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.     

Superstructure in the Metastable Intermediate‐Phase Li2/3FePO4 Accelerating the Lithium Battery Cathode Reaction

LiFePO₄ is an important cathode material for lithium‐ion batteries. Regardless of the biphasic reaction between the insulating end members, Li_xFePO₄, x≈0 and x≈1, optimization of the nanostructured architecture has substantially improved the power density of positive LiFePO₄ electrode. The charge transport that occurs in the interphase region across the biphasic boundary is the primary stage of solid‐state electrochemical reactions in which the Li concentrations and the valence state of Fe deviate significantly from the equilibrium end members. Complex interactions among Li ions and charges at the Fe sites have made understanding stability and transport properties of the intermediate domains difficult. Long‐range ordering at metastable intermediate eutectic composition of Li_2/3FePO₄ has now been discovered and its superstructure determined, which reflected predominant polaron crystallization at the Fe sites followed by Li⁺ redistribution to optimize the Li? Fe interactions.     

Synthesis of mono- and 1,3-diaminocalix[4]arenes via Ullmann-type amination and amidation of 1,3-bistriflate esters of calix[4]arenes

Practical methods are described for the preparation of monoamines 4 and 1,3-diamines 5, bearing one or two amino group(s) instead of the hydroxy group(s) at the 28-position or at both the 26- and 28-positions of p-tert-butylcalix[4]arene (1a) and p-tert-butylthiacalix[4]arene (1b), via the Ullmann-type amination or amidation. Thus, the copper-catalyzed or mediated amination of the 1,3-bistriflate ester (2a) of 1a with benzylamine affords either mono(benzylamino) triflate 7a or 1,3-bis(benzylamine) 8 in a high yield, depending on the reaction conditions. On the other hand, the 1,3-bistriflate ester (2b) of 1b resists disubstitution and produces, under stoichiometric conditions, mono(benzylamino) triflate 7b. The disubstitution of 2b is achieved by amidation with tosylamide, giving 1,3-bis(tosylamide) 17b. The hydrogenolysis of the benzylamino moiety of 7a, followed by the hydrolysis of the Tf moiety, affords monoamine 4a, while the hydrogenolysis of 8 affords 1,3-diamine 5a. The amino moiety of 7b can be deprotected under acidic conditions to give, after hydrolysis, monoamine 4b. The hydrolysis of 17b affords 1,3-diamine 5b. The overall yields of compounds 4a, 4b, 5a, and 5b are 72%, 45%, 78%, and 24%, respectively, based on commercially available compounds 1 and are much higher than the ones previously reported in the literature.     

Selective solvation effects in phase separation in aqueous mixtures

Selective solvation can be crucial in phase separation in polar binary mixtures (water–oil) with a small amount of hydrophilic ions or hydrophobic particles. They are preferentially attracted to one of the solvent components, leading to a number of intriguing effects coupled to phase separation. For example, if cations and anions interact differently with the two components, an electric double layer emerges at a liquid–liquid interface. The main aim of this paper is to show that a strongly hydrophilic (hydrophobic) solute induces precipitation of water-rich (oil-rich) domains above a critical solute density n_p outside the solvent coexistence curve.     

Critical exponent and critical blow-up for quasilinear parabolic equations

B ₂-groups are special (torsion-free) abelian Butler groups. The interest in this class of groups comes from representation theory. A particular functor, also called Butler functor, connects algebraic properties of the category of free abelian groups with (a few) distinguished subgroups with these Butler groups. This helps to understand Butler groups and caused lots of activities on Butler groups. Butler groups were originally defined for finite rank, however a homological connection discovered by Bican and Salce opened the investigation of Butler groups of infinite rank. Despite the fact that classifications of Butler groups are possible under restriction even for infinite rank (see a forthcoming paper by Files and Göbel [Mathematische Zeitschrift]), general structure theorems are impossible. This is supported by the following very special case of the Main Theorem of this paper, showing that any ring with a free additive group is an endomorphism ring of a Butler group. The result implies the existence of large indecomposable or of large superdecomposable Butler groups as well as the existence of counter-examples for Kaplansky’s test problems.     

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.     

Field Analysis of Pillbox Resonator with Radiation Loss Utilizing Finite-Element Beam Propagation Method Formulation

To deal with the radiation from an axially symmetric pillbox resonator embedded in LiNbO₃ substrate, an eigenvalue equation is derived from the expression of the finite-element beam propagation method in a cylindrical coordinate system. By solving the eigenvalue equation iteratively, the field distributions and the angular phase constants of the pillbox resonator are evaluated. The iterative scheme of solving the eigenvalue equation is an extension of our previous direct solution method which was described for a lossless case. The results obtained by this method are compared with those obtained analytically, and excellent agreement is found.