Global convergence of modified multiplicative updates for nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is the problem of approximating a given nonnegative matrix by the product of two nonnegative matrices. The multiplicative updates proposed by Lee and Seung are widely used as efficient computational methods for NMF. However, the global convergence of these updates is not formally guaranteed because they are not defined for all pairs of nonnegative matrices. In this paper, we consider slightly modified versions of the original multiplicative updates and study their global convergence properties. The only difference between the modified updates and the original ones is that the former do not allow variables to take values less than a user-specified positive constant. Using Zangwill’s global convergence theorem, we prove that any sequence of solutions generated by either of those modified updates has at least one convergent subsequence and the limit of any convergent subsequence is a stationary point of the corresponding optimization problem. Furthermore, we propose algorithms based on the modified updates that always stop within a finite number of iterations.     

Two- and Three-dimensional Frameworks with (6,3) and (10,3)-a Topology from Self-assembly of Three-connecting Organic Ligands with Cadmium(II) and Silver(I) Salts

Assembly of three-connecting ligands 1,3,5-tris(1-imidazolyl)benzene (tib) and 1,3,5-tris(imidazol-1-ylmethyl)-2,4,6-trimethylbenzene (titmb) with cadmium(II) and silver(I) salts provide new metal-organic frameworks, [Cd(tib)₂](NO₃)₂·4H₂O (1), [Ag(tib)(PPh₃)](CF₃SO₃) (2) and [Ag(titmb)(PPh₃)](CF₃SO₃)·1.5H₂O (3) (PPh₃=triphenylphosphine). Single-crystal X-ray diffraction studies reveal that complexes 1 and 3 are two-dimensional honeycomb networks, while complex 2 is a noninterpenetrated three-dimensional architecture with (10,3)-a topology. The results indicate that the nature (structure and flexibility) of the organic ligands and the bulky auxiliary ligand have great impact on the assembly and structure of metal-organic frameworks. The photoluminescent properties of the synthesized complexes were studied in the solid state at room temperature.     

Boc-L-甲基苯丙氨酸的合成与拆分

利用Boc-2-氨基丙二酸二乙酯和甲基苄溴为原料, 合成邻位、间位、对位甲基取代的Boc-苯丙氨酸乙酯, 经枯草杆菌蛋白酶拆分得到对应的Boc-L-甲基苯丙氨酸. 通过红外光谱、核磁共振、质谱及旋光度分析对3种物质的结构进行了表征.     

Ethylaluminum oxide catalysts from Et2AlOLi–Et2AlCl binary system in relation to species of AlEt3–water catalyst

Equimolar reaction of Et₂AlOLi and Et₂AlCl gave Et₂AlOAlEt₂. The catalyst behavior for polymerization of acetaldehyde, propylene oxide, and epichlorohydrin was compared with that of the AlEt₃–H₂O (1:0.5) catalyst system. The thermal disproportionation product of Et₂AlOAlEt₂ derived from Et₂AlOLi–Et₂AlCl had the structure, ? (EtAlO)_n? , and it showed catalyst behavior quite similar to that of the product obtained by the same treatment of AlEt₃–H₂O (1:0.5). These ethylaluminum oxides can be regarded as species predominating in AlEt₃–H₂O (1:0.5) and AlEt₃–H₂O (1:1), respectively. Stereospecific or high molecular weight polymerizations of these species were investigated.     

Optically active cyclohexene derivative as a new antisepsis agent: an efficient synthesis of ethyl (6R)-6-[N-(2-chloro-4-fluorophenyl)sulfamoyl]cyclohex-1-ene-1-carboxylate (TAK-242)

Two new synthetic methods were established for the efficient synthesis of optically active cyclohexene antisepsis agent, ethyl (6R)-6-[N-(2-chloro-4-fluorophenyl)sulfamoyl]cyclohex-1-ene-1-carboxylate [(R)-1: TAK-242)]. The first method involved recrystallization from methanol of the diastereomeric mixture (6RS,1'R)-7, obtained by esterification of carboxylic acid 3 with (S)-1-(4-nitrophenyl)ethanol [(S)-5)] to give the desired isomer (6R,1'R)-7 with 99% de in 32% yield. Subsequent catalytic hydrogenolysis and esterification gave (R)-1 with >99% ee. The second method employed enantioselective hydrolysis of acetoxymethyl ester 9a (prepared by alkylation of 3 with bromomethyl acetate) with Lipase PS-D to give the eutomeric enantiomer (R)-9a with excellent enantioselectivity (>99% ee) and high yield (48%). The desired (R)-1 was then obtained by transesterification with ethanol in the presence of concentrated sulfuric acid without loss of ee. Of these, the procedure employing enzymatic kinetic resolution using Lipase PS-D is the more efficient and practical preparation of (R)-1.     

Metal-organic architectures of silver(I), cadmium(II), and copper(II) with a flexible tricarboxylate ligand

Three novel metal-organic architectures, [Ag3(bta)].1.5H2O (1), [Cd3(bta)2(H2O)7].5H2O (2), and [Cu11(bta)6(Hbta)2(H2O)10].29H2O (3), were obtained by reactions of the corresponding metal salts with a flexible tripodal ligand, benzene-1,3,5-triacetic acid (H3bta), and their structures were determined by single-crystal X-ray diffraction studies. The results revealed that, in complexes 1 and 2, the carboxylate groups of the bta3- ligand adopted varied coordination modes to link metal atoms and further to form three-dimensional structures with open channels occupied by water molecules, while in complex 3, for the first time, the flexible H3bta acted as a secondary building unit to generate a novel nanometer-sized metallocage, which is composed of a Cu(II) paddle wheel (square secondary building units) and bta3-/Hbta2- organic links (triangular secondary building units). The photoluminescence properties of complexes 1 and 2 were investigated, and the results showed that 2 exhibited photoluminescence in the solid state at room temperature.     

A novel generation of indole-2,3-quinodimethanes

[reaction: see text] A novel and efficient procedure for the generation of the reactive indole-2,3-quinodimethane intermediates from the allenylanilines is described. The indole-2,3-quinodimethane intermediates were captured by several dienophiles to afford the corresponding tetrahydro- and dihydrocarbazole derivatives. This method is significantly different from the previously reported ones, which involve the 1,4-elimination or its related reactions of the indole derivatives that possess suitable substituents at both the C-2 and C-3 positions.