Topological degree of a family of convex functions and localization of nash equilibrium points

We define the degree of a family of convex functionsf _x, wherex , and prove that, if this degree is different from zero, one has at least one Nash equilibrium point in . This is a criterion of localization of such equilibrium points.     

The static dielectric constant of solutions of water inn-alcohols at 15, 25, 35, and 45°C

Dielectric constants (measured at 1 MHz) are reported for solutions of water (concentration range 0 to 0.2 mole fraction) in 1-propanol at 25°C, and in 1-butanol, 1-pentanol, and 1-hexanol at 15, 25, 35, and 45°C. These results, together with literature values for solutions of water in methanol, ethanol, 1-heptanol, and 1-octanol, show that water interacts with alcohols in at least two ways: (1) it can participate in the formation of dynamic hydrogen-bonded chains, thereby raising the polarizability; (2) it can form relatively stable structures such as H₂O(ROH)₄ which have zero net dipole moment and consequently diminish volume polarizability. For C _n H _2n+1 OH alcohols,n≥4, most of the initially added water forms complexes: consequently, addition of water to these alcohols lowers the dielectric constant.     

Density functional calculations of NMR chemical shifts and ESR g-tensors

An overview is given on recent advances of density functional theory (DFT) as applied to the calculation of nuclear magnetic resonance (NMR) chemical shifts and electron spin resonance (ESR) g-tensors. This is a new research area that has seen tremendous progress and success recently; we try to present some of these developments. DFT accounts for correlation effects efficiently. Therefore, it is the only first-principle method that can handle NMR calculations on large systems like transition-metal complexes. Relativistic effects become important for heavier element compounds; here we show how they can be accounted for. The ESR g-tensor is related conceptually to the NMR shielding, and results of g-tensor calculations are presented. DFT has been very successful in its application to magnetic properties, for metal complexes in particular. However, there are still certain shortcomings and limitations, e.g., in the exchange-correlation functional, that are discussed as well. Received: 24 October 1997 / Accepted: 19 December 1997     

Methylation of Chitosan with Iodomethane: Effect of Reaction Conditions on Chemoselectivity and Degree of Substitution

N,N,N‐trimethylchitosan (TMC) was prepared by reacting purified chitosan with iodomethane, in the presence of sodium hydroxide, water and sodium iodide, at room temperature. The reaction medium was N‐methyl‐2‐pyrrolidone. Different samples of TMC were obtained by adding to the reaction medium a fixed volume (5.5 mL) of aqueous NaOH solutions at different concentrations (15, 20, 30 and 40 wt.‐%) and carrying out the reactions for 9 or 24 h. The features observed in the ¹H and ¹³C NMR spectra of these chitosan derivatives, in respect of the chemical shift, number and relative intensity of the signals, depended strongly on the excess of NaOH and H₃CI added to the reaction medium, but when the lowest excess was employed, the characteristics of the derivative were not affected by the reaction time to the same extent. The average degree of quaternization of these N‐methylated derivatives of chitosan ranged from 10.5% to 44.8%, according to the reaction conditions. Increasing the excess of NaOH, in reactions carried out for 9 h, resulted in TMC samples with progressively higher content of trimethylated sites however, the reaction yields were correspondingly lower and O‐alkylation was favored in these cases.

¹H NMR spectrum of sample [A_X]^24h dissolved in D₂O (C_p = 10 g/L).     

The Mathematical Theory of Ito Diffusions on Hypersurfaces, with Applications to NMR Relaxation Problems

A mathematical framework for translational Brownian motion on hypersurfaces is presented, using an imbedding of the surface and Ito diffusions in the ambient space. This includes a survey of Ito calculus and differential geometry. Computational methods for time correlation functions relevant to spin relaxation studies on curved interfaces are given, and explicit calculations of time correlation functions and order parameters for a Rippled surface are presented.     

