Marginal electron density and density-difference functions

Summary For visual analysis of the density reorganization and distortion, the one-dimensional cut (x, y ₀,z ₀) and the two-dimensional cut (x, y, z ₀) of the three-dimensional electron density difference function (x, y, z) are frequently employed. However, these cut functions do not satisfy any sum rules in contrast to the original difference function (x, y, z). To avoid this difficulty, the use of the marginal electron density functions _x (x) and _xy (x, y) and their difference functions _x (x) and _xy (x, y) is proposed. The marginal densities are condensation of the three-dimensional density onto a particular plane or line of our interest, and they satisfy the sum rule (i.e., the conservation of the number of electrons) exactly. Some basic properties of the marginal electron density are clarified for typical diatomic molecular orbitals. An illustrative application is given for the bonding and antibonding processes in the H₂ system.     

Stability Relationships Among the Gibbs Sampler and its Subchains

The use of Gibbs samplers driven by improper posteriors has been a controversial issue in the statistics literature over the last few years. It has recently been demonstrated that it is possible to make valid statistical inferences through such Gibbs samplers. Furthermore, theoretical and empirical evidence has been given to support the idea that there are actually computational advantages to using these nonpositive recurrent Markov chains rather than more standard positive recurrent chains. These results provide motivation for a general study of the behavior of the Gibbs Markov chain when it is not positive recurrent. This article concerns stability relationships among the two-variable Gibbs sampler and its subchains. We show that these three Markov chains always share the same stability; that is, they are either all positive recurrent, all null recurrent, or all transient. In addition, we establish general results concerning the ways in which positive recurrent Markov chains can arise from null recurrent and transient Gibbs chains. Six examples of varying complexity are used to illustrate the results.     

高阶控制增稳系统等效拟配参数时域在线辨识方法

现代控制增稳飞机是高达几十阶的复杂系统,将其降阶为特定形式的低阶等效系统才能与已有飞行品质规范进行比较,进而预测和评价飞机飞行品质.从时域角度出发,提出了一种高阶控制增稳系统等效拟配参数在线辨识方法.分析了时域等效系统拟配的基本方法,提出了等效系统实时在线拟配的系统框架思路,基于Tustin变换法推导了由高阶系统时域响应获取四阶等效系统参数的数学公式,应用最小二乘法在MATLAB中实现了算法编程,并用试飞数据验证了所提方法的可行性.计算结果表明,所提方法可以快速准确地实现高阶控制增稳系统的等效系统参数辨识.     

A fast Petrov–Galerkin method for solving the generalized airfoil equation

In this paper we develop a fast Petrov–Galerkin method for solving the generalized airfoil equation using the Chebyshev polynomials. The conventional method for solving this equation leads to a linear system with a dense coefficient matrix. When the order of the linear system is large, the computational complexity for solving the corresponding linear system is huge. For this we propose the matrix truncation strategy, which compresses the dense coefficient matrix into a sparse matrix. We prove that the truncated method preserves the optimal order of the approximate solution for the conventional method. Moreover, we solve the truncated equation using the multilevel augmentation method. The computational complexity for solving this truncated linear system is estimated to be linear up to a logarithmic factor.     

The Stable Class of the Augmentation Ideal

In [F.E.A. Johnson, Stable Modules and the D(2)-Problem, LMS Lecture Notes In Mathematics, vol. 301, CUP (2003)], for finite groups G, we gave a parametrization of the stable class of the augmentation ideal of Z[G] in terms of stably free modules. Whilst the details of this parametrization break down immediately for infinite groups, nevertheless one may hope to find parallel arguments for restricted classes of infinite groups. Subject to the restriction that Ext¹(Z, Z[G]) = 0, we parametrize the minimal level in Ω₁(Z) by means of stably free modules and give a lower estimate for the size of Ω₁(Z).     

Rate optimal estimation with the integration method in the presence of many covariates

For multivariate regressors, integrating the Nadaraya–Watson regression smoother produces estimators of the lower-dimensional marginal components that are asymptotically normally distributed, at the optimal rate of convergence. Some heuristics, based on consistency of the pilot estimator, suggested that the estimator would not converge at the optimal rate of convergence in the presence of more than four covariates. This paper shows first that marginal integration with its internally normalized counterpart leads to rate-optimal estimators of the marginal components. We introduce the necessary modifications and give central limit theorems. Then, it is shown that the method apply also to more general models, in particular we discuss feasible estimation of partial linear models. The proofs reveal that the pilot estimator shall over-smooth the variables to be integrated, and, that the resulting estimator is itself a lower-dimensional regression smoother. Hence, finite sample properties of the estimator are comparable to those of low-dimensional nonparametric regression. Further advantages when starting with the internally normalized pilot estimator are its computational attractiveness and better performance (compared to its classical counterpart) when the covatiates are correlated and nonuniformly distributed. Simulation studies underline the excellent performance in comparison with so far known methods.     

Test sets of integer programs

This article is a survey about recent developments in the area of test sets of families of linear integer programs. Test sets are finite subsets of the integer lattice that allow to improve any given feasible non-optimal point of an integer program by one element in the set. There are various possible ways of defining test sets depending on the view that one takes: theGraver test set is naturally derived from a study of the integral vectors in cones; theScarf test set (neighbors of the origin) is strongly connected to the study of lattice point free convex bodies; the so-calledreduced Gröbner basis of an integer program is obtained from a study of generators of polynomial ideals. This explains why the study of test sets connects various branches of mathematics. We introduce in this paper these three kinds of test sets and discuss relations between them. We also illustrate on various examples such as the minimum cost flow problem, the knapsack problem and the matroid optimization problem how these test sets may be interpreted combinatorially. From the viewpoint of integer programming a major interest in test sets is their relation to the augmentation problem. This is discussed here in detail. In particular, we derive a complexity result of the augmentation problem, we discuss an algorithm for solving the augmentation problem by computing the Graver test set and show that, in the special case of an integer knapsack problem with 3 coefficients, the augmentation problem can be solved in polynomial time.Supported by a Gerhard-Hess-Forschungsförderpreis of the German Science Foundation (DFG).