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     They are that if J∶V→R is G-differentiable in V,either 1)and if for any u,v∈V with J(u)≠J(v),we have J~′(v,u-v)
     即如果泛函J∶V→R在V中是G-可微的,且其G-微分满足1)u,v∈V,J(u)≠J(v),有J′(v,u-v)
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     In this paper,it is proved that the Stancu operators S α n(f,x) have the property of preserving the modulus and if f∈H ω,then S α n(f,x)∈H 2ω .
     给出Stancu算子Sαn(f,x)的连续模保持性质 ,证明了如果f∈Hω,则Sαn(f,x)∈H2ω.
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     we obtain the following main results:Theorem A In category C,if the elements of Aut(X + Y) are all reduced and if Aut~X (X + Y) and Aut~Y(X + Y) are subgroups of Aut(X + Y),thenAut(X + Y) = Aut~Y(X + Y)Aut~X(X + Y).
     定理B 在范畴HNCWA中,如果Aut(X/A∨Y/A)中元素可对角化,则Aut(s+At)=Aut(iX)Aut(iY).
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     This paper proves that:let G be a 3-onnected K1.3graph,and if every inducde subgraph A, A of G satisfies (a1,a2),then G is panconnected(except for u and v (G)with d(u,v) = l, there may not be(u,v)- path for k=(2,3,4).
     本文证明了:如果G是3连通的无爪图且G的每个导出子图A,A~(?) 都满足ψ(a_1,a_2)则G是泛连通图(除了当u,v∈V(G),d(u,v)=1时,G中可能不存在(u,v)—k路,k∈(2,3,4)以外)
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     This paher used Gramer rule, give Unigue solution of singular eguation AX=b [ Ind(A)=k, b∈R(Ak) ] , and if A is nonsingular induction to general Cramer rule.
     本文用克莱姆法则,给出奇异方程AX=b [Ind(A)=k,b∈R(A~k)]的唯一解,并且如果A是非奇异,可归纳到通常的克莱姆法则。
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  “and if”译为未确定词的双语例句
     ENERGY TRANSFER BETWEEN N_2(A ) ,N(~2D )AND IF(X)
     N_2(A)及N(~2D)对IF(X)的传能研究
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     Some Characterizations of FP-injective Rings and IF Rings
     FP—内射环和IF环的几个特征
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     Coherent Rings and IF Rings
     凝聚环和IF环
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     D: 19 cycles and deoxycycline was given after IVF-ET(100mg,bid×10d). Hydrosalpinx was aspirated on the day when ovule was taken and if there was fluid in cavitary uteri the day before embryo transplantation,fluid was aspirated immediately;
     D组:19个周期,IVF-ET开始后口服强力霉素100 mg,每日两次10 d,取卵日抽吸输卵管积水,胚胎移植前一日发现宫腔有积液者立即行宫腔积液抽吸;
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     And if the thickness of crystal is reduced by 30%,the maximum thermal distortion is reduced by 23.5%.
     晶体厚度减小30%时,端面最大热形变量减少23.5%。
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     If M (?)
     V(M)={v∈V(G):存在一个顶点x∈V(G)使得vx∈M}.如果M?
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     (c) If s(M) (?)
     (c)若s(M)(?)
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     ItlocatedbytheCanobicRiveranditsmodernnameisKomGe'if.
     它位于尼罗河的支流卡诺比克河口,其遗址的现代名字是考姆盖伊夫(KomGe'if)。
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     If-Conditionals and Stereotype
     If条件句与常规关系
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With "hat" denoting the Banach envelope (of a quasi-Banach space) we prove that if 0>amp;lt;p>amp;lt;1, 0>amp;lt;q>amp;lt;1, ?, while if 0>amp;lt;p>amp;lt;1, 1≤q>amp;lt;+∞, ∝, and if 1≤p>amp;lt;+∞, 0>amp;lt;q>amp;lt;1, ?.
      
It is proved that every 3-connected loopless multigraph has maximum genus at least one-third of its cycle rank plus one if its cycle rank is not less than ten, and if its cycle rank is less than ten, it is upper-embeddable.
      
This paper investigates Buck's question about which class of spaces is strongly monotonically T2, and if other properties are combined with strongly monotonically T2, which class of spaces could be got.
      
The preliminary pharmacological tests show that the compounds have good hypoglycemic activity and can enhance the action of insulin, especially Ib, Id and If.
      
The latter method is the most convenient for practical problems, since it is effective, has a known accuracy, and if the extension is based on an analysis of simulation problems, it provides certain physical information.
      
