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Adaptive Random Effects/Coefficients Modeling
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作者 George J. Knafl 《Open Journal of Statistics》 2024年第2期179-206,共28页
Adaptive fractional polynomial modeling of general correlated outcomes is formulated to address nonlinearity in means, variances/dispersions, and correlations. Means and variances/dispersions are modeled using general... Adaptive fractional polynomial modeling of general correlated outcomes is formulated to address nonlinearity in means, variances/dispersions, and correlations. Means and variances/dispersions are modeled using generalized linear models in fixed effects/coefficients. Correlations are modeled using random effects/coefficients. Nonlinearity is addressed using power transforms of primary (untransformed) predictors. Parameter estimation is based on extended linear mixed modeling generalizing both generalized estimating equations and linear mixed modeling. Models are evaluated using likelihood cross-validation (LCV) scores and are generated adaptively using a heuristic search controlled by LCV scores. Cases covered include linear, Poisson, logistic, exponential, and discrete regression of correlated continuous, count/rate, dichotomous, positive continuous, and discrete numeric outcomes treated as normally, Poisson, Bernoulli, exponentially, and discrete numerically distributed, respectively. Example analyses are also generated for these five cases to compare adaptive random effects/coefficients modeling of correlated outcomes to previously developed adaptive modeling based on directly specified covariance structures. Adaptive random effects/coefficients modeling substantially outperforms direct covariance modeling in the linear, exponential, and discrete regression example analyses. It generates equivalent results in the logistic regression example analyses and it is substantially outperformed in the Poisson regression case. Random effects/coefficients modeling of correlated outcomes can provide substantial improvements in model selection compared to directly specified covariance modeling. However, directly specified covariance modeling can generate competitive or substantially better results in some cases while usually requiring less computation time. 展开更多
关键词 Adaptive Regression Correlated Outcomes Extended Linear Mixed Modeling Fractional polynomials Likelihood Cross-Validation Random Effects/coefficients
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ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS 被引量:3
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作者 Eid H.DOHA Waleed M.ABD-ELHAMEED Mahmoud A.BASSUONY 《Acta Mathematica Scientia》 SCIE CSCD 2015年第2期326-338,共13页
Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds t... Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given. 展开更多
关键词 Chebyshev polynomials of the third and fourth kinds expansion coefficients generalized hypergeometric functions boundary value problems
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APPROXIMATION BY RATIONAL FUNCTIONS WITH POLYNOMIALS OF POSITIVE COEFFICIENTS AS THE DENOMINATORS 被引量:1
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作者 Dansheng Yu Songping Zhou 《Analysis in Theory and Applications》 2006年第1期65-71,共7页
The present paper proves that if(x) ∈ C[0,1], changes its sign exactly l times at 0 〈 y1〈 y2 … 〈 y1 〈 1 in (0, 1), then there exists a pn(x) ∈ Пn(+), such that |f(x)- p(x)/pn(x)|≤ Cωφ(f,n^... The present paper proves that if(x) ∈ C[0,1], changes its sign exactly l times at 0 〈 y1〈 y2 … 〈 y1 〈 1 in (0, 1), then there exists a pn(x) ∈ Пn(+), such that |f(x)- p(x)/pn(x)|≤ Cωφ(f,n^(-1/2)), where ρ(x) is defined by ρ(x)={^lПi=1(x-yi),if f (x)≥0 for x ∈(y1,1), {-^lПi=1(x-yi),if f (x)〈0 for x ∈(y1,1), which improves and generalizes the result of . 展开更多
关键词 polynomial with positive coefficients rational approzimation Ditzian-Totik modulus of smoothness
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Coefficient Multipliers for Polynomials
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作者 申大维 《Journal of Beijing Institute of Technology》 EI CAS 2001年第2期113-118,共6页
Let Q n denote the class of all polynomials p(z) nonvanishing in the unit disk with deg p≤n and p (0)=1, and let W n denote the class of all polynomials s(z) satisfying deg s≤n and for all... Let Q n denote the class of all polynomials p(z) nonvanishing in the unit disk with deg p≤n and p (0)=1, and let W n denote the class of all polynomials s(z) satisfying deg s≤n and for all p∈Q n, s*p∈Q n , where * denotes the Hadamard product. Some properties for W n and Q n are obtained. 展开更多
关键词 polynomialS coefficientS MULTIPLIERS
