Irreducible compositions and the first return to the origin of a random walk

Let $n = b_1 + ... + b_k = b_1' + \cdot + b_k'$ be a pair of compositions of $n$ into $k$ positive parts. We say this pair is {\em irreducible} if there is no positive $j < k$ for which $b_1 + ... b_j = b_1' + ... b_j'$. The probability that a random pair of compositions of $n$ is irreducible is shown to be asymptotic to $8/n$. This problem leads to a problem in probability theory. Two players move along a game board by rolling a die, and we ask when the two players will first coincide. A natural extension is to show that the probability of a first return to the origin at time $n$ for any mean-zero variance $V$ random walk is asymptotic to $\sqrt{V/(2 \pi)} n^{-3/2}$. We prove this via two methods, one analytic and one probabilistic

A Discontinuity in the Distribution of Fixed Point Sums

The quantity $f(n,r)$, defined as the number of permutations of the set $[n]=\{1,2,... n\}$ whose fixed points sum to $r$, shows a sharp discontinuity in the neighborhood of $r=n$. We explain this discontinuity and study the possible existence of other discontinuities in $f(n,r)$ for permutations. We generalize our results to other families of structures that exhibit the same kind of discontinuities, by studying $f(n,r)$ when ``fixed points'' is replaced by ``components of size 1'' in a suitable graph of the structure. Among the objects considered are permutations, all functions and set partitions.Comment: 1 figur

On voxel-by-voxel accumulated dose for prostate radiation therapy using deformable image registration.

Since delivered dose is rarely the same with planned, we calculated the delivered total dose to ten prostate radiotherapy patients treated with rectal balloons using deformable dose accumulation (DDA) and compared it with the planned dose. The patients were treated with TomoTherapy using two rectal balloon designs: five patients had the Radiadyne balloon (balloon A), and five patients had the EZ-EM balloon (balloon B). Prostate and rectal wall contours were outlined on each pre-treatment MVCT for all patients. Delivered fractional doses were calculated using the MVCT taken immediately prior to delivery. Dose grids were accumulated to the last MVCT using DDA tools in Pinnacle3 TM (v9.100, Philips Radiation Oncology Systems, Fitchburg, USA). Delivered total doses were compared with planned doses using prostate and rectal wall DVHs. The rectal NTCP was calculated based on total delivered and planned doses for all patients using the Lyman model. For 8/10 patients, the rectal wall NTCP calculated using the delivered total dose was less than planned, with seven patients showing a decrease of more than 5% in NTCP. For 2/10 patients studied, the rectal wall NTCP calculated using total delivered dose was 2% higher than planned. This study indicates that for patients receiving hypofractionated radiotherapy for prostate cancer with a rectal balloon, total delivered doses to prostate is similar with planned while delivered dose to rectal walls may be significantly different from planned doses. 8/10 patients show significant correlation between rectal balloon anterior-posterior positions and some VD values

Submaps of maps. I. General 0–1 laws

AbstractLet Mn be the set of n edge maps of some class on a surface of genus g. When g = 0 (planar maps) we show how to prove that limn → ∞ |Mn|1/n exists for many classes of maps. Let P be a particular map that can appear as a submap of maps in our class. There is often a strong 0–1 law for the property that P is a submap of a randomly chosen map in Mn: If P is planar, then almost all Mn contain at least cn disjoint copies of P for small enough c; while if P is not planar, almost no Mn contain a copy of P. We show how to establish this for various classes of maps. For planar P, the existence of limn → ∞ |Mn|1n suffices. For nonplanar P, we require more detailed asymptotic information

Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part, number of distinct parts and number of parts of multiplicity k, all accurate to o(1).Comment: 28 page

The Asymptotic Number of Irreducible Partitions

A partition of [1, n] = {1,..., n} is called irreducible if no proper subinterval of [1, n] is a union of blocks. We determine the asymptotic relationship between the numbers of irreducible partitions, partitions without singleton blocks, and all partitions when the block sizes must lie in some specified set

Searching for Massive Black Hole Binaries in the first Mock LISA Data Challenge

The Mock LISA Data Challenge is a worldwide effort to solve the LISA data analysis problem. We present here our results for the Massive Black Hole Binary (BBH) section of Round 1. Our results cover Challenge 1.2.1, where the coalescence of the binary is seen, and Challenge 1.2.2, where the coalescence occurs after the simulated observational period. The data stream is composed of Gaussian instrumental noise plus an unknown BBH waveform. Our search algorithm is based on a variant of the Markov Chain Monte Carlo method that uses Metropolis-Hastings sampling and thermostated frequency annealing. We present results from the training data sets and the blind data sets. We demonstrate that our algorithm is able to rapidly locate the sources, accurately recover the source parameters, and provide error estimates for the recovered parameters.Comment: 11 pages, 6 figures, Submitted to CQG proceedings of GWDAW 11, AEI, Germany, Dec 200

Synthesis and Characterization of Erbia Doped Metal Oxide Nanofibers for Applications in the Rmophotovoltaics

Titania (TiO2) nanofibers doped with erbia (Er2O3) have been synthesized by electrospinning mixtures of polymers, titanium-containing materials, and erbia particles. These electrospun nanofibers are subsequently annealed at temperatures of 800, 900, 1000, and 1050 degrees C to remove the organics and leave behind the metal oxides. The crystal structure and optical properties of the metal oxides depend on the annealing temperature, and we characterize these nanofibers using x-ray diffraction and Fourier transform infrared spectroscopy (FTIR). An Er2Ti2O7 phase is formed in an amount which depends on the annealing temperature, and relationships between the nature of FTIR spectra and the relative amounts of different phases are demonstrated. Finally, the relevance of this work to thermophotovoltaics and other applications is discussed. (c) 2007 American Vacuum Society

The Cauchy convergence of T and P-approximant templates for test-mass Kerr binary systems

In this work we examine the Cauchy convergence of both post-Newtonian (T-approximant) and re-summed post-Newtonian (P-approximant) templates for the case of a test-mass orbiting a Kerr black hole along a circular equatorial orbit. The Cauchy criterion demands that the inner product between the $n$ and $n+1$ order approximation approaches unity, as we increase the order of approximation. In previous works, it has been shown that we achieve greater fitting factors and better parameter estimation using the P-approximant templates for both Schwarzschild and Kerr black holes. In this work, we show that the P-approximant templates also display a faster Cauchy convergence making them a superior template to the standard post-Newtonian templates.Comment: 5 pages, Replaced with shortened published versio