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Abstract.In this paper, the author establishes an isoperimetric inequality on surfaces with conical singularities, and by using it, proves a Trüdinger inequality with best constant on such surfaces. The best constants of the Trüdinger inequality are also found for a class of "symmetric" singular matrices. IntroductionRecently Troyanov [1] systematically studied the surfaces with conical singularities, the singularities Picard considered. In order to prescribe Gaussian curvature on such surfaces, he proved the following Trüdinger inequality:(0.1) feb"2dA

Successful application of the Hadamard transform (HT) technique to gas chromatography/mass spectrometry (GC/MS) and liquid chromatography/mass spectrometry (LC/MS) is described. Novel sample injection devices were developed to achieve multiple sample injections in both GC and LC instruments. Air pressure was controlled by an electromagnetic valve in GC, while a syringe pump and Tee connector were employed for the injection device in LC. Two well-known, abused drugs, 3,4-methylenedioxy-N-methylamphetamine (MDMA) and N, N-dimethyltryptamine (DMT), were employed as model samples. Both of the injection devices permitted precise successive injections, resulting in clearly modulated chromatograms encoded by Hadamard matrices. After inverse Hadamard transformation of the encoded chromatogram, the signal-to-noise (S/N) ratios of the signals were substantially improved compared with those expected from theoretical values. The S/N ratios were enhanced approximately 10-fold in HT-GC/MS and 6.8 in HT-LC/MS, using the matrices of 1023 and 511, respectively. The HT-GC/MS was successfully applied to the determination of MDMA in the urine sample of a suspect.

Characteristic mass fragmentation of 20 phenethylamine/tryptamine standards were investigated and compared by means of matrix assisted laser desorption/time-of-flight mass spectrometry (MALDI/TOFM), gas chromatography-electron ionization-mass spectrometry (GC-EI/MS) and liquid chromatography-electrospray ionization/mass spectrometry (LC-ESI/MS) methods. As a result, three characteristic peaks ([M] + and fragments from the C β-C α bond breakage) were found to be unique and contained information useful in identifying 2C series compounds based on the GC-EI/MS method. We found that the protonated molecular ion ([M+H] +) and two types of fragments produced from the α-cleavage and β-cleavage processes were useful mass spectral information in the rapid screening and confirmation of phenethylamine and tryptamine derivatives when ESI/MS and MALDI/TOFMS methods were applied. This assay was successfully used to determine samples that contain illicit drugs.