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On the Coefficients of Polynomials with Positive Real Part in the Unit Disk
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作者 申大维 《Journal of Beijing Institute of Technology》 EI CAS 1995年第2期110+106-110,共6页
Let Q be the class of real coefficient polynomials of degree 2 with positive real part in the unit disk and constant term equal to 1. aam coefficient region of polynomials in Q is found and some sharp coefficient esti... Let Q be the class of real coefficient polynomials of degree 2 with positive real part in the unit disk and constant term equal to 1. aam coefficient region of polynomials in Q is found and some sharp coefficient estimates for the polynomials with positive real part in the unit disk are established in this paper. 展开更多
关键词 polynomialS coefficients/positive real part SUBORDINATION
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POLYNOMIAL COEFFICIENT FINDING FOR ROOT-MUSIC 被引量:1
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作者 Liu Xiaojun Liu Congfeng Liao Guisheng 《Journal of Electronics(China)》 2009年第4期543-548,共6页
Abs Root-MUSIC (MUltiple Signal Classification) is the polynomial rooting form of MUSIC, namely, the spectrum peak searching is resplaced by the polynomial rooting in MUSIC implementation. The coefficients finding o... Abs Root-MUSIC (MUltiple Signal Classification) is the polynomial rooting form of MUSIC, namely, the spectrum peak searching is resplaced by the polynomial rooting in MUSIC implementation. The coefficients finding of the polynomial is the critical problem for Root-MUSIC and its improvements By analyzing the Root-MUSIC algorithm thoughly, the finding method of the polynomial coefficient is deduced and the concrete calculation formula is given, so that the speed of polynomial finding roots will get the bigger exaltation. The particular simulations are given and attest correctness of the theory analysis and also indicate that the proposed algorithm has preferable estimating performance. 展开更多
关键词 Direction Of Arrival (DOA) estimation Root-MUSIC (MUltiple Signal Classification) polynomial coefficient finding
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ON THE COEFFICIENTS OF A POLYNOMIAL WITH RESTRICTED ZEROS
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作者 Abdul Aziz W. M. Shah 《Analysis in Theory and Applications》 2009年第2期142-146,共5页
Let P(z)=anz^n+an-1z^n-1+…+a0be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coeff... Let P(z)=anz^n+an-1z^n-1+…+a0be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coefficients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z). 展开更多
关键词 polynomial inequality coefficient estimate separation of zeros
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Large-Integer Multiplication Based on Homogeneous Polynomials
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作者 Boris S. Verkhovsky 《International Journal of Communications, Network and System Sciences》 2012年第8期437-445,共9页
Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations an... Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations and elimination of necessity to evaluate polynomials in points with larger coordinates. It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation. A multistage implementation of these algorithms is also less efficient than their direct implementation. Although the proposed algorithms as well as the corresponding Toom-Cook algorithms require numerous algebraic additions, the Generalized Horner rule for evaluation of homogeneous polynomials, provided in the paper, decrease this number twice. 展开更多
关键词 Homogeneous polynomialS Toom-Cook Algorithm Multidigit integerS Multi-Stage MULTIPLICATION Generalized Horner Rule Large-integer MULTIPLICATION
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On the Growth and Polynomial Coefficients of Entire Series
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作者 Huzoor H. Khan Rifaqat Ali 《Applied Mathematics》 2011年第9期1124-1128,共5页
In this paper we have generalized some results of Rahman [1] by considering the maximum of |f(z)| over a certain lemniscate instead of considering the maximum of|f(z)|, for |z|=r and obtain the analogous results for t... In this paper we have generalized some results of Rahman [1] by considering the maximum of |f(z)| over a certain lemniscate instead of considering the maximum of|f(z)|, for |z|=r and obtain the analogous results for the entire function |f(z)|=Σpk(z) [q(z)]k-1 where q(z) is a polynomial of degree m and pk(z)is of degree m-1. Moreover, we have obtained some inequalities on the lover order, type and lower type in terms of polynomial coefficients. 展开更多
关键词 Lemniscate LOWER Order LOWER Type Slowly CHANGING FUNCTION polynomial coefficientS and Entire Functions.
