On the size of approximately convex sets in normed spaces

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let \Co(A) be the convex hull and \diam(A) the diameter of $A$. We prove that every $n$-dimensional normed space contains approximately convex sets $A$ with \mathcal{H}(A,\Co(A))\ge \log_2n-1 and \diam(A) \le C\sqrt n(\ln n)^2, where $\mathcal{H}$ denotes the Hausdorff distance. These estimates are reasonably sharp. For every $D>0$, we construct worst possible approximately convex sets in $C[0,1]$ such that \mathcal{H}(A,\Co(A))=\diam(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.Comment: 32 pages. See also http://www.math.sc.edu/~howard

Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$f((x+y)/2) \le (f(x)+f(y))/2 + 1.$$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function~$E$ vanishing on the vertices of a simplex. A set $A$ in a normed space is an approximately convex set iff for all $a,b\in A$ the distance of the midpoint $(a+b)/2$ to $A$ is $\le 1$. The bounds on approximately convex functions are used to show that in $\R^n$ with the Euclidean norm, for any approximately convex set $A$, any point $z$ of the convex hull of $A$ is at a distance of at most $[\log_2(n-1)]+1+(n-1)/2^{[\log_2(n-1)]}$ from $A$. Examples are given to show this is the sharp bound. Bounds for general norms on $R^n$ are also given.Comment: 39 pages. See also http://www.math.sc.edu/~howard

Mineral Trioxide Aggregate Material Use in Endodontic Treatment: A Review of the Literature

Objective The purpose of this paper was to review the composition, properties, biocompatibility, and the clinical results involving the use of mineral trioxide aggregate (MTA) materials in endodontic treatment. Methods Electronic search of scientific papers from January 1990 to August 2006 was accomplished using PubMed and Scopus search engines (search terms: MTA, GMTA, WMTA, mineral AND trioxide AND aggregate). Results Selected exclusion criteria resulted in 156 citations from the scientific, peer-reviewed dental literature. MTA materials are derived from a Portland cement parent compound and have been demonstrated to be biocompatible endodontic repair materials, with its biocompatible nature strongly suggested by its ability to form hydroxyappatite when exposed to physiologic solutions. With some exceptions, MTA materials provide better microleakage protection than traditional endodontic repair materials using dye, fluid filtration, and bacterial penetration leakage models. In both animal and human studies, MTA materials have been shown to have excellent potential as pulp-capping and pulpotomy medicaments but studies with long-term follow-up are limited. Preliminary studies suggested a favorable MTA material use as apical and furcation restorative materials as well as medicaments for apexogenesis and apexification treatments; however, long-term clinical studies are needed in these areas. Conclusion MTA materials have been shown to have a biocompatible nature and have excellent potential in endodontic use. MTA materials are a refined Portland cement material and the substitution of Portland cement for MTA products is presently discouraged. Existing human studies involving MTA materials are very promising, however, insufficient randomized, double-blind clinical studies of sufficient duration exist involving MTA for all of its clinical indications. Further clinical studies are needed in these areas

Characterization of a POROS\u3csup\u3eTM\u3c/sup\u3e-fumonisin B1 Affinity Column for Isolating Ceramide Synthase from Rat Liver

Fumonisin B1 is a mycotoxin produced by fungi of the genus Fusarium, common pathogens of corn and other grain plants. Toxic effects associated with fumonisin B1 include equine leukoencephalomacia, porcine pulmonary edema, rat renal carcinoma, and murine hepatocellular carcinoma. Increased risk for esophageal cancer in humans has been epidemiologically associated with consumption of corn contaminated with Fusarium, suggesting that fumonisin B1 may be involved. The biological effects of fumonisin B1 exposure result primarily from disruption of de novo sphingolipid biosynthesis via inhibition of ceramide synthase. Exposure of animals or cultured cells to fumonisin B1 results in the characteristic accumulation of sphinganine, a toxic sphingolipid intermediate, concomitant with depletion of essential complex sphingolipids. Ceramide synthase has not been purified to homogeniety and characterized. We prepared crude ceramide synthase from detergent-extracted rat liver homogenates using PEG-precipitation and cation exchange chromatography. Ceramide synthase activity was then sequestered, using fumonisin B1 covalently coupled to POROS-NH particles, and eluted selectively. The observed 119-fold enrichment in specific activity demonstrates the utility of fumonisin-POROS affinity chromatography in the purification of ceramide synthase

