An ordering of dependence for contingency tables

A majorization ordering is defined on matrices with the same row and column sums. This ordering is used as an ordering of dependence for contingency tables. Results are derived for maximal and minimal matrices with respect to the majorization ordering. This theory can be used to maximize and minimize Schur concave functions defined over matrices, when there are row and column sum constraints; in this paper, it is applied to the algorithm of Mehta and Patel (1983) for finding the P-value of Fisher's exact test.     

Scale-invariant correlations in dynamic bacterial clusters

In Bacillus subtilis colonies, motile bacteria move collectively, spontaneously forming dynamic clusters. These bacterial clusters share similarities with other systems exhibiting polarized collective motion, such as bird flocks or fish schools. Here we study experimentally how velocity and orientation fluctuations within clusters are spatially correlated. For a range of cell density and cluster size, the correlation length is shown to be 30% of the spatial size of clusters, and the correlation functions collapse onto a master curve after rescaling the separation with correlation length. Our results demonstrate that correlations of velocity and orientation fluctuations are scale invariant in dynamic bacterial clusters.     

Polynuclear lanthanide complexes of a series of bridging ligands containing two tridentate N,N',O-donor units: structures and luminescence properties

A set of three potentially bridging ligands containing two tridentate chelating N,N',O-donor (pyrazole-pyridine-amide) donors separated by an o, m, or p-phenylene spacer has been prepared and their coordination chemistry with lanthanide(III) ions investigated. Ligand L(1) (p-phenylene spacer) forms complexes with a 2:3 M:L ratio according to the proportions used in the reaction mixture; the Ln(2)(L(1))(3) complexes contain two 9-coordinate Ln(III) centres with all three bridging ligands spanning both metal ions, and have a cylindrical (non-helical) 'mesocate' architecture. The 1:1 complexes display a range of structural types depending on the conditions used, including a cyclic Ln(4)(L(1))(4) tetranuclear helicate, a Ln(2)(L(1))(2) dinuclear mesocate, and an infinite one-dimensional coordination polymer in which metal ions and bridging ligands alternate along the sequence. ESMS studies indicate that the 1:1 complexes form a mixture of oligonuclear species {Ln(L(1))}(n) in solution (n up to 5) which are likely to be cyclic helicates. In contrast, ligands L(2) and L(3) (with o- and m-phenylene spacers, respectively) generally form dinuclear Ln(2)L(2) Ln(III) complexes in which the two ligands may be arranged in a helical or non-helical architecture about the two metal ions. These complexes also contain an additional exogenous bidentate bridging ligand, either acetate or formate, which has arisen from hydrolysis of solvent molecules promoted by the Lewis-acidity of the Ln(III) ions. Luminescence studies on some of the Nd(III) complexes showed that excitation into ligand-centred pi-pi* transitions result in the characteristic near-infrared luminescence from Nd(III) at 1060 nm.     

Some cyclization reactions of 1,3-diphenylbenzo[e][1,2,4]triazin-7(1H)-one: preparation and computational analysis of non symmetrical zwitterionic biscyanines

Regioselective nucleophilic addition of bisnucleophiles 1,2-benzenediamine, 2-amino-benzenethiol, and N-phenyl-1,2-benzenediamine to 1,3-diphenylbenzo[e][1,2,4]triazin-7(1H)-one (1) at C6 followed by intramolecular cyclocondensation at the C7 carbonyl afforded highly coloured tetracenes 1,3-diphenyl-1,6-dihydro-[1,2,4]triazino[5,6-b]phenazin-4-ium 4-methylbenzenesulfonate (12), 1,3-diphenyl-1H-[1,2,4]triazino[6,5-b]phenothiazine (14) and 1,3,11-triphenyl-1,6-dihydro-[1,2,4]triazino[5,6-b]phenazin-11-ium 4-methylbenzenesulfonate (15), respectively. Neutralization of the latter with alkali gave the free base 1,3,11-triphenyl-1H-[1,2,4]triazino[5,6-b]phenazin-11-ium-6-ide (16). Furthermore, the benzotriazinone 1 reacts with dimethyl malonate to give 6-(methoxycarbonyl)-7-oxo-1,3-diphenyl-7H-benzofuro[5,6-e][1,2,4]triazin-1-ium-4-ide (17) in 74% yield, while with S(4)N(4) [5,6-c]-thiadiazolo-7-oxo-1,3-diphenyl-1,2,4-benzotriazine (22) was formed in 15% yield. The free bases 16 and 17 display negative solvatochromism, which supports charge separated ground states similar to those of zwitterionic biscyanines, and DFT calculations at the UB3LYP/6-31G(d) level afford ΔE(ST) values of -13.6 and -18.7 kcal mol(-1), respectively that strongly favour the singlet ground state. All ring systems described are new and fully characterized.     

Stabilization of Large Adsorbates by Rotational Entropy: A Time‐Resolved Variable‐Temperature STM Study

Investigating the dynamics in an adlayer of the oligopyridine derivative 2‐phenyl‐4,6‐bis(6‐(pyridine‐2‐yl)‐4‐(pyridine‐4‐yl)pyridine‐2‐yl)pyrimidine (2,4′‐BTP) on Ag(111) by fast scanning tunneling microscopy (video‐STM), we found that rotating 2,4′‐BTP adsorbates coexist in a two‐dimensional (2D) liquid phase (β‐phase) in a dynamic equilibrium with static adsorbate molecules. Furthermore, exchange between an ordered phase (α‐phase) and β‐phase leads to fluctuations of the domain boundary on a time scale of seconds. Quantitative evaluation of the temperature‐dependent equilibrium between rotating and static adsorbates, evaluated from a large number of STM images, gains insight into energetic and entropic stabilization and underlines that the rotating adsorbate molecules are stabilized by an entropy contribution, which is compatible with that derived by using statistical mechanics. The general validity of the concept of entropic stabilization of rotating admolecules, favoring rotation already at room temperature, is tested for other typical small, mid‐size and large adsorbates.     

Synthesis and Properties of Porphyrin Nanotubes

Discrete π‐conjugated zinc porphyrin nanotubes are investigated as molecular analogues of carbon nanotubes. These porphyrin nanotubes have a diameter of 2.4 nm (Zn–Zn distance) and lengths of up to 3.6 nm, measured to the van der Waals surfaces of the outer β‐pyrrole hydrogen atoms, or 4.5 nm measured to the para hydrogen atoms of the aryl groups. We explore three different strategies for synthesizing these nanotubes. The first two strategies use a template to achieve direct or sequential stave‐joining, respectively, and proceed via linear oligomers that pre‐define the length of the nanotube. These strategies are applied to synthesize porphyrin nanotubes containing 12‐ or 18‐porphyrin subunits, with ethynylene (C₂) or butadiynylene (C₄) links between the 6‐porphyrin nanorings. The third strategy involves the covalent stacking of pre‐formed 6‐porphyrin nanorings to form a 12‐porphyrin nanotube, without using a template to guide this coupling reaction. The nanotubes show strongly red‐shifted absorption spectra and low fluorescence quantum yields, indicating structural rigidity and extensive π‐conjugation.