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In 1940, Liapounoff [8] showed that the range of a bounded nonatomic vector measure, with values in a finite-dimensional space, is convex and compact. Since then, Halmos [5] and Lindenstrauss [10] have given simplified proofs, while Olech [I 1] has generalized this result to unbounded measures, still with values in a finite dimensional space. A direct(More)
Denoting the nonnegative (resp. signed) integers byN (resp.Z) and the real numbers byR, letK ⊂ R m andf: R m → R. Thenf is astoring function (resp.packing function) onK wheneverf|(Z m ⊒ K) is an injection into (resp. bijection onto)N. Unit translations gm of some P. Chowla [1961] polynomials are packing functions on the correspondingN m , and all(More)
HereN = {0, 1, 2, ...}, while a functionf onN m or a larger domain is apacking function if its restrictionf¦N m is a bijection ontoN. (Packing functions generalize Cantor's [1]pairing polynomials, and yield multidimensional-array storage schemes.) We call two functionsequivalent if permuting arguments makes them equal. Alsos(x) =x 1 + ... +x m when x = (x(More)
HereR andN denote respectively the real numbers and the nonnegative integers. Also 0 <n εN, ands(x) =x 1+...+x n when x = (x 1,...,x n) εR n. Adiagonal function of dimensionn is a mapf onN n (or any larger set) that takesN n bijectively ontoN and, for all x, y inN n, hasf(x) <f(y) whenevers(x) <s(y). We show that diagonalpolynomials f of dimensionn all have(More)
Treatment of POME has become more challenging with stringent effluent discharge standards and difficulty of current treatment processes to meet the standards due to the complex nature of POME. Utilization of high-rate anaerobic reactor operated under thermophilic condition is proposed for primary POME treatment due to low energy requirements and biogas(More)
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