3-(2,4-Dichloro­phen­yl)-5-(4-fluoro­phen­yl)-2-methyl-7-(trifluoro­meth­yl)pyrazolo­[1,5-a]pyrimidine

In the title compound, C20H11Cl2F4N3, the central pyrazolo­[1,5-a]pyrimidine unit is almost planar [the mean deviation from the best least-square plane through the nine atoms is 0.006 (2) Å]. The fluoro­benzene ring is rotated out of this plane by 10.3 (3)°, whereas the dichloro­benzene ring is rotated by 46.2 (3)°. The crystal packing is dominated by Cl⋯Cl inter­actions of 3.475 (3) Å and van der Waals inter­actions

(Dimethyl­formamide-κO){4,4′,6,6′-tetra­bromo-2,2′-[o-phenyl­enebis(nitrilo­methyl­idyne)]diphenolato-κ4 O,N,N′,O′}copper(II) dimethyl­formamide solvate

In the title compound, [Cu(C20H10Br4N2O2)(C3H7NO)]·C3H7NO, the CuII ion is coordinated by two N atoms and two O atoms from a tetra­dentate Schiff base ligand and the O atom of one dimethyl­formamide ligand in an almost square-pyramidal geometry. The uncoordinated dimethyl­formamide solvent mol­ecule is disordered over two sets of positions with occupancies of 0.741 (4) and 0.259 (4). The crystal packing is stabilized by C—H⋯O inter­actions

Packing chromatic vertex-critical graphs

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{\rho}$-critical for short, are introduced as the graphs $G$ for which $\chi_{\rho}(G-x) < \chi_{\rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{\rho}(G) = k$, then $G$ is $k$-$\chi_{\rho}$-critical. It is shown that if $G$ is $\chi_{\rho}$-critical, then the set $\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$\chi_{\rho}$-critical graphs are characterized, and $4$-$\chi_{\rho}$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$\chi_{\rho}$-critical tree and that a $k$-$\chi_{\rho}$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G)$, then $G\,\square\, H$ is $\chi_{\rho}$-critical

Cross-sectional Structure of the Central Spindle of Diatoma vulgare Evidence for Specific Interactions between Antiparallel Microtubules

During the transition from prometaphase to metaphase, the cross-sectional area of the central spindle of Diatoma decreases by a factor of nearly two, both at the poles and at the region of overlapping microtubules (MTs) near the spindle equator. The density of spindle MT packing stays approximately constant throughout mitosis. Optical diffraction analysis of electron micrographs shows that the packing of the MTs at the poles at all stages of mitosis is similar to that expected for a two-dimensional liquid. Analysis of the region of overlap reveals more packing regularity: during prometaphase, a square packing emerges that displays sufficient organization by late metaphase to generate five orders of diffraction; during anaphase the packing in the overlap region shifts to hexagonal; at telophase, it returns to square. From the data provided by serial section reconstructions of the central spindle, it is possible to identify the polarity of almost every spindle MT, that is, to identify one pole with which the MT is associated. Near neighbor analyses of MTs in cross sections of the overlap region show that MTs prefer antiparallel near neighbors. These near neighbors are most often found at a spacing of approximately 40 nm center-to-center, while parallel near neighbors in the zone of overlap are spaced essentially at random. These results are evidence for a specific interaction between antiparallel MTs. In some sections definite bridges between MTs can be seen. Our findings show that certain necessary conditions for a sliding filament model of anaphase spindle elongation are met

(2,3,5,10,12,13,15,20-Octa­phenyl­porphinato)copper(II) 1,1,2,2-tetra­chloro­ethane solvate

The title complex, [Cu(C68H44N4)]·C2H2Cl4, exhibits nearly square-planar geometry around the CuII centre and the macrocyclic ring is almost planar. The porphyrin mol­ecule has an approximate non-crystallographic inversion centre (Ci), and a non-crystallographic twofold rotation axis (C 2) within the CuII–porphyrin ring plane. Further, it has non-crystallographic twofold rotation axis and mirror plane (Cs) symmetry perpendicular to the mol­ecular plane. The mol­ecular packing of the complexes and the solvent molecules shows weak inter­molecular C—H⋯π, C—H⋯Cl and C—H⋯N inter­actions, forming a clathrate-like structure

Packing index of subsets in Polish groups

For a subset $A$ of a Polish group $G$, we study the (almost) packing index \ind_P(A) (resp. \Ind_P(A)) of $A$, equal to the supremum of cardinalities $|S|$ of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$ is (almost) disjoint (in the sense that $|xA\cap yA|<|A|$ for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are small in a geometric sense. We show that \ind_P(A)\in \IN\cup\{\aleph_0,\cc\} for any $\sigma$-compact subset $A$ of a Polish group. If $A\subset G$ is Borel, then the packing indices \ind_P(A) and \Ind_P(A) cannot take values in the half-interval [\sq(\Pi^1_1),\cc) where \sq(\Pi^1_1) is a certain uncountable cardinal that is smaller than \cc in some models of ZFC. In each non-discrete Polish Abelian group $G$ we construct two closed subsets $A,B\subset G$ with \ind_P(A)=\ind_P(B)=\cc and \Ind_P(A\cup B)=1 and then apply this result to show that $G$ contains a nowhere dense Haar null subset $C\subset G$ with \ind_P(C)=\Ind_P(C)=\kappa for any given cardinal number \kappa\in[4,\cc]

Confirmation of Anomalous Dynamical Arrest in attractive colloids: a molecular dynamics study

Previous theoretical, along with early simulation and experimental, studies have indicated that particles with a short-ranged attraction exhibit a range of new dynamical arrest phenomena. These include very pronounced reentrance in the dynamical arrest curve, a logarithmic singularity in the density correlation functions, and the existence of `attractive' and `repulsive' glasses. Here we carry out extensive molecular dynamics calculations on dense systems interacting via a square-well potential. This is one of the simplest systems with the required properties, and may be regarded as canonical for interpreting the phase diagram, and now also the dynamical arrest. We confirm the theoretical predictions for re-entrance, logarithmic singularity, and give the first direct evidence of the coexistence, independent of theory, of the two coexisting glasses. We now regard the previous predictions of these phenomena as having been established.Comment: 15 pages,15 figures; submitted to Phys. Rev.