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partial
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    Application of Wu-Ritt Elimination Method in the Partial Differential Algebraic Equation
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    Long Time Behavior of the Solutions of Nonlinear Functional Differential Equations and Partial Functional Differential Equations
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    Numerical Methods for Some Time Dependent Nonlinear Partial Differential Equations
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  partial
We also compute the Euler characteristic of the space of partial flags containingnt and give a connection with hyperplane arrangements.
      
Then theG0-orbit structure ofZ is described explicitly by the partial Cayley transforms of a certain hermitian symmetric sub-flagF?Z.
      
Semi-invariants of quivers can be constructed by taking admissible partial polarizations of the determinant of matrices containing sums of matrix components of the representation and the identity matrix as blocks.
      
We define a partial order on the set No,c of pairs (O,C), where O is a nilpotent orbit and C is a conjugacy class in A(O), Lusztig's canonical quotient of A(O).
      
We also give formulas for intersection pairings on resolutions of singularities (or more precisely partial resolutions, since orbifold singularities are allowed) of the quotients.
      
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In the first part of this paper we consider the partial differential equa-tion as a generalized Euler-Poisson equation:(?) (1.1)where β,β′are constants, and a(x,y),b(x,y),c(x,y),d(x,y)are all regularfunctions in Hadamard's sense.Therefore x=y is the singular line of thecoefficients.The behaviors of the solutions of(1.1)in the neighborhood ofthe singular line x=y are described by introducing the concepts of“index”and the“regular part”:Let ρ be a constant and υ(x,y)be a regularfunction(υ(x,x)≠0)such thatu(x,y)=(x-y)~ρυ(x,y)is...

In the first part of this paper we consider the partial differential equa-tion as a generalized Euler-Poisson equation:(?) (1.1)where β,β′are constants, and a(x,y),b(x,y),c(x,y),d(x,y)are all regularfunctions in Hadamard's sense.Therefore x=y is the singular line of thecoefficients.The behaviors of the solutions of(1.1)in the neighborhood ofthe singular line x=y are described by introducing the concepts of“index”and the“regular part”:Let ρ be a constant and υ(x,y)be a regularfunction(υ(x,x)≠0)such thatu(x,y)=(x-y)~ρυ(x,y)is a solution of(1.1),then the constant ρ is said to be the“index”andρ(x,y)the“regular part”of the solution.It is shown that all the possibleindexes must satisfy the indicial equation(?)and if F(ρ+1)≠0,then the normal derivative of the regular part on thesingular line x=y is determined completely by the value itself,i.e.(?)The regular part υ(x,y)satisfies the equation of a particular form of(1.1),in which γ=0,and therefore it is sufficient to study the equation of theform(?) (?) (3.2)We define the singular Cauchy prob em as follows:to find a functionυ(x,y)continuous together with its first derivatives and twice differentiablein the region ACBD(cf.figure 1 p.518),and satisfying the equation(3.2)in the region ACBD,except the singular line AB,on which it takes anygiven regular funtion u_0(2x)as its initial value.We give the existence proof of such singular Cauchy problem in thegeneral case(β+β′≠0),and it follow that,the solution of the equation(1.1)may,in general,be expressed as.(?)where ρ_1 and ρ_2 are different roots of the indicial equation;or(?)where ρ_1 is the double root of indicial equation.The second part of this paper,deals with the singular equation in spa-ce,especially the equation of the following form:(?) (15.5)where A_σ is any linear operator which (?)epends only on the variables σ==(σ_1,…,σ_n),such that,the Cauchy problem for the associated regular equation(?) (15.6)and the initial data(?)has a unique soluion υ(x,σ_,…,σ_n).The solution of singular Cauchy pro-blem for equation(15.5),with initial data(?)can be expressed by υ(x,σ_1,…,σ_n)in the form(?)where K(τ,t)is a kernel well defined by the operator(?)For example,the kerne for Euler-Poisson-Darboux opera-tor(?)is(?). The same method can be applied to solve the Cauchy problem for thegeneralized Chapligin equation(?)(where K(t)is an increasing function,and K(0)=0),with initial data(?)The solution is given explicitly by(17.12).(p.550).

