An introduction to continuum mechanics and some numerical methods used for simulations of continuous phenomena, with a brief overview of their use in the numerical modeling of impact cratering phenomena. References are provided for more details.
An examination of the problem of assessing the validity of numerical simulations of continuous-time dynamical systems by treating numerical methods as discrete-time dynamical systems. Simulations of simple explicit and implicit Runge-Kutta methods, along with their Matlab code, are included.
An exposition of the paper "Finite combinatory processes-formulation 1" by Emil Post (1936), where he presents a model of computing based on operations on a symbol space determined by a given set of directions. Post hypothesizes, but does not prove, that his concept of computability is logically equivalent to the concept of recursion in the sense given by Gödel and Church. This paper was published the same year that Turing published his famous paper "On Computable Numbers, with an application to the Entscheidungsproblem", where he introduces the concept of computability, also logically equivalent to recursion, that now bears his name.
An exposition of the addition of real numbers defined as cuts as presented in Dedekind's "Continuity and irrational numbers, in essays on the theory of numbers" (1872). The proof that the addition of two cuts determine a unique cut in the case of the sum of a rational and an irrational number. This shows how the reals inherit the additive structure of the rational numbers.
Answers to five questions concerning issues in the philosophy of quantum mechanics from a course I took with Wayne Myrvold in 2004. The questions concern constraints on hidden variable theories, the Bell-Kochen-Specker theorem, the measurement problem and some proposed solutions, the EPR argument and entanglement.
An expository paper on the fibre bundle formalism in coordinate-free differential geometry (based on Gauge Fields, Knots and Gravity by John Baez and Javier Munian) as it is applied to gauge field theory in physics together with an examination of the role that the fibre bundle formalism plays in addressing philosophical, particularly ontological, issues that are raised by the standard model and general relativity and in the interplay between physics and mathematics.
A set of introductory notes on topos theory, which is an abstract theory of generalized sets (though toposes, particularly Grothendieck toposes, can also be regarded as generalized topological spaces). After a brief review of some basic concepts from category theory, toposes are defined and some simple examples are given. The category of sets is described and characterized as the topos of constant sets. Topos logic is then described from the internal and external perspectives and then the internal logic of a topos is presented in the language of local set theories. The notes conclude with a consideration of number systems and arithmetic in the context of topos theory.
A set of introductory notes on axiomatic set theory presented in an informal way (which could be called "naive axiomatic set theory", since it is informal as naive set theory is but also presented in terms of axioms). The sections included are: fundamentals of sets and classes; products and coproducts; functions; relations; quotients; finite and infinite sets; well-order and induction; ordinals and cardinals; the axiom of choice; and the number systems (from the natural numbers to the real and complex numbers).
The first part of a basic introduction to algebraic Galois theory. After a summary of the basic concepts and definitions of group theory and field theory, the basics of field extensions and some basic results from group theory are presented. This part concludes with a statement of the beautiful fundamental theorem.
The second part of a basic introduction to algebraic Galois theory. After a review of the concepts and definitions from Part I, a galois extension is defined and several examples are given, followed by an illustration of the fundamental theorem. The notes conclude with a more complete version of the beautiful fundamental theorem.