Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

Ultracompact and fabrication-tolerant integrated polarization splitter

Design and fabrication of a 2×2 two-mode interference (TMI) coupler based on-chip polarization splitter is presented. By changing the angle between the access waveguides, one can tune the effective TMI length for the mode with less optical confinement (transverse magnetic, TM) to coincide with the target TMI length for a desired transmission of the mode with higher optical confinement (transverse electric, TE). The fabricated 0.94?μm long 2×2 TMI splits the input power into TM (bar) and TE (cross) outputs with splitting ratio over 15?dB over 50?nm bandwidth. Fabrication tolerance analysis shows that the device is tolerant to fabrication errors as large as 60?nm.     

Iterative in situ click chemistry assembles a branched capture agent and allosteric inhibitor for Akt1

We describe the use of iterative in situ click chemistry to design an Akt-specific branched peptide triligand that is a drop-in replacement for monoclonal antibodies in multiple biochemical assays. Each peptide module in the branched structure makes unique contributions to affinity and/or specificity resulting in a 200 nM affinity ligand that efficiently immunoprecipitates Akt from cancer cell lysates and labels Akt in fixed cells. Our use of a small molecule to preinhibit Akt prior to screening resulted in low micromolar inhibitory potency and an allosteric mode of inhibition, which is evidenced through a series of competitive enzyme kinetic assays. To demonstrate the efficiency and selectivity of the protein-templated in situ click reaction, we developed a novel QPCR-based methodology that enabled a quantitative assessment of its yield. These results point to the potential for iterative in situ click chemistry to generate potent, synthetically accessible antibody replacements with novel inhibitory properties.     

薄层色谱法鉴定APC药片组分的实验条件优化

对薄层色谱法鉴定复方乙酰水杨酸(APC)药片组分的展开剂进行了探索实验,用石油醚和乙酸乙酯替换了乙醚和二氯甲烷。新的展开剂对药片组分的分离更容易观察,并且比原展开剂挥发性低、毒性小,既提高了教学效果又减少了实验室污染。     

印度大学自主招生考试一例——印度理工学院招生考试化学试题评介

印度理工学院的联合招生考试是其招生的主要凭据,也是其他很多大学招生的重要参考标准。通过分析2011年印度理工学院招生考试的化学试题及其特点,为我国高考和高校自主招生考试提供参考。     

关于次甲基蓝水溶液吸附法测定颗粒活性炭比表面积实验的探讨

次甲基蓝水溶液吸附法测定颗粒活性炭的比表面积是大学物理化学实验教学中一个重要的基础实验.本文对次甲基蓝吸附取向、端基吸附的投影面积、次甲基蓝标准溶液的配制以及吸附时间和吸光度的关系等几个方面进行了细致的讨论,并针对存在的问题提出了改进方法.     

A highly specific BODIPY-based fluorescent probe for the detection of hypochlorous acid

A fluorescent probe, HKOCl-1, has been successfully developed for the detection of hypochlorous acid on the basis of a specific reaction with p-methoxyphenol. The formation of HOCl has been successfully detected not only in an abiotic system but also in an enzymatic system (myeloperoxidase/H2O2/Cl(-) system) and in living macrophage cells upon stimulation. This new probe might be used as an efficient tool for probing the roles HOCl plays in biological systems.