On determining whether two polynomials are relatively prime

A comparison is made of the computational effort required for four different algorithms which establish whether two monic polynomials have a common zero. Two of the algorithms are based on the vanishing of the resultant and the other two on a recent theorem of Vogt and Bose.     

Some polynomials that are hard to compute

Using the result of Heintz and Sieveking [1], we show that the polynomials $\text{Σ}{}_{\mathrm{1?j?d}}\text{b}{}^{\mathrm{<mtext>1</mtext><mtext>i</mtext>}}\text{X}{}^{\mathrm{j}}$ with b positive real different from one, and Σ_1?j?dj^rX^j with r rational not integer, are hard to compute.     

Random low-degree polynomials are hard to approximate

We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials over mathbb F{mathbb F} . We prove that almost all degree d polynomials have only an exponentially small correlation with all polynomials of degree at most d − 1, for all degrees d up to Θ(n). That is, a random degree d polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low-degree polynomial. Recently, several results regarding the weight distribution of Reed–Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed–Muller codes.     

快速得到大周期多项式的方法

在密码学和编码学中,经常需要构造周期很大的多项式,然而多项式周期的计算是亚指数级别的算法,因此确定一个多项式的周期是非常复杂的事情,在实践中一般只是估计多项式的周期.利用分析根的方法,给出了一类特殊多项式因子的周期估计定理的一种新的证明方法,该方法简单直观,可以用来解决其它类似问题,最后对定理进行了推广,得出了一个一般性的结论,并根据结论给出一个得到大周期多项式的算法.     

A method to determine the required number of neural-networktraining repetitions

Conventional neural-network training algorithms often get stuck in local minima. To find the global optimum, training is conventionally repeated with ten, or so, random starting values for the weights. Here we develop an analytical procedure to determine how many times a neural network needs to be trained, with random starting weights, to ensure that the best of those is within a desirable lower percentile of all possible trainings, with a certain level of confidence. The theoretical developments are validated by experimental results. While applied to neural-network training, the method is generally applicable to nonlinear optimization.     

A non-parametric method to determine basic probability assignment for classification problems

As an important tool for knowledge representation and decision-making under uncertainty, Dempster-Shafer evidence theory (D-S theory) has been used in many fields. The application of D-S theory is critically dependent on the availability of the basic probability assignment (BPA). The determination of BPA is still an open issue. A non-parametric method to obtain BPA is proposed in this paper. This method can handle multi-attribute datasets in classification problems. Each attribute value of the dataset sample is treated as a stochastic quantity. Its non-parametric probability density function (PDF) is calculated using the training data, which can be regarded as the probability model for the corresponding attribute. The BPA function is then constructed based on the relationship between the test sample and the probability models. The missing attribute values in datasets are treated as ignorance in the framework of the evidence theory. This method does not have the assumption of any particular distribution. As a result, it can be flexibly used in many engineering applications. The obtained BPA can avoid high conflict between evidence, which is desired in data fusion. Several benchmark classification problems are used to demonstrate the proposed method and to compare against existing methods. The constructed classifier based on the proposed method compares well to the state-of-the-art algorithms.     

A zonal RANS-LES method to determine the flow over a high-lift configuration

High Reynolds number flows are particularly challenging problems for large-eddy simulations (LES) since small-scale structures in thin and often transitional boundary layers are to be resolved. The range of the turbulent scales is enormous, especially when high-lift configuration flows are considered. For this reason, the prediction of high Reynolds number flow over the entire airfoil using LES requires huge computer resources. To remedy this problem a zonal RANS-LES method for the flow over an airfoil in high-lift configuration at Re_c=1.0×10⁶ is presented. In a first step, a 2D RANS solution is sought, from which boundary conditions are formulated for an embedded LES domain, which comprises the flap and a sub-part of the main airfoil. The turbulent fluctuations in the boundary layers at the inflow region of the LES domain are generated by controlled forcing terms, which use the turbulent shear stress profiles obtained from the RANS solution. The comparison with an LES solution for the full domain and with experimental data shows likewise results for the velocity profiles and wall pressure distributions. The zonal RANS-LES method reduces the computational effort of a full domain LES by approx. 50%.     

一种新的GBPA生成方法及其在模式识别中的应用

广义证据理论是一种在不完备识别框架中处理多传感器信息融合问题的实用方法.由于时代环境的影响,人们的认知存在局限性,难免会将不完备的识别框架认为是完备的,经典证据理论在这种情况下并不完全适用.因此,根据广义证据理论提出一种新的广义基本概率赋值(generalized basic probability assignment,GBPA)生成方法.该方法首先根据训练数据分别构造样本类别和测试样本的广义三角模糊数模型;然后通过计算样本和类别间的广义三角模糊距离生成GBPA;最后使用广义组合规则融合所有证据并得出最终的结论.Iris数据集的实验结果表明所提方法合理有效,即使在样本不足的情况下仍有较高的分类精度.     

A novel access control method using Morton number and prime factorization

A novel scheme used for controlling access requests in security information system is proposed. In the proposed method, the system administrator chooses distinct prime numbers representing each atomic access right as well as four large prime numbers for encryption. By setting these representative prime numbers as input parameters, the proposed method applies a one-way function combining the Morton number theory transferring into a single value to derive the encrypted compound privilege (ECP). With ECP, verification of right of access can be achieved easily and secretly. Meanwhile, the proposed scheme provides the following advantages: (1) the verification of right of access can be effectively implemented using the Morton sequence with coordinate transformation; (2) the problem of dynamic access control also can be effectively implemented; (3) integrity and confidentiality while controlling system resources can be ensured; (4) the proposed method can decrease the redundancy of the access matrix in some specific circumstances.     

