Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain

Letd(;z, t) be the smallest diameter of the arcs of a Jordan curve with endsz andt. Consider the rapidity of decreasing ofd(;)=sup{d(;z, t):z, t , ¦z–t¦} (as 0,0) as a measure of nicety of . Letg(x) (x0) be a continuous and nondecreasing function such thatg(x)x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(x))². LetH(x) be an arbitrary primitive of 1/h ^–1(x). Note that the functionH ^–1 x is positive and increasing on (–, +),H ^–1 0 asx– andH ^–1+ asx +. The following statement is proved in the paper.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 176–184, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

Mass shifting roles of negative kernel mass in density estimation

f(x) is a univariate density in C ⁴ with bounded support. For any n and sufficiently small kernel bandwidths, the symmetric appendage of any negative mass, –U, to any smooth unimodal symmetric kernel of order p=2 shifts expected estimator mass from regions where f(x)>0 to regions where f(x)<0. For large n, the mean automatic kernel adaptation induced by –U is analyzed in the simplest MISE reduction scenario: The symmetric appendage of –U to the uniform kernel K(x, X) over MISE-optimal bandwidths reduces MISE by shifting K(x, X) mass asymmetrically across the observation X in the direction of decreasing |f(x)|.     

Selective synthesis of isobutanol by means of the Guerbet reaction: Part 2. Reaction of methanol/ethanol and methanol/ethanol/n-propanol mixtures over copper based/MeONa catalytic systems

The synthesis of isobutanol via the Guerbet condensation between methanol and ethanol was studied by using sodium methoxide (MeONa) as soluble basic component and copper-based catalysts as heterogeneous dehydrogenating/hydrogenating metal species. The effect of the nature of the catalyst and the relative amount of its individual components with respect to the reacting alcohols as well as of temperature on productivity and selectivity of the process was investigated. The collected data indicated that the copper chromite/MeONa was more active than Cu-Raney/MeONa system. The reaction was shown to proceed with the formation only of n-propanol and isobutanol. Ethanol conversion up to 61% with selectivity to isobutanol up to 98.4% was obtained. The same catalytic systems were also employed in the reaction of the methanol/ethanol/n-propanol ternary mixture. Again copper chromite/MeONa resulted more active than the Cu-Raney/MeONa system. Finally, experiments were carried out on methanol/n-propanol mixtures in the presence of the copper chromite/MeONa catalytic system by recycling both the recovered solid copper component and the liquid reaction mixture for evidencing eventual copper leaching by MeONa. On the basis of the obtained results it was concluded that in the Guerbet reaction copper chromite works as heterogeneous catalyst.     

一阶导数光谱K系数法同时测定硝基酚钠三组分

研究和建立了同时测定硝基酚钠混合三组份的新方法——一阶导数光谱K系数法。确定了最佳实验条件和测定波长,方法简便、快速、准确。通过对动植物细胞赋活增进剂中邻硝基酚钠、对硝基酚钠、5-硝基愈创木酚钠三组份的直接测定,其相对误差小于2%,相对标准偏差小于3%,回收率为99～103%。     

倍数光度滴定法测定微量过氧化苯甲酰

联合应用BPO与KIO3(在KI存在下)的倍增反应和光度滴定数据的二阶导数法处理,使面粉中微量BPO测定的灵敏度较常规碘量法提高了200倍,方法的检出限可达0.1 mg.L-1。在面粉实样分析中,测定结果的RSD值在1.3%~2.8%之间,加标回收率在98.6%~102.2%之间。提出的方法所得测定结果与国标法(GB/T 18415-2001)所得结果一致。     

EFFECTS OF COUPLING AGENTS ON THE CRYSTALLIZATION BEHAVIOR OF PP/T-ZnOw COMPOSITES

The objectives of this paper are to understand the crystallization behavior of polypropylene(PP)composites with surface modified tetra-needle-shaped zinc oxide whisker(T-ZnOw).T-ZnOw was surface modified with different coupling agents,such as silane coupling agents(KH-550,KH-560)and titanate coupling agent(NDZ-105),in order to improve the compatibility between PP and T-ZnOw.DSC and POM were used to characterize the melt and crystallization behavior and the crystalline structures of the composites,respectively.The results show that the surface modified T-ZnOw acts as a nucleating agent of PP crystallization,depending on the coupling agent used for modification.KH-550 and KH-560 have more apparent role in improving the interfacial interaction than NDZ-105 and induce PP crystallization at higher temperature and with smaller spherulites size.The results also suggest that the crystallization behavior depends on not only the content of coupling agent,but also the content of the surface modified T-ZnOw used in the composites.     

Complete graph conjecture for inner-core electrons: homogeneous index case

The complete graph conjecture that encodes the inner-core electrons of atoms with principal quantum number n >or= 2 with complete graphs, and especially with odd complete graphs, is discussed. This conjecture is used to derive new values for the molecular connectivity and pseudoconnectivity basis indices of hydrogen-suppressed chemical pseudographs. For atoms with n = 2 the new values derived with this conjecture are coincident with the old ones. The modeling ability of the new homogeneous basis indices, and of the higher-order terms, is tested and compared with previous modeling studies, which are centered on basis indices that are either based on quantum concepts or partially based on this new conjecture for the inner-core electrons. Two similar algorithms have been proposed with this conjecture, and they parallel the two "quantum" algorithms put forward by molecular connectivity for atoms with n > 2. Nine properties of five classes of compounds have been tested: the molecular polarizabilities of a class of organic compounds, the dipole moment, molar refraction, boiling points, ionization energies, and parachor of a series of halomethanes, the lattice enthalpy of metal halides, the rates of hydrogen abstraction of chlorofluorocarbons, and the pED(50) of phenylalkylamines. The two tested algorithms based on the odd complete graph conjecture give rise to a highly interesting model of the nine properties, and three of them can even be modeled by the same set of basis indices. Interesting is the role of some basis indices all along the model.     

‘New mathematics’ and teaching

Inside the scientific world it is not always understood that the mood of mathematics, which is a product and a part of culture, can change with time. This is partly why many have been surprised by the coming of the so‐called new mathematics.

In the truly creative mathematical mind two opposite tendencies coexist: the logical and the imaginative. Apparently it seems that new mathematics can be reduced to a purely logical machinery. In fact it contains as much imaginative contributions as classical mathematics. But it is difficult to show simultaneously the logical sequence of propositions and the clumsy progression of research itself. Mathematical exposition does not always follow the ‘ most natural slopes’ of the mind. Unfamiliar presentations often give an impression of ‘ abstraction ‘, more familiar ones an impression of concreteness ‘.

So it appears that difficulties with new mathematics are mostly of psychological origin. Misuses of it can easily raise up intolerance reactions and emotional blocks. Perhaps insisting upon the fact that, here as elsewhere, it is important to be able to guess, to realize that intuition and imagination are essential, could help to make new mathematics better understood, more useful and more able to be considered as a unifing element among sciences.     

An anti‐diffusive volume of fluid method for interfacial fluid flows

A volume of fluid (VOF) method is developed combining a first‐order limited downwind scheme with higher order accurate schemes. The method is characterized by retaining a sharp fluid interface and a reduction in numerical diffusion near the interface, but avoids complicated geometrical reconstruction as occurs in most volume tracing algorithms. To demonstrate the accuracy and robustness of the method, a selection of numerical experiments are presented involving a pure advection problem, a water wave impact caused by a dam breaking and liquid sloshing in a partially filled tank. Copyright © 2011 John Wiley & Sons, Ltd.