Questions and conjectures about multinomial coefficients

The purpose of this thesis is to try to answer some of the questions in Dr. Bachman\u27s paper On Divisibility Properties of Certain Multinomial Coefficients . First we let {ai} be any sequence (finite or infinite) of positive integers such that i1ai ≤1 . It is clear that n!&sqbl0;na1 &sqbr0;!&sqbl0;na2&sqbr0; !&sqbl0;na3&sqbr0;!&ldots; is an integer because it is a multiple of a certain multinomial coefficient. We let fan=n! Ln&sqbl0;n a1&sqbr0;!&sqbl0;na 2&sqbr0;!&sqbl0;na3 &sqbr0;!&ldots; where L(n) = lcm(1, 2, 3, .., n). It is easy to show that fa(n) is integer-valued. In particular, we would like to study the sequence a1 = b1 = 2 and ak+1 = bk+1 = Pki=1 bi + 1. The first goal of my thesis was to prove the following conjecture by computer for all m up to 100; Conjecture 1. For every positive integer m there exists a number n0 such that m divides f( n) for all n \u3e n0 where fn=n!L n&sqbl0;n2&sqbr0; !&sqbl0;n3&sqbr0;!&sqbl0;n 7&sqbr0;!&ldots I did this by using Theorem 1 of Dr. Bachman\u27s paper; Theorem 1. pv|| f(n) if and only if there are exactly v pairs of integers (k,l),k,l ≥ 1, such that Rk&parl0;&sqbl0;npl &sqbr0;&parr0;Bk\u3c Rk+1&parl0;&sqbl0;npl &sqbr0;&parr0;Bk+1 with Rk(m) defined as m ≡ Rk(m) mod Bk and 0 \u3c Rk( m) ≤ Bk where Bk = bk+1 - 1; The second part of my thesis is concerned with attacking Conjecture 1 as it was written in Dr. Bachman\u27s paper. Before we can restate Conjecture 1 we need to define the base p expansion of a positive integer. We write nj = a0pj + a1pj -1 +..+ aj where 0 ≤ ai ≤ p - 1. Now we restate Conjecture 1 as Conjecture 2; Conjecture 2. Let {nj} be defined above. Then there exist infinitely many integers j for which the inequality Rk&parl0;nj&parr0;B

Looking forward to making predictions

As described in the preceding pages, since the BGS was established in 1835, the British population has coped with many challenges. These have ranged from finding resources to fuel the Industrial Revolution, understanding and combating water-borne diseases such as typhoid, the threat of invasion and aerial bombardment, through to modern-day environmental problems and climate change. To help deal with these problems, decisionmakers from governments and other organisations have required our help and advice

Results from the Supernova Photometric Classification Challenge

We report results from the Supernova Photometric Classification Challenge (SNPhotCC), a publicly released mix of simulated supernovae (SNe), with types (Ia, Ibc, and II) selected in proportion to their expected rates. The simulation was realized in the griz filters of the Dark Energy Survey (DES) with realistic observing conditions (sky noise, point-spread function, and atmospheric transparency) based on years of recorded conditions at the DES site. Simulations of non–Ia-type SNe are based on spectroscopically confirmed light curves that include unpublished non-Ia samples donated from the Carnegie Supernova Project (CSP), the Supernova Legacy Survey (SNLS), and the Sloan Digital Sky Survey-II (SDSS-II). A spectroscopically confirmed subset was provided for training. We challenged scientists to run their classification algorithms and report a type and photo-z for each SN. Participants from 10 groups contributed 13 entries for the sample that included a host-galaxy photo-z for each SN and nine entries for the sample that had no redshift information. Several different classification strategies resulted in similar performance, and for all entries the performance was significantly better for the training subset than for the unconfirmed sample. For the spectroscopically unconfirmed subset, the entry with the highest average figure of merit for classifying SNe Ia has an efficiency of 0.96 and an SN Ia purity of 0.79. As a public resource for the future development of photometric SN classification and photo-z estimators, we have released updated simulations with improvements based on our experience from the SNPhotCC, added samples corresponding to the Large Synoptic Survey Telescope (LSST) and the SDSS-II, and provided the answer keys so that developers can evaluate their own analysis

