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).

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).

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.

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;

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.

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).

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.

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.