Twist-bend nematics,liquid crystal dimers,structure–property relations

ABSTRACT

One of the current challenges in liquid crystal science is to understand the molecular factors leading to the formation of the intriguing twist-bend nematic phase (N_TB) and determine its properties. During our earlier hunt for the N_TB phase created on cooling directly from the isotropic phase and not the nematic phase, we had prepared 30 symmetric liquid crystal dimers. These had odd spacers and methylene links to the two mesogenic groups; desirable but clearly not essential features for the formation of the N_TB. Here, we report the phases that the dimers exhibit and their transition temperatures as functions of both the lengths of the spacer and the terminal chains. In addition we describe the transitional entropies, their optical textures, the X-ray scattering patterns and the ²H NMR spectra employed in characterising the phases. All of which may lead to important properties of the twist-bend nematic phase.     

Curvature computations for a two-component Camassa-Holm equation with vorticity

In the present paper, a two-component Camassa-Holm (2CH) system with vorticity is studied as a geodesic flow on a suitable Lie group. The paper aims at presenting various details of the geometric formalism and a major result is the computation of the sectional curvature K of the underlying configuration manifold. As a further result, we show that there are directions for which K is strictly positive and bounded away from zero.     

CR Geometry in 3-D

CR geometry studies the boundary of pseudo-convex manifolds.By concentrating on a choice of a contact form,the local geometry bears strong resemblence to conformal geometry.This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three.While the sign of this operator is important in the embedding problem,the kernel of this operator is also closely connected with the stability of CR structures.The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed.The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem.The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension,and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms.Finally,an isoperimetric constant determined by the Q-prime curvature integral is discussed.     

Minimum cost homomorphisms to semicomplete multipartite digraphs

For digraphs D and H, a mapping f:V(D)→V(H) is a homomorphism ofDtoH if uv∈A(D) implies f(u)f(v)∈A(H). For a fixed directed or undirected graph H and an input graph D, the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by a real-world problem in defence logistics and was introduced recently by the authors and M. Tso.Suppose we are given a pair of digraphs D,H and a cost c_i(u) for each u∈V(D) and i∈V(H). The cost of a homomorphism f of D to H is ∑_u∈V(D)c_f(u)(u). Let H be a fixed digraph. The minimum cost homomorphism problem for H, MinHOMP(H), is stated as follows: For input digraph D and costs c_i(u) for each u∈V(D) and i∈V(H), verify whether there is a homomorphism of D to H and, if it does exist, find such a homomorphism of minimum cost. In our previous paper we obtained a dichotomy classification of the time complexity of when H is a semicomplete digraph. In this paper we extend the classification to semicomplete k-partite digraphs, k≥3, and obtain such a classification for bipartite tournaments.     

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.     

Almost contact Riemannian three-manifolds with Reeb flow symmetry

In this paper we give a complete classification of simply connected homogeneous almost α-Kenmotsu three-manifolds M whose Ricci operator is invariant along the Reeb flow. We get this classification by using the Gaussian and the extrinsic curvature associated with the canonical foliation of M.     

复射影空间CP~((n+p)/2)中具有2-调和的一般子流形

本文研究了复射影空间中具有2-调和的一般子流形问题.利用活动标架法,获得了这类子流形成为极小子流形的Pinching定理和Simons型积分不等式,此外还得到关于2-调和伪脐一般子流形的一个刚性定理,推广了复射影空间中具有2-调和全实子流形的一些相应结果.     

A class of Finsler metrics of scalar flag curvature

It is known that every locally projectively flat Finsler metric is of scalar flag curvature. Conversely, it may not be true. In this paper, for a certain class of Finsler metrics, we prove that it is locally projectively flat if and only if it is of scalar flag curvature. Moreover, we establish a class of new non-trivial examples.     

局部对称Lorentz空间中的类空超曲面

研究了局部对称Lorentz空间中具有常平均曲率或常数量曲率的类空超曲面.利用丘成桐的广义极大值原理和自伴随算子得到了两个重要的内蕴刚性定理,其分别推广了欧阳崇祯和刘新民的主要结果.