The relation between the number of leaves of a tree and its diameter

Czechoslovak Mathematical Journal - Let L(n, d) denote the minimum possible number of leaves in a tree of order n and diameter d. Lesniak (1975) gave the lower bound B(n,d) = ⌈2(n −...     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

茶碱对聚苯胺尿酸酶电极的生物电化学活性的抑制

用聚苯胺尿酸酶电极研究了茶碱对固定酶的影响，结果表明，茶碱对固定尿酶酶有明显的抑制作用，但这种抑制作用是可逆的，在茶碱的存在下，ＰＨ对酶电极的响应电流影响与无茶碱存在时不同；在０．２－０．５Ｖ之间，酶电极的响应电流随电位加而迅速升高，当电位进一步升高时，其变化速度减慢，茶碱使尿酶酶催化反应的活化能从无抑制剂存在的时的２９．９ｋＪ．ｍｏｌ＾－１提高到４７．８ｋＪ．ｍｏｌ＾－１，即抑制剂改变了尿酸催化     

System-oriented dispersion models of general-shaped electrophoresis microchannels

This paper presents a system-oriented model for analyzing the dispersion of electrophoretic transport of charged analyte molecules in a general-shaped microchannel, which is represented as a system of serially connected elemental channels of simple geometry. Parameterized analytical models that hold for analyte bands of virtually arbitrary initial shape are derived to describe analyte dispersion, including both the skew and broadening of the band, in elemental channels. These models are then integrated to describe dispersion in the general-shaped channel using appropriate parameters to represent interfaces of adjacent elements. This lumped-parameter system model offers orders-of-magnitude improvement in computational efficiency over full numerical simulations, and is verified by results from experiments and numerical simulations. The model is used to perform a systematic parametric study of serpentine channels consisting of a pair of complementary turn microchannels, and the results indicate that dispersion in a particular turn can contribute to either an increase or decrease of the overall band broadening. The efficiency and accuracy of the system model is further demonstrated by its application to general-shaped channels that occur in practice, including a serpentine channel with multiple complementary turns and a multi-turn spiral-shaped channel. The results indicate that our model is an accurate and efficient simulation tool useful for designing optimal electrophoretic separation microchips.     

Asymmetric synthesis of dihydro-1,3-dioxepines by Rh(ii)/Sm(iii) relay catalytic three-component tandem [4 + 3]-cycloaddition

Heterocycles have been widely used in organic synthesis, agrochemical, pharmaceutical and materials science industries. Catalytic three-component ylide formation/cycloaddition enables the assembly of complex heterocycles from simple starting materials in a highly efficient manner. However, asymmetric versions remain a yet-unsolved task. Here, we present a new bimetallic catalytic system for tackling this challenge. A combined system of Rh(ii) salt and chiral N,N′-dioxide–Sm(iii) complex was established for promoting the unprecedented tandem carbonyl ylide formation/asymmetric [4 + 3]-cycloaddition of aldehydes and α-diazoacetates with β,γ-unsaturated α-ketoesters smoothly, affording various chiral 4,5-dihydro-1,3-dioxepines in up to 97% yield, with 99% ee. The utility of the current method was demonstrated by conversion of products to optically active multi-substituted tetrahydrofuran derivatives. A possible reaction mechanism was provided to elucidate the origin of chiral induction based on experimental studies and X-ray structures of catalysts and products.

Catalytic asymmetric tandem carbonyl ylide formation/[4 + 3]-cycloaddition of β,γ-unsaturated α-ketoesters, aldehydes and α-diazoacetates was achieved by using a bimetallic rhodium(ii)/chiral N,N′-dioxide–Sm(iii) complex catalyst.