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For some special cases, the paper gives some important identical theorems, and then establishes a valuable relation between the uniformly almost periodic functions and the trigonometric polynomials.Secondly, on the basis of the identical theorem, the paper investigates the Fourier series of the uniformly B2 almost periodic functions, and further proves that the series is unique.Thirdly, the paper discusses the Parseval equation of the uniformly B2 almost periodic functions, which establishes the relation between these functions and the coefficients of their Fourier series; and next investigates an important approximation theorem-Riesc-Fischer theorem, about the uniformly B2 almost periodic functions and the trigonometric polynomials.

并给出了特殊情况下的几个重要的恒同定理，将一致概周期函数与有限三角多项式联系起来；第二，在恒同定理的基础上，给出了一致B~2概周期函数的Fourier级数，并且级数是唯一的；第三，讨论了一致B~2概周期函数的Parseval方程，建立了函数与其Fourier级数的系数之间的联系；接着给出了关于一致B~2概周期函数和三角多项式之间的一个重要近似定理—Riesc-Fischer定理。

Prove the formulas with related the Bernoulli polynomials of higher order and Euler polynomials of higher order.

证明了与高阶Bernoulli多项式和高阶Euler多项式相关的无穷积分的计算公式。

In this article,we study mainly the following chapters:In the first chapter,we introduce simply definitions of the generalization of Bernoulli polynomials and Euler polynomials,and some related knowledge.

本文主要分以下内容：第一章，简要介绍广义Bernoulli多项式和广义Euler多项式的定义以及相关的知识。

Hence, a necessary and sufficient condition for the positivity of polynomials of arbitrary degree is presented in this paper. A practical algorithm to express this condition in terms of the coefficients of the polynomials is also given.

由此提出了任意阶多项式为正的一个充要条件，也提出了一个实用的算法，从而可以只用此多项式的系数来表示得到的充要条件。

In order to solve the problem,We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential manifold. The travel time are expressed as polynomials of the horizontal offset between the two points, and the single-square-root operator in frequency-wavenumber domain are expressed as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddle-point method.

针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量(地面点到地下相点的水平距离)的多项式，将频率波数域单平方根算子表示成波数的多项式，运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。

In order to solve the problem, We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddlepoint method.

针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量（地面点到地下相点的水平距离）的多项式，将频率波数域单平方根算子表示成波数的多项式，运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。

Furthermore, we express a real quaternionic polynomial with four general polynomials which have real coefficients, thus build direct relation between real quaternionic polynomials and general polynomials.

新方法以四元数的复表示及其可对角化的特征结构为理论基础，将问题等价为求解2×2矩阵多项式的可对角化解；进一步将四元数多项式用四个实系数多项式表示，建立了四元数多项式与一般多项式的直接关联。

It is proved that all monic polynomials of order n with roots lying in some open region on the complex plane forms a simply connected set in the polynomial parameter space. Based on this result, Edge Theorems for D-stability of general polyhedrons of polynomials and Boundary Theorems for D-stability of compact sets of polynomials are obtained.

证明了其根位于复平面中某开域的所有n阶多项式的全体构成多项式参数空间中的一个单连通集，并基于这一结果进一步提出了关于一般多面体多项式族D-稳定性的棱边定理和关于一般多项式紧集D-稳定性的边界定理。

The theory of orthogonal symmetric polynomials (Jack polynomials, Macdonald's polynomials and others) and multivariate hypergeometric functions associated to symmetric polynomials are treated.

对称的正交多项式理论（杰克多项式，麦克唐纳的多项式等）及相关的多元对称多项式超几何函数的待遇。

A bstract: This paper focus on virtual instrument design of modal parameters identification with curve fitting theory based on rational fraction orthogonal polynomials. Using Forsythe orthogonal polynomials solves the coefficient equation instead of normal power polynomials in order to decrease the morbidity of it, making the orders of coefficient equation reduce to half and increasing computing efficiency.

本文基于有理分式正交多项式曲线拟合理论对振动系统的模态参数辨识进行了虚拟仪器的设计，为了减少系数矩阵的病态，采用 Forsythe 正交多项式代替普通的幂多项式进行求解，使系数矩阵阶次降低近一半，提高了运算速度。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失