A combinatorial perspective on the non-Radon partitions

Let E be a finite set of points in $\text{R}$^d. Then {A, E ? A} is a non-Radon partition of E iff there is a hyperplane H separating A strictly from E?A. Or equivalently iff ${}_{\mathrm{<mtext>A</mtext>}}\text{O}$ is an acyclic reorientation of (M_Aff(E), O), the oriented matroid canonically determined by E. If (M(E), O) is an oriented matroid without loops then the set $\text{NR(E, O) = {(A, E ? A):}{}_{\mathrm{<mtext>A</mtext>}}\text{O}\text{is acyclic}\text{}}$ determines (M(E), O). In particular the matroidal properties of a finite set of points in $\text{R}$^d are precisely the properties which can be formulated in non-Radon partitions terms. The Möbius function of the poset $\text{A = {A: A ? E,}{}_{\mathrm{<mtext>A</mtext>}}\text{O}\text{is acyclic}\text{}}$ and in a special case its homotopy type are computed. This paper generalizes recent results of P. Edelman (A partial order on the regions of $\text{R}$ⁿ dissected by hyperplanes     

Euler's relation,möbius functions and matroid identities

We give a short combinatorial proof of the Euler relation for convex polytopes in the context of oriented matroids. Using counting arguments we derive from the Euler relation several identities holding in the lattice of flats of an oriented matroid. These identities are proven for any matroid by Möbius inversion.     

An efficient, PIFA mediated approach to benzo-, naphtho-, and heterocycle-fused pyrrolo[2,1-c][1,4]diazepines. An advantageous access to the antitumor antibiotic DC-81

[reaction: see text] The synthesis of a series of optically pure benzo-, naphtho-, and heterocycle-fused pyrrolo[2,1-c][1,4]-diazepin-5,11-dione derivatives starting from l-proline methyl ester is presented. The synthetic plan includes an aroylation step at the proline nitrogen followed by transformation of the ester residue into a N-methoxyamide group. The subsequent key cyclization step embraces the PIFA mediated formation of a N-acylnitrenium intermediate and its succeeding intramolecular trapping by the aromatic ring. The presented general approach solves the need of starting from not very accessible amino (or a related functionality) aromatic starting materials, and its effectiveness is demonstrated in the synthesis of the antitumor antibiotic DC-81.     

Electron energy loss measurements on PbF2, PbCl2, PbBr2 and PbI2

The results of electron energy loss measurements on polycrystalline films of PbF₂, PbC1₂, PbBr₂ and PbI₂ are present and the calculated dielectric functions are discussed in relation to their band structures and optical data.     

Electronic and geometric properties of ETS-10: QM/MM studies of cluster models

Hybrid DFT/MM methods have been used to investigate the electronic and geometric properties of the microporous titanosilicate ETS-10. A comparison of finite length and periodic models demonstrates that band gap energies for ETS-10 can be well represented with relatively small cluster models. Optimization of finite clusters leads to different local geometries for bulk and end sites, where the local bulk TiO6 geometry is in good agreement with recent experimental results. Geometry optimizations reveal that any asymmetry within the axial O-Ti-O chain is negligible. The band gap in the optimized model corresponds to a O(2p) --> Tibulk(3d) transition. The results suggest that the three Ti atom, single chain, symmetric, finite cluster is an effective model for the geometric and electronic properties of bulk and end TiO6 groups in ETS-10.     

Magnetic properties of ferromagnetic quasi-1D copper-peptide compounds: exchange interactions and very low temperature phase transitions

The magnetic properties of the Cu(II)-peptide compounds (L-tyrosyl-L-leucinato)Cu(II) and (L-tryptophyl-glycinato)Cu(II), to be identified as Cu(II)Tyr-Leu and Cu(II)Trp-Gly, respectively, have been investigated by specific heat (0.08 < T < 28 K), dc magnetization (2 < T < 80 K, with B(0) = mu(o)H < or = 9 T), and ac magnetic susceptibility (with B(0) = 0 for 0.03 < T < 3 K and B(0) up to 9 T for 2 < T < 80 K) measurements. Above approximately 1 K, the specific heat and magnetization of both compounds display a ferromagnetic (FM) spin chain behavior sustained by syn-anti carboxylate bridges connecting equatorially Cu(II) ions at about 5 A. To model this behavior, we calculated the eigenvalues of Heisenberg chains with up to 20 spins 1/2 and used the method of Bonner and Fisher. A global fit of the model to the specific heat and magnetization data gives 2J(0)/k(B) = 3.60(5) K and 2.59(5) K for the intrachain exchange interactions in Cu(II)Tyr-Leu and Cu(II)Trp-Gly, respectively (H(ex)(i,j) = -2J(0) S(i).S(j)). These values of 2J(0) are discussed in terms of structural properties of the carboxylate bridges in the two compounds. Using the parameters obtained from the global fit, we calculated isothermal susceptibilities in agreement with the ac susceptibilities measured with small applied dc magnetic fields. However, the ac susceptibility measured with applied dc fields larger than 1 T lie between the values calculated for the isothermal and adiabatic susceptibilities. At 0.16 K for Cu(II)Tyr-Leu and 0.53 K for Cu(II)Trp-Gly, the observed specific heat and magnetic susceptibility display peaks associated to three-dimensional magnetic phase transitions. The interchain exchange couplings 2J(1) producing the 3D magnetic order are ferromagnetic and have magnitudes 2J(1)/k(B) approximately 0.015 and 0.073 K for Cu(II)Tyr-Leu and Cu(II)Trp-Gly, respectively.