The Schottky problem in genus five

In this paper, we present a solution to the Schottky problem in the spirit of Schottky and Jung for genus five curves. To do so, we exploit natural incidence structures on the fibers of several maps to reduce all questions to statements about the Prym map for genus six curves. This allows us to find all components of the big Schottky locus and thus, to show that the small Schottky locus introduced by Donagi is irreducible.Comment: 20 page

Universities and the Success of Entrepreneurial Ventures: Evidence from the Small Business Innovation Research Program

There has been little direct, systematic empirical analysis of the role that universities play in enhancing the success of entrepreneurial ventures. We attempt to fill this gap by analyzing data from the SBIR program, a set-aside program that requires key federal agencies (e.g., Department of Defense) to allocate 2.5 percent of their research budget to small firms that attempt to commercialize new technologies. Based on estimation of Tobit and negative binomial regressions of the determinants of commercial success, we find that start-ups with closer ties to universities achieve higher levels of performance.

Parsimoneous Modeling of Yield Curves for U.S. Treasury Bills

A new model is proposed for representinq the term to maturity structure of interest rates at a point in time.The model produces humped, monotonic and S-shaped yield curves using four parameters. Conditional on a time decay parameter, estimates of the other three are obtained by least squares. Yield curves for thirty-seven sets of U.S. Treasury bill yields with maturities up to one year are presented. The median standard deviation of fit is just over seven basis points and the corresponding median R-squared is .96. Study of residuals suggests the existence of specific maturity effects not previously identified. Using the models to predict the price of a long term bond provides a diagnostic check and suggests directions for further research.

What does fault tolerant Deep Learning need from MPI?

Deep Learning (DL) algorithms have become the de facto Machine Learning (ML) algorithm for large scale data analysis. DL algorithms are computationally expensive - even distributed DL implementations which use MPI require days of training (model learning) time on commonly studied datasets. Long running DL applications become susceptible to faults - requiring development of a fault tolerant system infrastructure, in addition to fault tolerant DL algorithms. This raises an important question: What is needed from MPI for de- signing fault tolerant DL implementations? In this paper, we address this problem for permanent faults. We motivate the need for a fault tolerant MPI specification by an in-depth consideration of recent innovations in DL algorithms and their properties, which drive the need for specific fault tolerance features. We present an in-depth discussion on the suitability of different parallelism types (model, data and hybrid); a need (or lack thereof) for check-pointing of any critical data structures; and most importantly, consideration for several fault tolerance proposals (user-level fault mitigation (ULFM), Reinit) in MPI and their applicability to fault tolerant DL implementations. We leverage a distributed memory implementation of Caffe, currently available under the Machine Learning Toolkit for Extreme Scale (MaTEx). We implement our approaches by ex- tending MaTEx-Caffe for using ULFM-based implementation. Our evaluation using the ImageNet dataset and AlexNet, and GoogLeNet neural network topologies demonstrates the effectiveness of the proposed fault tolerant DL implementation using OpenMPI based ULFM

Syracuse Random Variables and the Periodic Points of Collatz-type maps

Let $p$ be an odd prime, and consider the map $H_{p}$ which sends an integer $x$ to either $\frac{x}{2}$ or $\frac{px+1}{2}$ depending on whether $x$ is even or odd. The values at $x=0$ of arbitrary composition sequences of the maps $\frac{x}{2}$ and $\frac{px+1}{2}$ can be parameterized over the $2$-adic integers ($\mathbb{Z}_{2}$) leading to a continuous function from $\mathbb{Z}_2$ to $\mathbb{Z}_p$ which the author calls the numen of $H_{p}$, denoted $\chi_p$; the $p=3$ case turns out to be an alternative version of the Syracuse Random Variables constructed by Tao [arXiv:1909.03562]. This paper establishes the Correspondence Theorem, which shows that an odd integer $\omega$ is a periodic point of $H_p$ if and only if $\omega=\chi_{p}\left(n\right)/\left(1-r_{p}\left(n\right)\right)$ for some integer $n\geq1$, where $r_{p}\left(n\right) = p^{\#_{1}\left(n\right)} / 2^{\lambda\left(n\right)}$, where $\#_1 \left(n\right)$ is the number of $1$s digits in the binary expansion of $n$ and $\lambda\left(n\right)$ is the number of digits in the binary expansion of $n$. Using this fact, the bulk of the paper is devoted to examining the Dirichlet series associated to $\chi_p$ and $r_p$, which are used along with Perron's Formula to reformulate the Correspondence Theorem in terms of contour integrals, to which Residue calculus is applied so as to obtain asymptotic formulae for the quantities therein. A sampling of other minor results on $\chi_p$ are also discussed in the paper's final section.Comment: 92 pages, 5 figures. arXiv admin note: text overlap with arXiv:2002.1276