Notes on the optimal variational iteration method

Here, various variational iteration algorithms are compared. An auxiliary parameter can be introduced in the iteration procedure, and can be identified optimally, which results in Turkyilmazoglu’s optimal variational iteration method. Some unknown auxiliary parameters can be also included in the initial solution, and can be optimally determined, that is Heri?anu and Marinca’s optimal variational iteration method.     

某些积分的方程(组)解法

通过变量替换或分部积分可建立关于某些积分的方程,通过引入辅助积分可建立关于某些积分的方程组,解这些方程或方程组可得所求积分.     

Optimal induced norm computation of discrete <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control systems with time-delays

Time-delays in state or control can never be eliminated in many discrete systems, like computer controlled systems. Introducing an extended state vector, the original equations with time-delays can be transformed into standard equations without time-delays. Then the theory and methods of usual discrete system can be applied. Based on analogies between structural mechanics and optimal control theory, the optimal norm corresponding to the fundamental frequency of structural vibration, which is a Rayleigh-quotient problem, can be solved by extended Wittrick–Williams algorithm. Numerical results disclose that the optimal norm does not increase monotonously with time-delays and can be decreased effectively by selecting appropriate time delays.     

“五家共井”所承载数学史信息的探索

通过对一道古算题的探索,发现公元一世纪人们就形成了完整的对线段的度量的概念,当线段可公度(成比例)的时候,度量的结果可以用十进制循环小数精确表出;当线段不可公度(不成比例)时,度量的结果只能用十进制非循环小数足够精确的近似表出[3],扩大了人们对数的存在的认知,为后来用数轴表示数创造了历史条件,对连续统理论奠定了基础.     

Some remarks on blvarlational bounds and the pointwise estimation of solutions

When a given equation can be realised in some Hilbert space H as an operator equation in the from A[d] then it is known that complementary variational bounds can be obtained for the inner product [d]. Recently it has been shown that complementary bivariational bounds can be obtained for the inner product [d]associated with the equation A[d]= f and an arbitary elementy [d]. These latter bounds are a natural starting point for investigatng pointwise estimatesd of solutions provided thta certain questions relating, on the one hand, to the self-adjointness and the invertibility of associated operators and on the other to the availability of a suitable large Hilbert space can be resolved. We show that if the underlying problem can be given an operator realisation in a suitably equipped Hilbert space then pointwise estimates of solution can be obtained     

Data set generation for rectangular placement problems

This paper describes a recursive process for generating data sets of rigid rectangles that can be placed into rectangular regions with zero waste. The generation procedure can be modified to guarantee that the aspect and area ratios of the rectangles in the generated data sets satisfy user-specified parameters. This recursive process can thus be employed to create a variety of data sets that can be used to evaluate the efficiency and scalability of rectangular cutting and packing algorithms.     

On optimality conditions of relaxed non-convex variational problems

Non-convex variational problems in many situations lack a classical solution. Still they can be solved in a generalized sense, e.g., they can be relaxed by means of Young measures. Various sets of optimality conditions of the relaxed non-convex variational problems can be introduced. For example, the so-called “variations” of Young measures lead to a set of optimality conditions, or the Weierstrass maximum principle can be the base of another set of optimality conditions. Moreover the second order necessary and sufficient optimality conditions can be derived from the geometry of the relaxed problem. In this article the sets of optimality conditions are compared. Illustrative examples are included.     

Irreducibly inconsistent systems of linear inequalities

Irreducibly inconsistent systems of linear inequalities are considered from the point of view of applying simplex-like methods for investigation.We give necessary and sufficient conditions, for a system to be irreducibly inconsistent, which can be found by applying a simplex algorithm. Next we consider general inconsistent systems. The simplex method can be used to identify irreducibly inconsistent subsystem. Such subsystems can give structural insight in the inconsistency and are used in trying to make the overall system consistent. This can be done in an easy way e.g. by right-hand side manipulations. Finally the problem of finding a ‘cheapest’ way of doing this can be formulated and solved as an extended linear program.     

An optimal approach for hypothesis testing in the presence of incomplete data

The adverse effect of small sample sizes, excessive nonresponse rate, and high dimensionality on likelihood ratio test statistic can be reduced by integrating with respect to a prior distribution. If information regarding the prior is too general (for example, only a parametric family can be specified), this distribution can be chosen from a principle of the most powerful testing. We propose the integrated most powerful test in the presence of missing data. This test can be used as a viable alternative to the maximum likelihood.     

Dual gradient method for linearly constrained,strongly convex,separable mathematical programming problems

A new dual gradient method is given to solve linearly constrained, strongly convex, separable mathematical programming problems. The dual problem can be decomposed into one-dimensional problems whose solutions can be computed extremely easily. The dual objective function is shown to have a Lipschitz continuous gradient, and therefore a gradient-type algorithm can be used for solving the dual problem. The primal optimal solution can be obtained from the dual optimal solution in a straightforward way. Convergence proofs and computational results are given.     