An investigation of heparinase immobilization

A systematic investigation of the parameters that affect the efficiency of immobilizing heparinase onto cyanogen bromide activated crosslinked 8% agarose beads was conducted. Two experimental measures, the “fraction bound” and the “fraction retained,” were used to monitor the coupling efficiency. The fraction bound is the portion of the total initial enzyme that is bound to the agarose gel. The fraction retained is the fraction of bound enzyme that is active. The product of the two measures indicates the coupling efficiency. The activity of the immobilized heparinase was measured under conditions free of both internal and external mass transfer limitations, and thus, the fraction retained represents the true immobilized enzyme activity. Increasing the degree of activation of the beads results in an increase in the fraction bound, the fraction retained, and consequently, the coupling efficiency. As the ratio of enzyme solution to gel volume increases from 1.5 to 2.2, the fraction bound remains constant but the fraction retained decreases (heparinase concentration; 0.15 mg/mL and degree of activation; 9.5 μmol of cyanate esters/g of gel). At volume ratios greater than 2.2, both the fraction bound and the fraction retained decline continuously. Changing the heparinase concentration in the coupling solution changes the coupling efficiency in a manner similar to that of the volume ratio change. When heparin is added during the coupling process, the fraction bound declines as the heparin concentration increases, whereas the fraction retained increases up to a heparin concentration of 12 mg/mL and decreases thereafter. When arginine, lysine, and glycine are used to block the unreacted cyanate ester groups after the coupling process, the immobilized heparinase shows different pH optima of 6.5, 6.9, and 7.2, respectively. Based upon these findings, a protocol to optimize heparinase immobilization is developed.     

Small-angle X-ray study of air-dried elastoidin using three- and one- dimensional correlation functions obtained from slit-smeared intensity

In the present paper an isotropic sample of air-dried elastoidin has been studied considering it to be a non-ideal two-phase densely packed system after the theories by Vonk [1]. The relevant important physical parameters obtained for the sample areE, the width of the transition layer,D, the average periodicity transverse to the layers, S/V, the specific inner surface ₁ and ₂, the volume fraction of two phases, i. e. matter and void,l ₁ andl ₂ the transversal lengths,l _r, the range of inhomogeneity,l _c, the length of coherence and 2E/D, the volume fraction of the transition layer. The values ofE obtained by two approches as given by Vonk [1] and Ruland [2] show but a small difference indicating the correctness of the analysis.     

CTMAB胶束体系中反离子缔合度的测定

对十六烷基三甲基淡化按（CTMAB）水胶束体系，不同方法测得其反离子缔合度产值不同【‘－＼且很高于由股团电化学理论所得q值0．67问．本文以离子选择性电极法测定CTMAB水胶束体系中反离子缩合度（仁0．65），并研究了外加盐和乙醇的影响·1实验部分1．工试剂和仪器十六烷基三甲基漠化技（CP）在无水乙醇中重结晶两次，纯化物经帕环法测其表面张力在CMC附近无最低声、·淡化钾（AR），水为。次蒸馏水，PXJIC型离子计，302型PB，电极（响应范围为PB，＝1．6～3．3，响应时间为5分钟）222型甘汞电极·1．2实验方法在CMCV／de的稀…     

Raman spectra and structural features of ClOF2 + cation in a solution of anhydrous HF

The Raman spectra of ClOF₂ ⁺ cation in solutions of anhydrous HF were studied. In the ClOF₂ ⁺HF₂ ⁻ and ClOF₂ ⁺BF₄ ⁻−HF systems, this cation exists as a pyramidal structure (C _s symmetry), while in the ClOF₂ ⁺AuF₆ ⁻−HF system, it exists as a planar structure (C _2v symmetry). Based on nonempirical calculations by the Hartree-Fock-Roothaan method, an explanation for the dependence of the structure of the ClOF₂ ⁺ cation on the nature of the anion was proposed. For the Cl−O bond vibrations, the correlation functions of vibrational and rotational relaxations were calculated, and the characteristic times of these processes were determined. The main contribution to the formation of the band contours corresponding to the above-mentioned modes is made by the vibrational dephasing. Translated fromIzvestiya Akademii Nauk, Seriya Khimicheskaya, No. 3, pp. 432–437, March, 1998.