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A proof of maximum contention-free property of interleavers for Turbo codes using permutation polynomials over integer rings
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作者 MA Xin-rui XU You-yun ZHANG Le 《Journal of Zhejiang University-Science A(Applied Physics & Engineering)》 SCIE EI CAS CSCD 2007年第1期24-27,共4页
It is well known that interleavers play a critical role in Turbo coding/decoding schemes, and contention-free interleaver design has become a serious problem in the paraUelization of Turbo decoding, which is indispens... It is well known that interleavers play a critical role in Turbo coding/decoding schemes, and contention-free interleaver design has become a serious problem in the paraUelization of Turbo decoding, which is indispensable to meet the demands for high throughput and low latency in next generation mobile communication systems. This paper unveils the fact that interleavers based on permutation polynomials modulo N are contention-free for every window size W, a factor of the intedeaver length N, which, also called maximum contention-free interleavers. 展开更多
关键词 Turbo codes integer ring Permutation polynomial INTERLEAVER Maximum contention-free (MCF)
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Three-Variable Shifted Jacobi Polynomials Approach for Numerically Solving Three-Dimensional Multi-Term Fractional-Order PDEs with Variable Coefficients
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作者 Jiaquan Xie Fuqiang Zhao +1 位作者 Zhibin Yao Jun Zhang 《Computer Modeling in Engineering & Sciences》 SCIE EI 2018年第4期67-84,共18页
In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs wit... In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method. 展开更多
关键词 Three-variable shifted Jacobi polynomials multi-term FRACTIONAL-ORDER PDES VARIABLE coefficients numerical solution convergence analysis
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A REMARK ON THE COEFFICIENTS OF BOUNDED POLYNOMIALS
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作者 Wirths K.-J. 《Analysis in Theory and Applications》 1996年第3期98-99,共2页
Let p(z)=akzk be such that |p(eip)|≤1 for R and |p(1)|=a[0,1].An inequality of Dewan and Cavil for the sum |av|+|au|,0≤u<v≤n is sharpened.Let p(z)=abzk be such that for R,
关键词 A REMARK ON THE coefficientS OF BOUNDED polynomialS
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STANCU POLYNOMIALS BASED ON THE Q-INTEGERS
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作者 Xueyan Xiang 《Analysis in Theory and Applications》 2012年第3期232-241,共10页
A new generalization of Stancu polynomials based on the q-integers and a nonnegative integer s is firstly introduced in this paper. Moreover, the shape-preserving and convergence properties of these polynomials are al... A new generalization of Stancu polynomials based on the q-integers and a nonnegative integer s is firstly introduced in this paper. Moreover, the shape-preserving and convergence properties of these polynomials are also investigated. 展开更多
关键词 Stancu polynomial q-integer Q-DERIVATIVE shape-preserving property con-vergence rate modulus of continuity
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Chebyshev Polynomial-Based Analytic Solution Algorithm with Efficiency, Stability and Sensitivity for Classic Vibrational Constant Coefficient Homogeneous IVPs with Derivative Orders n, n-1, n-2
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作者 David P. Stapleton 《American Journal of Computational Mathematics》 2022年第4期331-340,共10页
The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant... The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant coefficients and exhibit vibration. The use of the Chebyshev polynomials allows calculation of the analytic solutions for arbitrary n in terms of the orthogonal Chebyshev polynomials to provide a more stable solution form and natural sensitivity analysis in terms of one parameter and the initial conditions in 6n + 7 arithmetic operations and one square root. 展开更多
关键词 Differential Equation STABILITY Sensitivity Analysis Chebyshev polynomials coefficient Formula
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Partial Time-Varying Coefficient Regression and Autoregressive Mixed Model
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作者 Hui Li Zhiqiang Cao 《Open Journal of Statistics》 2023年第4期514-533,共20页
