Molecular Ordering in Liquid Crystals and the Effect of End-Chains on the Even-Odd Effect

The second and fourth orientational order parameters [Pbar]₂, [Pbar]₄ of the homologous series of 4-n-alkyl-4′-cyanobiphenyls (nCB, n = 5 ～ 8) in the nematic liquid crystalline state have been determined from the Raman depolarization ratios by the resonance Raman probe method. It was found out that not only [Pbar]₂ but also [Pbar]₄ exhibits evident even-odd effect when they are compared at the same reduced temperature. For understanding the mechanism of the even-odd effect, a theoretical calculation has been carried out on the basis of the mean field theory of Marcelja, in which the effect of the end alkyl-chain on the liquid crystalline state is taken explicitly into consideration. The observed trend in [Pbar]₂, [Pbar]₄, nematic-isotropic transition temperature were well reproduced. Detailed inspection of the results shows that the presence of anisotropic molecular field plays an important role in the appearance of the even-odd effect. The anisotropic molecular field produces redistribution of the statistical weight of various conformers in such a way that the end chains of odd-members align better along the direction of the rigid core part than those of even-members.     

Numerical Studies of Ground State Phase Diagrams for the MMX Chains

Abstract

We study ground state phase diagrams for the MMX chains, using a one-dimensional dimerized 3/4-filled extended Hubbard-Peierls model with site diagonal and off-diagonal electron-lattice interactions. The ground states are obtained mainly in the Hartree-Fock approximation, and their accuracy is checked by the exact diagonalization of small clusters. We find a new phase in addition to frequently considered phases and compare our results with experimental results.     

Measurement-induced disturbance and thermal negativity in 1D optical lattice chain

We study the measurement-induced disturbance (MID) in a 1D optical lattice chain with nonlinear coupling. Special attention is paid to the difference between the thermal entanglement and MID when considering the influences of the linear coupling constant, nonlinear coupling constant and external magnetic field. It is shown that MID is more robust than thermal entanglement against temperature T$T$ and external magnetic field B$B$, and MID may reveal more properties about quantum correlations of the system, which can be seen from the point of view that MID can be nonzero when there is no thermal entanglement and MID can detect the critical point of quantum phase transition at finite temperature.     

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

Employing the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the XXX spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated T–Q$T\text{–}Q$ relation and the Bethe ansatz equations are derived.     

Comprehensive Characterization of Chain End Groups of Vinylidene Fluoride Based Polymers

Summary: In this work an investigation of the chain end groups produced in the free radical copolymerization of vinilydene fluoride (VDF) and hexafluoropropylene (HFP) is performed. Type and amount of chain end groups are evaluated by a meticulous analytical characterization of VDF/HFP copolymer. At first pulsed gradient spin-echo nuclear magnetic resonance (spin-echo NMR) is used to identify all the chain end groups also at very low concentration (equal to 0.1 mmol · Kg⁻¹). The instrument sensitivity is increased of an order of magnitude in comparison with the traditional NMR. Moreover potentiometric titration and ion chromatography (IC) are also used to study the chain end groups and, as a consequence, the nature and the amount of the acidity showed by the polymer chains. In details two intensity of acidity are detected by potentiometric titration, namely strong and weak. The strong acidity is associated to the presence of residual surfactant and can be removed washing the polymer, while the weak acidity is due to free molecules of fluoride acid (HF). The standard ion chromatography facility is properly modified to quantify the fluoride in the polymer matrix without any pre extraction in water. Thanks to this the HF concentration in the polymer is evaluated with high accuracy. A detailed kinetic scheme for the VDF/HFP polymerization is also proposed taking into account all the findings obtained studying the chain end groups.     

Freshman rules in calculus

In this note we present a class of functions (f, g) that satisfy ‘freshman rules’ in calculus.     

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.     

On sub‐Riemannian geodesics on the Engel groups: Hamilton's equations

We study the sub‐Riemannian geometry on the Engel group which is a step 3 nilpotent Lie group on . Our main result is to solve the Hamiltonian equations associated with the bi‐characteristic curves and express the solutions in terms of elliptic functions. Our model covers both the Heisenberg group and the Martinet case when setting certain parameters to be zero.