Quadrangulations on 3-Colored Point Sets with Steiner Points and Their Winding Numbers

Let P be a point set on the plane, and consider whether P is quadrangulatable, that is, whether there exists a 2-connected plane graph G with each edge a straight segment such that V(G) = P, that the outer cycle of G coincides with the convex hull Conv(P) of P, and that each finite face of G is quadrilateral. It is easy to see that it is possible if and only if an even number of points of P lie on Conv(P). Hence we give a k-coloring to P, and consider the same problem, avoiding edges joining two vertices of P with the same color. In this case, we always assume that the number of points of P lying on Conv(P) is even and that any two consecutive points on Conv(P) have distinct colors. However, for every k ≥ 2, there is a k-colored non-quadrangulatable point set P. So we introduce Steiner points, which can be put in any position of the interior of Conv(P) and each of which may be colored by any of the k colors. When k = 2, Alvarez et al. proved that if a point set P on the plane consists of \({\frac{n}{2}}\) red and \({\frac{n}{2}}\) blue points in general position, then adding Steiner points Q with \({|Q| \leq \lfloor \frac{n-2}{6} \rfloor + \lfloor \frac{n}{4} \rfloor +1}\) , P ∪ Q is quadrangulatable, but there exists a non-quadrangulatable 3-colored point set for which no matter how many Steiner points are added. In this paper, we define the winding number for a 3-colored point set P, and prove that a 3-colored point set P in general position with a finite set Q of Steiner points added is quadrangulatable if and only if the winding number of P is zero. When P ∪ Q is quadrangulatable, we prove \({|Q| \leq \frac{7n+34m-48}{18}}\) , where |P| = n and the number of points of P in Conv(P) is 2m.     

Study of molecular motion of block copolymers in solution by high-resolution proton magnetic resonance

Molecular motions of hydrophobic–hydrophilic water-soluble block copolymers in solution were investigated by high-resolution proton magnetic resonance (NMR). Samples studied include block copolymers of polystyrene–poly(ethylene oxide), polybutadiene–poly(ethylene oxide), and poly(ethylene oxide)–poly(propylene oxide)–poly(ethylene oxide). NMR measurements were carried out varying molecular weight, temperature, and solvent composition. For AB copolymers of polystyrene and poly(ethylene oxide), two peaks caused by the phenyl protons of low-molecular-weight (M?_n = 3,300) copolymer were clearly resolved in D₂O at 100°C, but the phenyl proton peaks of high-molecular-weight (M?_n = 13,500 and 36,000) copolymers were too broad to observe in the same solvent, even at 100°C. It is concluded that polystyrene blocks are more mobile in low-molecular-weight copolymer in water than in high-molecular-weight copolymer in the same solvent because the molecular weight of the polystyrene block of the low-molecular-weight copolymer is itself small. In the mixed solvent D₂O and deuterated tetrahydrofuran (THF-d₈), two peaks caused by the phenyl protons of the high-molecular-weight (M?_n = 36,000) copolymer were clearly resolved at 67°C. It is thought that the molecular motions of the polystyrene blocks are activated by the interaction between these blocks and THF in the mixed solvent.     

Mechanical relaxations in some ionene polymers. II. Influence of counterions and absorbed water

Six 6,10-ionenes with different counterions were prepared by ion exchange reactions in aqueous solutions. The counterions were Br, I, CIO₄, BF₄, SCN, and B(C₆H₅)₄. The dynamic mechanical properties of these polymers were investigated by use of a torsional braid analyser. Three relaxations α (25–140°C), β (?30–0°C), and γ (?140–120°C) were observed at the frequencies of 0.3–0.8 Hz. The temperature of the α and β relaxations were largely dependent on the size of counterions, but those of the γ relaxations had little variation. The effects of electrostatic forces in the polymers on each relaxation was discussed. The influence of absorbed water on the α, β, and γ relaxations was examined. The absorbed water in the polymers greatly depressed the temperature of the α relaxations and this phenomenon was interpreted to be the result of the specific hydration on ionic portions.     

Synthesis and dynamic mechanical properties of heterogeneous network polymers from poly(L-glutamic acid) and poly(oxyethylene glycol)

Heterogeneous network polymers composed of rigid polypeptide chains and flexible polyether chains were synthesized. That is, poly(L -glutamic acid) (PLGA) was crosslinked with poly(oxyethylene glycol) (PEG) at various carboxy/hydroxyl mole ratios K. The solubility tests and hydrolysis of heterogeneous network polymers suggest that the crosslinking reaction proceeds by esterification. The dynamic mechanical properties of these polymers(100 Hz, ?100–200°C) are greatly influenced by the presence of a trace of water and the weight per cent of PLGA. In addition, some of these polymers show only one maximum in the temperature dispersion of dynamic loss modulus E″ and tan δ, although their shape is rather broad. The x-ray photographs of these polymers show an amorphous halo or weak Debye-Sherrer rings. These findings suggest that these polymers are not simple adducts; neverthless PLGA and/or PEG domains exist.