Number Theory Commons^™

A (Not So) Complex Solution To A² + B² = Cⁿ, Arnold M. Adelberg, Arthur T. Benjamin, David I. Rudel '99

All HMC Faculty Publications and Research

No abstract provided in this article.

Group Actions In Number Theory, Tyler J. Evans

Tyler Evans

Nonic 3-Adic Fields, John W. Jones, David P. Roberts

Mathematics Publications

We compute all nonic extensions of Q₃ and find that there are 795 of them up to isomorphism. We describe how to compute the associated Galois group of such a field, and also the slopes measuring wild ramification. We present summarizing tables and a sample application to number fields.

An Abc Construction Of Number Fields, David P. Roberts

Mathematics Publications

We describe a general three step method for constructing number fields with Lie-type Galois groups and discriminants factoring into powers of specified primes. The first step involves extremal solutions of the matrix equation ABC = I. The second step involves extremal polynomial solutions of the equation A(x) + B(x) + C(x) = 0. The third step involves integer solutions of the generalized Fermat equation ax^p + by^q + cz^r = 0. We concentrate here on details associated to the third step and give examples where the field discriminants have the form ±2^a3^b .

Pencils Of Quadratic Forms Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas

Articles and Preprints

A formula for the number of common zeros of a non-degenerate pencil of quadratic forms is given. This is applied to pencils which count binary strings with an even number of 1's prescribed distances apart.

Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin

Mathematics Faculty Publications

In this paper we give a new formula for the n-th power of a 2 × 2 matrix. More precisely, we prove the following: Let A = (a b c d) be an arbitrary 2 × 2 matrix, T = a + d its trace, D = ad − bc its determinant and define yn : = b X n/2c i=0 (n − i i )T n−2i (−D) i . Then, for n ≥ 1, A n = (yn − d yn−1 b yn−1 c yn−1 yn − a yn−1) . We use this formula together with an existing formula …

Random Walks With Badly Approximable Numbers, Doug Hensley, Francis Su

All HMC Faculty Publications and Research

Using the discrepancy metric, we analyze the rate of convergence of a random walk on the circle generated by d rotations, and establish sharp rates that show that badly approximable d-tuples in Rd give rise to walks with the fastest convergence.

Statistical Mechanics Of Farey Fraction Spin Chain Models, Jan Fiala

Electronic Theses and Dissertations

This thesis considers several statistical models defined on the Farey fractions. Two of these models, considered first, may be regarded as "spin chains", with long-range interactions, another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-F'robenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that all these models have a second-order phase transition with a specific …