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polynomials
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  多项式
     Research on Polynomial Functions and Permutation Polynomials
     关于多项式函数与置换多项式的研究
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     Approximation of Functions in Z_(a,s) by Means of Trigonometric Polynomials
     用线性正多项式算子来迫近Z_(a,s)中的函数
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     The Geometric Method for Finding Roots of Polynomials
     寻求多项式根的几何方法
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     I_(01) APPROXIMATION AND THE ROUND-OFF PROBLEM FOR COEFFICIENTS IN COMPUTATION OF POLYNOMIALS
     I_(01)逼近和多项式计算中的系数舍入
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     On Modified Landau Polynomials of Order k
     关于K阶改进的Landau多项式
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  多项式的
     0, then there exists a qn(x)∈Πn(+) such that ||f-1/qn||L1≤Cω(f,n-1/2)L1, where Πn(+) indicates all polynomials of degree n with positive coefficients.
     0,则存在一个次数不超过n正系数多项式qn(x)∈Πn(+),使得||f-1/qn||L1≤Cω(f,n-1/2)L1,其中Πn(+)表示所有次数不超过n的正系数多项式的全体.
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     On the RIeationship of Polynomials{g_(2j+1)(t)}with Bernoulli's and Euler's Polynomial
     函数序列{g_(2j+1)(t)}及{f_(2j+1)(t)}与Bernonlli多项式及Euler多项式的关系
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     In this paper, Some identities involving Euler number and Bernoulli polynomials are given:E 2n=(-1) n+1B 2n+1(1/4)2n+14 2n+1, et al;
     给出了欧拉数与贝努利多项式的几个恒等式 ,即E2n=(- 1) n + 1 B2n + 1 (1 4)2n + 142n + 1 等 ;
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     The Factorization of Adjoint Polynomials of Graphs G_(j_1j_2…j_t)~(s~* (i))(p,tkm) and Chromatically Equivalence Analysis
     图簇G_(j_1j_2…j_t)~(S~* (i))(p,tkm)的伴随多项式的因式分解及色性分析
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     In this paper,we study the convergence of the generalized Lagrange interpolatingpolynomials at roots of unity in the complex plane,and obtain the order of approximationof the polynomials to f(z)∈A(|z|≤1) in the space L_p(|z|=1), 0<p<+∞.
     研究了复平面单位根上广义Lagrange插值多项式的收敛性,得到了其在L_p(|z|=1)空间上(0<p<+∞)收敛于f(z)∈A(|z|≤1)的逼近阶。
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  “polynomials”译为未确定词的双语例句
     ALTERNATION THEORY IN APPROXIMATION BY POLYNOMIALS HAVING BOUNDED COEFFICIENTS
     ALTERNATION THEORY IN APPROXIMATION BY POLYNOMIALS HAVING BOUNDED COEFFICIENTS
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     On the Interpolation Problems of Hermite and Hermite-Fejer polynomials Bosed on the Zeros of (1-x~2) P′_(n-1)(x)
     关于以(1-x~2)_(p′_(n-1))(x)的零点为结点的Hermite和Hermite-Fejer插值问题
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     +(e~P+Q)f=0(1) (where P and Q are polynomials with deg P>0) possesses a non-trivial solution with λ(f)
     若方程f~(11)+(e~(P(z))+Q(z))f=0存在一非平凡解f,使得λ(f)
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     Stechkin-Marchaud-type Inequalities for Bernstein Polynomials
     关于Bernstein算子的Stechkin-Marchaud型不等式
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     (a|ˉ), R=F [λ] is a ring of polynomials in one variable over F, then a_mλ~m+…+a_1λ +a_0 (?)
     是F的一个二阶自同构,它诱导出F的一个二阶自同构:a_mλ~m+…+a_1λ+a_0(?)
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  polynomials
We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups).
      
BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials
      
For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation.
      
We also prove a binomial formula for 6-parametric Koornwinder polynomials.
      
We also use known results about canonical bases forUq2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schützenberger and Zelevinsky.
      