Torsional tests on microstructures: two methods to determine shear-moduli

In this paper a new static torsional test is presented. The experimental setup is explained and first test results on silicon and metallic LIGA (from the German Lithographie, Galvanik, Abformung) microstructures are shown. By performing experiments on specimens with different cross sectional dimensions the determination of the governing shear-moduli is possible. This analysis can be carried out using two different procedures applied to the experimental data. The first procedure involves numerical and analytical methods whereas the second one is solely based on numerical methods. Both procedures are explained and first results on both silicon and LIGA specimens are presented. The shear-moduli found for silicon are in good agreement with values known from bulk material. The shear-moduli found for LIGA material show an anisotropic behaviour.     

A Bayesian approach to system reliability when two components are dependent

A Bayesian approach is used to estimate and to find highest posterior density intervals for R(t)=P(X₁2#62;t, X₂2#62;t) when (X₁, X₂) follow the Gumbel bivariate exponential distribution. Because of the complexity of the likelihood function, numerical integration must be used and, for this setting, Jacobi and Laguerre rules are employed as they arise naturally. A data set from an application is used to illustrate the procedures.     

确保鲁棒性能的PID控制器参数域的一种确定方法

研究了将混合灵敏度的H∞范数作为PID控制器的鲁棒设计指标时,控制器参数的可行域确定问题．首先将混合灵敏度的H∞范数要求转化为一族复系数多项式的稳定性要求,再加上基本闭环特征多项式的稳定性,通过求解这些含参数多项式的稳定性,解出参数的可行域．在求解多项式稳定性的过程中,采用了推广的Hermite—Biehler定理．     

A semi-infinite programming method for approximating load duration curves by polynomials

Load duration curves play an important role in the planning practice of electric power systems. In the paper, we consider the problem of approximating a load duration curve by a polynomial under monotonicity and some other constraints. We show that semi-infinite programming techniques can be applied for solving this problem. A convergent inner-outer method and a finite ε-optimal algorithm is proposed.     

基于核密度估计的基本概率指派生成方法

D-S合成方法作用的对象是基本概率指派（basic probability assign,BPA）,如何生成BPA是D-S理论应用中重要且有待解决的首要步骤。针对生成BPA提出一种基于核密度估计KDE（kernel density estimation）的BPA生成方法：训练数据用于构建基于最优化窗宽的核密度估计的数据属性模型;然后利用训练数据的核密度模型计算测试数据的密度—距离—分布值Tri-D（density-distance-distribution）,通过嵌套式的方法分配Tri-D值获取测试数据对应的BPA;最后D-S合成BPA得到最终判断,通过分类准确率来判断BPA生成方法的有效性。实验通过在UCI数据集上与其他方法的分类准确率对比验证了提出方法的有效性。     

A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint

In this paper, a new and simple approach is presented to exactly calculate the critical buckling loads of beams with arbitrarily axial inhomogeneity. For various end boundary conditions, we transform the governing equation with varying coefficients to linear algebraic equations; then a characteristic equation in critical buckling loads will be obtained. Several examples of estimating buckling loads under typical end supports are discussed. By comparing our numerical results with the exact and existing results for homogeneous and nonhomogeneous beams, it can be found that our method has fast convergence and the obtained numerical results have high accuracy. Moreover, the buckling behavior of a functionally graded beam composed of aluminum and zirconia as two constituent phases is investigated for axially varying material properties. The effects of gradient parameters on the critical buckling loads are elucidated. Finally, we give an example to illustrate the enhancement of the load-carrying capacity of tapered beams for admissible shape profiles with constant volume or weight. The proposed method is of benefit to optimum design of beams against buckling in engineering applications.     

A new method to determine the periodic orbit of a nonlinear dynamic system and its period

Periodic motion is an important steady-state motion in the real world. In this paper, a new generalized shooting method for determining the periodic orbit of a nonlinear dynamic system and its period is presented by rebuilding the traditional shooting method. First, by changing the time scale, the period of the periodic orbit of a nonlinear system is drawn into the governing equation of this system explicitly. Then, the period is used as a parameter in the iteration procedure of the shooting method. The periodic orbit of the system and the period can be determined rapidly and precisely. The requirement of this method for the initial iteration conditions is not rigorous. This method can be used to analyze the forced nonlinear system and the parameter exciting system. As an example, the results of the Rössler equation for an eight-dimensional, nonlinear, flexible, rotor-bearing system are compared with those obtained by the Runge-Kutta integration algorithm. The validity of this method is verified by the numerical results obtained in the two examples.     

An unsupervised method to determine the optimal number of independent components

In this paper, we present a new method to determine the optimal number of independent components after applying an Independent Component Analysis (ICA) to a set of mixed signals. The proposed method, called Linear Correlations between Components (LCC), uses the JADE algorithm to calculate the independent components. The LCC method allows to automatically select the optimal number of independent components in an unsupervised way without any previous knowledge. It has been tested using synthetic mixed signals where the number of pure (or independent) signals is known. This method is very simple, fast and easy to implement.