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.     

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.     

Recyclable hydrophilic-hydrophobic micropatterns on glass for microarray applications

A novel method for fabricating recyclable hydrophilic-hydrophobic micropatterns on glass chips is presented. TiOx patterns (100-2000 microm) were sputtered on glass chips via a through-hole mask. The patterned chips were then vapor-coated with fluoroalkylsilane, for example, (heptadecafluoro-1,1,2,2-tetrahydrodecyl)triethoxysilane (FTES) to form a hydrophobic coating layer. The fluoroalkyl chain of FTES film on TiOx patterns was photocleaved under UV irradiation, exposing the fresh hydrophilic TiOx patterns. The resulting chip could be used multiple times by repeating the coating and photocleaving processes with negligible deterioration of the hydrophobic FTES film coated on glass. If desired, bare glass patterns could also be generated by removing the TiOx patterns with KOH. The patterned glass chips have been successfully used for microarray fabrication.     

Triterpenoids from Eugenia grandis: structure elucidation by NMR spectroscopy

A new pentacyclic triterpenoid, 2alpha,3beta-dihydroxylup-12-en-28-oic acid (1) and a rarely encountered pentacyclic triterpenoid, 3beta-hydroxylup-12-en-28-oic acid (2), together with five known compounds, friedelin (3), 3beta-friedelinol (4), betulinic acid (5), oleanolic acid (6) and beta-sitosterol (7) were isolated from the chloroform extract of stem bark of Eugenia grandis (Syn: Syzygium grande). The structure and stereochemistry of the new compound (1) and the rarely encountered compound (2) were established by 1D and 2D NMR spectroscopic techniques. All the above isolated compounds from this plant are reported for the first time.     

Determination of ochratoxin A at parts-per-trillion levels in cereal products by immunoaffinity column cleanup and high-performance liquid chromatography/mass spectrometry

Ochratoxin A (OTA) is a toxic and potentially carcinogenic fungal toxin found in a variety of food commodities. A new sensitive method has been developed to quantify OTA in cereal products by reversed-phase liquid chromatography (LC) with mass spectrometric (MS) detection. Ochratoxin B was used as the internal standard. OTA was extracted from cereal products with acetonitrile-water, and the extract was diluted with a buffer; the diluted extract was cleaned up on an immunoaffinity column before LC/MS analysis. Two multiple-reaction monitoring transitions were used, one for quantification of OTA and one for confirmation of identity. The method was shown to be highly sensitive, with a low decision limit (CCalpha) of 0.012 microg/kg and a detection capability (CCbeta) of 0.021 microg/kg. Within-laboratory repeatability coefficient of variation values were 7.1, 3.7, and 3.1%, and the corresponding recoveries were 104, 106, and 103% for rice samples fortified with OTA at 0.05, 0.10, and 0.15 microg/kg, respectively. Method validation was performed according to the criteria of European Commission Decision 2002/657/EC. All criteria as presented in the Commission Decision were fulfilled. This method is the first fully validated method using immunoaffinity chromatography for cleanup and MS for detection in the analysis of cereals for OTA. The method was also successfully applied to cereal-derived products. The analytical results for determination of the OTA content of cereal products commercially available in Hong Kong are also reported.