Abstract:
Let $\mathcal P_X$ and $\mathcal S_X$ be the partition monoid and symmetric group on an infinite set $X$. We show that $\mathcal P_X$ may be generated by $\mathcal S_X$ together with two (but no fewer) additional partitions, and we classify the pairs $\alpha,\beta\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal S_X\cup\{\alpha,\beta\}$. We also show that $\mathcal P_X$ may be generated by the set $\mathcal E_X$ of all idempotent partitions together with two (but no fewer) additional partitions. In fact, $\mathcal P_X$ is generated by $\mathcal E_X\cup\{\alpha,\beta\}$ if and only if it is generated by $\mathcal E_X\cup\mathcal S_X\cup\{\alpha,\beta\}$. We also classify the pairs $\alpha,\beta\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal E_X\cup\{\alpha,\beta\}$. Among other results, we show that any countable subset of $\mathcal P_X$ is contained in a $4$-generated subsemigroup of $\mathcal P_X$, and that the length function on $\mathcal P_X$ is bounded with respect to any generating set.

Abstract:
This study compiled a wide range of modern and historic geospatial datasets to examine ecological and geomorphic change at Diego Garcia Atoll across a 38-year period (1967–2005). This remarkable collection of spatially referenced information offered an opportunity to advance our understanding of the nature and extent of environmental change that has taken place with the construction of the military airbase at Diego Garcia. Changes assessed included movements of the lagoon rim shorelines, changes in the terrestrial vegetation on the lagoon rim and amendments to the bathymetry of the lagoon basin through dredging activities. Data compiled included detailed shoreline and vegetation maps produced as part of the H.M.S. Vidal Indian Ocean Expedition (1967), three Ikonos satellite images acquired in 2005 that collectively covered the complete Atoll area, a ground truthing field dataset collected in the northern section of the lagoon for the purpose of seafloor mapping (2005), observational evidence of shoreline erosion including photographs and descriptions of seawater inundations and bathymetric soundings from five independent surveys of the lagoon floor (1967, 1985, 1987, 1988 and 1997). Results indicated that much of the change along the lagoon rim is associated with the expansion of the inner lagoon shoreline as a result of the construction of the military airbase, with an estimated increase in land area of 3.01？km2 in this portion of the atoll rim. Comparisons of 69 rim width transects measured from 1967 and 2005 indicated that shorelines are both eroding (26 transects) and accreting (43 transects). Within a total vegetated area of 24？km2, there was a notable transition from Cocos Woodland to Broadleaf Woodland for a land area of 5.6？km2. From the hydrographic surveys, it was estimated that approximately 0.55？km3 of carbonate sediment material has been removed from the northwest quadrant of the lagoon, particularly in the vicinity of the Main Passage. As no previous record of benthic character exists, a complete benthic habitat map of the atoll was derived through classification of the three IKONOS satellite images. Management implications arising from this overall appraisal of geomorphic and ecological change at Diego Garcia included the need for ongoing monitoring of shoreline change at a representative set of sites around the atoll rim, monitoring of the water flow regime through the northern channels between the open ocean and the lagoon basin and an ongoing mapping campaign to record periodic changes in the character of the benthic surface ecology.

Abstract:
A new intraluminal stiffening device has become available in two grades of stiffness. However, there is no published evidence of its effectiveness. This randomized, controlled trial was designed to determine the effectiveness of the stiffening wires in improving cecal intubation rate and time following routine application. A secondary analysis determines effectiveness of application only after intractable failure with the unaided colonoscope.The colonoscope tested was an Olympus CF-100TL, approximately fifteen years old. Patients were randomly assigned to the unaided colonoscope or the standard or firm wire introduced routinely on entry into transverse colon. Each phase of colonoscopy was timed. Failure to advance the colonoscope for 5？minutes (despite usual manipulations to minimize looping) required switching to another intervention according to a prescribed methodology and the originally assigned intervention was recorded as failed.The study was terminated after accrual of 112 participants (target sample size 480) because the colonoscope required repairs (no damage attributable to stiffening wires) which would have been uneconomical. There were no statistically significant differences between per-protocol cecal intubation rates (81.1, 71.1 and 70.3 percent respectively), a finding which persisted after multiple imputation for a virtual sample size of 480. Similarly, there were no statistically significant differences between per-protocol cecal intubation times (15, 16.2 and 13.9？minutes). However, a statistically significant improvement in cecal intubation rate (from 81.1% to 97.3%, P？=？0.0313) was achieved when the wires were applied after intractable failure of the unaided colonoscope in the first intervention group.Routine application of either stiffening wire does not improve caecal intubation rate nor time compared to the unaided colonoscope. However, application of the stiffening wires after intractable failure of the unaided colonoscope enabled a statistic

