在代数多重网格法中的新插值公式

推导出代数多重网格法的一个新的插值公式。理论分析和数值计算表明这个公式很有效,且适用性强。推广了原代数多重网格法的应用范围,能够求解一些很病态的代数方程组。     

On a point of order

A method of using algebraic curves to obtain estimates of critical points accurate enough to identify them as simple algebraic numbers (if that is what they are) is discussed and illustrated with an application to the (q-state Potts model on the triangular lattice for cases of pure two-site interactions and pure three-site interactions. In the latter case the critical point is conjectured to be . In a similar conjecture for the critical percolation probability on thedirected square lattice,q _c ^1/2 (q _c +3)=2(q _c +p _c =1).     

A ‘tree’ for generation and identification of the colour symmetry point groups

A flow chart (inverted ‘tree’) for generating and identifying the 58 magnetic and 18 polychromatic point groups using a classification for the 32 generating crystallographic point groups is suggested. The idea of colour generator is explored for generating the colour symmetry point groups. The advantages in presenting the identification of colour groups through a tree are discussed.     

Schrödinger Operators with Random Sparse Potentials. Existence of Wave Operators

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) (No. 15340005) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject classifications (2000). primary, 34M35, secondary, 33E20.This revised version was published online in March 2005 with corrections to the cover date.     

On <Emphasis Type="Italic">osp</Emphasis>(<Emphasis Type="Italic">M</Emphasis>∣2<Emphasis Type="Italic">n</Emphasis>) Integrable Open Spin Chains

We consider open spin chains based on osp(M2n) Yangians and solve the reflection equations for some classes of reflection matrices, including the diagonal ones. Having then integrable open spin chains, we write the analytical Bethe Ansatz equations. More details and references can be found in D. Arnaudon et al.: Nucl. Phys B 668 (2003) 469 and 687 (2004) 257.     

Algebraic Formulation of Quantum Decoherence

An algebraic formalism for quantum decoherence in systems with continuous evolution spectrum is introduced. A certain subalgebra, dense in the characteristic algebra of the system, is defined in such a way that Riemann–Lebesgue theorem can be used to explain decoherence in a well defined final pointer basis.     

Studies on heteronuclear diatomic molecular dissociation energies using algebraic energy method

The algebraic energy method (AEM) is applied to the study of molecular dissociation energy $D_e$ for 11 heteronuclear diatomic electronic states: $a^3\Sigma^+$ state of NaK, $X^2\Sigma^+$ state of XeBr, $X^2\Sigma^+$ state of HgI, $X^1\Sigma^+$ state of LiH, $A^3\Pi(1)$ state of ICl, $X^1\Sigma^+$ state of CsH, $A(^3\Pi_1)$ and $B0^+(^3\Pi)$ states of ClF, $2^1\Pi$ state of KRb, $X^1\Sigma^+$ state of CO, and $c^3\Sigma^+$ state of NaK molecule. The results show that the values of $D_e$ computed by using the AEM are satisfactorily accurate compared with experimental ones. The AEM can serve as an economic and useful tool to generate a reliable $D_e$ within an allowed experimental error for the electronic states whose molecular dissociation energies are unavailable from the existing literature.     

Public perception of nanotechnology

While several studies on the public opinion of nanotechnology have pointed to a rather enthusiastic U.S. public, the public uptake of nanotechnology in Europe is more contained. The results of the Swiss publifocus on nanotechnology reveal a pragmatic attitude of citizens toward the emerging technologies, thus confirming what has been identified as a “balanced approach” in the NanoJury UK.

用节点变分的代数方法研究双原子体系的完全振动能谱和离解能

对于使用实验数据作为原数据进行的数值计算, 由于实验误差的普遍存在, 在数值计算过程中可能存在对实验误差的放大效应, 使得微小的实验误差对数值计算的结果产生明显影响. 因此本文通过在AM (algebraic method) 方法中加入用以抵消实验误差的微小变分项δE, 从而将AM改进为节点变分的代数方法VAM (variational algebraic method). 该方法具有更广泛的适用范围, 尤其对处理那些实验数据较少、 误差较大、 已知的实验振动能级远离体系离解能的双原子体系效果明显. 本文利用VAM方法研究了AM方法难以处理的5¹Π_u⁷Li₂, (6d)¹Δ_g Na₂, (7d)¹Δ_gNa₂ 和5¹∑⁺ NaK 等不同碱金属双原子分子的完全振动能谱与离解能, 不但得到了与实验数据精确相符的理论结果, 还正确地预言了许多由于实验条件与技术原因而未能测得的物理数据. 充分表明了VAM 方法的可行性与正确性. 此处对数值误差的分析和物理思考对其他精确的数值计算 或数值模拟研究也有积极的参考意义.     

关于伽罗瓦理论中几个问题的讨论

讨论了伽罗瓦理论中的几个问题，给出了伽办瓦群关于子群的陪集分解定理和实数域的真子域上的质数次既约多项式的可解性与其零点分布之间的关系，改正了原文献中的一些错误，给出了相关定理的正确形式。