1.16. Again, I’ll only do the triangle inequality. Let x = (a1, . . . , an), y = (b1, . . . , bn), z = (c1, . . . , cn) be three points in Rn, then there is some integer j, 1 ≤ j ≤ n where the max occurs, so that d∞(x, y) = |aj − bj |. Hence, d∞(x, y) = |(aj−cj)+(cj−bj)| ≤ |aj−cj |+|cj−bj | But, |aj−cj | ≤ max{|a1− c1|, . . . , |an − cn|} = d∞(x, z), since it is one of the terms occurring in th...

Let R be a commutative ring with identity and M be a unitary R-module. Let : S(M) −! S(M) [ {;} be a function, where S(M) is the set of submodules ofM. Suppose n 2 is a positive integer. A proper submodule P of M is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 R and x 2 M and a1 . . . an−1x 2P(P), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x 2 P...

Ephrin-A1, the prototypic ligand for EphA receptor tyrosine kinases, is overexpressed in vascularized tumors relative to normal tissue. Moreover, ephrin-A1-Fc fusion proteins induce endothelial cell sprouting, migration, and assembly in vitro, and s.c. vascular remodeling in vivo. Based on these data, we hypothesized that native, membrane-bound ephrin-A1 regulates tumor angiogenesis and progres...

In this study, we isolated and characterized a human cyclin A-like gene that we named cyclin A1. Cyclin A1 has 48% identity with human cyclin A and is more related to the recently cloned murine cyclin A1 (84% identity). The human cyclin A1 is specifically expressed in testis and brain among all of the normal tissues that we studied by Northern blot analysis; in addition, it is expressed in seve...

Some somatic single nucleotide variants (SNVs) are thought to be pathogenic, leading to neurological disease. We hypothesized that heterogeneous nuclear ribonuclear protein A1 (hnRNP A1), an autoantigen associated with multiple sclerosis (MS) would contain SNVs. MS patients develop antibodies to hnRNP A1 (293-304), an epitope within the M9 domain (AA (268-305)) of hnRNP A1. M9 is hnRNP A1's nuc...

Human cyclin A1, a newly discovered cyclin, is expressed in testis and is thought to function in the meiotic cell cycle. Here, we show that the expression of human cyclin A1 and cyclin A1-associated kinase activities was regulated during the mitotic cell cycle. In the osteosarcoma cell line MG63, cyclin A1 mRNA and protein were present at very low levels in cells at the G0 phase. They increased...

Urea transport, mediated by the urea transporter A1 (UT-A1) and/or UT-A3, is important for the production of concentrated urine. Vasopressin rapidly increases urea transport in rat terminal inner medullary collecting ducts (IMCD). A previous study showed that one mechanism for rapid regulation of urea transport is a vasopressin-induced increase in UT-A1 phosphorylation. This study tests whether...