本文的第一部分研究了含奇线方程的解在奇线附近的性质;引进了“指数”的概念,从而给出了关于这类方程的“奇型郭西问题”的正确提法;并且通过一种特殊的积分-征分方程的研究,证明了这种“奇型郭西问题”的解的存在性,并且给出其近似解法;最后,就一般的情形,给出了方程一般解的表达式,从而说明了在β+β′<0时,郭西问题的多解性。本文的第二部分研究了空间含奇面方程(?)其中 A_σ是任一祇与变元σ=(σ_1…,σ_n)有关的算子,并且关于(15.5)的奇型郭西问题的解可以用关于方程(不合奇面)(?)(15.6)的郭西问题的解表示出来。同样的方法可用来解决空间却普里金方程(17.1)的郭西问题。

The boundary-value problems of the linearized supersonic conical fields of thin wings are reduced to the boundary-value problems of analytical functions. Instead of using perturbation velocity potentials as unknown functions[6,7], the n-th partial derivatives of perturbed velocity potentials are used as unknown functions[2]. It is found that the present method is mathematically simple and direct and, moreover, can be used when the boundary-value is an arbitrary homogeneous function.The cases of plane delta...

The boundary-value problems of the linearized supersonic conical fields of thin wings are reduced to the boundary-value problems of analytical functions. Instead of using perturbation velocity potentials as unknown functions[6,7], the n-th partial derivatives of perturbed velocity potentials are used as unknown functions[2]. It is found that the present method is mathematically simple and direct and, moreover, can be used when the boundary-value is an arbitrary homogeneous function.The cases of plane delta wings are discussed in detail. The lift and pitching and rolling moments of plane delta wings are calculated by theorem of residues. This technique is much simpler than direct integrations of the pressure distributions. The cases of cruciform triangular wings are also discussed.The present method can be generally used in the linearized supersonic conical flow problems of the first and higher order for the plane and cruciform wings, while the boundary conditions may be cither those of the direct problems, the inverse problems or the mixed problems.

本文在已有的超声速薄翼錐形流問題解法的基础上,提出了一种比較普遍适用而計算又比較簡便的方法。本文的方法是以扰动速度势函数对座标的n阶偏导数为未知函数,将問題化为解析函数的边值問題求解。利用解析函数的性貭,討論了利用留数定理来比較簡便地計算超声速后緣的三角形机翼的升力、俯仰力垣和滾轉力矩的方法。簡要討論了本文方法对十字形三角机翼問題的应用。 本文方法可以計算边界条件中齐次函数是任意齐次函数的情形,而不象某些文献中主要仅限于研究齐次函数是n-1次多項式的情形。这点在实际問題中也是有意义的,例如:具有錐形扭轉的三角机翼的一阶錐形流問題,机翼上各种操纵面的偏轉和运动問題,超声速边緣的問題以及某些升力抵消流場的問題等。 本文方法适用于比較普遍的情形:錐形流可以是一阶的,也可以是高阶的;机翼形状可以是平面形的,也可以是十字形的;边界条件可以是已知翼剖面形状,也可以是已知压力分布,或者一部分已知翼剖面形状而另一部分已知压力分布。 附录中討論了和正文有关的解析函数边值問題及其某些簡单的推广。

The partial wave analysis for the reaction x+N→π+π+N has been made, i. e., the general expressions for angular operators, which characterise the dependence of scattering marries on spin and directions of momenta have been given, in the case of any value of angular momenta. For the given energy and directions of momenta, it is showed that, by what observables one can reconstructs the scattering matrics uniguely. The ambiguity in phase shift analysis, which remains the differeutial cross section ivariant...

The partial wave analysis for the reaction x+N→π+π+N has been made, i. e., the general expressions for angular operators, which characterise the dependence of scattering marries on spin and directions of momenta have been given, in the case of any value of angular momenta. For the given energy and directions of momenta, it is showed that, by what observables one can reconstructs the scattering matrics uniguely. The ambiguity in phase shift analysis, which remains the differeutial cross section ivariant and satisfies the other physical conditions has been further discussed. Starting from the general conditions which should be satisfied by phase sbift, the correspedent Minami transformations have been obtained. It is also pointed out that, by what polarizations, the ambiguity can be eliminated. All the results in this paper can he extend to the preduction of other bosoms.

本文給出了反应π+N→π+π+N的分波分析,即在角动量为任意值的情形下,求出了决定散射矩陣对于动量方向与自旋之依賴关系的角算符的普遍表达式。对于給定的能量与动量方向,指出了何种观測量可以唯一地决定散射矩陣。进一步討論了保持微分截面不变与滿足其他物理要求的相移的不定性。从对散射矩陣的普遍要求出发,求出了相应的南氏变換。并指出測量何种极化可以消除相移的不定性。本文的所有結果同样适用于其他玻色子的产生过程。

 
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