The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains.

Suppose that Ω⊂Rn+1, n≥1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Ω. We show that the corresponding elliptic measure ωL is quantitatively absolutely continuous with respect to surface measure of ∂Ω in the sense that ωL∈A∞(σ) if and only if any bounded solution u to Lu=0 in Ω is ε-approximable for any ε∈(0,1). By ε-approximability of u we mean that there exists a function Φ=Φε such that ‖u-Φ‖L∞(Ω)≤ε‖u‖L∞(Ω) and the measure µ~Φ with dµ~=|∇Φ(Y)|dY is a Carleson measure with L∞ control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L1-type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.

Nutritional practices of elite athletes. Practical recommendations.

The nutritional intake of elite athletes is a critical determinant of their athletic performance and ability to compete both physically and mentally. However, their demanding training and travel schedules in addition to a possible lack of nutritional knowledge may prohibit them from maintaining an optimal dietary intake. Sound scientific data about the nutritional habits of elite athletes are limited and, therefore, it is not clear as to whether elite athletes are following nutritional recommendations and maintaining nutritionally sound diets. This review takes a comprehensive look at 22 recent dietary intake studies, including 50 groups of elite athletes. The time period for food record collection ranged from 3 to 7 days except for 2 studies which collected records for 21 and 22 days. Energy intakes of > 50 kcal/kg/day for male athletes who train for > 90 min/day and 45 to 50 kcal/kg/day for female athletes training for > 90 min/day are recommended. Bodyweight should be monitored frequently as a check on calorie intake. With a sufficient calorie intake (1.2 to 2.0 kg/kg/day) protein supplementation is not necessary. Ingested carbohydrate stored as glycogen serves as the primary fuel for muscle performance. Athletes in training should consume 70% of total calories as carbohydrate. Athletes on low energy diets (< 2200 kcal/day) should have a diet of < 25% fat, and athletes with large energy needs should consume 30% fat in their diet. In general, fat intakes should be reduced and carbohydrate intakes increased. Athletes should also restrict alcohol intake during training and competition periods. Athletes with low calorie intakes should consume foods with high contents of iron, calcium, magnesium, zinc and vitamin B12. Athletes with high calorie intakes should consume foods that are naturally high in or fortified with B-group vitamins. Fluid, electrolyte and energy supplementation is desirable to support circulatory, metabolic and thermoregulatory functions. There is no special food that will help elite athletes perform better; the most important aspect of the diet of elite athletes is that it follows the basic guidelines for healthy eating.

Zonal changes in renal structure and phospholipid metabolism in potassium-deficient rats.

Morphologic alterations and membrane metabolism were studied in the kidneys of rats fed a low potassium diet. Transmission and scanning electron microscopy following perfusion-fixation of kidneys revealed that the earliest morphologic change occurs in cells of the papillary tip in which multivesicular bodies, a specific type to lysosome, appear after 1 day. Increased depletion leads to extension of the lesion to all cells of the papilla. After 1 week, a narrow band of hyperplasia in the inner red medulla appears; this band is characterized by adenomatous proliferation of intercalated and light cells and partial obstruction of collecting tubules. These alterations and cortical growth in the normal pattern result in increased renal weight. New membrane formation for lysosomes and growing cells was studied by measuring the rate of [14C]choline incorporation into phospholipid in slices from five zones of the kidney. In the papilla the rate increased 39 per cent after 18 hours, the earliest change detected. After 36 hours the rate increased in inner red medulla by 28 per cent, inner cortex by 25 per cent and outer cortex by 40 per cent. [14C]choline was a specific precursor of the three renal phospholipids, phosphatidylcholine, lysophosphatidylcholine, and sphingomyelin. The relative distribution of the label did not change with growth induced by potassium depletion. The results indicate that potassium depletion induces early increases in the formation of cell membrane phospholipid which correlate with specific morphologic changes in different zones within the kidney.