## Lecture Notes

Algebra and Trigonometry Notes on the basic aspects of algebra and trigonometry with the objective of preparing for the study of calculus. Concentrates on understanding concepts. The development occurs in the context of applied problems. Problems, exercises, and solutions.

Calculus Notes on the basic aspects of calculus emphasizing the concepts underlying the computational formulas. Problems, exercises, and solutions.

Multivariate Calculus Notes on aspects of multivariate calculus emphasizing the role of linearity. Included is a discussion of differential forms. Problems, exercises, and solutions.

Ordinary Differential Equations A collection of notes, used as a text, on the basics of solving first and second order differential equations. Concentrates on linear equations. Includes a discussion of basic qualitative techniques. Problems and solutions are included.

Mathematical Modeling Notes on the process of constructing a mathematical model. Use of differential and difference equations and systems, dimensional analysis, simulation, and empirical data to construct models are all examined. Emphasis is on constructing the model rather than the development of new mathematical tools. Problems, exercises, and laboratory exercises are included.

Mathematics of Finance Notes on the material on the mathematics of finance as preparation for the Society of Actuaries examination on the subject. Exercises, problems, and solutions to the exercises and problems are included.

Actuarial Mathematics Notes on the basic aspects of actuarial mathematics, concentrating on the theory of life insurance. Exercises, problems, and solutions to the problems are included.

## Scholia

Quadrature Rules from an Advanced Perspective provides a simple, extensible method for computing the error when using a numerical quadrature method. The quadrature rule is viewed as integration with respect to a purely atomic measure.

The spectrum of an element in a complex Banach algebra provides a simple self-contained proof of the fact that any element of a complex Banach algebra has non-empty spectrum.

Stirling's Formula with Error Bounds provides a self contained derivation of Stirling's approximation along with error bounds in the form attributed to Gosper. The relationship between the factorial function and the gamma function is proved, then Laplace's method is applied to the gamma function integral. A simple derivation of lower and upper bounds on the factorial function can then be obtained. The bounds are of the form often attributed to Gosper.

Convergence of Series gives a proof that if a series of independent random variables all defined on the same probability space converges in distribution, then the series converges almost surely.

Hypergeometic Representation proves that any hypergeometric random variable can be written as a sum of independent, but usually not identically distributed, Bernoulli variables. The proof fills in some details of existing published proofs.

Hoeffding's Inequality is proved and some of its applications are developed here. The inequality provides tail probability bounds that decrease exponentially. Applications are given to estimating the tail probability for a hypergeometric random variable, a derivation of McDiarmid's inequality, and a proof of the multivariate Glivenko-Cantelli Theorem.