Most Subsets are Balanced in Finite Groups

The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset $S + S = \{x+y : x,y\in S\}$ of a set of integers $S$. A finite set of integers $A$ is sum-dominated if $|A+A| > |A-A|$. Though it was believed that the percentage of subsets of $\{0,...,n\}$ that are sum-dominated tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominated if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately $4.5 \cdot 10^{-4}$). While most sets are difference-dominated in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominated tend to zero but the probability that $|A+A|=|A-A|$ tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of $\{0,..., n\}$ have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.Comment: Version 2.0, 11 pages, 2 figure

The Pythagorean Won-Loss Formula and Hockey: A Statistical Justification for Using the Classic Baseball Formula as an Evaluative Tool in Hockey

Originally devised for baseball, the Pythagorean Won-Loss formula estimates the percentage of games a team should have won at a particular point in a season. For decades, this formula had no mathematical justification. In 2006, Steven Miller provided a statistical derivation by making some heuristic assumptions about the distributions of runs scored and allowed by baseball teams. We make a similar set of assumptions about hockey teams and show that the formula is just as applicable to hockey as it is to baseball. We hope that this work spurs research in the use of the Pythagorean Won-Loss formula as an evaluative tool for sports outside baseball.Comment: 21 pages, 4 figures; Forthcoming in The Hockey Research Journal: A Publication of the Society for International Hockey Research, 2012/1

A software architecture for automating operations processes

The Operations Engineering Lab (OEL) at JPL has developed a software architecture based on an integrated toolkit approach for simplifying and automating mission operations tasks. The toolkit approach is based on building adaptable, reusable graphical tools that are integrated through a combination of libraries, scripts, and system-level user interface shells. The graphical interface shells are designed to integrate and visually guide a user through the complex steps in an operations process. They provide a user with an integrated system-level picture of an overall process, defining the required inputs and possible output through interactive on-screen graphics. The OEL has developed the software for building these process-oriented graphical user interface (GUI) shells. The OEL Shell development system (OEL Shell) is an extension of JPL's Widget Creation Library (WCL). The OEL Shell system can be used to easily build user interfaces for running complex processes, applications with extensive command-line interfaces, and tool-integration tasks. The interface shells display a logical process flow using arrows and box graphics. They also allow a user to select which output products are desired and which input sources are needed, eliminating the need to know which program and its associated command-line parameters must be executed in each case. The shells have also proved valuable for use as operations training tools because of the OEL Shell hypertext help environment. The OEL toolkit approach is guided by several principles, including the use of ASCII text file interfaces with a multimission format, Perl scripts for mission-specific adaptation code, and programs that include a simple command-line interface for batch mode processing. Projects can adapt the interface shells by simple changes to the resources configuration file. This approach has allowed the development of sophisticated, automated software systems that are easy, cheap, and fast to build. This paper will discuss our toolkit approach and the OEL Shell interface builder in the context of a real operations process example. The paper will discuss the design and implementation of a Ulysses toolkit for generating the mission sequence of events. The Sequence of Events Generation (SEG) system provides an adaptable multimission toolkit for producing a time-ordered listing and timeline display of spacecraft commands, state changes, and required ground activities

The Simple Truth About the Gender Pay Gap - Fall 2018 Edition

Updated regularly with the most current statistics from the Bureau of Labor Statistics and the Census Bureau, this report is a common-sense guide that provides key facts about the gender pay gap in the United States. Topics covered in the report include: the definition of the pay gap and its history; the pay gap in each state; the pay gap by age, race/ethnicity, and education; guidance for women facing workplace discrimination; and resources for fair pay advocates

First Order Approximations of the Pythagorean Won-Loss Formula for Predicting MLB Teams' Winning Percentages

We mathematically prove that an existing linear predictor of baseball teams' winning percentages (Jones and Tappin 2005) is simply just a first-order approximation to Bill James' Pythagorean Won-Loss formula and can thus be written in terms of the formula's well-known exponent. We estimate the linear model on twenty seasons of Major League Baseball data and are able to verify that the resulting coefficient estimate, with 95% confidence, is virtually identical to the empirically accepted value of 1.82. Our work thus helps explain why this simple and elegant model is such a strong linear predictor.Comment: 7 pages, 1 Table, Appendix with Alternative Proof; By the Numbers 21, 201

Distribution of Missing Sums in Sumsets

For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They also conjectured the existence of a limiting distribution for |A+A| and showed that the expectation of |A+A| is 2n - 11 + O((3/4)^{n/2}). Zhao proved that the limits m(k) := lim_{n --> oo} Prob(2n-1-|A+A|=k) exist, and that sum_{k >= 0} m(k)=1. We continue this program and give exponentially decaying upper and lower bounds on m(k), and sharp bounds on m(k) for small k. Surprisingly, the distribution is at least bimodal; sumsets have an unexpected bias against missing exactly 7 sums. The proof of the latter is by reduction to questions on the distribution of related random variables, with large scale numerical computations a key ingredient in the analysis. We also derive an explicit formula for the variance of |A+A| in terms of Fibonacci numbers, finding Var(|A+A|) is approximately 35.9658. New difficulties arise in the form of weak dependence between events of the form {x in A+A}, {y in A+A}. We surmount these obstructions by translating the problem to graph theory. This approach also yields good bounds on the probability for A+A missing a consecutive block of length k.Comment: To appear in Experimental Mathematics. Version 3. Larger computations than before, conclusively proving the divot exists. 40 pages, 15 figure