Sliding Dynamics of Ring Chains on Two Asymmetric/Symmetric Chains in a Simple Slide-Ring Gel

The sliding dynamics along two asymmetric/symmetric axial chains of ring chains linked by a linear chainis investigated using molecular dynamics (MD) simulations. A novel sub-diffusion behavior is observed for ring chains sliding along eithera fixed rod-like chain or fluctuating axial chain on asymmetric/symmetric axial chainsat the intermediate time range due to their strongly interplay between two ring chains. However, two ring chains slide in the normal diffusion at along time range because their sliding dynamics can be regarded as an overall motion of two ring chains. For ring chains sliding on two symmetric/asymmetricaxial chains, the diffusion coefficient D of ring chains relies on the bending energy of axial chains (Kb) as well as the distance of two axial chains (d). There exists a maximum diffusion coefficient Dmax at d = d* in which ring chains slide at the fastest velocity due to the maximum conformational entropy for the linking chain between two ring chainsat d = d*. Ring chain slide on fixed rod-like axial chainsfaster in the symmetric axial chain case than that in the asymmetric axial chain case. However, ring chains slide on fluctuatingaxial chainsslower in the symmetric axial chain case than that in the asymmetric axial chain case. This investigation can provide insights into the effects of the linked chain conformation on the sliding dynamics of ring chains in a slide-ring gel.

Decompositions of the Boolean Lattice into Rank-symmetric Chains

The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset\cdots\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,\ldots,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We show that there exist a bijection $$p: [n]^{(\geq n/2)}\rightarrow [n]^{(\leq n/2)}$$ and a partial ordering $<$ on $[n]^{(\geq n/2)}$ satisfying the following properties:$\subset$ is an extension of $<$ on $[n]^{(\geq n/2)}$;if $C\subset [n]^{(\geq n/2)}$ is a chain with respect to $<$, then $p(C)\cup C$ is a rank-symmetric chain in $2^{[n]}$, where $p(C)=\{p(x): x\in C\}$;the poset $([n]^{(\geq n/2)},<)$ has the so called normalized matching property.We show two applications of this result.A conjecture of Füredi asks if $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that the size of any two chains differ by at most 1. We prove an asymptotic version of this conjecture with the additional condition that every chain in the partition is rank-symmetric: $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ rank-symmetric chains, each of size $\Theta(\sqrt{n})$.Our second application gives a lower bound for the number of symmetric chain partitions of $2^{[n]}$. We show that $2^{[n]}$ has at least $2^{\Omega(2^{n}\log n/\sqrt{n})}$ symmetric chain partitions.

Synthesis and structural features of complexes [(P∩P)2Rh][hfacac] containing hexafluoroacetyl- acetonate as a noncoordinating anion

The synthesis and characterization of complexes [(P∩P)2Rh][hfacac] (P∩P = chelating bidentate phosphine ligand R2P(CH2)nPR2 (2a-g), hfacac = hexafluoroacetylacetonate anion) (4) is reported. The molecular structures of 4a (R = Ph, n = 1) and 4f (R = Cy, n = 2) in the solid state were determined by single-crystal X-ray diffraction. The complexes crystallize in the monoclinic space groups C2/c (No. 15) and P21/n (No. 14), respectively. No coordinative interaction between the rhodium center of the cation [(P∩P)2Rh]+ (4a+, 4f+) and the hfacac anion is evident in either cases. In the crystal structure of 4a, hydrogen bonds between the oxygen atoms of the hfacac anion and methylene protons of the CH2 bridges of the phosphine ligand lead to highly symmetric chains of regularly alternating cations and anions. The coordination geometry around the rhodium center in 4a+ is ideally square-planar, whereas 4f+ is significantly distorted towards a tetrahedron with an angle between the two P2Rh moieties of 18.6°. The cation 4b+ (R = Cy, n = 1) was investigated in form of the tetrafluoroborate salt for comparison. The compound [{Cy2P(CH2)PCy2}2Rh][BF4] crystallizes as a THF solvate (4b′) in the triclinic space group P[Formula: see text] (No. 2) containing ideally square-planar [(P∩P)2Rh]+ cations. Key words: rhodium, chelating ligands, coordination modes, 1,3-diketonates, phosphorus ligands.

Symmetric Chain Decompositions of Linear Lattices

It has been known for several years that the lattice of subspaces of a finite vector space has a decomposition into symmetric chains, i.e. a decomposition into disjoint chains that are symmetric with respect to the rank function of the lattice. This paper gives a positive answer to the long-standing open problem of providing an explicit construction of such a symmetric chain decomposition for a given lattice of subspaces of a finite (dimensional) vector space. The construction is done inductively using Schubert normal forms and results in a bracketing algorithm similar to the well-known algorithm for Boolean lattices.