sabermetrics”, which noted practitioner Bill James has defined as “the search for objective knowledge about baseball.” Mathematics, by its very nature, is objective, but baseball opinion is often based on a subjective interpretation of events. Thus, a one-credit course in Sabermetrics team-taught by two professors allows for a broader range of opinion to be introduced in class. Students also benefit from the two different teaching styles offered