PhD Candidate in Mathematics
University of Colorado, Boulder,
Boulder, CO 80309-0395
|Email:||sebastian.bozlee AT colorado.edu|
These notes derive the fundamental solutions associated to linear homogeneous recurrence relations using linear algebra, including the repeated eigenvalue case. The approach in these notes differs from the usual approach in that I do not use generating functions. Link
Weibel's "homological algebra" is a book with a lot of content, but also a lot left to the reader. Here I attempted to fill in some of what's left to the reader: filling in gaps in proofs, performing checks, correcting errors, and doing exercises.
These notes are very much a work in progress, with only solutions for parts of chapters 3, 4, and 8. Feel free to contact me with any corrections. Last updated 1/19/17. Link
Notes for a series of lectures on category theory for first year graduate students.Day 1
Notes for a 2 hour talk introducing sheaves and abelian categories. Aimed at graduate students who have completed a graduate course or two in algebra and topology.Link
Notes I wrote while taking these courses:
Displays algebraic curves in the real projective plane. Available under the terms of the GPL.
Source (Qt Creator Project)
Abstract: A recurrence matrix is a matrix whose terms are sequential members of a linear homogeneous recurrence sequence of order k and whose dimensions are both greater than or equal to k. In this paper, the ranks of recurrence matrices are determined. In particular, it is shown that the rank of such a matrix differs from the previously found upper bound of k in only two situations: When (a_j) satisfies a recurrence relation of order less than k, and when the nth powers of distinct eigenvalues of (a_j) coincide.
Abstract: The topological data of a group action on a compact Riemann surface can be encoded using a tuple (h; m_1, ..., m_s) called its signature. There are two number theoretic conditions on a tuple necessary for it to be a signature: the Riemann-Hurwitz formula is satisfied and each m_i equals the order of a non-trivial group element. We show on the genus spectrum of a group that asymptotically, satisfaction of these conditions is in fact sufficient. We also describe the order of growth for the number of tuples which satisfy these conditions but are not signatures in the case of cyclic groups.