Limiting Lagrangians: A primal approach

In this paper, we consider a convex program with either a finite or an infinite number of constraints and its formal Lagrangian dual. We show that either the primal program satisfies a general condition which implies there is no duality gap or that there is a nonzero vectord with the following properties: First, wheneverd is added to the objective function, where is a positive number not greater than one, the resulting program satisfies the general sufficient condition cited above for no duality gap. Second, the optimal value of this perturbed program is attained and tends to the optimal value of the original program as tends to zero. Third, the optimal solutions of the perturbed programs form a minimizing sequence of the original program. As a consequence of the above, we derive the limiting Lagrangian theory of Borwein, Duffin, and Jeroslow.The authors are indebted to an unknown referee who suggested the very short and elegant proofs of Lemma 2.3 and Theorem 2.3.This work was completed while the first author was a member of the College of Management, Georgia Institute of Technology, Atlanta, Georgia.     

Molecular structure and nanosecond optical limiting property of a typical silver cubane-like cluster containing tetraselenotungstate

A typical silver cubane-like heteroselenometallic cluster, (μ₃-WSe₄)Ag₃(PPh₃)₃Cl·0.5SePPh₃, was synthesized from reaction of [Et₄N]₂[WSe₄] and Ag(PPh₃)₂(NO₃) in CH₂Cl₂ solution and structurally characterized by X-ray crystallography. The compound crystallizes in the trigonal space group R3 with a=16.258(4), b=22.80(3) Å, V=5217(7) ų, and Z=3. The structure contains a strongly distorted cubane-like {WAg₃Se₃Cl} core. The coordination geometries of the W center and each Ag atom are tetrahedral. The mean W-Ag separation is 3.010(3) Å. Optical nonlinearity of this cubane-like cluster was studied. Large optical limiting effect with threshold of 0.45 J/cm² was observed with the laser pulses of 7 ns at 532 nm.     

Limiting Partial Molar Volumes of Electrolytes in 2-Methyl-2-Butanol <Emphasis Type="Bold">+</Emphasis> Water Mixtures at 298.15 K

Partial molar volumes at infinite dilution, V⁰₂, of alkali–metal halides (LiCl, NaCl KCl RbCl CsCl, NaBr, KBr, KI), tetra-n-alkylammonium bromides, R₄NBr (R=Me, Et, n-Pr, n-Bu, n-Pen), NaBPh₄, and Ph₄PCl have been determined in binary solvent mixtures of water with 2-methyl-2-butanol covering the water-rich region and the alcohol-rich region at 298.15 K. V⁰₂ for alkali–metal halides show relatively little dependence on the solvent composition. However, in the case of hydrophobic electrolytes the observed effects are more pronounced. A good linear dependence between V⁰₂(R₄NBr) and the molecular weight of the tetra-n-alkylammonium cation is found. Limiting single-ion volumes have been obtained using the assumption that V⁰(Ph₄P⁺)–V⁰(BPh^–₄)=2.0 cm³-mol^–1. The trends in the single-ion volumes are discussed in both solvent regions.     

Measurement of acid-catalyzed isomerization of unsaturated aldehyde-2,4-dinitrophenylhydrazone derivatives by high-performance liquid chromatography analysis

Aldehyde-2,4-dinitrophenylhydrazones exist as (E)- and (Z)-geometrical isomers, and adventitious isomerization during sample preparation can cause analytical errors. Purified alkenal-2,4-dinitrophenylhydrazone derivatives comprise only the (E)-isomer. However, partial isomerization to the (Z)-isomer occurs upon addition of acid to attain an equilibrium isomer ratio. The UV-visible spectral properties of the isomers differ; the (Z)-isomer exhibiting a 6-10 nm lower absorption maximum compared to the (E)-isomer. Alkenal-2,4-dinitrophenylhydrazones having a CC double bond at the 2- or 3-position of the alkenal exhibited similar absorption maxima with an equilibrium isomer ratio (0.035) that was much lower than those of other alkenals. The CC double bond at the 3-position migrates to a position of conjugation with the CN double bond during hydrazone synthesis to form a stabilized molecular structure. Alkenal-2,4-dinitrophenylhydrazones having a double bond at the 4-position or greater exhibited a similar absorption maxima equilibrium isomer ratio (0.14) to alkanal-2,4-dinitrophenylhydrazones. The quantitative analysis of carbonyl compounds in air or water using DNPH is usually conducted in the presence of an acid catalyst. Consequently, the solution of the direct extract prepared for HPLC or GC analysis contains both (E)- and (Z)-isomers.     

