Whiteley, W., Weavings, sections and projections of spherical polyhedra, Discrete Applied Mathematics 32 (1991) 275-294. In this paper we give simultaneous answers to three questions: (a) When does a plane picture of lines, weaving over and under in the plane, lift and separate into a configuration of disjoint lines in 3-space? (b) When is a configuration of lines in the plane the cross-section...

Hard helices can be regarded as a paradigmatic elementary model for a number of natural and synthetic soft matter systems, all featuring the helix as their basic structural unit, from natural polynucleotides and polypeptides to synthetic helical polymers, and from bacterial flagella to colloidal helices. Here we present an extensive investigation of the phase diagram of hard helices using a var...

Skyrmions are whirl-like topological spin objects with high potential for future magnetic data storage. A fundamental question that is relevant to both basic research and application is whether ferroelectric (FE) polarization can be associated with skyrmions' magnetic texture and whether these objects can be manipulated by electric fields. We study the interplay between magnetism and electric p...

Cell-membrane glycerolipids exhibit a common structural backbone of asymmetric 1,2-diacyl-sn-glycerol bearing polar head groups in the sn-3 position. In this study, the possible effects of sn-3 head groups on the helical conformational property around the 1,2-diacyl moiety in the solution state were examined. 1H NMR Karplus relation studies were carried out using a series of 1,2-dipalmitoyl-sn-...

A new type of twin boundary was found when two order parameters interact by linear-quadratic coupling QP(2). In this solution, we find that a domain wall consists of two layers in which in one layer both order parameters Q and P are active while in the second layer only Q is active. The adjacent domains are equally asymmetric (Q, P) and (Q, 0) so that one phase could be polar and/or magnetic an...

In this paper, which is the second in a series of papers, we analyse what parameters can determine width radio pulsar 'death valley' $P$-${\dot P}$ diagram. Using exact expression for maximum potential drop, be realised over magnetic polar caps and corresponding threshold secondary plasma production determined Paper I, detail observed distribution pulsars taking into account all possible (radiu...

In this paper we are concerned with producing exhaustive lists of simple monotone Venn diagrams that have some symmetry (non-trivial isometry) when drawn on the sphere. A diagram is simple if at most two curves intersect at any point, and it is monotone if it has some embedding on the plane in which all curves are convex. We show that there are 23 such 7-Venn diagrams with a 7-fold rotational s...

Inverse patchy colloids (IPCs) differ from conventional patchy particles because their patches repel (rather than attract) each other and attract (rather than repel) the part of the colloidal surface that is free of patches. These particular features occur, e.g. in heterogeneously charged colloidal systems. Here we consider overall neutral IPCs carrying two, relatively small, polar patches. Pre...

The understanding of quantal many-body systems is a challenging task for far-from-equilibrium conditions as appearing in high energy heavy-ion collisions or cosmological problems. It is of relevance to describe ab-initio thermalization by taking into account all quantum aspects of the particles rather than invoking semi-classical approximation schemes from the beginning [1, 2]. We here consider...

Cartesian and polar form of a complex number. The Argand diagram. Roots of unity. The relationship between exponential and trigonometric functions. The geometry of the Argand diagram. 1 The Need For Complex Numbers All of you will know that the two roots of the quadratic equation ax + bx+ c = 0 are x = −b± √ b2 − 4ac 2a (1) and solving quadratic equations is something that mathematicians have b...