A B3LYP study on counterpoise-corrected geometry optimizations for hydrated complexes of [K(H2O)n] and [Na(H2O)n]

We report the basis set dependencies and the basis set superposition errors for the hydrated complexes of K⁺ and Na⁺ ions in relation to the recent studies of the KcsA potassium channel. The basis set superposition errors are estimated by the geometry optimizations at the counterpoise-corrected B3LYP level. The counterpoise optimizations alter the hydration distances by about 0.02–0.03 Å. The enthalpies and free energies for K⁺ + n(H₂O) → [K(H₂O)_n]⁺ and Na⁺ + n(H₂O) → [Na(H₂O)_n]⁺ (n = 1–6) are compared between the theoretical and experimental values. The results show that the addition of diffuse functions to K, Na, and O species are effective. However, it is also found that the counterpoise corrections using diffuse functions work so as to underestimate the free energies for the complexes with increasing the hydration number. The stabilization energies in aqueous solution are larger for a Na⁺ ion than for a K⁺ ion, suggesting the contributions of their dehydration processes to the ion selectivity of the KcsA potassium channel. The changes in coordination distance between the isolated [K(H₂O)₈]⁺ and the [K(H₂O)₈]⁺ in the KcsA potassium channel indicate the importance of hydrogen bondings between the first hydration shell and the outer hydration shells.     

泛函中的两个对偶定理

本文利用＊隔离定理,证明了Ｍａｚｕｒ定理和凸集的最佳逼近元存在性定理的对偶定理．     

On bifurcation points of a complex polynomial

Let be a polynomial of degree . Assume that the set there is a sequence s.t. and is finite. We prove that the set of generalized critical values of (hence in particular the set of bifurcation points of ) has at most points. Moreover, We also compute the set effectively.

     

仿紧集上的拟—似变分不等式

借助于文（１）中的结论研究了仿紧集上的拟－似变分不等式，统一和发展了文（２－６）中的相应结果。     

Dynamics of a family of transcendental meromorphic functions having rational Schwarzian derivative

In the present paper, a class F of critically finite transcendental meromorphic functions having rational Schwarzian derivative is introduced and the dynamics of functions in one parameter family is investigated. It is found that there exist two parameter values λ^∗=?(0)>0 and , where and is the real root of ?^′(x)=0, such that the Fatou sets of fλ(z) for λ=λ^∗ and λ=λ∗∗ contain parabolic domains. A computationally useful characterization of the Julia set of the function fλ(z) as the complement of the basin of attraction of an attracting real fixed point of fλ(z) is established and applied for the generation of the images of the Julia sets of fλ(z). Further, it is observed that the Julia set of fλ∈K explodes to whole complex plane for λ>λ∗∗. Finally, our results found in the present paper are compared with the recent results on dynamics of one parameter families λtanz, [R.L. Devaney, L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. 22 (4) (1989) 55-79; L. Keen, J. Kotus, Dynamics of the family λtan(z), Conform. Geom. Dynam. 1 (1997) 28-57; G.M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. Soc. 49 (1994) 281-295] and , λ>0 [G.P. Kapoor, M. Guru Prem Prasad, Dynamics of : The Julia set and bifurcation, Ergodic Theory Dynam. Systems 18 (1998) 1363-1383].     

r×t阶Kirkman三连系构造的一种方法

发现了 6n+3的高阶 Kirkman三连系构造方法 .阐明了 r× t阶 Kirkman三连系构造的基本理论 .给出了完全三分图的定义 ,并证明了关于 Kirkman三连系构造的命题 .介绍了 1 3 5阶 Kirkman三连系的构造过程 .     

Absolute fixed point sets for multi-valued maps

The notion of a multi-valued absolute fixed point set (MAFS) will be defined and characterized in the setting of set-valued maps with images containing multiple components.

     

Intramolecular basis set superposition errors

The effects of intramolecular basis set superposition errors are less well documented than the corresponding intermolecular effects. The intramolecular basis set superposition errors are examined, using the approach of Jensen, for several basis sets developed by Pople and his co‐workers, which are widely used in studies of larger molecules. Prototype calculations are reported for the ground state of the water molecule using both the matrix Hartree–Fock method and the many‐body perturbation expansion for the correlation energy taken through second order. A similar investigation is carried out for some of the correlation consistent basis sets published by Dunning and his collaborators. Specifically, the following aspects are investigated: (i) the magnitude of the intramolecular basis set superposition error, (ii) the nonadditivity of intramolecular counterpoise corrections when applied in a pairwise fashion, and (iii) the use of multiple “ghost” centers. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 282–292, 2001     

On Selfsimilar Jordan Curves on the Plane

We study the attractors of a finite system of planar contraction similarities S _j (j=1,...,n) satisfying the coupling condition: for a set {x ₀,...,x _n} of points and a binary vector (s ₁,...,s _n ), called the signature, the mapping S _j takes the pair {x ₀,x _n} either into the pair {x _j-1,x _j } (if s _j =0) or into the pair {x _j , x _j-1} (if s _j =1). We describe the situations in which the Jordan property of such attractor implies that the attractor has bounded turning, i.e., is a quasiconformal image of an interval of the real axis.     

HYPERFINITE TRANSVERSAL THEORY. II

We continue the investigation of validity of Hall's theorem in the case of the Loeb space of an internal, uniformly distributed, hyperfinite measure space initiated in1992 by the author. Some new classes of graphs are introduced for which the measure theoretic version of Hall's theorem still holds.