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

The development of mass spectrometry-based methodologies for the high throughput quantitation of peptides in biological matrices

The aim of this research was the development of mass spectrometry-based methodologies for the high-throughput quantitation of peptides in biological matrices. Glucagon and GLP-1, which are of interest as biomarkers and in the development of therapeutics, were chosen as model peptides. Immunoassays that are traditionally used to quantify these often perform poorly; therefore, necessitating the development of alternative methodologies. Application of mass spectrometry-based methodologies to these analytes has, however, been limited, primarily due to sensitivity challenges, but also due to analytical challenges associated with their endogenous nature and instability in biological matrices. Chapter 2 describes the development and qualification of the first liquid-chromatography coupled tandem mass spectrometry (LC-MS/MS) method for the quantitation of endogenous glucagon from human plasma. A novel 2D extraction procedure was developed to ensure robustness and sensitivity, whilst a novel surrogate matrix quantitation strategy took into account the endogenous nature of the analyte. A lower limit of quantitation (LLOQ) of 25 pg/mL was qualified, which was a considerable improvement over that previously reported in the literature (250 pg/mL) for a LC-MS/MS method. Clinical samples were cross-validated against a conventional radioimmunoassay (RIA), and similar pharmacokinetic (PK) profiles resulted, demonstrating that the methods were complementary. In Chapter 2 glucagon instability in biological matrix was noted. To characterise this further, in Chapter 3 in vitro glucagon metabolites were identified using high-resolution mass spectrometry (HRMS). Metabolites observed by others (glucagon19-29, glucagon3 29 and [pGlu]3glucagon3 29) in alternative matrices were identified, alongside novel metabolites (glucagon20-29 and glucagon21-29). Cross-interference of these metabolites in immunoassays may help to explain their poor performance, whilst knowledge of metabolism may also aid the development of future stabilisation strategies. The method developed in Chapter 2 was refined in Chapter 4 to improve sensitivity, robustness and throughput, and to add GLP-1 as a secondary analyte. The sensitivity achieved (glucagon: 15 pg/mL LLOQ, GLP-1: 25 pg/mL LLOQ) is the highest reported for both peptides for an extraction avoiding immunoenrichment. Specificity of endogenous glucagon quantitation was assured using a novel approach with a supercharging mobile phase additive to access a sensitive qualifier transition. A cross-validation against established immunoassays using physiological study samples demonstrated some similarities between the methods. Differences between the immunoassay results exemplified the need to develop alternative methodologies. The resulting LC-MS/MS method is considered a viable alternative to immunoassays, for the quantitation of endogenous glucagon, dosed glucagon and/or dosed GLP-1 in human plasma

A Strategic Approach to Agricultural Research Program Planning in Sub-Saharan Africa

Research and Development/Tech Change/Emerging Technologies, Downloads May 2008-July 2009: 13,

A Strategic Approach to Agricultural Research Program Planning in Sub-Saharan Africa

Recent studies have shown that agricultural research can have high payoffs in Africa, but impact depends on how well technology fits with evolving needs and capacity in the agricultural sector and the rest of the economy. Structural adjustment policies (e.g., market liberalization, currency devaluation) and political change are transforming user demands for new technology and the economic environment in which technology must perform. The challenge is how to design agricultural research as a strategic input to promote broad-based economic growth, structural transformation, and food security in the increasingly market-driven, but fragile, economies of Africa.Food Security, Food Policy, Agricultural Research, Research and Development/Tech Change/Emerging Technologies, Downloads May 2008-July 2009: 44, Q18,

Chromospheric Activity of HAT-P-11: an Unusually Active Planet-Hosting K Star

Kepler photometry of the hot Neptune host star HAT-P-11 suggests that its spot latitude distribution is comparable to the Sun's near solar maximum. We search for evidence of an activity cycle in the CaII H & K chromospheric emission $S$-index with archival Keck/HIRES spectra and observations from the echelle spectrograph on the ARC 3.5 m Telescope at APO. The chromospheric emission of HAT-P-11 is consistent with a $\gtrsim 10$ year activity cycle, which plateaued near maximum during the Kepler mission. In the cycle that we observed, the star seemed to spend more time near active maximum than minimum. We compare the $\log R^\prime_{HK}$ normalized chromospheric emission index of HAT-P-11 with other stars. HAT-P-11 has unusually strong chromospheric emission compared to planet-hosting stars of similar effective temperature and rotation period, perhaps due to tides raised by its planet.Comment: 16 pages, 8 figures; accepted to the Astrophysical Journa