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Fractalized Cyclotomic Polynomials, David P. Roberts

Mathematics Publications

For each prime power p^m, we realize the classical cyclotomic polynomial Φ_p^m(x) as one of a collection of 3^m different polynomials in Z[x]. We show that the new polynomials are similar to Φ_p^m(x) in many ways, including that their discriminants all have the form ±p^c. We show also that the new polynomials are more complicated than Φ_p^m(x) in other ways, including that their complex roots are generally fractal in appearance.

Galois Number Fields With Small Root Discriminant, John W. Jones, David P. Roberts

Mathematics Publications

We pose the problem of identifying the set K(G,Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632. We definitively treat the cases G = A₄. A₅, A₆, and S₄, S₅, S₆, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois groups SL₃(2), A₇, S₇, PGL₂(7), SL₂(8), ΣL₂(8), PGL₂(9), PSL₂(11), and …

Cosmogenic 10be And 36cl Ages From Late Pleistocene Terminal Moraine Complexes In The Taylor River Drainage Basin, Central Colorado, U.S.A., Keith A. Brugger

Geology Publications

Cosmogenic surface-exposure ages from boulders on a terminal moraine complex establish the timing of the local last glacial maximum (LGM) in the Taylor River drainage basin, central Colorado. Five zero-erosion ¹⁰Be ages have a mean of 19.5±1.8 ka while that for three ³⁶Cl ages is 20.7±2.3 ka. Corrections for modest rates (∼1 mm ka⁻¹) of boulder surface erosion result in individual and mean ages that are generally within 2% of their zero-erosion values. Both the means and the range in ages of individual boulders are consistent with those reported for late Pleistocene moraines elsewhere in the …

Enumerating Building Block Semantics In Genetic Programming, Nicholas Freitag Mcphee, Brian Ohs, Tyler Hutchinson

Faculty Working Papers

This report provides a collection of definitions for the semantics of sub-trees and contexts as manipulated by standard sub-tree crossover in genetic programming (GP). These definitions allow us to completely and compactly describe the exact semantics of the components manipulated by sub-tree crossover, and the semantic results of those interactions. Sub- sequent work shows how these definitions can be used to collect valuable data about the available diversity in a GP population and the opportunities available to sub-tree crossover.

Semantic Building Blocks In Genetic Programming, Nicholas Freitag Mcphee, Brian Ohs, Tyler Hutchinson

Faculty Working Papers

In this paper we present a new mechanism for studying the impact of subtree crossover in terms of semantic building blocks. This approach allows us to completely and compactly describe the semantic action of crossover, and provide insight into what does (or doesn’t) make crossover effective. Our results make it clear that a very high proportion of crossover events (typically over 75% in our experiments) are guaranteed to perform no immediately useful search in the semantic space. Our findings also indicate a strong correlation between lack of progress and high proportions of fixed contexts. These results then suggest several new, …

Wild Partitions And Number Theory, David P. Roberts

Mathematics Publications

We introduce the notion of wild partition to describe in combinatorial language an important situation in the theory of p-adic fields. For Q a power of p, we get a sequence of numbers λ_Q,n counting the number of certain wild partitions of n. We give an explicit formula for the corresponding generating function Λ_Q(x) = Σλ_Q,nx^_n and use it to show that λ^1/n _Q,n tends to Q^1/(p-1). We apply this asymptotic result to support a finiteness conjecture about number fields. Our finiteness conjecture …

The Non-Synchronous Response To Rabots Glaciar And Storglaciaren, Northern Sweden, To Recent Climate Change: A Comparative Study, Keith A. Brugger

Geology Publications

Rabots Glaciar and Storglaciaren, two small valley glaciers in the Swedish Arctic, have not behaved synchronously in response to recent climate change. Both glaciers advanced late in the 19th century and then began to retreat in response to an approximately 1 degree C warming that occurred around 1910. By the mid-1980s the terminus and volume of Storglaciaren had essentially stabilized, so it may have completed its response to the earlier warming. In contrast, ongoing thinning and retreat of Rabots Glaciar are substantial and suggest its response time is considerably longer. A time-dependent numerical model was used to investigate each glacier’s …

Rock Glaciers In Central Colorado, U.S.A., As Indicators Of Holocene Climate Change, Kurt A. Refsnider, Keith A. Brugger

Geology Publications

We measured thalli diameters of the lichen Rhizocarpon subgenus Rhizocarpon on 48 individual lobes of 18 rock glaciers and rock glacier complexes in the Elk Mountains and Sawatch Range of central Colorado. Cumulative probability distribution and K-means clustering analyses were used to separate lichen thalli measurements into statistically-distinct groups, each interpreted as representing a discrete episode of rock glacier activity driven by an interval of cooler climate. Lichen ages for these episodes were assigned using a growth curve developed for Rhizocarpon geographicum in the nearby Front Range. An early Neoglacial episode, ca. 3080 yr BP, is correlative to other glacial …

Computational Soundness Of A Call By Name Calculus Of Recursively-Scoped Records, Elena Machkasova

Faculty Working Papers

The paper presents a calculus of recursively-scoped records: a two-level calculus with a traditional call-by-name lambda-calculus at a lower level and unordered collections of labeled lambda-calculus terms at a higher level. Terms in records may reference each other, possibly in a mutually recursive manner, by means of labels. We define two relations: a rewriting relation that models program transformations and an evaluation relation that defines a small-step operational semantics of records. Both relations follow a call-by-name strategy. We use a special symbol called a black hole to model cyclic dependencies that lead to infinite substitution.

Computational soundness is a property …