Ordinary and Generalized Circulation Algebras for Regular Matroids

Let E be a finite set, and let R(E) denote the algebra of polynomials in indeterminates (x_e)_{e in E}, modulo the squares of these indeterminates. Subalgebras of R(E) generated by homogeneous elements of degree 1 have been studied by many authors and can be understood combinatorially in terms of the matroid represented by the linear equations satisfied by these generators. Such an algebra is related to algebras associated to deletions and contractions of the matroid by a short exact sequence, and can also be written as the quotient of a polynomial algebra by certain powers of linear forms. We study such algebras in the case that the matroid is regular, which we term circulation algebras following Wagner. In addition to surveying the existing results on these algebras, we give a new proof of Wagner's result that the structure of the algebra determines the matroid, and construct an explicit basis in terms of basis activities in the matroid. We then consider generalized circulation algebras in which we mod out by a fixed power of each variable, not necessarily equal to 2. We show that such an algebra is isomorphic to the circulation algebra of a "subdivided" matroid, a variation on a result of Nenashev, and derive from this generalized versions of many of the results on ordinary circulation algebras, including our basis result. We also construct a family of short exact sequences generalizing the deletion-contraction decomposition

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

A comprehensive functional map of the hepatitis C virus genome provides a resource for probing viral proteins.

UnlabelledPairing high-throughput sequencing technologies with high-throughput mutagenesis enables genome-wide investigations of pathogenic organisms. Knowledge of the specific functions of protein domains encoded by the genome of the hepatitis C virus (HCV), a major human pathogen that contributes to liver disease worldwide, remains limited to insight from small-scale studies. To enhance the capabilities of HCV researchers, we have obtained a high-resolution functional map of the entire viral genome by combining transposon-based insertional mutagenesis with next-generation sequencing. We generated a library of 8,398 mutagenized HCV clones, each containing one 15-nucleotide sequence inserted at a unique genomic position. We passaged this library in hepatic cells, recovered virus pools, and simultaneously assayed the abundance of mutant viruses in each pool by next-generation sequencing. To illustrate the validity of the functional profile, we compared the genetic footprints of viral proteins with previously solved protein structures. Moreover, we show the utility of these genetic footprints in the identification of candidate regions for epitope tag insertion. In a second application, we screened the genetic footprints for phenotypes that reflected defects in later steps of the viral life cycle. We confirmed that viruses with insertions in a region of the nonstructural protein NS4B had a defect in infectivity while maintaining genome replication. Overall, our genome-wide HCV mutant library and the genetic footprints obtained by high-resolution profiling represent valuable new resources for the research community that can direct the attention of investigators toward unidentified roles of individual protein domains.ImportanceOur insertional mutagenesis library provides a resource that illustrates the effects of relatively small insertions on local protein structure and HCV viability. We have also generated complementary resources, including a website (http://hangfei.bol.ucla.edu) and a panel of epitope-tagged mutant viruses that should enhance the research capabilities of investigators studying HCV. Researchers can now detect epitope-tagged viral proteins by established antibodies, which will allow biochemical studies of HCV proteins for which antibodies are not readily available. Furthermore, researchers can now quickly look up genotype-phenotype relationships and base further mechanistic studies on the residue-by-residue information from the functional profile. More broadly, this approach offers a general strategy for the systematic functional characterization of viruses on the genome scale

Guidance, Navigation, and Control for NASA Lunar Pallet Lander

The NASA Lander Technology project is leading the development and integration of the Lunar Pallet Lander (LPL) concept. The objective is to demonstrate precision landing by delivering a payload to the lunar surface within 100 meters of a landing target. Potential landing sites are selected near the lunar pole where water may be present in permanently shadowed regions that could enable future in-situ resource utilization. The LPL is part of a sequence of missions aimed at maturing the necessary technologies, such as lunar precision landing sensors, that will enable the next generation of multi-ton lunar payloads and human landers. This paper provides an overview of the Mission Design, Guidance Navigation and Control (GNC) algorithms, and sensor suite. The results show the LPL simulated trajectory and landing precision performance under nominal and dispersed conditions. The landing precision simulation confirms the need to rely on high-accuracy navigation techniques and sensors such as Terrain Relative Navigation (TRN) and the Navigation Doppler Lidar (NDL), currently being developed for space applications. The results also demonstrate the ability of the guidance and control system to perform a soft lunar touchdown by combining thrust vector control during the solid rocket motor deceleration phase, and pulse engine control, for the liquid powered descent phase