Brad Jackson

Studies in Astronomical Time Series Analysis. VI. Bayesian Block Representations

This paper addresses the problem of detecting and characterizing local variability in time series and other forms of sequential data. The goal is to identify and characterize statistically significant variations, at the same time suppressing the inevitable corrupting observational errors. We present a simple nonparametric modeling technique and an algorithm implementing it—an improved and generalized version of Bayesian Blocks [Scargle 1998]—that finds the optimal segmentation of the data in the observation interval. The structure of the algorithm allows it to be used in either a real-time trigger mode, or a retrospective mode. Maximum likelihood or marginal posterior functions to measure model fitness are presented for events, binned counts, and measurements at arbitrary times with known error distributions. Problems addressed include those connected with data gaps, variable exposure, extension to piecewise linear and piecewise exponential representations, multi-variate time series data, analysis of variance, data on the circle, other data modes, and dispersed data. Simulations provide evidence that the detection efficiency for weak signals is close to a theoretical asymptotic limit derived by [Arias-Castro, Donoho and Huo 2003]. In the spirit of Reproducible Research [Donoho et al. (2008)] all of the code and data necessary to reproduce all of the figures in this paper are included as auxiliary material.

The splitting number of the complete graph

If a given graphG can be obtained bys vertex identifications from a suitable planar graph ands is the minimum number for which this is possible thens is called the splitting number ofG. Here a formula for the splitting number of the complete graph is derived.

Heawood's empire problem

We consider the coloring of maps of empires on surfaces. A country is by convention assumed to be connected. On the other hand in a map we call a certain collection of m disjoint countries an empire. If we wish to specify that an empire has exactly m components, we call it an m-pire. Two empires are adjacent if they share a common boundary edge. For instance Fig. 1 illustrates a map with 19 “3-pires” on the Klein bottle. In the rectangle of Fig. 1 the two horizontal boundary lines have to be identified to obtain first a cylinder; then the two vertical boundary lines are identified in opposite directions to obtain the Klein bottle. In this example each 3-pire consists of one quadrilateral, one pentagon, and one 9- gon; and the 19 “3-pires” are mutually adjacent. In coloring a map all components of a given empire must receive the same color and, of course, two adjacent empires must receive different colors. Let S be a closed surface with Euler characteristic E. And let x(S, M) be the smallest number k of colors such that every map on S, where each empire has at most M components, is colorable by k colors. x(S, M) is called the M-pire chromatic number of S. With the exception of M= 1 and E= 2 (four color problem) Heawood [S] showed in 1890 that x(X M)<L;(6M+ 1 +,/(6M+ l)“-24E))_I (1) holds. As usual the symbol LX] means the greatest integer <x. An alter- native proof is also given in Ringel’s first book [9, p. 761. Haken and Appel [ 1,2] have shown (1) in the remaining case M = 1 and E = 2 (four color theorem). Heawood conjectured that equality always holds in (1). In the following special cases equality in (1) has been proven: 168

Maps of m-pires on the projective plane

In this paper the authors study maps of m-pires (empires). It is shown that Heawoods m-pire formula holds, for all values of m, for maps on the projective plane using the theory of current graphs.

Research problems on Gray codes and universal cycles

Open problems arising from the Workshop on Generalizations of de Bruijn cycles and Gray Codes at the Banff International Research Station in December, 2004.

A shortness exponent for r-regular r-connected graphs

Let r≧ 3 be an integer. It is shown that there exists e= e(r), 0 0 such that for all n ≧ N (if r is even) or for all even n ≧ N(if r is odd), there is an r-connected regular graph of valency r on exactly n vertices whose longest cycles have fewer than ne vertices.

Variations on Ringel's earth-moon problem

Abstract We use current graphs to find decompositions of complete graphs into subgraphs with certain embeddability properties. These decompositions provide solutions to various extensions of Ringels earth–moon problem to other surfaces and to more than two surfaces. In particular, we find a decomposition of K 6 n +1 into n toroidal graphs, and from this get a decomposition of K 6 n into n projective-plane graphs. We find a decomposition of K 19 into three Klein bottle graphs and for n >3, we also conjecture that there exist decompositions of K 6 n +1 into n Klein bottle graphs. Finally, we find the 3-chromatic number for infinitely many different orientable surfaces.

Hamilton surfaces for the complete even symmetric bipartite graph

Abstract The authors discuss a hamilton surface of a graph, which is a two-dimensional analog of a hamilton cycle. Hamilton surface decompositions are given for K 2n , 2n . In addition a few hamilton surface decompositions are given for K 2n .