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We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various $\mathscr R$-, $\mathscr L$- and $\mathscr D$-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras., Comment: Small changes to intro and references. To appear in Journal of Combinatorial Theory, Series A. 26 pages, 4 figures, 10 tables

We show that every countable set of partial bijections from an infinite set to itself can be obtained as a composition of just two such partial bijections. This strengthens a result by Higgins, Howie, Mitchell and Ru\v{s}kuc stating that every such countable set of partial bijections may be obtained as the composition of two partial bijections and their inverses.

We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, $X$, into $\mathbb{R}^n$. The key question is `what is the generic dimension of $f(X)$?' and we consider two different approaches to answering it: Baire category and prevalence. In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, $n$, but find that the behaviour of the Hausdorff, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of $n$ and the topological dimension of $X$. We also study the typical Hausdorff and packing measures of $f(X)$ and, in particular, give necessary and sufficient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of $f(X)$ and give some simple applications concerning the prevalent dimensions of graphs of real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux., Comment: 14 pages

We consider the Banach space consisting of real-valued continuous functions on an arbitrary compact metric space. It is known that for a prevalent (in the sense of Hunt, Sauer and Yorke) set of functions the Hausdorff dimension of the image is as large as possible, namely 1. We extend this result by showing that `prevalent' can be replaced by `1-prevalent', i.e. it is possible to \emph{witness} this prevalence using a measure supported on a one dimensional subspace. Such one dimensional measures are called \emph{probes} and their existence indicates that the structure and nature of the prevalence is simpler than if a more complicated `infinite dimensional' witnessing measure has to be used., Comment: 8 pages

We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on $[0,1]^d$ for $d \in \mathbb{N}$ and use this to obtain results on the `horizon problem' for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function.

A well-known hazard associated with exposure to the space environment is the risk of failure from an impact from a meteoroid and orbital debris (MMOD) particle. An extravehicular mobility unit (EMU) “spacesuit” impact during a US extravehicular activity (EVA) is of great concern as a large leak could prevent an astronaut from safely reaching the airlock in time resulting in a loss of life. A risk assessment is provided to the EVA office at the Johnson Space Center (JSC) by the Hypervelocity Impact Technology (HVIT) group prior to certification of readiness for each US EVA. Need to understand the effect of updated meteoroid and orbital debris environment models to EMU risk.

This paper describes meteoroid and orbital debris (MMOD) damage observations on the International Space Station (ISS). Several hundred MMOD damage sites on ISS have been documented using imagery taken from ISS windows. MMOD damage sites visible from ISS windows are typically larger – approximately 5mm diameter and greater – due to the larger viewer-to-surface distance. Closer inspection of these surfaces by astronauts during spacewalks reveals many smaller features that are typically less distinct. Characterization of these features as MMOD or non- MMOD is difficult, but can be partially accomplished by matching physical characteristics of the damage against typical MMOD impact damage observed on ground-based impact tests. Numerous pieces of space-exposed ISS hardware were returned during space shuttle missions. Subsequent ground inspection of this hardware has also contributed to the database of ISS MMOD impact damage. A handful of orbital replacement units (ORUs) from the ISS active thermal control and electrical power subsystems were swapped out and returned during the Space Shuttle program. In addition, a reusable logistics module was deployed on ISS for a total 59.4 days on 11 shuttle missions between 2001 and 2011 and then brought back in the shuttle payload bay. All of this returned hardware was subjected to detailed post-flight inspections for MMOD damage, and a database with over 1,400 impact records has been collected. A description of the largest observed damage features is provided in the paper. In addition, a discussion of significant MMOD impact sites with operational or design aspects is presented. MMOD impact damage to the following ISS modules/subsystems is described: (1) Solar Arrays, (2) US and Russian windows, (3) Extravehicular Activity (EVA) handrails, (4) Radiators, and (5) Russian Functional Cargo Block (FGB) module.

A well-known hazard associated with exposure to the space environment is the risk of vehicle failure due to an impact from a micrometeoroid and orbital debris (MMOD) particle. Among the vehicles of importance to NASA is the extravehicular mobility unit (EMU) “spacesuit” used while performing a US extravehicular activity (EVA). An EMU impact is of great concern as a large leak could prevent an astronaut from safely reaching the airlock in time resulting in a loss of life. For this reason, a risk assessment is provided to the EVA office at the Johnson Space Center (JSC) prior to certification of readiness for each US EVA.

