Rank 72 high minimum norm lattices

Given a polarization of an even unimodular lattice and integer k?1, we define a family of unimodular lattices L(M,N,k). Of special interest are certain L(M,N,3) of rank 72. Their minimum norms lie in {4,6,8}. Norms 4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a L(M,N,3). Our method constructs such L(M,N,k) in dimensions 96, 120 and 128 with minimum norms 8.     

Tricyclic graphs with maximum Merrifield-Simmons index

It is well known that the graph invariant, ‘the Merrifield-Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield-Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16-21]. The sharp upper and lower bounds for the Merrifield-Simmons indices of unicyclic graphs are characterized by Pedersen and Vestergaard [A.S. Pedersen, P.D. Vestergaard, The number of independent sets in unicyclic graphs, Discrete Appl. Math. 152 (2005) 246-256]. The sharp upper bound for the Merrifield-Simmons index of bicyclic graphs is obtained by Deng, Chen and Zhang [H. Deng, S. Chen, J. Zhang, The Merrifield-Simmons index in (n,n+1)-graphs, J. Math. Chem. 43 (1) (2008) 75-91]. The sharp lower bound for the Merrifield-Simmons index of bicyclic graphs is determined by Jing and Li [W. Jing, S. Li, The number of independent sets in bicyclic graphs, Ars Combin, 2008 (in press)]. In this paper, we will consider the tricyclic graph, i.e., a connected graph with cyclomatic number 3. The tricyclic graph with n vertices having maximum Merrifield-Simmons index is determined.     

Textbook representations of radian angle measure: The need to build on the quantitative view of angle

Empirical studies of angle, angle measure, and radian angle measure have primarily focused on documenting students' and teachers' conceptions. Representations of angle measure have the potential to inform and build on these conceptions. To gain insight into the concept of radian angle measure from a curricular perspective, I investigated the representations of radian angle measure in eight widely used U.S. high school mathematics textbooks. The data for this study are the 20 figures and the 8 definitions of radian angle measure in the textbooks. Inductive content analysis was used to identify the frequency of observable characteristics in the representations. Textbook figures of radian angle measure most frequently focus on a one-radian angle. Textbook figures also demonstrated radian angle measures in terms of π, in standard position, even when the Cartesian-coordinate plane was not in the figure. The frequently used characteristics of radian representations could inadvertently reinforce a qualitative concept of angle, instead of fostering the quantitative attributes of radian angle measure. Moving forward, analyzing conceptions that are evoked by the frequently used representations of radian angle measure could build a link between evidence from this study and studies of learning.     

Maximality of the signless Laplacian energy

We study the problem of determining the graph with $n$ vertices having largest signless Laplacian energy. We conjecture it is the complete split graph whose independent set has (roughly) $2n∕3$ vertices. We show that the conjecture is true for several classes of graphs. In particular, the conjecture holds for the set of all complete split graphs of order $n$, for trees, for unicyclic and bicyclic graphs. We also give conditions on the number of edges, number of cycles and number of small eigenvalues so the graph satisfies the conjecture.     

A New Green Thermoplastic Polymer with Improved Hydrophobic Character

Summary: A green thermoplastic polymer based on wheat flour was modified by the addition of a natural crosslinker genipin. Films of the polymer modified with different composition of genipin were prepared by extrusion. Free surface energy using contact angle method, moisture absorption test and hardness test were used to characterize the products. From moisture absorption and contact angle measurements, it was seen that for genipin amounts higher than 0.2% w/w, a hydrophobic character is achieved. Results obtained indicate enhancement in hydrophobic properties of the films.     

不连续泛函微分方程周期边值问题解的单调迭代技巧(英文)

文章通过运用上下解的方法证明了一类带有周期边值条件的不连续泛函微分方程解的存在性.并且采用不连续的上下解显著扩大了上下解的选择范围.进而,通过构造具有一致收敛性的上下解的单调迭代序列,得到了该周期边值问题的极值解,即最大、小解.     

