Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

We study a nonlinear equation in the half-space {x ₁ > 0} with a Hardy potential, specifically

$$ - \Delta u - \frac{\mu }{{x_1^2}}u + {u^p} = 0in\mathbb{R}_ + ^n,$$

where p > 1 and ?∞ < μ < 1/4. The admissible boundary behavior of the positive solutions is either O(x ₁ ^?2/(p?1)) as x ₁ → 0, or is determined by the solutions of the linear problem \( - \Delta h - \frac{\mu }{{x_1^2}}h = 0\). In the first part we study in full detail the separable solutions of the linear equations for the whole range of μ. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like O(x ₁ ^?2/(p?1)) near the origin and which, away from the origin, have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like O(x ₁ ^?2/(p?1)) at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.

     

Tiled-grating compression of multiterawatt laser pulses

High-energy petawatt lasers require large diffraction gratings for pulse compression. As an alternative to meter-sized gratings, we demonstrate the capability of a tiled-grating system to compress multiterawatt subpicosecond laser pulses. Using a 100 TW-class Nd:glass chirped-pulse amplification laser facility, we report on the performance of a two-grating mosaic to compress high-energy pulses to 2.5 J, 450 fs (5.5 TW) in air with a beam size of 50 mm and energy transmission of 63%. Stability of the grating mosaic alignment was realized by use of an accurate nanopositioning optomechanical system. The output Gaussian spectrum was preserved from grating-gap spectral clipping and was free of modulation.     

Dense mode clustering in brain maps

A mode-based clustering method is developed for identifying spatially dense clusters in brain maps. This type of clustering focuses on identifying clusters in brain maps independent of their shape or overall variance. This can be useful for both localization in terms of interpretation and for subsequent graphical analysis that might require more coherent or dense regions of interest as starting points. The method automatically does signal/noise sharpening through density mode seeking. We also discuss the problem of parameter selection with this method and propose a new method involving 2-parameter control surface, in which we show that the same cluster solution results from tradeoff of these 2 parameters (the local density k and the radius r of the spherical kernel). We benchmark the new dense mode clustering by using several artificially created data sets and brain imaging data sets from an event perception task by perturbing the data set with noise and measuring three kinds of deviation from the original cluster solution. We present benchmark results that demonstrate that the mode clustering method consistently outperforms the commonly used single-linkage clustering, k means method (centroid method) and Ward's method (variance method).     

New pseudan (2-alkyl-4(1H)-quinolinone) alkaloids from Boronia ternata var. elongata and Boronia alulata (Rutaceae)

The aerial parts of Boronia ternata var. elongata from western Australia has yielded three new 2-acyl-4(1H)-quinolinone alkaloids, characterised as 2-n-pentyl-4(1H)-quinolinone, 1-methyl-2-n-pentyl-4(1H)-quinolinone and 1-methyl-2-(1 xi-methyl)-propyl-4(1H)-quinolinone, as well as known alkaloids of this class and a furoquinoline alkaloid. Boronia alulata, from northeast Queensland, also yielded 2-n-pentyl-4(1H)-quinolinone together with the known 2-n-propyl-4(1H)-quinolinone. Both species are assigned to Boronia section Valvatae which, alone among the sections of Boronia, appears to be characterised by the presence of alkaloids.     

A modular approach to marine macrolide construction. 4. Assembly of C36-C51 and C29-C44 building blocks and evaluation of key coupling reactions targeting spongistatin 1 (altohyrtin A)

Routes have been developed for the stereocontrolled elaboration of two highly functionalized sectors of spongistatin 1. The approach to ring F takes advantage of B-alkyl Suzuki-Miyaura coupling to install the C44-C45 bond. The E-ring pyran moiety was generated by acylation of an alpha-sulfonyl carbanion, the stereogenic centers of which were incorporated by sequential asymmetric aldol reactions. [structure: see text].     

Enantioselective catalytic ring expansion of methylenecyclopropane carboxamides promoted by a chiral magnesium Lewis acid

A catalytic enantioselective ring expansion of monoactivated methylenecyclopropanes (MCP) in the presence of N-tosyl aldimines was developed using a chiral bis(oxazoline) ligand-MgI2 complex. After evaluation of ligands and optimization of the reaction conditions, the reaction has been applied to a variety of aromatic and heteroaromatic aldimines providing the corresponding trans-C2,C3-disubstituted methylenepyrrolidines in generally good yields (greater than 52%) and up to 86% ee. [reaction: see text].     

Determination of bitter orange alkaloids in dietary supplement Standard Reference Materials by liquid chromatography with atmospheric-pressure ionization mass spectrometry

A liquid chromatographic atmospheric-pressure ionization electrospray mass spectrometry (LC–API–ES–MS) method has been developed for the determination of five bitter orange alkaloids (synephrine, octopamine, n-methyltyramine, tyramine, and hordenine) in bitter orange-containing dietary supplement standard reference materials (SRMs). The materials represent a variety of natural, extracted, and processed sample matrices. Two extraction techniques were evaluated: pressurized-fluid extraction (PFE) and sonication extraction. The influence of different solvents, extraction temperatures, and pH were investigated for a plant material and a processed sample. The LC method uses a new approach for the separation of highly polar alkaloids. A fluorinated, silica-based stationary phase separated the five alkaloids and the internal standard terbutaline in less than 20 min. This method enabled the determination of the dominant alkaloid synephrine and other minor alkaloids in a variety of dietary supplement SRMs.     

Biosynthetic production of universally <Superscript>13</Superscript>C-labelled polyunsaturated fatty acids as reference materials for natural health product research

Long-chain polyunsaturated fatty acids (LCPUFA) including eicosapentaenoic acid (EPA, 20:5n-3) and docosahexaenoic acid (DHA, 22:6n-3) have become important natural health products with numerous proven benefits related to brain function and cardiovascular health. Not only are omega-3 fatty acids available in a plethora of dietary supplements, but they are also increasingly being incorporated as triglycerides into conventional foods, including bread, milk, yoghurt and confectionaries. Recently, transgenic oil seed crops and livestock have been developed that enhance omega-3 fatty acid content. This diverse array of matrices presents a difficult analytical challenge and is compounded further by samples generated through clinical research. Stable isotope ¹³C-labelled LCPUFA standards offer many advantages as research tools because they may be distinguished from their naturally abundant counterparts by mass spectrometry and directly incorporated as internal standards into analytical procedures. Further, ¹³C-labelled LCPUFAs are safe to use as metabolic tracers to study uptake and metabolism in humans. Currently, ¹³C-labelled LCPUFAs are expensive, available in limited supply and not in triglyceride form. To resolve these issues, marine heterotrophic microorganisms are being isolated and screened for LCPUFA production with a view to the efficient biosynthetic production of U-¹³C-labelled fatty acids using U-¹³C glucose as a carbon source. Of 37 isolates obtained, most were thraustochytrids, and either DHA or omega-6 docosapentaenoic acid (22:5n-6) were produced as the major LCPUFA. The marine protist Hyalochlorella marina was identified as a novel source of EPA and omega-3 docosapentaenoic acid (22:5n-3). As proof of principle, gram-level production of ¹³C-labelled DHA has been achieved with high chemical purity ( >99%) and high ¹³C incorporation levels (>90%), as confirmed by NMR and MS analyses. Finally, U-¹³C-DHA was enzymatically re-esterified to glycerol to yield a ¹³C-labelled tridocosahexaenoin.