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The purpose of this note is to prove, as Lusztig stated, that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nilelliptic elementnt istcl wherel is the rank of the associated finite type Lie algebra.


In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number.


In all these cases we actually show that Γ=π1(M) has a finite index subgroup which is mapped onto a nonabelian free group.


The presentations are given in the form of graphs resembling Dynkin diagrams and very similar to the presentations for finite complex reflection groups given in [2].


We study the multiplicative structure of rings of coinvariants for finite groups.

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 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 Wplane with a finite number of analytic cuts. For the cases k=4 and k=5 SchaefferSpencer [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.... 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 Wplane with a finite number of analytic cuts. For the cases k=4 and k=5 SchaefferSpencer [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_3αa_2~2=1+2 exp(2α/(1α))(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_3αa_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:  The purpose of this short paper is to compare the two existing theories of quantization of equations of motion containing high derivatives. As well known, when the order of the derivatives of field quantities q are finite, it is possible in certain cases to express q as a linear combination of quantities Q, each of which satisfies an equation of the second order. Quantization proceeds as if the various Q are independent. On the other hand, one may, following Ostrogradski, put the equations of motion for... The purpose of this short paper is to compare the two existing theories of quantization of equations of motion containing high derivatives. As well known, when the order of the derivatives of field quantities q are finite, it is possible in certain cases to express q as a linear combination of quantities Q, each of which satisfies an equation of the second order. Quantization proceeds as if the various Q are independent. On the other hand, one may, following Ostrogradski, put the equations of motion for the variables q in canonical form and then perform a subsequent quantization. (Such a theory was also discussed by the author in an earlier paper.) It is obviously worthwhile to see if the two theories are identical.  这篇短文比较了两种含有高次微商的量子理论。一个是在某些情形下可用的,它将变数q表为许多适合二阶方程的Q的线性组合,而在量子化时,各个Q分别地被量子化。另一个是先将q的运动方程正则化,再引入量子条件。我们证明了两个理论,无论就各种量的对易关系而言,或就总哈密顿而言,是等效的。  The finite difference equations are applied in this paper for the analysis ofcaissonbeam, which is composed of two intersecting sets of parallel beams,namely the longitudinal beams and the transversal beams. Two cases of caissonbeam have been discussed in this paper. In the first case, all the 1ongitudinalbeams are arranged in equidistance and have the same stiffness, all the transversal beams are also arranged in equidistance and have the same stiffness. Byusing the method of redundant forces and deformations,... The finite difference equations are applied in this paper for the analysis ofcaissonbeam, which is composed of two intersecting sets of parallel beams,namely the longitudinal beams and the transversal beams. Two cases of caissonbeam have been discussed in this paper. In the first case, all the 1ongitudinalbeams are arranged in equidistance and have the same stiffness, all the transversal beams are also arranged in equidistance and have the same stiffness. Byusing the method of redundant forces and deformations, we obtain a set ofsimultaneous partial finite difference equations, where the deflections of the jointsand the bending moments of longitudinal beams and transversal beams at jointsare unknown functions. The finite sine series are used in the solution of theseequations. In the second case, all the transversal beams are arranged inequidistance and have the same stiffness, while the longitudinal beams arearraseed in arbitrary distances and have different stiffness. By using the methodof redundant forces, we obtain a set of simultaneous ordinary finite differenceequations, where the bending moments of the longitudinal beams at joints areunknown functions. The general method for solving these equations has beendiscussed.  本文用差分方程解交叉梁系(井字梁)。交叉梁系中的一组平行梁称为横梁,而另一组平行梁称为纵梁。本文解两种交叉梁系。第一种是所有横梁断面惯性矩相同、间距相等,所有纵梁断面亦然。用结构力学中的混合法,取结点挠度和横梁、纵梁在结点的弯矩为未知数,得到一个联立偏差分方程组。解这个联立偏差分方程组采用了有限三角级数,并分成三种情况讨论它们的解,即四边简支,两对边简支其他两对边任意支承,四边任意支承等。第二种交叉梁是所有横梁断面惯性矩相同、间矩相等而纵梁的断面惯性矩、间距则是任意的。用结构力学中的冗力法,取纵梁在结点处的弯知为未知函数,得到一个联立常差分方程组。本文讨论了这个方程组的一般解和纵梁两端为简支时的有限三角级数解。   << 更多相关文摘 
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