基于素性检验思想的不可约多项式判断

论文将Fermat素性检验的思想运用于不可约多项式的判断,给出了一个对于不可约判断问题的Monte Carlo 算法,分析了该算法的计算复杂度问题,并且给出了次数在200以内的检验结果。     

基于Zernike多项式的波前像差研究

波前像差的大小直接影响着视网膜的成像质量。就波前像差的Zernike多项式的单索引表示方法,双索引表示方法进行介绍,并探究波前像差重建的原理和波前像差重建的方法,以及Zernike函数项的改变与视觉质量之间的关系。基于波前像差的重建方法的优缺点进行分析,并探讨基于波前像差的数学统计模型的研究方向以及在临床应用上的广阔前景。     

Results on rotation symmetric polynomials over GF(p)

In this paper, we generalize the recent counting results about rotation symmetric Boolean functions to the rotation symmetric polynomials over finite fields GF(p). By using Möbius function, we obtain some formulas for more general n, the number of variables. Some known formula in Boolean case are simplified.     

Weight distribution and dual distance

Several results in coding theory (e.g. the Carlitz-Uchiyama bound) show that the weight distributions of certain algebraic codes of lengthn are concentrated aroundn/2 within a range of width n. It is proved in this article that the extreme weights of a linear binary code of sufficiently high dual distance cannot be too close ton/2, the gap being of order n. The tools used involve the Pless identities and the orthogonality properties of Krawtchouk polynomials, as well as estimates on their zeroes. As a by-product upper bounds on the minimum distance of self-dual binary codes are derived.     

Comparison between cellular automata and linear feedback shift registers based pseudo-random number generators

A detailed comparison between pseudo-random number generators (PRNGs) based on cellular automata (CA) and linear feedback shift registers (LFSRs) is presented in this paper. Various statistical tests have been applied in order to reveal the advantages and disadvantages of both approaches. Both LFSRs and hybrid additive cellular automata (HACA) produce satisfactory PRNGs. HACA operate at higher speeds than LFSRs with the same characteristic polynomials. Regarding the silicon area, direct comparisons between the two approaches cannot be made since it depends on the PRNG length. However, the inherent modularity of HACA reduces the silicon area occupied by them and, when long feedback paths are used, the silicon area occupied by LFSRs increases.     

Least-Squares, Moment-Based, and Hybrid Polynomializations of Drag Forces

In order to carry out nonlinear dynamic analyses of fixed offshore structures in the frequency domain, polynomial approximations of the distributed drag term, u∣u∣, and inundation drag term, ηu∣u∣, of the Morison forces are studied. The methods of least-squares and moment-based approximations are considered for cases with and without current. Numerical results and analytical expressions of the polynomial coefficients are presented for the cubic approximations of u∣u∣ and quartic approximations of ηu∣u∣. The curve shapes, first four central moments, and probability density functions of the different approximations are evaluated and compared with the corresponding exact solutions. For the nonmonotonic inundation drag term with the current effect included, a hybrid polynomialization, based on the least-squares approximation for the odd-order polynomial coefficients and the moment-based approximation for the even-order coefficients, is proposed.     

Single-factor coefficient bounds

In a 1993 paper Beauzamy, Trevisan and Wang derived a single-factor coefficient bound, one which limits the max norm (height) of at least one irreducible factor of any univariate integral polynomial A. Their bound is a function of the degree and the weighted norm of A. In the conclusion of their paper they ask whether the max norm of A might already be a single-factor coefficient bound. In 1998 Knuth, citing these authors, asked instead whether there is a constant c such that c times the max norm of A is a single-factor coefficient bound. We present the results of extensive calculations relating to this question. We show that c, if it exists, must be greater than 2 and accrue evidence in support of a conjecture that the answer to Knuth’s question is “no”.     

The Decomposition Method For Solving Of A Class Of Singular Two-Point Boundary Value Problems

In this paper, a class of nonlinear singular two-point boundary value problems are solved by using Adomian's decomposition methods. The approximate solution of this problem is calculated in the form of series with easily computable components. Finally, we give a nonlinear numerical example.     

一种基于泛函网络的多项式Euclidean算法

提出一种基于泛函网络的多项式Euclidean计算新模型,给出一种基于泛函网络的多项式Euclidean新算法。网络的泛函参数利用解线性方程组方法来完成。相对于传统方法,该方法不但能够快速地获得所求多项式问题的精确解,而且可获得所求多项式问题的近似解。计算机仿真结果表明,该算法十分有效、可行,可以看作是对传统的Euclidean算法的一种推广。该算法将在计算机数学、代数密码学等方面有着广泛的应用。     

Training a reciprocal-sigmoid classifier by feature scaling-space

This paper presents a reciprocal-sigmoid model for pattern classification. This proposed classifier can be considered as a Φ-machine since it preserves the theoretical advantage of linear machines where the weight parameters can be estimated in a single step. The model can also be considered as an approximation to logistic regression under the framework of Generalized Linear Models. While inheriting the necessary classification capability from logistic regression, the problems of local minima and tedious recursive search no longer exist in the proposed formulation. To handle possible over-fitting when using high order models, the classifier is trained using multiple samples of uniformly scaled pattern features. Empirically, the classifier is evaluated using a benchmark synthetic data from random sampling runs for initial statistical evidence regarding its classification accuracy and computational efficiency. Additional experiments based on ten runs of 10-fold cross validations on 40 data sets further support the effectiveness of the reciprocal-sigmoid model, where its classification accuracy is seen to be comparable to several top classifiers in the literature. Main reasons for the good performance are attributed to effective use of reciprocal sigmoid for embedding nonlinearities and effective use of bundled feature sets for smoothing the training error hyper-surface. Editor: Risto Miikkulainen