Continuity properties of the normal cone to the level sets of a quasiconvex function

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)^–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x _n)}, where {x _n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Optimality for set functions with values in ordered vector spaces

Let (X, , ) be a finite atomless measure space,L a convex subfamily of , andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:LY beC-convex andG:LZ beD-convex set functions. Consider the following optimization problem (P): minimizeF(), subject to L andG() _D . The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.This work was partially supported by National Science Council, Taipei, Taiwan. This paper was written while the first author was visiting at the University of Iowa, 1987-88.The authors would like to express their gratitude to the two anonymous referees for their valuable comments. Also, they would like to thank Professor P. L. Yu for his encouragement and suggestions which improved the material presented here considerably.     

纳米氧化铜的制备及常温脱硫效能研究

采用硝酸铜为原料、氢氧化钠溶液为沉淀剂、使用体积比为1∶1的乙醇-水溶液为分散剂的直接沉淀法制备，通过改变工艺条件得到了三种纳米氧化铜产品，平均粒径分别为25.63，10.74和7.57 nm。采用TG-DTA、FIR、XRD和TEM等对纳米氧化铜进行表征。对产品的常温脱硫活性进行了穿透试验，并与已开发的其他常温精脱硫产品进行了比较，结果表明纳米氧化铜应用于H₂S脱硫，可在常温条件下具有优异的脱硫活性，对H₂S的脱除精度可以达到0.05 mg·m^-3以下。经过优选，产品在3 000 h^-1空速下穿透硫容达到25.3%，高于同类型的其他脱硫剂产品。     

Vibrational assignment, <Emphasis Type="Italic">ab initio</Emphasis> calculation and normal coordinate analysis of 2,2′-biquinoline

The vibrational wavenumbers and the fundamental modes of 2,2′-biquinoline were obtained by density functional theory (DFT) with the B3LYP functional using the 6-31G(d,p) basis set. The calculated wavenumbers were scaled by a single factor of 0.965 to correct them for vibrational anharmonicity, but the force constants were overestimated. Normal coordinate analysis of the molecule was also carried out by using the force field of the quinoline molecule and the force field parameters of quinoline are shown to be transferable to 2,2′-biquinoline. The potential energy distribution associated with the normal modes is also given. The theoretical wavenumbers are found to be in good agreement with the experimental data.     

Synthesis and Structure of Ag_3PSe_4

The title compound Ag3PSe4 was synthesized by the reaction of Ag powder,P2Se5 and Se in a molar ratio of 1:1:1 at 500℃ and structurally characterized by X-ray crystallography.The crystal belongs to orthorhombic,space group Pmn21 with cell parameters:a=7.689(4),b=6.660(3),c=6.379(4)A,V=326.7(3)A^3,Z=2,Dc=6.816g/cm^3,Mr=670.42,F(000)=584,μ=31.302mm^-1,R=0.0606,wR=0.1289 and S=1.012,The 3-D structure can be regarded as constructed from the stacking of puckered Ag-P-Se honeycomb-like sheets along the c direction,in which the Ag,P and Se atoms are bonded to each other to form a chair-like six-membered ring,and the rings then build the sheets by sharing edges.     

On a characterization of the normal distribution by means of identically distributed linear forms

Let X₁, X₂,…, be independent, identically distributed random variables. Suppose that the linear forms L₁ = Σ_j=1^∞a_jX_j and L₂ = Σ_j=1^∞b_jX_j exist with probability one and are identically distributed; necessary and sufficient conditions assuring that X₁ is normally distributed are presented. The result is an extension of a theorem of Linnik (Ukrainian Math. J.5 (1953), 207–243, 247–290) concerning the case that the linear forms L₁ and L₂ have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.     

FOURIER TRANSFORM FAR INFRARED SPECTRA AND NORMAL COORDINATE ANALYSIS OF FIFTEEN RARE-EARTH OXALATE HYDRATES

The Fourier transform far infrared spectra of oxalate hydrates of whole rare-earth series except Pm and Sc are investigated in the range of 100-400 cm~(-1). The assignment of vibrational frequencies is made on the basis of normal coordinate analysis using our NORVIB program. The model used for the hghter elements of the lanthanide series (La to Eu) is based on the crystal structure of Nd_2(C_2O_4)_3. 10H_2O, in which each lanthanide ion is surrounded by nine O atoms, six from three oxalate ions and three from Water molecules.The model adopted for the heavier elements of the lanthanide series (Gd to Lu) and Ytterbium is based on the crystal structure of Yb_2(C_2O_4)_3· 6H_2O, in which each ion is surrounded by eight atoms, six from three oxalate ions and two from water molecules. The variation of the metal-ligand frequencies and force constants with the atomic numbers of lanthanides is plotted and discussed.     

二组分混合物的最小流化特性 Ⅰ.混合/分离状态与最小流化空隙率

分析了二组分混合物的３种混合／分离状态及在完全混合时发生连续相转变的极限组成。提出一个判别非等密度体系混合／分离状态的法则与确定混合物空隙率简易可行的随意松散堆积方法。     

STRUCTURING ＆ RHEOLOGY OF MOLTEN POLYMER／CLAY NANOCOMPOSITES

The evolution and the origin of “solid-like state” in molten polymer/clay nanocomposites are studied. Using polypropylene/clay hybrid (PPCH) with sufficient maleic anhydride modified PP (PP-MA) as compatibilizer, well exfoliation yet solid-like state was achieved after annealing in molten state. Comprehensive linear viscoelasticity and non-linear rheological behaviors together with WAXD and TEM are studied on PPCH at various dispersion stages focusing on time,temperature and deformation dependencies of the “solid-like” state in molten nanocomposites. Based on these, it is revealed that the solid-structure is developed gradually along with annealing through the stages of inter-layer expansion by PP-MA,the diffusion and association of exfoliated silicate platelets, the formation of band/chain structure and, finally, a percolated clay associated network, which is responsible for the melt rigidity or solid-like state. The network will be broken down by melt frozen/crystallization and weakened at large shear or strong flow and, even more surprisingly, may be disrupted by using trace amount of silane coupling agent which may block the edge interaction of platelets. The solid-like structure causes characteristic non-linear rheological behaviors, e.g. residual stress after step shear, abnormal huge stress overshoots in step flows and, most remarkably, the negative first normal stress functions in steady shear or step flows. The rheological and structural arguments challenge the existing models of strengthened entangled polymer network by tethered polymer chains connecting clay particles or by chains in confined melts or frictional interaction among tactoids. A scheme of percolated networking of associated clay platelets, which may in band form of edge connecting exfoliated platelets, is suggested to explain previous experimental results.