反相液相色谱中固定相和流动相对柱相比的贡献

依据柱相比的热力学定义和反相液相色谱中溶质的计量置换保留理论(the stoichiometric dispheement heory of solute for retention，SDT-R)，对反相液相色谱中固定相和流动相性质、温度对柱相比的影响进行了研究。结果表明：固定相的种类和配基的疏水性对柱相比影响较大，而流动相中有机溶剂的种类，特别是脂肪酸作为置换剂时，对柱相比的影响更大，而柱相比受温度的影响较小。此外，通过用27种小分子溶质对柱相比的测定，其logI和Z良好的线性关系，进一步证明柱相比是一个与溶质性质无关的常数。     

1,3-二烯基-1,1,3,3-四甲基二硅氧烷/苯乙烯共聚合的环化率与竞聚率

对１３０℃下１，３ 二烯基 １，１，３，３ 四甲基二硅氧烷（ＤＶＤＳ）／苯乙烯（Ｓｔ）的溶液环化共聚合动力学进行了研究．根据低转化率（＜１０％）的共聚物组成及环状结构比例，通过参数估计方法，求得环化率与竞聚率，分别为ｒ１＝０１０２、ｒ２＝３４８、ｒｃ＝００２１０、Ｋｃ＝３８１ｍｏｌ／Ｌ、Ｋ′ｃ＝０３８９ｍｏｌ／Ｌ，同时得到它们９５％置信概率的联合置信区间及其交互影响的联合置信区间．表明ＤＶＤＳ的反应活性较Ｓｔ低，其非环状结构自由基具有较强的分子内环化反应能力；环的位阻效应使环状结构ＤＶＤＳ自由基的活性低于其非环状结构自由基．     

Selective Oxidation of Methane to Methanol Using Molecular Oxygen on MoOx／（LaCoO3＋Co3O4） Catalysts

Comparatively high CH3OH selectivity (60.0%) and yield (6.7%) were obtained on MoOx/(LaCoO3 Co3O4) catalysts in selective oxidation of methane to methanol using molecular oxygen as oxidant. The interaction between MoOx and La-Co-oxide modified the molecular structure of molybdenum oxide and the ratio of O^-/O^2- on the catalyst surface, which controlled the catalytic performance of MoOx/(LaCoO3 Co3O4) catalysts.     

Transport of glycine in systems with cation-exchange membranes in hydrochloric acid solutions: Effect of a heterogeneous protonation reaction

The electrical mass transfer of cations in electromembrane systems (EMS) with an MK-40 cation-exchange membrane and glycine in aqueous solutions of hydrochloric acid is studied using the method of a rotating membrane disk. Limiting current densities and limiting steps of the transport of cations in such systems are determined. Shown is the possibility of an increase in the electrical mass transfer of glycine as a consequence of the occurrence of a heterogeneous reaction of protonation of its zwitterions. The effect of the membrane surface state on the kinetic regularities of transport of cations in EMS with glycine in solutions of hydrochloric acid is exposed.     

四氟乙烯-乙烯溶液沉淀聚合

本文编制了一个简易计算机程序,测定了氟氯烷溶液沉淀聚合体系中四氟乙烯-乙烯共聚竞聚率:40℃:γC_2H_4=0.156±0.001,γC_2F_4=0.022±0.01670℃:γC_2H_4=0.197±0.001,γC_2F_4=0.055±0.020在此聚合体系中,聚合温度和共聚物组成比对链的规整性都有明显影响。当乙烯在45～55mol%范围内时,共聚物的IR随组成比而变化,利用此规律可简便地测定共聚物的组成比。     

丙烯酸甲酯/4-丙烯酰胺基-2,2,6,6-四甲基哌啶光共聚合体系的竞聚率和序列分布研究

以丙烯酸甲酯(MA,M1)和4-丙烯酰胺基-2,2,6,6-四甲基哌啶(AATP,M2)溶液光共聚合体系为研究对象,采用1H-NMR手段测定了MA/AATP共聚物的组成,用Mayo-Lewis积分法和扩展Kelen-Tüdos方法计算的竞聚率分别为0.88相似文献