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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.

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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 Approximation Examined provides a self contained derivation of Stirling's approximation of the factorial function,
using Laplace's method applied to the gamma function integral, and also a simple derivation
of lower and upper bounds on the factorial function. The bounds are of the form often attributed to Gosper.