This paper gives a definition and axiomatization of higher order twisted cohomology operations and an application to the problem of classifying cross sections .of fibrations. "Phe axioms are analogous to, and essentially include, those given by Maunder
Part III. Proofs of the Theorems of I.
The minimax inequality
min
x
sup
y
f
(
x
,
y
)
⩽
sup
x
f
(
x
,
x
)
{\min _x}{\sup _y}f(x,y) \leqslant {\sup _x}f(x,x)
,proved by K. Fan for convex spaces, is proved here for certain contractible and acyclic spaces. Some variational inequality and fixed point theorems are deduced.
The burning of 1 g monosaccharide yields 3.74 large calories, disaocharide 3.95, animal or vegetable fat 9.5, animal protein 5.7. All of this heat is produced by their burning in the body, except in case of protein, which produces only 4.5 calories as it is not completely burned, being eliminated as urea and uric acid and ammonia, etc. The mineral constituents are apparently for the most part dissolved in the water contained in the cell and will be considered later in regard to their effect on the solubilities (aggregation states) of the proteins and phospholipines. Among those constituents occurring only in traces the most important are the enzymes, thermolabile substances of unknown composition which accelerate certain chemical reactions. Some of the enzymes are poured out of the cells (as pepsin, for instance) so that they may be easily obtained, but many enzymes that are supposed to exist in all cells have been looked for in vain. Such are the enzymes that are supposed to accelerate the oxidation of the foodstuffs. It now appears that such oxidations are inseparably connected with the cell structure or to some substance that is destroyed when the cell structure is destroyed. The Archiv de Biologie (Van Beneden).
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