Proof of Claim 1.1: Of course, we assume that {nj}j=1 are all distinct. Let n1 = 1. Then there must exist An2 such that An2 ∩An1 6= ∅. If not, then An1 ∩Aj = ∅ for all j 6= n1. Thus, An1 ⊆ Aj for all j 6= n1. Now, fix some j0 6= n1 such thatAj0 6= An1 . Then, sinceF is a σ-algebra, we have thatAk0 = A c j0 for some k0 6= n1. Thus, setting j = k0 6= n1, An1 ⊆ Ack0 = ( Acj0 )c = Aj0 . The only wa...

Two nicotinamide adenine dinucleotide (NAD(+)) analogues modified at the 6 position of the purine ring were synthesized, and their substrate properties toward Aplysia californica ADP-ribosyl cyclase were investigated. 6-N-Methyl NAD(+) (6-N-methyl nicotinamide adenosine 5'-dinucleotide 10) hydrolyzes to give the linear 6-N-methyl ADPR (adenosine 5'-diphosphoribose, 11), whereas 6-thio NHD(+) (n...

We provide the proofs for the theorems in the main paper. 1 Proofs for Planted Clustering In this section, Theorems 1–6 refer to the theorems in the main paper. Equations are numbered continuously from the main paper. We let n1 := rK and n2 := n − rK be the numbers of nonisolated and isolated nodes, respectively. 1.1 Proof of Theorem 1 The proof relies on information theoretical arguments and t...

Treating 1,5-difluoro-2,4-dinitrobenzene (1) with N1-phenyl-5-(trifluoromethyl)benzene-1,2diamine (4) and N,N-diisopropylethylamine in EtOH at ca. 0 ◦C for 4 h affords a mixture of N1-(5-ethoxy-2,4-dinitrophenyl)-N2-phenyl-4-(trifluoromethyl)benzene-1,2-diamine (5) (38%) and N1-(5-fluoro-2,4-dinitrophenyl)-N2-phenyl-4-(trifluoromethyl)benzene-1,2-diamine (6) (51%) that can be separated by chrom...

• P = {1, 2, 3, . . .} • Seq = {< n1, . . . , nk > | k > 0, n1, . . . , nk ∈ P}. p, q, r and s, possibly decorated, are metavariables ranging over Seq. • If p =< m1, . . . ,mk > and q =< n1, . . . , nl > then p ∗ q = 〈m1, . . . ,mk, n1, . . . , nl〉. • p ∗ 〈 〉 = 〈 〉 ∗ p = p for every p ∈ Seq. • p 6 q iff there is s such that p ∗ s = q. Definition 1.1 (Trees). T ⊆ Seq is a tree iff: 1. q ∈ T and ...