Abstract:
We use simulations of hydrodynamics coupled with full general relativity to investigate the gravitational waves produced by a star colliding with a massive black hole when the star's tidal disruption radius lies far outside of the black hole horizon. We consider both main-sequence and white-dwarf compaction stars, and nonspinning black holes, as well as those with near-extremal spin. We study the regime in between where the star can be accurately modeled by a point particle, and where tidal effects completely suppress the gravitational wave signal. We find that nonnegligible gravitational waves can be produced even when the star is strongly affected by tidal forces, as well as when it collides with large angular momentum. We discuss the implications that these results have for the potential observation of gravitational waves from these sources with future detectors.

Abstract:
Denote by $\mathcal T_n$ and $\mathcal S_n$ the full transformation semigroup and the symmetric group on the set $\{1,\ldots,n\}$, and $\mathcal E_n=\{1\}\cup(\mathcal T_n\setminus \mathcal S_n)$. Let $\mathcal T(X,\mathcal P)$ denote the set of all transformations of the finite set $X$ preserving a uniform partition $\mathcal P$ of $X$ into $m$ subsets of size $n$, where $m,n\geq2$. We enumerate the idempotents of $\mathcal T(X,\mathcal P)$, and describe the subsemigroup $S=\langle E\rangle$ generated by the idempotents $E=E(\mathcal T(X,\mathcal P))$. We show that $S=S_1\cup S_2$, where $S_1$ is a direct product of $m$ copies of $\mathcal E_n$, and $S_2$ is a wreath product of $\mathcal T_n$ with $\mathcal T_m\setminus \mathcal S_m$. We calculate the rank and idempotent rank of $S$, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of $\mathcal E_n$.

Abstract:
We investigate the structure of the twisted Brauer monoid $\mathcal B_n^\tau$, comparing and contrasting it to the structure of the (untwisted) Brauer monoid $\mathcal B_n$. We characterise Green's relations and pre-orders on $\mathcal B_n^\tau$, describe the lattice of ideals, and give necessary and sufficient conditions for an ideal to be idempotent-generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent-generated, its rank and idempotent rank are equal. As an application of our results, we also describe the idempotent-generated subsemigroup of $\mathcal B_n^\tau$ (which is not an ideal) as well as the singular ideal of $\mathcal B_n^\tau$ (which is neither principal nor idempotent-generated), and we deduce a result of Maltcev and Mazorchuk that the singular part of the Brauer monoid $\mathcal B_n$ is idempotent-generated.

Abstract:
We study the ideals of the partition, Brauer, and Jones monoids, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham-Houghton graphs. We show that each proper ideal of the partition monoid P_n is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and we characterize the generating sets of this minimal cardinality. We also characterize and enumerate the minimal idempotent generating sets for the largest proper ideal of P_n, which coincides with the singular part of P_n. Analogous results are proved for the ideals of the Brauer and Jones monoids; in each case, the rank and idempotent rank turn out to be equal, and all the minimal generating sets are described. We also show how the rank and idempotent rank results obtained, when applied to the corresponding twisted semigroup algebras (the partition, Brauer, and Temperley-Lieb algebras), allow one to recover formulas for the dimensions of their cell modules (viewed as cellular algebras) which, in the semisimple case, are formulas for the dimensions of the irreducible representations of the algebras. As well as being of algebraic interest, our results relate to several well-studied topics in graph theory including the problem of counting perfect matchings (which relates to the problem of computing permanents of {0,1}-matrices and the theory of Pfaffian orientations), and the problem of finding factorizations of Johnson graphs. Our results also bring together several well-known number sequences such as Stirling, Bell, Catalan and Fibonacci numbers.

Abstract:
The variant of a semigroup S with respect to an element a in S, denoted S^a, is the semigroup with underlying set S and operation * defined by x*y=xay for x,y in S. In this article, we study variants T_X^a of the full transformation semigroup T_X on a finite set X. We explore the structure of T_X^a as well as its subsemigroups Reg(T_X^a) (consisting of all regular elements) and E_X^a (consisting of all products of idempotents), and the ideals of Reg(T_X^a). Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.