The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh L ² (X × Y) by sums _k=1 ⁿ f _k f _k, with a prescribed numbern of products of arbitrary functionsf _k L ² (X) andg _k L ² (Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL ² (X × Y).     

Ramsey partitions of integers and pair divisions

Ifk ₁ andk ₂ are positive integers, the partitionP = (₁,₂,..., _n ) ofk ₁+k ₂ is said to be a Ramsey partition for the pairk ₁,k ₂ if for any sublistL ofP, either there is a sublist ofL which sums tok ₁ or a sublist ofP –L which sums tok ₂. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork ₁,k ₂ having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork ₁ andk ₂.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok ₁k ₂. This can be done trivially usingk ₁+k ₂-1 cuts. However, every Ramsey partition ofk ₁+k ₂ also yields a fair division algorithm. This method yields fewer cuts except whenk ₁=1 andk ₂=1, 2 or 4.     

The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem

Summary The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL ²,H ¹ andL , provided that certain relationships hold between the frequency, mesh size and outer radius.     

Conditional solubility versus pO2− of cerium(III) oxide in molten equimolar NaCl+KCl at 727°C

Reactions between cerium trichloride and oxide ions were studied in NaCl+KCl (1/1) at 1000°K, by potentiometry with a calcia-stabilized zirconia membrane electrode. Titration curves clearly demonstrated the existence of soluble cerium oxychloride (CeO⁺) and precipitated cerium oxide (Ce₂O₃), with respective dissociation constants 10^?11 and 10^?30 (molality scale). The corresponding conditional solubility diagram {log S (Ce^III)=f(pO^2?)} is presented and discussed.     

Ein Neues,tricyclisches schwefelreiches Phosphan

A New Tricyclic Sulfur-rich Phosphane From the products of the reaction of P₄S₃ with tert-Butyliodide the compound (t-Bu(S)P)₂P₄S₃ was isolated in low yield. The compound is stable and has a structure related to the hydrocarbon brexane. By reaction with triphenylphosphin the compounds (t-BuP)(t-Bu(S)P)P₄S₃ and (t-BuP)₂P₄S₃ were obtained. The ³¹P-nmr spectra of all compounds were solved and used to determine the structure of the molecules. A complete set of ³¹P-nmr data is given.     

Synthesis of PNIPAAm-g-P4VP Microgel as Draw Agent in Forward Osmosis by RAFT Polymerization and Reverse Suspension Polymerization to Improve Water Flux

Microgels have unique and versatile properties allowing their use in forward osmosis areas as a draw agent. In this contribution, poly(4-vinylpyridine) (P4VP) was synthesized via RAFT polymerization and then grafted to a poly(N-Isopropylacrylamide) (PNIPAAm) crosslinking network by reverse suspension polymerization. P4VP was successfully obtained by the quasiliving polymerization with the result of nuclear magnetic resonance and gel permeation chromatography characterization. The particle size and particle size distribution of the PNIPAAm-g-P4VP microgels containing 0, 5, 10, 15 and 20 wt% P4VP were measured by means of a laser particle size analyzer. It was found that all the microgels were of micrometer scale and the particle size was increased with the P4VP load. Inter/intra-molecular-specific interactions, i.e., hydrogen bond interactions were then investigated by Fourier infrared spectroscopy. In addition, the water flux measurements showed that all the PNIPAAm-g-P4VP microgels can draw water more effectively than a blank PNIPAAm microgel. For the copolymer microgel incorporating 20 wt% P4VP, the water flux was measured to be 7.48 L∙m⁻²∙h⁻¹.     

P2X7 Receptors in Astrocytes: A Switch for Ischemic Tolerance

A sub-lethal ischemic episode (preconditioning [PC]) protects neurons against a subsequent lethal ischemic injury. This phenomenon is known as ischemic tolerance. PC itself does not cause brain damage, but affects glial responses, especially astrocytes, and transforms them into an ischemia-resistant phenotype. P2X7 receptors (P2X7Rs) in astrocytes play essential roles in PC. Although P2X7Rs trigger inflammatory and toxic responses, PC-induced P2X7Rs in astrocytes function as a switch to protect the brain against ischemia. In this review, we focus on P2X7Rs and summarize recent developments on how astrocytes control P2X7Rs and what molecular mechanisms they use to induce ischemic tolerance.     

