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sufficient
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    On the sufficient conditions for the complete continuity of noninear integral operators in Orlicz spaces
    Orlicz空间中非线性积分算子全连续性的充分条件
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    The Sufficient Conditions of Stability of Differential Difference Fquation of Neutral Typc
    中立型微分差分方程稳定性的充分条件
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    COMPUTATIONALLY VERIFYING SUFFICIENT CONDITION FOR THE IMPLICIT FUNCTION EXISTENCE THEOREM
    隐函数存在定理的计算可检验充分条件
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    Sevaral Sufficient Conditions for the Local Solvability of Linear PDO with Multiple Characteristics
    具多重特征的线性偏微分算子局部可解的充分条件
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    ON THE SUFFICIENT ALGEBRAIC CRITERIA FOR ALL NULL POINTS OF Det(a_(ij)+b_(ij)e~(λτ)-σ_(ij)λ)_(n×n)……DISTRIBUTION ON THE LEFT HALF PART OF THE COMPLEX PLANE
    超越函数Det(a_(ij)+b_(ij)e~(λr)-δ_(ij)λ)_n×n零点全分布在复平面左半部的代数充分准则
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  “sufficient”译为未确定词的双语例句
    ON THE NECESSARY AND SUFFICIENT CONDITION FOR THE VALIDITY OF FREQUENCY FIDELITY CRITERION IN SHANNON'S THEOREM
    关于Shannon定理中频率准则成立的充要条件
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    NECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL ASYMPTOTIC STABILITY OF SECOND ORDER DIFFERENTIAL EQUATIONS
    二阶微分方程全局渐近稳定性的充要条件
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    Necessary and Sufficient Conditions for the Uniqueness of the Global Solutions of Ordinary Differential Equations
    微分方程整体解的唯一性的充要条件
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    Fisher Information Quantity and Sufficient σ-algebra
    Fisher信息量和充分性σ-代数
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    The Necessary and Sufficient Condition for Quasi-leftcontinuity of Natural σ-fields of Jump Processes
    跳跃过程的自然σ域为拟左连续的充要条件
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  sufficient
We give a simple necessary and sufficient condition for a Schubert
      
A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one.
      
In the spirit of work of Kerman and Sawyer, a condition is given that is necessary and sufficient for the Fourier transform norm inequality
      
A sufficient condition for the sequence of sampling functions to constitute a frame is derived.
      
Furthermore, we generalize the sufficient and necessary conditions of orthogonality given by Lawton and Cohen to the case of several dimensions and arbitrary dilation matrix A.
      
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In this paper some cases of optimum control are studied when the conditions at the ends of trajectories are constrainted. The boundary conditions of the system of the differential equations (19), (22), etc. are determined; the formulae of functional variation (20) are derived; and the sufficient conditions of optimality and necessary conditions in some cases are proved.

当軌线末端受有条件限制时,将使最优控制系統有关的微分方程的边界条件变得复杂。本文考虑了各种受限情形,提出了确定边界条件的方法,得到泛函改变量公式,討論了控制最优性的充分条件以及在某些情形下的必要条件。

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。

It has long been realised that for laminar flow through a circuclar pipe, the Poiseulle's velocity distribution exits only at a sufficient distance downstream from the entrance to the pipe. For a well-rounded entrance, the velocily distribution at the entrance is approximately uniform, it then varies gradually with the distance from the entrance until the parabolic distribution is fully developed. The distance from the entrance required for practically complete transition has been calculated by Boussiuesq...

It has long been realised that for laminar flow through a circuclar pipe, the Poiseulle's velocity distribution exits only at a sufficient distance downstream from the entrance to the pipe. For a well-rounded entrance, the velocily distribution at the entrance is approximately uniform, it then varies gradually with the distance from the entrance until the parabolic distribution is fully developed. The distance from the entrance required for practically complete transition has been calculated by Boussiuesq and olso by Schiller, and formulas for calculating tle transition length have been proposed. For laminar flow between parallel plates, however, no information for the length of transition appears to be available. An attempt is thus made, in this paper, to calculate this lenqth by applying Schiller's theory; and the result obtained is given by L=0.0259bR in which L is the transition length, b is half the distance between the plates and R is the Reynolds number.

我們很早就知道流體在圓管中作層流運動時,泊西里(Poiseulle)的流速分佈只是在距進口處相當長的一段距離處才能實現。如管子進口是圓滑的,則在進口處斷面上的流速是均匀分佈的,此後流速分佈即逐漸轉變直至抛物線形的流速分佈完全形成爲止。由進口處至完全轉化成抛物線形流速分佈所需的距離曾先後經鮑辛尼斯克(Boussinesq)和錫拉(Schiller)進行過研究,他們求得了計算此轉化長度的公式。但是對於在二平行平板間的層流運動,其轉化長度的計算尚無公式可資利用。爲此本文應用錫拉理論對此進行了計算,求得了此轉化長度的公式爲L=0.0259bR,式中,L爲轉化長度,R爲雷諾數,b爲二平板間的距離之半。

 
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