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This paper attempts to answer the following two questions: (1) Is it possible to derive the law of distribution of hydrological frequency theoretically(2) What type of distribution curve should be adopted as the model of hydrological frequency curve and how to determine their parameters? The results obtained may be summarized as follows: 1. Hydrological phenomena are time series with concealed periodic fluctuations. The results from statistical analysis based upon the current assumption that hydrological phenomena...

This paper attempts to answer the following two questions: (1) Is it possible to derive the law of distribution of hydrological frequency theoretically(2) What type of distribution curve should be adopted as the model of hydrological frequency curve and how to determine their parameters? The results obtained may be summarized as follows: 1. Hydrological phenomena are time series with concealed periodic fluctuations. The results from statistical analysis based upon the current assumption that hydrological phenomena are independent stochastic variables should be accepted with due considerations. 2. In view of the regional nature of hydrological phenomena, the current parctice of analyzing samples taking from a single station only is, in effect, to narrow the sampling field arbitrarily from a larger area to a point, thus reducing the accuracy of the statistical results. Hence, the synthetic utilization of the data of all stations within the hydrologically homo- geneous region is an important measure to increase the accuracy of statistical analysis. 3. The belief that the flood frequency obeys the binomial theorem or Poisson's theorem is but to mix up the priori with the empirical probability problem. The binomial theorem, being a powerful weapon to deal with the problems of priori probability, has not been adquately and properly utilized in the hydrologieal frequency analysis. 4. Analyses have been made of the nature of distribution of shydrologieal series on the basis of Kaptyen's derivation of the skew distribution, which indicate: (1) That the theoretical interpretation of the log-probability law of the hydrologic phenomena by V. T. Chow is not sound; (2) that hydrologic phenomena being results of very complicated meteorological and hydrological processes, it is impossible to derive theoretically the law of distribution for the hydrological series. 5. The view that the flood frequency obeys the Gumbel's distribution is theoretically not sound and also not verified by actual data. 6. According to the nature of the mathematical treatments applied, the method of description of the empirical probability can be classified into three systems: (1) The methods of the generalization of the characteristic factors of the distributions, such as Pearson's curves, Goodrich's curves, etc.; (2) The methods of the modification of a fundamental distribution by series and polynomials, such as Gram-Charlier curves. curves, etc.; (3) The methods of transformed functions, such as the log-probability law, curves, etc. It should be remarked that not only Pearson's and Goodrich's curves are frequency curves of empirical nature, but even the theoretical laws, such as the normal law and the log-probability law, will be aceepted as curves of empirical nature, when used as models for empirical probability problem. 7. Hydrological frequency analysis should not be mystified and made absolute. Instead of free selections, the models of hydrological frequency curve should be uniquely selected and specified. Statistical parameters should be determined not solely by the short period data of single station, but also by the synthetic utilization of the data of possible more stations. 8. It is recommended that one of the two types of distribution, i.e. the log-normal frequency curve with both sides limited and the Pearson's type Ⅲ curve, may be selected as unified models. The author suggests that the K-value corresponding to recurrence intervals of say 10~4, 10~5, or 10~6 years may be selected as the upper and lower limits for the log-normal curve. For Pearson's type III curves, C_s should be treated not as independent but as dependent variables of C_v. 9. The proper way to select and determine the model frequency curve is to see whether it fits well with the actual data of grouped stations (stations to be grouped by regions for rainfall data and by C_v for runoff data) and the reasonableness of the extrapolating part. 10. Suggestions on the method of determination of x and C_v: For point rainfall, iso-x map may be utilized, and the mean C_v for each hydrologicregion may be adopted in order to minimize the errors from single stations and to avoid the discrepancies in results obtained from the same region. With regard to flood frequency analysis, flood mark reconnaissance must be utilized to determine the magnitude and the recurrence interval of the unusual flood. The x and C_v values of the floods and runoffs of hydrologically similiar river basins may be compared. Besides, the reasonableness of the results of frequency calculations as well as of the statistical parameters adopted therein may be checked by comparing runoffs and point-rainfall values of the same frequency.