In Silico Analysis of PORD Mutations on the 3D Structure of P450 Oxidoreductase

Cytochrome P450 oxidoreductase (POR) is a membrane-bound flavoprotein that helps in transferring electrons from its NADPH domain to all cytochrome P450 (CYP450) enzymes. Mutations in the POR gene could severely affect the metabolism of steroid hormones and the development of skeletal muscles, a condition known as Cytochrome P450 oxidoreductase deficiency (PORD). PORD is associated with clinical presentations of disorders of sex development, Antley and Bixler’s syndrome (ABS), as well as an abnormal steroid hormone profile. We have performed an in silico analysis of POR 3D X-ray protein crystal structure to study the effects of reported mutations on the POR enzyme structure. A total of 32 missense mutations were identified, from 170 PORD patients, and mapped on the 3D crystal structure of the POR enzyme. In addition, five of the missense mutations (R457H, A287P, D210G, Y181D and Y607C) were further selected for an in-depth in silico analysis to correlate the observed changes in POR protein structure with the clinical phenotypes observed in PORD patients. Overall, missense mutations found in the binding sites of POR cofactors could lead to a severe form of PORD, emphasizing the importance of POR cofactor binding domains in transferring electrons to the CYP450 enzyme family.     

SiH3+O(3P)反应机理的理论研究

利用abinitio方法对SiH3+O(3P)反应进行了理论研究,在MP2/6-311+G(d,p)水平上优化得到了反应途径上的反应物、中间体、过渡态和产物的几何构型,并在QCISD(T)/6-311+G(d,p)水平上进行单点能计算.计算结果表明,SiH3+O(3P)→IM1→TS3→IM2→TS8→HOSi+H2为主反应通道,其他可能存在的次要产物有HSiOH+H、H2SiO+H和HSiO+H2.HOSi、HSiO和HSiOH(cis)还可能进一步解离生成SiO.另外,计算结果对SiH4+O(3P)反应机理中存在的争议给出了可能的解释,认为Withnall等人在实验中观察到的产物HSiOH、H2SiO和SiO并不是SiH4+O(3P)反应的直接产物,而是来自副反应SiH3+O(3P).     

Computational Insight into Biotransformation Profiles of Organophosphorus Flame Retardants to Their Diester Metabolites by Cytochrome P450

Biotransformation of organophosphorus flame retardants (OPFRs) mediated by cytochrome P450 enzymes (CYPs) has a potential correlation with their toxicological effects on humans. In this work, we employed five typical OPFRs including tris(1,3-dichloro-2-propyl) phosphate (TDCIPP), tris(1-chloro-2-propyl) phosphate (TCIPP), tri(2-chloroethyl) phosphate (TCEP), triethyl phosphate (TEP), and 2-ethylhexyl diphenyl phosphate (EHDPHP), and performed density functional theory (DFT) calculations to clarify the CYP-catalyzed biotransformation of five OPFRs to their diester metabolites. The DFT results show that the reaction mechanism consists of Cα-hydroxylation and O-dealkylation steps, and the biotransformation activities of five OPFRs may follow the order of TCEP ≈ TEP ≈ EHDPHP > TCIPP > TDCIPP. We further performed molecular dynamics (MD) simulations to unravel the binding interactions of five OPFRs in the CYP3A4 isoform. Binding mode analyses demonstrate that CYP3A4-mediated metabolism of TDCIPP, TCIPP, TCEP, and TEP can produce the diester metabolites, while EHDPHP metabolism may generate para-hydroxyEHDPHP as the primary metabolite. Moreover, the EHDPHP and TDCIPP have higher binding potential to CYP3A4 than TCIPP, TCEP, and TEP. This work reports the biotransformation profiles and binding features of five OPFRs in CYP, which can provide meaningful clues for the further studies of the metabolic fates of OPFRs and toxicological effects associated with the relevant metabolites.