Non-regular simplicial matroids

Let (ks) denote the set of all k-element-subsets of a finite set S. A k-simplical matroid on a subset E of (ks) is a binary matroid the circuit of which are simplicial complexes {X1,…Xm} ? E with boundary 0 (mod 2). The k-simplical matroid on (ks) is called the full simplicial matroid Gk(S). The polygon matroid on the edges of a finite graph is 2-simplicial. Polygon-matroids and their duals are regular. The dual of Gk(S) is Gn?k(S) if the cardinnlity of S is n. More details on simplicial matroids can be found in [3, Chapter 6] and also in [4, pp. 180–181].Welsh asked if every simplicial matroid is regular. We prove that this is not the case, for all full k-simplicial matroids Gk(S) with 3?k?n?3 are non-regular (n is the cardinality of S). This result has also been proved σy R. Cordovil and M. Las Vergnas recently. Their proof is different from our proof, which is somewhat shorter.     

Brownian Approximation and Monte Carlo Simulation of the Non-Cutoff Kac Equation

The non-cutoff Boltzmann equation can be simulated using the DSMC method, by a truncation of the collision term. However, even for computing stationary solutions this may be very time consuming, in particular in situations far from equilibrium. By adding an appropriate diffusion, to the DSMC-method, the rate of convergence when the truncation is removed, may be greatly improved. We illustrate the technique on a toy model, the Kac equation, as well as on the full Boltzmann equation in a special case.     

On the Distribution of Free Path Lengths¶for the Periodic Lorentz Gas

Consider the domain

Zinc and Cobalt Complexes Derived from Tetradentate Tripodal Ligands with N2O2 Donorset as Model Compounds for Phosphatases

Starting from the tripodal tetradentate ligands ‐(3,5‐dibromo‐2‐hydroxybenzyl)(2‐hydroxybenzyl)(2‐pyridyl)methylamine (H₂L1), (3,5‐dibromo‐2‐hydroxybenzyl)(2‐hydroxy‐5‐nitrobenzyl)(2‐pyridyl)methylamine (H₂L2), and (3,5‐dichloro2‐hydroxybenzyl)(2‐hydroxy‐5‐nitrobenzyl)(2‐pyridyl)methylamine (H₂L3) the new isostructural dinuclear zinc compounds [Zn₂(L1)₂]·N(CH₂CH₃)₃ ( 1 ), [Zn₂(L2)₂]·2CH₃OH ( 2 ) and [Zn₂(L3)₂]·C₄H₁₀O ( 3 ) were synthesized. Due to their enzyme‐like trigonal bipyramidal N₂O₃ coordination environment of the zinc ions and the similar Zn···Zn distances the complexes can be considered to be structural models for the active sites in phospholipase C and nuclease P1. With H₂L3 also the dinuclear complex [Co₂(L2)₂(CH₃OH)]·2CH₃OH·0.5C₄H₁₀O ( 4 ) could be prepared. The new compounds were isolated and characterized by single crystal X‐ray crystallography as well as infrared spectroscopy. The cobalt compound 4 was additionally characterized by UV‐Vis spectroscopy and magnetic measurements. 1 crystallizes in the monoclinic space group P2₁/n with a = 11.2814(2), b = 28.6154(2), c = 13.1866(3) Å, β = 96.995(1)°, V = 4225.2(2) ų, Z = 4. 2 and 3 are monoclinic, space group C2/c with a = 23.084(5), b = 9.232(2), c = 21.849(4) Å, &β; = 96.83(3)°, V = 4623(2) ų, Z = 4, and a = 22.7834(3), b = 9.2463(1), c = 21.6351(3) Å, &β; = 97.592(1)°, V = 4517.7(2) ų, Z = 4, respectively. 4 crystallizes in the monoclinic space group I2/a with a = 22.4680(4), b = 20.5517(4), c = 22.8910(6) Å, &β; = 111.938(1)°, V = 9804.7(4) ų, Z = 8. 4 shows an effective magnetic moment of 6.72 μ_B at 300 K which clearly indicates the presence of two cobalt(II) high spin ions with Curie‐Weiss behaviour above 80 K. At lower temperatures a decrease of the effective magnetic moment was observed.