The Hardy—Littlewood circle method

Author(s):  
Timothy D. Browning
Keyword(s):  
Author(s):  
S.L. White ◽  
C.B. Jensen ◽  
D.D. Giera ◽  
D.A. Laska ◽  
M.N. Novilla ◽  
...  

In vitro exposure to LY237216 (9-Deoxo-11-deoxy-9,11-{imino[2-(2-methoxyethoxy)ethylidene]-oxy}-(9S)-erythromycin), a macrolide antibiotic, was found to induce cytoplasmic vacuolation in L6 skeletal muscle myoblast cultures (White, S.L., unpubl). The present study was done to determine, by autoradiographic quantitative analysis, the subcellular distribution of 3H-LY237216 in L6 cells.L6 cells (ATCC, CRL 1458) were cultured to confluency on polycarbonate membrane filters (Millipore Corp., Bedford, MA) in M-199 medium (GIBCO® Labs) with 10% fetal bovine serum. The cells were exposed from the apical surface for 1-hour to unlabelled-compound (0 μCi/ml) or 50 (μCi/ml of 3H-LY237216 at a compound concentration of 0.25 mg/ml. Following a rapid rinse in compound-free growth medium, the cells were slam-frozen against a liquid nitrogen cooled, polished copper block in a CF-100 cryofixation unit (LifeCell Corp., The Woodlands, TX). Specimens were dried in the MDD-C Molecular Distillation Drier (LifeCell Corp.), vapor osmicated and embedded in Spurrs low viscosity resin. Ultrathin sections collected on formvar coated stainless steel grids were counter-stained, then individually mounted on corks. A monolayer of Ilford L4 nuclear emulsion (Polysciences, Inc., Warrington, PA) was placed on the sections, utilizing a modified “loop method”. The emulsions were exposed for 7-weeks in a light-tight box at 4°C. Autoradiographs were developed in Microdol-X developer and examined on a Philips EM410LS transmission electron microscope. Quantitative analysis of compound localization employed the point and circle approach of Williams; incorporating the probability circle method of Salpeter and McHenry.


Author(s):  
Tim Browning ◽  
Shuntaro Yamagishi

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.


2016 ◽  
Vol 12 (01) ◽  
pp. 205-217 ◽  
Author(s):  
Taiyu Li

In this short note, we treat the enlarged major arcs of circle method in the Waring–Goldbach problem.


1994 ◽  
Vol 19 (1) ◽  
pp. 29-34
Author(s):  
A. Hartmann ◽  
O. Neidel ◽  
R. Pfab ◽  
G. Schmitt
Keyword(s):  

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