A simple game to derive lognormal distribution

In the paper we present a simple game that students can play in the classroom. The game can be used to show that random variables can behave in an unexpected way: the expected mean can tend to zero or to infinity; the variance can tend to zero or to infinity. The game can also be used to introduce the lognormal distribution.     

Fourier analysis on trapezoids with curved sides

It is well known that smooth periodic functions can be expanded into Fourier series and can be approximated by trigonometric polynomials. The purpose of this paper is to do Fourier analysis for smooth functions on planar domains. A planar domain can often be divided into some trapezoids with curved sides, so first we do the Fourier analysis for smooth functions on trapezoids with curved sides. We will show that any smooth function on a trapezoid with curved sides can be expanded into Fourier sine series with simple polynomial factors, and so it can be well approximated by a combination of sine polynomials and simple polynomials. Then we consider the Fourier analysis on the global domain. Finally, we extend these results to the three-dimensional case.     

Efficient Spatial Modeling Using the SPDE Approach With Bivariate Splines

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to nonstationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency. Supplementary materials are available online.     

Non-product form equilibrium probabilities in a class of two-station closed reentrant queueing networks

While many single station queues possess explicit forms for their equilibrium probabilities, queueing networks are more problematic. Outside of the class of product form networks (e.g., Jackson, Kelly, and BCMP networks), one must resort to bounds, simulation, asymptotic studies or approximations. By focusing on a class of two-station closed reentrant queueing networks under the last buffer first served (LBFS) policy, we show that non-product form equilibrium probabilities can be obtained. When the number of customer classes in the network is five or fewer, explicit solutions can be obtained. Otherwise, we require the roots of a characteristic polynomial and a matrix inversion that depend only on the network topology. The approach relies on two key points. First, under LBFS, the state space can be reduced to four dimensions independent of the number of buffers in the system. Second, there is a sense of spatial causality in the global balance equations that can then be exploited. To our knowledge, these two-station closed reentrant queueing networks under LBFS represent the first class of queueing networks for which explicit non-product form equilibrium probabilities can be constructed (for five customer classes or less), the generic form of the equilibrium probabilities can be deduced and matrix analytic approaches can be applied. As discussed via example, there may be other networks for which related observations can be exploited.     

非系统风险的回报率及其计量

现代金融理论认为,系统风险无法通过组合投资进行规避,承担系统风险被市场承认从而可以获得风险报酬;非系统风险可以通过组合投资进行有效分散,因而承担非系统风险不应获得风险回报.试图阐述系统风险完全可以规避,指出承担非系统风险也应获得风险报酬,给出计量非系统风险回报率的规划方法,该规划的最优解同样满足两基金分离定理.     

非线性动力系统中两鞍-结分岔点间非稳定曲线的确定

采用将伪弧长延拓法与Poincaré映射法相结合的方法,确定非自治动力系统中两鞍-结分岔点间非稳定曲线,并对采用一般延拓法时出现的奇异性进行了证明。该方法引入了一正则化方程,避免了在求解过程中出现的奇异问题,并给出了相应的迭代格式。在曲线的延拓过程中,由于存在两个延拓方向,为保证将曲线延拓出来,给出了一种确定切线方向的方法,该方法在分析非线性振动系统中的双稳态现象等问题是很有效的。     

A fast genetic method for inducting descriptive fuzzy models

Under certain inference mechanisms, fuzzy rule bases can be regarded as extended additive models. This relationship can be applied to extend some statistical techniques to learn fuzzy models from data. The interest in this parallelism is twofold: theoretical and practical. First, extended additive models can be estimated by means of the matching pursuit algorithm, which has been related to Support Vector Machines, Boosting and Radial Basis neural networks learning; this connection can be exploited to better understand the learning of fuzzy models. In particular, the technique we propose here can be regarded as the counterpart to boosting fuzzy classifiers in the field of fuzzy modeling. Second, since matching pursuit is very efficient in time, we can expect to obtain faster algorithms to learn fuzzy rules from data. We show that the combination of a genetic algorithm and the backfitting process learns faster than ad hoc methods in certain datasets.     

On the computational complexity of cut-reduction

Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.     

基于区间分析的不等式自动证明

提出了一种基于区间分析的不等式自动证明方法,这一方法可以处理类型较为一般的不等式, 只需对应的函数具有所需的高阶连续可微性质, 而传统的不等式自动证明方法一般仅处理代数类型, 或可最终转化为代数类型的不等式.实际例子显示, 该方法可以解决一些其他方法无法解决的问题.     

Fundamental Geometric Structures for the Dirac Equation in General Relativity

We present an axiomatic approach to Dirac's equation in General Relativity based on intrinsically covariant geometric structures. Structure groups and the related principal bundle formulation can be recovered by studying the automorphisms of the theory. Various aspects can be most neatly understood within this context, and a number of questions can be most properly addressed (specifically in view of the formulation of QFT on a curved background). In particular, we clarify the fact that the usual spinor structure can be weakened while retaining all essential physical aspects of the theory.