Infinitely Many Critical Points of Non-Differentiable Functions and Applications to a Neumann-Type Problem Involving the p-Laplacian

For a family of functionals in a Banach space, which are possibly non-smooth and depend also on a positive real parameter, the existence of a sequence of critical points (according to Motreanu and Panagiotopoulos (“Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,” Nonconvex Optimization Applications, Vol. 29, Kluwer, Dordrecht, 1998, Chap. 3)) is established by mainly adapting a new technique due to Ricceri (2000, J. Comput. Appl. Math.113, 401-410). Two applications are then presented. Both of them treat the Neumann problem for an elliptic variational-hemivariational inequality with p-Laplacian.     

一类二阶常微分方程初值问题的无穷多解性

应用分离变量法,得到了一类二阶微分方程初值问题存在无穷多个非负解的充分必要条件,并给出了所有的无穷多个非负解．。     

Infinitely many solutions for some nonlinear scalar system of two elliptic equations

In this paper, we consider the elliptic system of two equations in H¹(RN)×H¹(RN):

     

Low sets without subsets of higher many‐one degree

Given a reducibility ?_r, we say that an infinite set A is r‐introimmune if A is not r‐reducible to any of its subsets B with |A\B| = ∞. We consider the many‐one reducibility ?_m and we prove the existence of a low₁ m‐introimmune set in Π⁰₁ and the existence of a low₁ bi‐m‐introimmune set.     

最小二乘法在液体粘滞系数实验中的应用

本文用多管落球法测得原始实验数据,运用最小二乘法进行处理,求得无限广延液体粘滞系数,并与直接作图法进行比较。     

Coexistence of antiferromagnetic and ferromagnetic phase for ferromagnetic Kondo lattice model

To study the proposed phase separations in doped manganites, we performed Monte-Carlo calculations for the ferromagnetic Kondo lattice model with strong Hund's coupling between conduction electrons and localized spins. For the practical calculations, we adopted a one dimensional lattice and treated the spins of the localized t_2g electrons semi-classically. A direct evidence of the phase separation is observed from a snapshot of the spatial dependence of localized spins. No indication of the canted or spiral phases is found in the results of simulations. Further, the calculated results of the spin structure factor in the phase separation region are well compared with recent experiments. Received: 1st September 1998 / Revised: 30 October 1998 / Accepted: 27 November 1998     

On certain correspondences among various coupled-cluster theories for closed-shell systems

In this paper certain correspondences have been shown among various formulations of coupled-cluster theories for many electron closed-shell systems. Specifically it is shown that (i) the energy functional using unitary ansatz of the form exp (T−T ⁺) is exactly sameorder by order inT with the size-consistent energy functional 〈ψ|H|ψ〉/〈ψ|ψ〉 recently obtained by us in coupled-cluster framework; (ii) in the framework of unitary ansatz of the form exp (T−T ⁺), both non-variational and variational approaches lead to identical equations upto any given order inT andT ⁺ inT∼T ₂ approximation; (iii) variational procedure using our size-consistent energy functional or using the functional obtained in the framework of unitary ansatz (as envisaged by Kutzelnigg) leads to energy in both cases, inT∼T ₂ approximation, for a total of quadratic powers inT andT ⁺, same as Cizek’s linearised coupled pair many electron theory energy; (iv) in case of practical calculation of the energy through the variational procedures using our size-consistent energy functional and the functional in unitary ansatz framework, there is a loss of upper bound.     

An asymptotic minimax risk bound for estimation of a linear functional relationship

We consider estimation of the parameter B in a multivariate linear functional relationship X_i=ξ_i+ξ_1i, Y_i=Bξ_i+ξ_2i, i=1,…,n, where the errors (ζ_1i^′, ζ_2i^′) are independent standard normal and (ξ_i, i ) is a sequence of unknown nonrandom vectors (incidental parameters). If there are no substantial a priori restrictions on the infinite sequence of incidental parameters then asymptotically the model is nonparametric but does not fit into common settings presupposing a parameter from a metric function space. A special result of the local asymptotic minimax type for the m.1.e. of B is proved. The accuracy of the normal approximation for the m.l.e. of order n^−1/2 is also established.     

Comparison of CEPA and CP-MET methods

The known CEPA variants CEPA (v) withv = 0,1,2,3 and two new ones withv = 4, 5 are compared both formally and for various numerical examples with CP-MET. The main conclusions are: 1. In those situations where both CP-MET and the CEPA variants are justified (i.e. for “good” closed shell states) the correlation energies obtained with the 7 different schemes differ very little (by something like ±2%), with CEPA (1) closest to CP-MET (difference usually a fraction of 1%) and CEPA (4) nearly as close; this is rather insensitive to whether one uses canonical or localized orbitals. Even CEPA (3) is not too far from CP-MET, which confirms an earlier suggestion of Kelly. 2. In those cases where one of the 7 schemes fails (e.g. due to near degeneracy as in covalent molecules at large internuclear distances) the other 6 usually fail as well, though CEPA (0) is then somewhat poorer than the other schemes. Then no longer CEPA (1) but rather CEPA (3) is closest to CP-MET and then all schemes converge much better in a localized representation. 3. CEPA (2) usually leads to best agreement with experiment since it simulates to some extent triple substitutions. In none of the studied examples does CP-MET show a significant superiority as compared to the other schemes. Possible improvements to extend the domain of applicability of these methods are discussed.