Inverses of Boolean matrices

For a Boolean matrix A, a g-inverse of A is a Boolean matrix G satisfying AGA=A, and a Vagner inverse is a g-inverse which in addition satisfies GAG=G. We give algorithms for finding all g-inverses, all Vagner inverses, and all of several other types of inverses including Moore-Penrose inverses. We give a criterion for a Boolean matrix to be regular, and criteria for the various types of inverse to exist. We count the numbers of Boolean matrices having Moore-Penrose and related types of inverses.     

Enantio- and stereoselective syntheses of the dihydroxyoctadienoic acid framents of the roridins and trichoverrins

Syntheses of protected versions of the dihydroxyoctadienoic acid fragments corresponding to roridin A and trichoverrin B are described.     

Total synthesis of the formamicin aglycon,formamicinone

[structure: see text] The total synthesis of formamicinone (2), the aglycone of formamicin (1), has been accomplished via the late-stage Suzuki cross-coupling of fragments 5 and 6, the macrolactonization of seco ester 14, and the Mukaiyama aldol reaction of aldehyde 3 and methyl ketone 4. An efficient and highly stereoselective second generation synthesis of vinyl iodide 6 is also described.     

Lewis acid catalyzed intramolecular Diels-Alder reactions of acyclic (Z)-substituted 1,3-dienes

Lewis acid catalyzed intramolecular Diels-Alder reactions of trienes (E,E,Z)-1a-d, (E,E,Z)-4a-d, and (E,Z,Z)-7a,b are described. Trienes containing enal or enone dienophiles cyclize in excellent yield under mild conditions using substoichiometric amounts of MeAlCl(2), in most cases with high levels of diastereoselectivity. The thermal IMDA reactions of 1a, 4a, and 7a require forcing conditions and proceed in low yield with reversed stereoselectivity in the cases of 1a and 4a.     

The spectra of nonnegative integer matrices via formal power series

We characterize the possible nonzero spectra of primitive integer matrices (the integer case of Boyle and Handelman's Spectral Conjecture). Characterizations of nonzero spectra of nonnegative matrices over and follow from this result. For the proof of the main theorem we use polynomial matrices to reduce the problem of realizing a candidate spectrum to factoring the polynomial as a product where the 's are polynomials in satisfying some technical conditions and is a formal power series in . To obtain the factorization, we present a hierarchy of estimates on coefficients of power series of the form to ensure nonpositivity in nonzero degree terms.

     

Enantio- and diastereoselective synthesis of (E)-1,5-syn-diols: application to the synthesis of the C(23)-C(40) fragment of tetrafibricin

A highly stereoselective synthesis of (E)-1,5-syn-diols 6 is described. The kinetically controlled hydroboration of allenyltrifluoroborate 8 with Soderquist borane 2 provides the (Z)-allylic trifluoroborate 9, which undergoes sequential allylboration with two different aldehydes to provide (E)-1,5-syn-diols 6 in 72-98% yields with >95% ee and >20:1 dr. Application of this method to the synthesis of the tetrafibricin C(23)-C(40) fragment 19 is described.