Hydraulic fracturing tests in anhydrite interbeds in the WIPP, Marker Beds 139 and 140

Hydraulic fracturing tests were integrated with hydrologic tests to estimate the conditions under which gas pressure in the disposal rooms in the Waste Isolation Pilot Plant, Carlsbad, NM (WIPP) will initiate and advance fracturing in nearby anhydrite interbeds. The measurements were made in two marker beds in the Salado formation, MB139 and MB140, to explore the consequences of existing excavations for the extrapolation of results to undisturbed ground. The interpretation of these measurements is based on the pressure-time records in two injection boreholes and several nearby hydrologic observation holes. Data interpretations were aided by post-test borehole video surveys of fracture traces that were made visible by ultraviolet illumination of fluorescent dye in the hydraulic fracturing fluid. The conclusions of this report relate to the upper- and lower-bound gas pressures in the WIPP, the paths of hydraulically and gas-driven fractures in MB139 and MB140, the stress states in MB139 and MB140, and the probable in situ stress states in these interbeds in undisturbed ground far away from the WIPP

An inland sea high nitrate-low chlorophyll (HNLC) region with naturally high pCO2

© The Author(s), 2015. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Limnology and Oceanography 60 (2015): 957–966, doi:10.1002/lno.10062.We present a time series of data for temperature, salinity, nitrate, and carbonate chemistry from September 2011 to July 2013 at the University of Washington's Friday Harbor Laboratories. Samples were collected at the Friday Harbor dock and pump house. Seawater conditions at Friday Harbor were high nitrate-low chlorophyll, with average nitrate and pCO2 concentrations of ∼ 25 ± 5 μmol L−1 and ∼ 700 ± 103 μatm (pH 7.80 ± 0.06). Transient decreases in surface water nitrate and pCO2 corresponded with the timing of a spring bloom (April through June). The high nitrate and pCO2 originate from the high values for these parameters in the source waters to the Salish Sea from the California Undercurrent (CU). These properties are due to natural aerobic respiration in the region where the CU originates, which is the oxygen minimum zone in the eastern tropical North Pacific. Alkalinity varies little so the increase in pCO2 is due to inputs of dissolved inorganic carbon (DIC). This increase in DIC can come from both natural aerobic respiration within the ocean and input of anthropogenic CO2 from the atmosphere when the water was last at the sea surface. We calculated that the anthropogenic “ocean acidification” contribution to DIC in the source waters of the CU was 36 μmol L−1. This contribution ranged from 13% to 22% of the total increase in DIC, depending on which stoichiometry was used for C/O2 ratio (Redfield vs. Hedges). The remaining increase in DIC was due to natural aerobic respiration.We thank The Educational Foundation of America (EFA) and National Science Foundation Field Station Marine Lab Program (FSML) (NSF DBI 0829486) for essential initial funding to JWM to develop the Ocean Acidification Experimental Lab (OAEL). Additional support was provided by NSF award EF1041213 to E. Carrington Ken Sebens for encouragement to involve students in this research as part of a FHL mini-apprenticeship course

The Wicked Machinery of Government: Malta and the Problems of Continuity under the New Model Administration

This is a study focused on the early years of British rule in Malta (1800-1813). It explores the application to the island of the “new model” of colonial government, one based on direct rule from London mediated by the continuation of existing laws and institutions. Systemic deficiencies are identified. These tended to undermine the effectiveness of direct British rule. This study also reveals, in the context of legal and constitutional continuity, unresolved tensions between modernity and tradition. The political stability of the island was damaged and the possibility of continued British possession was threatened