The SpaceX Cargo Dragon has been flying supply missions for NASA to the International Space Station (ISS) since 2012. At the time of this writing, SpaceX has completed 14 missions to ISS with each returning to Earth via water landing. After each mission, a team from the NASA Johnson Space Center performs an inspection of the exterior surfaces of the Cargo Dragon spacecraft for hypervelocity impact damage features. Data from these inspections are collected into a database that includes impact feature dimensions as well as the location on the vehicle. Additional details on the type and size of particle that produced the damage site are pro-vided when sampling data and definitive spectroscopic analysis results are available. Observation data can be compared with impact estimates from risk assessment codes as a check on the micro-meteoroid and orbital debris (MMOD) environment predictions.

“Bumper” is NASA’s computer program for analyzing spacecraft micrometeoroid and orbital debris (MMOD) risk. Bumper was developed in the late-1980s and has been continuously used and maintained since. The user base has grown from a few government entities to now include numerous commercial entities as well. The NASA Johnson Space Center (JSC) Hypervelocity Impact Technology (HVIT) Team is responsible for all aspects of the Bumper software. Bumper has been used to characterize MMOD risk on hundreds of spacecraft. All of the International Space Station (ISS) modules, visiting vehicles and numerous external components and systems have been analyzed. Bumper was used to analyze each of the Space Shuttle missions since STS-50. The Orion Multi-Purpose Crew Vehicle (MPCV) MMOD shielding is being developed using Bumper as well. Bumper has also been used on numerous telescopes (Hubble, James Webb, and Fermi Gamma-ray Space Telescopes), scientific probes (Stardust, New Horizons, Parker Solar Probe), and Earth observation satellites (Landsat, Joint Polar Satellite System). Bumper is also being used to analyze the micrometeoroid risk and support design of the Deep Space Gateway (DSG) and Mars Sample Return (MSR) missions. The HVIT Bumper Configuration Control Board (CCB) ensures that all changes to the code are approved, reviewed, and documented. Most of the changes are made to add new MMOD damage “ballistic limit equations” (BLEs). BLEs are typically added in response to completion of a hypervelocity impact (HVI) test series and development of an associated BLE. Other less frequent changes include updates of the debris or meteoroid environment models, feature enhancements, and feature retirement. Some BLEs are commercially sensitive and/or proprietary, so the CCB also manages code user-version control and software distribution. The current version – “Bumper 3” – is a FORTRAN executable that utilizes a 64-bit architecture. Bumper 3 has numerous features that make it a powerful tool for analyzing spacecraft MMOD risk. Bumper uses the latest orbital debris and micrometeoroid environment models. Bumper also easily processes large spacecraft geometry models, recognizes hidden surfaces, permits BLE assignment by name or number, and conducts quality checks of the spacecraft geometry model. Bumper 3 can also be used to estimate the effects of particle penetration through thin, high-standoff distance hardware components such as solar arrays and radiators. This is done using a special HVIT-developed technique know as the “3-Part Analysis.” The paper introduces the Bumper 3 MMOD risk analysis code and provides an example MMOD risk assessment showing Bumper’s role in the overall MMOD protection design process.

Risk from micrometeoroid and orbital debris (MMOD) impacts on space vehicles is often quantified in terms of the probability of no penetration (PNP). However, for large spacecraft, especially those with multiple compartments, a penetration may have a number of possible outcomes. The extent of the damage (diameter of hole, crack length or penetration depth), the location of the damage relative to critical equipment or crew, crew response, and even the time of day of the penetration are among the many factors that can affect the outcome. For the International Space Station (ISS), a Monte-Carlo style software code called Manned Spacecraft Crew Survivability (MSCSurv) is used to predict the probability of several outcomes of an MMOD penetration—broadly classified as loss of crew (LOC), crew evacuation (Evac), loss of escape vehicle (LEV), and nominal end of mission (NEOM). By generating large numbers of MMOD impacts (typically in the billions) and tracking the consequences, MSCSurv allows for the inclusion of a large number of parameters and models as well as enabling the consideration of uncertainties in the models and parameters. MSCSurv builds upon the results from NASA’s Bumper software (which provides the probability of penetration and critical input data to MSCSurv) to allow analysts to estimate the probability of LOC, Evac, LEV, and NEOM. This paper provides an overview of the methodology used by NASA to quantify LOC, Evac, LEV, and NEOM with particular emphasis on describing in broad terms how MSCSurv works and its capabilities and most significant models.