Surface structure and properties of poly-(ethylene terephthalate) hydrolyzed by alkali and cutinase

This study was aimed at comparatively investigating the hydrolysis of crystalline and amorphous poly-(ethylene terephthalate) films by alkali and cutinase. Changes of surface properties were investigated by FTIR spectroscopy (ATR mode). The A₁₃₄₁/A₁₄₁₀ and I₁₁₂₀/I₁₁₀₀ absorbance ratios, and the full width at half maximum of the carbonyl stretching band (FWHM₁₇₁₅) were used to evaluate the polymer crystallinity and its changes upon hydrolysis. The effect of different treatments on chain orientation was evaluated by calculating R ratios of appropriate bands. The spectroscopic indexes showed that both alkali and enzyme treatments induced structural and conformational rearrangements with a consequent increase in crystallinity in both amorphous and crystalline films. The crystalline PET film was modified more strongly by alkali than by cutinase, while the opposite occurred for the amorphous one. The trend of the water contact angle (WCA) clearly indicates that alkali is more effective than cutinase in enhancing hydrophilicity of PET films and that the effect is stronger on amorphous than on crystalline films. The values of WCA correlate well with the FTIR indexes calculated from the spectra of hydrolyzed crystalline PET films. The mechanism of the surface hydrolysis of PET by alkali and cutinase is discussed.     

Effect of L64 on the Phase Behavior of 1-Dodecyl-3-methylimidazolium Chloride/Water System

Phase behavior of ternary system involving surfactant‐like ionic liquid 1‐dodecyl‐3‐methylimidazolium chloride ([C₁₂mim]Cl), water, and nonionic surfactant PEO‐PPO‐PEO block copolymer (Pluronic L64) is investigated at 25°C. Hexagonal (H₁) and lamellar liquid crystal phase (Lα) are found in [C₁₂mim]Cl/H₂O/L64 system by using polarized optical microscopy (POM), small‐angle X‐ray scattering (SAXS) techniques and ²H NMR spectra. The phase structure (H₁ phase), which is formed in [C₁₂mim]Cl/H₂O binary system, is not changed when L64 with a low concentration is added. However, phase transitions will occur from hexagonal to multiphases of H₁ and cubic phases (C), then to Lα+C phases with constant [C₁₂mim]Cl/H₂O ratio and increasing L64 concentration. Moreover, at given L64 (5%, 20%) concentration, the lattice parameter of H₁ or Lα phase decreases with increasing [C₁₂mim]Cl/H₂O ratio. Fourier transform infrared (FTIR) spectra indicate that the H‐bonded network comprising an imidazolium ring, chloride ion and water formed in [C₁₂mim]Cl/H₂O binary system is disrupted upon addition of L64. This is helpful to the phase transition, due to the decreasing of interfacial curvature induced by dehydration of hydrated layer after the addition of PEO block of L64.     

Functional limit theorems for Galton–Watson processes with very active immigration

We prove weak convergence on the Skorokhod space of Galton–Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes (1979).     

Design Criterion for Micro-Machine Scanner Optical Switch Based on Beam Steering Angle Error Analysis

The micro-machine scanner optical switch is gaining attention for its ability to switch over 100 ports. A beam steering error for devices using tilted mirrors or a moving lens is analyzed and schemes to overcome the limit are reported. The channel number is limited by the achievable beam steering error. A device with several hundred ports is possible with the beam steering error under 1#x0025;. Optimized focal length or fiber mode divergence angle is beneficial for increasing the channel number. To minimize the thermal expansion effect, large focal length is required indicating need for a compensation scheme to attain larger channel number. A new type of scanning optics is proposed to overcome the limit set by the beam steering error, and a new switch network described can further expand the channel number.This paper was originally presented at the 2nd International Conference on Optical Design and Fabrication, ODF2000 which was held on November 15-17, 2000 at the International Conference Center, Tokyo, Waseda University, Japan.