What are the Longest Ropes on the Unit Sphere?

We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between 0 and 1, we explicitly construct for each given thickness $${\Theta_n:= {\rm sin}\, \pi/(2n),}$$ $${n\in\mathbb{N}}$$, exactly $${\varphi(n)}$$ solutions, where $${\varphi}$$ is Euler's totient function from number theory. Then we prove that these solutions are unique, and also provide a complete characterisation of sphere filling curves on the unit sphere; that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once. All of these results carry over to open curves as well, as indicated in the last sectio

Collections for people: museums' stored collections as a public resource

Collections in UK museums grew enormously in the latter half of the 20th century yet museum collections, mostly maintained at public expense, are perceived as an underused resource. The Museums Association’s 2005 report, Collections for the Future1, together with press comments and books such as Treasures on Earth (2002)2 and Fragments of the World (2005)3, brought this issue into sharp focus. Collections for People set out to understand the scale of museum stored collections, and the main parameters of their access and use: • What is the size and nature of collections as a resource? How are they distributed, geographically and among different types of museum? • How much are different types of collection used by people other than museum staff? What sort of people use collections? What do they use them for: research, teaching and learning, creative activities, visits for enjoyment such as store tours? • How do users perceive this service? Do museums actively market collections access? Do they publicise what is in their collections? • How do museums facilitate collections use? What are the factors associated with greater use of collections? What do museums see as the barriers to more use

What are the Odds at the Russia 2018 FIFA World Cup?

In this note we explore the group stage competition format in a standard FIFA World Cup Soccer Championship group phase. The group stage involves thirty-two teams divided into eight groups of four teams each, based on a draw that takes the national teams’ seeding and geographical location into consideration. Each of the four teams in a given group is scheduled to play one match against every other team in the same group. Upon the completion of six games in each of the eight groups (for a total of 48 games), the top two highest scoring teams (the winner and the runner-up) advance to the knockout stage. In this note we focus on the forty different ways (sequential configurations or states) that a group stage in an arbitrary group can result at the end of the group stage upon the completion of the six games in a typical World Cup Championship. We generate simulations for these configurations on spreadsheets. We use the Russia 2018 FIFA World Cup as an example, along with other relevant historical data, to compare and contrast theoretical versus actual configurations and their probabilities

What Caused the Crime Decline?

Crime across the United States has steadily declined over the last two decades. Today, the crime rate is about half of what it was at its height in 1991. What was once seen as a plague, especially in urban areas, is now at least manageable in most places. Rarely has there been such a rapid change in mass behavior. This observation begs two central questions: Why has crime fallen? And to what degree is incarceration, or other criminal justice policy, responsible? Social scientists and policy experts have searched for answers. Various explanations have been offered: expanded police forces, an aging population, employment rates, and even legalized abortion. Most likely, there is no one cause for such widespread, dramatic change. Many factors are responsible.This report isolates two criminal justice policies -- incarceration and one policing approach -- and provides new findings on their effects on crime reduction using a regression analysis. This report issues three central findings, which are summarized: Increased incarceration at today's levels has a negligible crime control benefit:One policing approach that helps police gather data used to identify crime patterns and target resources, a technique called CompStat, played a role in bringing down crime in cities:Certain social, economic, and environmental factors also played a role in the crime drop

Variables : what are they and why are they important?

Includes bibliographical references.As a student in MATH 412 last semester, I encountered literature that suggested Algebra I students’ difficulty with that subject stemmed from their misconceptions concerning the concept of variable. Indeed, the transition from junior high school to high school mathematics is challenging enough and these things that we call variables truly form the base from which the concepts of Algebra develop. A variable can be identified as a special type of mapping, from a set of objects onto a number system. This mapping is based upon the measurement of some characteristic of the objects. Frequently, a variable is not named using a word or a phrase, but rather with an abstract symbol (i.e. a letter). Most of the Algebra textbooks I have examined have defined a variable as a letter that stands for a number. This seemingly simplistic treatment of the concept of variable may form the basis for why students’ understanding of variables is too narrow.B.S. (Bachelor of Science

Reliability training

Discussed here is failure physics, the study of how products, hardware, software, and systems fail and what can be done about it. The intent is to impart useful information, to extend the limits of production capability, and to assist in achieving low cost reliable products. A review of reliability for the years 1940 to 2000 is given. Next, a review of mathematics is given as well as a description of what elements contribute to product failures. Basic reliability theory and the disciplines that allow us to control and eliminate failures are elucidated

Testing strong gravity with gravitational waves and Love numbers

The LIGO observation of GW150914 has inaugurated the gravitational-wave astronomy era and the possibility of testing gravity in extreme regimes. While distorted black holes are the most convincing sources of gravitational waves, similar signals might be produced also by other compact objects. In particular, we discuss what the gravitational-wave ringdown could tell us about the nature of the emitting object, and how measurements of the tidal Love numbers could help us in understanding the internal structure of compact dark objects