Indestructibility and measurable cardinals with few and many measures

If κ < λ are such that κ is indestructibly supercompact and λ is measurable, then we show that both A = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries the maximal number of normal measures} and B = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries fewer than the maximal number of normal measures} are unbounded in κ. The two aforementioned phenomena, however, need not occur in a universe with an indestructibly supercompact cardinal and sufficiently few large cardinals. In particular, we show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry fewer than the maximal number of normal measures. We also, however, show how to construct a model with an indestructibly supercompact cardinal κ in which if δ < κ is a measurable cardinal which is not a limit of measurable cardinals, then δ must carry the maximal number of normal measures. If we weaken the requirements on indestructibility, then this last result can be improved to obtain a model with an indestructibly supercompact cardinal κ in which every measurable cardinal δ < κ carries the maximal number of normal measures. A. W. Apter’s research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. In addition, the author wishes to thank the referee, for helpful comments, corrections, and suggestions which have been incorporated into the current version of the paper.     

Graph family operations

In previous papers, Catlin introduced four functions, denoted , , , and , between sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k-flows for a fixed integer k3. Unfortunately, a subtle error caused several theorems previously published in Catlin (Discrete Math. 160 (1996) 67–80) to be incorrect. In this paper we correct those errors and further explore the relations between these functions, showing that there is a sort of duality between them and that they act as inverses of one another on certain sets of graphs.     

Synthesis and antitumor evaluation of some 1,3-disubstituted tetrahydropyrimido[4,5-c]isoquinolines

Cyclocondensation of 6-amino-2,4-dioxopyrimidine or 2,4,6-triaminopyrimidine with 1-cyclohexenecarbox-aldehyde 13 afforded regiospecifically, tricyclic, angular 1,3-disubstituted tetrahydropyrimido[4,5-c]isoquin-olines 5 and 6 respectively. In addition, 2,4,6-triaminopyrimidine when condensed with 2-chloro-1-cyclohex-enecarboxaldehyde 14 , regiospecifically afforded the angular isomer 6 . However, the cyclocondensation of 2,6-diamino-4-oxopyrimidine with 13 was regioselective and afforded a mixture of the linear and angular tetrahydropyrimidoisoquinolines 2 and 4 . The growth of leukemia L-1210 cells in culture were inhibited 50% by 6 at 9 × 10^?8 M. Compounds 4 and 5 were not significantly active.     

Are linear algorithms always good for linear problems?

We exhibit linear problems for which every linear algorithm has infinite error, and show a (mildly) nonlinear algorithm with finite error. The error of this nonlinear algorithm can be arbitrarily small if appropriate information is used. We illustrate these examples by the inversion of a finite Laplace transform, a problem arising in remote sensing.