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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。

Let f(z)=z+sum from n=2 to ∞ a_nz~n be regular and schlicht in the unit circle. M. Schiffer proved that the function w=f(z) in the class of such functions, which renders |a_κ| the maximum, maps |z|<1 onto the whole W-plane with a finite number of analytic cuts. For the cases k=4 and k=5 Schaeffer-Spencer [3] and Golusin [5] proved respectively that there is only one cut for the extremal domain. The principal object of the present paper is to show that the same thing holds true for the cases k=6 and k=7. Our...

Let f(z)=z+sum from n=2 to ∞ a_nz~n be regular and schlicht in the unit circle. M. Schiffer proved that the function w=f(z) in the class of such functions, which renders |a_κ| the maximum, maps |z|<1 onto the whole W-plane with a finite number of analytic cuts. For the cases k=4 and k=5 Schaeffer-Spencer [3] and Golusin [5] proved respectively that there is only one cut for the extremal domain. The principal object of the present paper is to show that the same thing holds true for the cases k=6 and k=7. Our proof depends upon the following lemmas: Lemma A. If{f(z)~2}_6=0; then |a_2|<1.63; and if {f(z)~2}_7=0; then |a_2|<1.77; Where {g(z)}_n denotes g~((n))(0). Lemma B. If |a_6|≥6 and {f(z)~2}6=0, than |a_2|>1.95, If |a_7|≥7 and {f(z)~2}_7=0, then |a_2|>1.85. Using merely the method of variation, without appealing to L(?)wner's method as done by M. Fekete and G. Szeg [6], we can prove the known theorem that (?)|a_3-αa_2~2|=1+2 exp(-2α/(1-α))(0≤α<1) with the "uniqueness" of the extremal function. For the functions f(z) satisfying the pair of conditions R(a_3)>0 and R(a_2)<0, we can pnove that the greatest value of R(a_2+a_3)is 1.03…,and that the correspondiong extremal function is of real coefficients.

S表示單位圆|z|<1上單葉且正則的函數 f(z)=z+α_2z~2+α_3z~3+… (1.1)的全體所成之族。設S′是S的一個子族,S′中任一函數满足條件 R(α_3)>0,R(α_2)<0。對於S′中的函數,本文證明R(α_2+α_3)之最大值是可以達到的,其值是1.03…。達到此值的極值函數的一切係數都是實數,極值函數只有一個。舍勾和飛克得[6]謝缶和斯賓塞爾[3]以及沙拉烏洛夫先後用樓五納的參數表示法和變分法,求出 |a_3-αa_2~2|(0≤α<1)的值,並指出達到此值的極值函數的一切係數都是實數,而且極值函數只有一個。本篇僅用變分法来建立他們的定理。惜缶[4]指出使|a_n|達到最大值的函數(1.1),其映象區域的境界是一組伸展到無窮遠處的解析若當曲綫。謝缶和斯賓塞爾[3],戈魯辛[5]分別證明對於|a_4|和|a_5|的極值區域,其境界綫只有一根。本篇對於|a_6|和|a_7|證明同樣的事實。證明是靠着如下的引理:

After the construction of reservior in an alluvial stream, the equilibrium condition of the downstream channel will be upsetted due to the reduction of sediment supply. This will result in a change in the form and gradient of the channel. The direction of the displacement will lead to a reduction of the sediment carrying capacity of the channel, through which the river regains its equilibrium. The whole process is accomplished by the flattening of the bed slope, as suggested by the theories generally accepted...

