On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the \emph{weak separation property} is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the \emph{weak separation property} is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the \emph{weak separation property} is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a \emph{weak tangent} that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the `equality/maximal' dichotomy does not extend to this setting.Comment: 24 pages, 2 figure

Sbf/MTMR13 coordinates PI(3)P and Rab21 regulation in endocytic control of cellular remodeling.

Cells rely on the coordinated regulation of lipid phosphoinositides and Rab GTPases to define membrane compartment fates along distinct trafficking routes. The family of disease-related myotubularin (MTM) phosphoinositide phosphatases includes catalytically inactive members, or pseudophosphatases, with poorly understood functions. We found that Drosophila MTM pseudophosphatase Sbf coordinates both phosphatidylinositol 3-phosphate (PI(3)P) turnover and Rab21 GTPase activation in an endosomal pathway that controls macrophage remodeling. Sbf dynamically interacts with class II phosphatidylinositol 3-kinase and stably recruits Mtm to promote turnover of a PI(3)P subpool essential for endosomal trafficking. Sbf also functions as a guanine nucleotide exchange factor that promotes Rab21 GTPase activation associated with PI(3)P endosomes. Of importance, Sbf, Mtm, and Rab21 function together, along with Rab11-mediated endosomal trafficking, to control macrophage protrusion formation. This identifies Sbf as a critical coordinator of PI(3)P and Rab21 regulation, which specifies an endosomal pathway and cortical control

Constraining Primordial Non-Gaussianity With the Abundance of High Redshift Clusters

We show how observations of the evolution of the galaxy cluster number abundance can be used to constrain primordial non-Gaussianity in the universe. We carry out a maximum likelihood analysis incorporating a number of current datasets and accounting for a wide range of sources of systematic error. Under the assumption of Gaussianity, the current data prefer a universe with matter density $\Omega_m\simeq 0.3$ and are inconsistent with $\Omega_m=1$ at the $2\sigma$ level. If we assume $\Omega_m=1$, the predicted degree of cluster evolution is consistent with the data for non-Gaussian models where the primordial fluctuations have at least two times as many peaks of height $3\sigma$ or more as a Gaussian distribution does. These results are robust to almost all sources of systematic error considered: in particular, the $\Omega_m=1$ Gaussian case can only be reconciled with the data if a number of systematic effects conspire to modify the analysis in the right direction. Given an independent measurement of $\Omega_m$, the techniques described here represent a powerful tool with which to constrain non-Gaussianity in the primordial universe, independent of specific details of the non-Gaussian physics. We discuss the prospects and strategies for improving the constraints with future observations.Comment: Minor revisions to match published ApJ version, 14 pages emulateap

Generalised Cantor sets and the dimension of products

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ≥ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α

Test Results for a Non-toxic, Dual Thrust Reaction Control Engine

A non-toxic, dual thrust reaction control engine (RCE) was successfully tested over a broad range of operating conditions at the Aerojet Sacramento facility. The RCE utilized LOX/Ethanol propellants; and was tested in steady state and pulsing modes at 25-lbf thrust (vernier) and at 870-lbf thrust (primary). Steady state vernier tests vaned chamber pressure (Pc) from 0.78 to 5.96 psia, and mixture ratio (MR) from 0.73 to 1.82, while primary steady state tests vaned Pc from 103 to 179 psia and MR from 1.33 to 1.76. Pulsing tests explored EPW from 0.080 to 10 seconds and DC from 5 to 50 percent at both thrust levels. Vernier testing accumulated a total of 6,670 seconds of firing time, and 7,215 pulses, and primary testing accumulated a total of 2,060 seconds of firing time and 3,646 pulses

Ignition Characterization Test Results for the LO2/Ethanol Propellant Combination

A series of contracts were issued by the Marshall Space Flight Center (MSFC) of the National Aeronautics and Space Administration (NASA) under the auspices of the Exploration Systems Mission Directorate to develop and expand the maturity of candidate technologies considered to be important for future space exploration. One such technology was to determine the viability of incorporating non-toxic propellants for Reaction Control Subsystems (RCS). Contract NAS8-01109 was issued to Aerojet to develop a dual thrust Reaction Control Engine (RCE) that utilized liquid oxygen and ethanol as the propellants. The dual thrust RCE incorporated a primary thrust level of 870 lbf, and a vernier thrust level of 10 - 30 lbf. The preferred RCS approach for the dual thrust RCE was to utilize pressure-fed liquid oxygen (LOX) and ethanol propellants; however, previous dual thrust feasibility testing incorporated GOX/Ethanol igniters as opposed to LOX/Ethanol igniters in the design. GOX/Ethanol was easier to ignite, but this combination had system design implications of providing GOX for the igniters. A LOX/Ethanol igniter was desired; however, extensive LOX/Ethanol ignition data over the anticipated operating range for the dual thrust RCE did not exist. Therefore, Aerojet designed and tested a workhorse LOX igniter to determine LOX/Ethanol ignition characteristics as part of a risk mitigation effort for the dual thrust RCE design. LOX, encompassing potential two-phase flow conditions anticipated being present in real mission applications. A workhorse igniter was designed to accommodate the hll LOX design flowrate, as well as a reduced GOX flowrate. It was reasoned that the initial LOX flow through the igniter would flash to GOX due to the latent heat stored in the hardware, causing a reduced oxygen flowrate because of a choked, or sonic, flow condition through the injection elements. As LOX flow continued, the hardware would chill-in, with the injected oxygen flow transitioning from cold GOX through two-phase flow to subcooled LOX. permitted oxygen state points to be determined in the igniter oxidizer manifold, and gas-side igniter chamber thermocouples provided chamber thermal profile characteristics. The cold flow chamber pressure (P(sub c)) for each test was determined and coupled with the igniter chamber diameter (D(sub c)) to calculate the characteristic quench parameter (P(sub c) x D(sub c)), which was plotted as a function of core mixture ratio, m. Ignition limits were determined over a broad range of valve inlet conditions, and ignition was demonstrated with oxygen inlet conditions that ranged from subcooled 210 R LOX to 486 R GOX. Once ignited at cold GOX conditions, combustion was continuous as the hardware chilled in and the core mixture ratio transitioned from values near 1.0 to over 12.5. Pulsing is required in typical RCS engines; therefore, the workhorse igniter was pulse tested to verify the ability to provide the required ignition for a pulsing RCE. The minimum electrical pulse width (EPW) of the dual thrust RCE was 0.080 seconds