Regression and autoregressive mixed models are classical models used to analyze the relationship between time series response variable and other covariates. The coefficients in traditional regression and autoregressiv... Regression and autoregressive mixed models are classical models used to analyze the relationship between time series response variable and other covariates. The coefficients in traditional regression and autoregressive mixed models are constants. However, for complicated data, the coefficients of covariates may change with time. In this article, we propose a kind of partial time-varying coefficient regression and autoregressive mixed model and obtain the local weighted least-square estimators of coefficient functions by the local polynomial technique. The asymptotic normality properties of estimators are derived under regularity conditions, and simulation studies are conducted to empirically examine the finite-sample performances of the proposed estimators. Finally, we use real data about Lake Shasta inflow to illustrate the application of the proposed model. 展开更多
关键词 Regression and Autoregressive Time Series Partial Time-Varying coefficient Local polynomial
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Partial Time-Varying Coefficient Regression and Autoregressive Mixed Model
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作者 Hui Li Zhiqiang Cao 《Open Journal of Endocrine and Metabolic Diseases》 2023年第4期514-533,共20页
Regression and autoregressive mixed models are classical models used to analyze the relationship between time series response variable and other covariates. The coefficients in traditional regression and autoregressiv... Regression and autoregressive mixed models are classical models used to analyze the relationship between time series response variable and other covariates. The coefficients in traditional regression and autoregressive mixed models are constants. However, for complicated data, the coefficients of covariates may change with time. In this article, we propose a kind of partial time-varying coefficient regression and autoregressive mixed model and obtain the local weighted least-square estimators of coefficient functions by the local polynomial technique. The asymptotic normality properties of estimators are derived under regularity conditions, and simulation studies are conducted to empirically examine the finite-sample performances of the proposed estimators. Finally, we use real data about Lake Shasta inflow to illustrate the application of the proposed model. 展开更多
关键词 Regression and Autoregressive Time Series Partial Time-Varying coefficient Local polynomial
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纽结琼斯多项式与整系数多项式
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作者 韩友发 马凯千 +1 位作者 李欣璐 燕佳玉 《辽宁师范大学学报(自然科学版)》 CAS 2024年第3期289-293,共5页
主要研究纽结琼斯多项式与整系数多项式之间的关系.利用纽结琼斯多项式的性质以及在某些点的特殊值,给出了宽度不同的整系数多项式为纽结琼斯多项式的成立条件.首先,给出了整系数多项式是某纽结Jones多项式的充分必要条件,给出了宽度为... 主要研究纽结琼斯多项式与整系数多项式之间的关系.利用纽结琼斯多项式的性质以及在某些点的特殊值,给出了宽度不同的整系数多项式为纽结琼斯多项式的成立条件.首先,给出了整系数多项式是某纽结Jones多项式的充分必要条件,给出了宽度为6的多项式是某一组结Jones多项式的充分必要条件.其次,主要研究Jones多项式与十一次整系数多项式的关系,研究宽度为9的十一次整系数多项式是琼斯多项式的必要条件,进而给出了某些纽结的Arf不变量. 展开更多
关键词 纽结琼斯多项式 宽度 整系数多项式 Arf不变量
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n 的多项式为系数的幂级数和函数注记
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作者 张奇业 王创坤 徐宇鑫 《大学数学》 2024年第1期99-107,共9页
研究了以n的多项式为系数的幂级数的和函数的求解问题,给出了这类幂级数和函数的一种求解方法,并得到了其和函数的通用计算公式,通过实例验证了公式的正确性.文中还得到了两个关于组合数的等式.
关键词 幂级数 多项式系数 和函数 通用形式 数学归纳法
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从一道概率题的错解说起
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作者 李卫高 《高等数学研究》 2024年第1期58-59,共2页
从一道概率题的错解与正解出发,利用线性代数、初等数论的有关知识,给出了二者相等的几种情况,以及有关概率计算的一点启示.
关键词 全概率公式 向量垂直 整系数不定方程
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摄像机畸变中心及畸变系数标定方法
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作者 王海 杨延竹 李恒 《机械设计与制造》 北大核心 2024年第8期38-43,共6页
针对传统针孔模型下摄像机畸变校正方法应用场景受限,和基于图像特征的畸变校正方法中将图像中心作为畸变中心造成标定精度不高的问题,提出一种基于角点共线特征的畸变中心及畸变系数标定方法。利用过光学中心直线畸变最小的原理,设计... 针对传统针孔模型下摄像机畸变校正方法应用场景受限,和基于图像特征的畸变校正方法中将图像中心作为畸变中心造成标定精度不高的问题,提出一种基于角点共线特征的畸变中心及畸变系数标定方法。利用过光学中心直线畸变最小的原理,设计迭代算法搜寻到畸变程度最小的两条直线,解算得到精确的畸变中心,进行畸变系数的标定。采用Levenberg-Marquardt算法和多项式畸变模型,利用整幅棋盘格角点拟合的直线特征建立非线性优化模型,标定出一阶径向畸变系数,仿真标定误差在e^(-12)~e^(-11)量级。通过实验验证了畸变中心和畸变系数标定算法的可靠性和准确性,实验结果表明,无论是FA镜头,还是Scheimpflug镜头和广角镜头的畸变,算法均能满足畸变校正的需求。 展开更多
关键词 畸变校正 畸变中心 多项式模型 LM算法 畸变系数
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