Mapona Volume 2
Sandrine Willert
The effect of 5 days administration of ACTH or DOC, was examined before and after chronic potassium (K) loading in sheep. K loading raised plasma [K], urine volume and K excretion but had no effect on mean arterial pressure (MAP). On a normal K intake DOC (5 mg/day i.v.) increased MAP and plasma sodium [Na]. Plasma [K], urinary Na (day 1) and K (day 1) excretion were decreased. On a high K intake (congruent to 800 mmol K/day), DOC lowered plasma [K] but had no effect on MAP or Na excretion. The hypertensive effects of ACTH were not affected by K intake.
On page 186, the authors are discussing that if $f:M\rightarrow M$ is a non-invertible smooth map on a manifold $M$, $\Omega$ is a volume form and $\rho:M\rightarrow \mathbbR^+$ is a density so that the measure $\mu(A) := \int_A \rho.\Omega$ defines an $f$-invariant measure, then
I was wondering why this should be true. I was guessing that probably you can still define something like a measure theoretic version of the pull back of the volume form for non-invertible smooth maps. Indeed, if $M$ is compact, almost every point has finitely many pre-images by Sard's and Regular value thoerems. Therefore, via summing up sum stuff over probably finitely many pre-images we should be able to get what we want via a few applications of the change of variables theorem. However, I wasn't able to define this pull back for some reason. Either if this is or is not the strategy, why is this formula correct?
ALTHOUGH Capri, the small island lying at the entrance to the Bay of Naples, has a total area of only some four square miles, it has an extensive literature. Its many interests have brought it fame since the days when it was the resort of Roman emperors. Prof. Friedlaender attempts with considerable success to survey all aspects of the island. The peculiar geological structure of the caves and caverns of the island is fully discussed, and the flora and fauna receive special notice. The second half of the book deals with the story of the people, the antiquities and the history, and concludes with a bibliography. If the volume contains little that is new, it collects much scattered matter in an accessible form. There is a map on a scale of 1:12,000 in an end pocket.
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