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

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 Simulations of non–Ia-type SNe are based on spectroscopically confirmed light curves that include unpublished non-Ia samples donated from the Carnegie Supernova Project (CSP), the Supernova Legacy Survey (SNLS), and the Sloan Digital Sky Survey-II (SDSS-II).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

Thermosetting polymers and composites from agricultural oils

Renewable, bio-based thermosetting copolymer resins, ranging from tough and ductile rubbers to hard and glassy plastics to durable waterborne latex coatings, have been prepared by the polymerization of soybean, corn and linseed oils with various co-monomers. The development of these resin formulations with the right combination of processing viscosity, cure kinetics, and ultimate thermal mechanical properties for various manufacturing processes will be discussed. As expected, the thermal and mechanical properties, as well as the long term environmental durability of the material, are shown to be highly dependent on vegetable-oil composition, processing conditions, and co-monomer chemistry. These new bioplastic composites are likely to have a tremendous impact economically, environmentally and energy-wise, since the oils are (1) readily available in huge quantities from a renewable natural resource, (2) much cheaper than petroleum-based resins used in many polymers and composites, and (3) able to provide properties not presently available in commercial plastics

Binning is Sinning: Redemption for Hubble Diagram using Photometrically Classified Type Ia Supernovae

Bayesian Estimation Applied to Multiple Species (BEAMS) has been implemented in the BEAMS with Bias Corrections (BBC) framework to produce a redshift-binned Hubble diagram (HD) for Type Ia Supernovae (SN Ia). The BBC method corrects for selection effects and non-SNIa contamination, and systematic uncertainties are described by a covariance matrix with dimension matching the number of BBC redshift bins. For spectroscopically confirmed SNIa samples, a recent "Binning is Sinning" article (BHS21, arxiv:2012.05900) showed that an unbinned HD and covariance matrix reduces the systematic uncertainty by a factor of 1.5 compared to the binned approach. Here we extend their analysis to obtain an unbinned HD for a photometrically identified sample processed with BBC. To test this new method, we simulate and analyze 50 samples corresponding to the Dark Energy Survey (DES) with a low-redshift anchor; the simulation includes SNe Ia, and contaminants from core collapse SNe and peculiar SNe Ia. The analysis includes systematic uncertainties for calibration, and measures the dark energy equation of state parameter (w). Compared to a redshift-binned HD, the unbinned HD with nearly 2,000 events results in a smaller systematic uncertainty, in qualitative agreement with BHS21, and averaging results among the 50 samples we find no evidence for bias in measured cosmological parameters. To reduce computation time for fitting an unbinned HD with large samples, we propose an HD-rebinning method that defines the HD in bins of redshift, color, and stretch; the rebinned HD results in similar uncertainty as the unbinned case, and shows no evidence for biased cosmology parameters

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