我国近期水文频率计算方法的研究工作在选择方法,经验频率公式,参数的误差和利用我国水文资料检验各种频率线型等方面有了一定的成果和实用的结论[1],但是下面两个问题还没有获得解决: (1)能否从机率理论证明水文频率属于何种分布律? (2)水文频率曲线应当采用什么线型?如何确定参数?本文试图解答以上两个问题。本文分析了水文系列的时序性质和区域性质,把机率问题按先验、极限和后验三种基本性质对水文频率问题进行了分析;利用开布屯推导偏态分布的方法分析了水文系列的分布性质,并从而批判了有关水文频率肯定属于对数正态律,耿贝尔极限律或二项式定理等等说法。认为属于后验机率性质的水文频率,不能从机率理论证明它属于何种分布律。最后提出联合利用各站水文资料来选择线型和确定参数的方法,并建议在两端有限对数正态和皮尔逊Ⅲ型两种线型中选择一种作为统一采用的线型,对两端有限曲线提出了简易可行的确定上下极限的方法,对皮尔逊Ⅲ型曲线认为应该把Cs作Cv的倚变参数。

In this paper the degree of approximation by n-dimensional Bernstein polynomials is discussed. By a k-dimensional Bemstein polynomial of a cominuous function f(x1,x2…xk) on the k-dimensional cube 0≤x1≤1,i=1,2,…k, it is defined by the following expression:whereThe following inequalities are established.Here the expressions ωf(δ1,δ2,…δk),ω2(δ) and ω4(δ) are moduli of continuity in different metrecs defined as followFurthermore, let us define the Bernstein polynomials on the k-dimensional simplex 0≤x1+x3+…xk≤1,xi≥0,i=1,2,…k,...

In this paper the degree of approximation by n-dimensional Bernstein polynomials is discussed. By a k-dimensional Bemstein polynomial of a cominuous function f(x1,x2…xk) on the k-dimensional cube 0≤x1≤1,i=1,2,…k, it is defined by the following expression:whereThe following inequalities are established.Here the expressions ωf(δ1,δ2,…δk),ω2(δ) and ω4(δ) are moduli of continuity in different metrecs defined as followFurthermore, let us define the Bernstein polynomials on the k-dimensional simplex 0≤x1+x3+…xk≤1,xi≥0,i=1,2,…k, by the following expression:Herethen the following inequality holds,

本文讨论了k维空间的伯恩斯坦多项式在不同的距离下的逼近度.所谓在k维单位区间上的伯恩斯坦多项式是指其中本文建立了下列关于连续函数的逼近度.式中ω_f(δ_1,δ_2,…δ_k)表f(x_1,x_2,…x_k)的连续模即此外建立了在单纯形0≤x_1+x_2+……+x_k≤1,x_i≥0,i=1,2,……k上的伯恩斯坦多项式即的逼近度,式中本文建立了下列关于连续函数的逼近度最后一式与维数k无关.

The assignment of aberration tolerances for optical systems suffering from wave-front aberrations greater than λ/4 is worth further studying. In this paper it is shown that "low" contrast resolving power of optical systems can be taken as an appropriate criterion for assessing image quality, where the optical transfer function can be evaluated approximately in terms of the sums of Legendre polynomials. Accordingly, the best program of aberration correction and tolerences in the case where certain aberration...

The assignment of aberration tolerances for optical systems suffering from wave-front aberrations greater than λ/4 is worth further studying. In this paper it is shown that "low" contrast resolving power of optical systems can be taken as an appropriate criterion for assessing image quality, where the optical transfer function can be evaluated approximately in terms of the sums of Legendre polynomials. Accordingly, the best program of aberration correction and tolerences in the case where certain aberration constants are inevitably present and others are adjustable in optical design can be obtained. The results so obtained as applied to aberrations of small magnitude comparable to λ/4 conform well to the Strehl Deffinitionshelligkeit method.

具有大象差的光学系统象差公差问题一直还未得到很好解决。本文中提出以低对比分辨本领作为光学系统的质量指标,并将光学系统的传递函数近似表示为正交多项式之和,象差任意大小的光学系统的最佳校正方案和公差之值就可以求出。在小象差情况下,所得的结果与Strehl判断的结果基本一致。

 
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