The vertex isoperimetric problem for the powers of the diamond graph

We introduce a new graph for all whose cartesian powers the vertex isoperimetric problem has nested solutions. This is the fourth kind of graphs with this property besides the well-studied graphs like hypercubes, grids, and tori. In contrast to the mentioned graphs, our graph is not bipartite. We present an exact solution to the vertex isoperimetric problem on our graph by introducing a new class of orders that unifies all known isoperimetric orders defined on the cartesian powers of graphs.     

trans‐1,2,3‐Tris(1‐hydroxycyclo­propyl)cyclopropane

The central three‐membered ring in the title compound, trans‐1,1′,1′′‐cyclo­propane‐1,2,3‐triyl­tris­(cyclo­propanol), C₁₂H₁₈O₃, shows pronounced asymmetry of the bond lengths, which is induced by the different orientations of the substituents. A network of hydrogen bonds links the mol­ecules into sheets.     

Palladium(II)–rare-earth metal(III) paddlewheel carboxylate complexes: Easy total acetate to pivalate metathesis

Bis-paddlewheel heterobimetallic complexes in which palladium(II) is connected to the rare-earth metals(III) [Pd(μ-OOCMe)₄Ln(OH₂)(μ,η²-OOCMe)]₂ × 2HOOCMe (Ln = Nd, Sm, Eu, Yb and Tm) by four acetate bridges were synthesised by the reaction of Pd₃(μ-OOCMe)₆ with the Ln^III acetates. The tetraacetate-bridged complexes were unexpectedly found to be readily transformed by the stoichiometric amount of pivalic acid into the mono-paddlewheel tetrapivalate-bridged analogues in which the paddlewheel structure [Pd(μ-OOCR)₄Ln] maintains as established by X-ray crystallography. The role of the intra- and intermolecular H-bonding in these complexes is discussed.     

Pushing the envelope of the test functions in the Szegö and Avram–Parter theorems

The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far.     

Kinetic study on the living/controlled cationic polymerization of p‐methoxystyrene coinitiated by tris(pentafluorophenyl)borane

A kinetic study of the living cationic polymerization of p‐methoxystyrene using 1‐(4‐methoxyphenyl)ethanol ( 1 )/B(C₆F₅)₃ initiating system in a mixture of CH₃CN with CH₂Cl₂ 1:1 (v/v) at room temperature was carried out utilizing a wide variety of conditions. The polymerization proceeded in a living fashion even in the presence of a large amount of water ([H₂O]/[B(C₆F₅)₃] ratio up to 20) to afford polymers whose M_n increased in direct proportion to monomer conversion with fairly narrow MWDs (M_w/M_n ≤ 1.3). The investigation revealed that the rate of polymerization was first‐order in B(C₆F₅)₃ concentration, while a negative order in H₂O concentration close to ?2 was obtained. It was also found that the rate of polymerization decreased with lowering temperature, which could be attributed to a decreased concentration in free Lewis acid, the true coinitiator of polymerization. A mechanistic scheme to explain the kinetic behavior of living p‐methoxystyrene polymerization is proposed, which has been validated by PREDICI simulation on multiple‐data curves obtained by ¹H NMR in situ polymerization experiment. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 6928–6939, 2008     

Fast-switching effect using viscous chiral-nematic materials

We report on chiral-nematic mixtures containing viscous chiral dopants, composed of rigid rod-like chiral molecules, in order to solve the problem of backflow in the middle layers of the liquid crystal cells. During the study, the viscoelastic properties of the liquid crystal compositions have been optimised, as well as the helical twisting power of the chiral compounds, the anchoring energy and the pre-tilt angle of alignment materials, the ratio between the cell gap and the helical pitch (d/P₀). After optimisation, we prepared test cells with low operating voltage – from 1.5 till 5.0 V, fast response time (τ_on + τ_off) – less than 1 ms, wide operating temperature range from ?40°C till +100°C, and good thermal stability. It should be noted that the response time of the new electro-optical mode does not strongly depend on the cell gap, which is in contrast to other known operating modes. We have achieved rather linear than quadratic dependence of the response time when changing the thickness of the cell.