SNANA: A Public Software Package for Supernova Analysis

We describe a general analysis package for supernova (SN) light curves, called SNANA, that contains a simulation, light curve fitter, and cosmology fitter. The software is designed with the primary goal of using SNe Ia as distance indicators for the determination of cosmological parameters, but it can also be used to study efficiencies for analyses of SN rates, estimate contamination from non-Ia SNe, and optimize future surveys. Several SN models are available within the same software architecture, allowing technical features such as K-corrections to be consistently used among multiple models, and thus making it easier to make detailed comparisons between models. New and improved light-curve models can be easily added. The software works with arbitrary surveys and telescopes and has already been used by several collaborations, leading to more robust and easy-to-use code. This software is not intended as a final product release, but rather it is designed to undergo continual improvements from the community as more is learned about SNe. Below we give an overview of the SNANA capabilities, as well as some of its limitations. Interested users can find software downloads and more detailed information from the manuals at http://www.sdss.org/supernova/SNANA.html .Comment: Accepted for publication in PAS

Testing Models of Intrinsic Brightness Variations in Type Ia Supernovae, and their Impact on Measuring Cosmological Parameters

For spectroscopically confirmed Type Ia supernovae we evaluate models of intrinsic brightness variations with detailed data/Monte Carlo comparisons of the dispersion in the following quantities: Hubble-diagram scatter, color difference (B-V-c) between the true B-V color and the fitted color (c) from the SALT-II light curve model, and photometric redshift residual. The data sample includes 251 ugriz light curves from the 3-season Sloan Digital Sky Survey-II, and 191 griz light curves from the Supernova Legacy Survey 3-year data release. We find that the simplest model of a wavelength-independent (coherent) scatter is not adequate, and that to describe the data the intrinsic scatter model must have wavelength-dependent variations. We use Monte Carlo simulations to examine the standard approach of adding a coherent scatter term in quadrature to the distance-modulus uncertainty in order to bring the reduced chi2 to unity when fitting a Hubble diagram. If the light curve fits include model uncertainties with the correct wavelength dependence of the scatter, we find that the bias on the dark energy equation of state parameter $w$ is negligible. However, incorrect model uncertainties can lead to a significant bias on the distance moduli, with up to ~0.05 mag redshift-dependent variation. For the recent SNLS3 cosmology results we estimate that this effect introduces an additional systematic uncertainty on $w$ of ~0.02, well below the total uncertainty. However, this uncertainty depends on the samples used, and thus this small $w$-uncertainty is not guaranteed in future cosmology results.Comment: accepted by Ap

Prevalence and patterns of antidepressant switching amongst Primary Care patients in the UK

Objective: Non-response to antidepressant treatment is a substantial problem in primary care, and many patients with depression require additional second-line treatments. This study aimed to examine the prevalence and patterns of antidepressant switching in the UK, and identify associated demographic and clinical factors. Method: Cohort analysis of antidepressant prescribing data from the Clinical Practice Research Datalink, a large, anonymised UK primary care database. The sample included 262,844 patients who initiated antidepressant therapy between 1 January 2005 and 31 June 2011. Results: 9.3% of patients switched to a different antidepressant product, with most switches (60%) occurring within 8 weeks of the index date. The proportion switching was similar for selective serotonin reuptake inhibitors (SSRIs), tricyclic antidepressants and other antidepressants (9.3%, 9.8% and 9.2%, respectively). Most switches were to an SSRI (64.5%), and this was the preferred option regardless of initial antidepressant class. Factors predictive of switching included male gender, age, and history of self-harm and psychiatric illness. Conclusion: Over one in every 11 patients who initiates antidepressant therapy will switch medication, suggesting that initial antidepressant treatment has been unsatisfactory. Evidence to guide choice of second-line treatment for individual patients is currently limited. Additional research comparing different pharmacological and psychological second-line treatment strategies is required in order to inform guidelines and improve patient outcomes. </jats:sec