After the construction of reservior in an alluvial stream, the equilibrium condition of the downstream channel will be upsetted due to the reduction of sediment supply. This will result in a change in the form and gradient of the channel. The direction of the displacement will lead to a reduction of the sediment carrying capacity of the channel, through which the river regains its equilibrium. The whole process is accomplished by the flattening of the bed slope, as suggested by the theories generally accepted at the present. From a study of the fluvial processes of Colorado River and some other streams, it appears that a decrease in channel gradient is one of the three possibilities only, and should not be regraded as an universal rule. In fact, the change in bed slope below an impounding reservoir depends essentially on the geologic and geographic conditions of that area. It is first considered that the channel width does not change materially in the downstream direction. In many of the sandy rivers, there exists, a layer of gravel or pebble below the finer surface material. The inclination of this layer is generally steeper than the present bed slope. After the releasing of clear water from the reservoir, the surface material is carried away and thereby exposing the coarse layer, first near the dam and gradually extended to the downstream. This will result in an uneven erosion along the direction of the flow, and the steepening of the channel gradient is the consequence of such a process. The Colorado Eiver below the Hoover Dam belongs to this case. If the distribution of the bed material in the downstream, direction is more or less uniform, and if there exists no coarse layer within a finite depth below the bed surface, the bed will be degraded as a whole and the channel gradient remains essentially at its initial value. Results of flume study seem to indicate that such is the case. The Colorado Eiver below both the Parker and the Imperial Dam also belongs to this group. Only when there is a base level within a short distance from the dam which controls the depth of scour at that neighborhood, or if ther is a tributary which brings coarser material into the lower part of the reach, will the channel slope below the dam become flatter. On the other hand, if the river is impounded in a gorge and enters a plain not far away from the dam site, the width of the channel on the alluvial fan becomes progressively wider downstream. It is then possible that the depth of degradation decreases in the downstream direction and the slope becomes flatter. This case is well illustrated by one of the rivers in U. S. S. R. It must be realized that no matter how the channel gradient below an impounding reservoir changes, it is mainly through the coarsening of the bed material which makes the channel regains its equilibrium. Two different types of the coarsening of the bed material can be distinguished from the field data available. One of which is the exposing of a gravel or pebble layer below the present alluvium. As soon as this layer is exposed, there will be a sudden jump in the bed material size, and the stream channel becomes stable again. The othertype of coarsening of bed material takes place continuously and at a much slower rate. One often fails to notice the occurence of such a phenomenon, as the size of the bed material only increases in a relatively small amount. Yet, such a small increase in bed material size results in a much significantly rise of the roughness coefficient, by remolding the sand bars on the bed surface. Both the velocity of the flow and the sediment carrying capacity of the channel will be reduced, and the river gradually re-establishes its equilibrium. It is now possible to calculate such a proeess according to the theories on sediment transport and river roughness. Following the coarsening of the bed material, the down-cutting rate of the stream channel decreases with time. Both the flume studies and field measurements indicate that the relationship between the down-cutting rate and the cumulative time of erosion is an exponential one. This study demonstrates that the fluvial processes of an alluvial channel depend very much on the hydrographic conditions of the basin, of which the stream forms a part. One can't visualize the whole aspect of the problem (?)y studying the fluvial proteases through hydraulics, view-point only.

冲积河流上修建水库以后,下游河道将失去平衡。河流为了重新建立平衡,必将发生一系列的变化。本文对这—变化的过程,提出了一些新的概念。根据枢纽下游所在地区的地质地理条件,河流的坡降可以有各种不同的变化。坡降的渐趋和缓只是三种可能性中的一种,不能当做普遍规律来看待。河槽的重新建立平衡,主要是通过河床质的粗化作用。河床质的粗化现象具有两种不同的类型。一种是河床表层复盖物因清水冲刷而外移,露出了底下属于另一地质时代的较粗的卵石层,这时河床质的粒径会发生突变。另一种是由于水流的分迭作用,河床质的变化是连续的,徐缓的。在第二种情况下,河床质的粒径虽然变化不多,但是,通过河底沙垅的加多,却可以使曼宁糙率系数有很快的增加,这就使水流的流速降低,挟沙能力减小,河流逐渐建立起新的平衡。随着河床质的渐次粗化,河床的冲刷下切率与时俱減。野外实测结果及水槽试验成果都指出冲刷率与冲刷历时之间成一指数关系。河床质中不能为水流所挟运的颗粒所占百分比愈大,冲刷率下降得愈快,河道重新恢复平衡所需要的时间也愈短。从黄河下游的具体情况来判断,将来修建枢纽以后,下游河道在承受清水冲刷时可能作整体下降,坡降的变化不会很大。黄河下游的河床质虽然很细,同样也...

冲积河流上修建水库以后,下游河道将失去平衡。河流为了重新建立平衡,必将发生一系列的变化。本文对这—变化的过程,提出了一些新的概念。根据枢纽下游所在地区的地质地理条件,河流的坡降可以有各种不同的变化。坡降的渐趋和缓只是三种可能性中的一种,不能当做普遍规律来看待。河槽的重新建立平衡,主要是通过河床质的粗化作用。河床质的粗化现象具有两种不同的类型。一种是河床表层复盖物因清水冲刷而外移,露出了底下属于另一地质时代的较粗的卵石层,这时河床质的粒径会发生突变。另一种是由于水流的分迭作用,河床质的变化是连续的,徐缓的。在第二种情况下,河床质的粒径虽然变化不多,但是,通过河底沙垅的加多,却可以使曼宁糙率系数有很快的增加,这就使水流的流速降低,挟沙能力减小,河流逐渐建立起新的平衡。随着河床质的渐次粗化,河床的冲刷下切率与时俱減。野外实测结果及水槽试验成果都指出冲刷率与冲刷历时之间成一指数关系。河床质中不能为水流所挟运的颗粒所占百分比愈大,冲刷率下降得愈快,河道重新恢复平衡所需要的时间也愈短。从黄河下游的具体情况来判断,将来修建枢纽以后,下游河道在承受清水冲刷时可能作整体下降,坡降的变化不会很大。黄河下游的河床质虽然很细,同样也存在着粗化的可能。通过粗化作用,可以使河床下切的深度大大减低。

 
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