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Reductive group actions on affine quadrics with 1dimensional quotient: Linearization when a linear model exists


A linear model for a given action is a linear orthogonal action with the same orbit types and equivalent slice representations.


We prove that if a reductive group action on an affine quadric with a 1dimensional quotient has a linear model, then the action is linearizable.


Using the path model and the theory of crystals, we generalize the concept of patterns to arbitrary complex semisimple algebraic groups.


In the model situationD is the Siegel disc,S is the manifold of Lagrangian subspaces andG is the symplectic group.

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 A simplified wavemechanical method of deriving the temperature factor in the reflection of Xrays from crystals is given(1). The general expression for the function M is the same as calculated by Zener and Jauncey. Hence the discrepency between Zener's theory for hexagonal crystals and the experiments may be due to the special model used in his calculation.  由简单之波动力学计算,求得DebyeWaller函数结果与Zener及Jauncey者相同故Zener对六角结晶之理论与实验结果之差异,或因其计算中所用之特殊模形所致。  1. The conformity of the pattern of Cicindela anchoralis to what has been considered typical of the genus becomes in telligible if the middle band, which is undoubtedly the most variable of the markings of this species, is assumed to have shifted forward and to have developed extensions both cephaled and caudad along the lateral margin and on the disc of the eIytron.2. The variation of the pattern is of the continuous type and proceeds in one principal direction. As expected, some in dividuals depart from... 1. The conformity of the pattern of Cicindela anchoralis to what has been considered typical of the genus becomes in telligible if the middle band, which is undoubtedly the most variable of the markings of this species, is assumed to have shifted forward and to have developed extensions both cephaled and caudad along the lateral margin and on the disc of the eIytron.2. The variation of the pattern is of the continuous type and proceeds in one principal direction. As expected, some in dividuals depart from this main series, in one way or another and ten such subtypes have been recognized in this study. The range of variation is very considerable.3. The male sex is found to have a greater tendency to lose its pigmentation, and this fact is interpreted as indicating its having a more variable middle band.4. Mating between the different types is entirely at random, the frequency of any given combination is apparently a function of the abundance of the individuals in a given class.5. In describing the elytral pattern of this species, Chevrolat (1845) used these words: "vitta anchorali, cum lienola pone humerum (femina amplioribus); cupreis". He evidently had regarded as typical the pattern showing fullest pigmentation. The present study shows, on the other hand, that such a pat tern, while most primitive, is not the most representative. The typical pattern is to be sought in Class Ⅳ, the model class. This last remark certainly has wider application and merits the consideration of taxonomists describing highly variable forms.  斑蝥科甲虫翅鞘色型之差異,在一部份種類中,極为顯著。本篇所論,根據中國种錨紋斑蝥,於廿二年夏採於厦門北岸,計雌雄共九百餘個,內有交配者六十二對,所得結論如下: (一)本種色型,甚为異特。若以美人薛福特氏所立之本屬理想模式色型推解,則其中間淡色横斑,較为移前,其左右則向上下分枝成H形。 (二)此種色型之變異為連續式仝趨一主要方向,但依其變異之分岐,可别为十種副式色型,詳情已於西文中加以討論。 (三)雄虫翅鞘之中間淡色横斑變異較大,其深色紋斑之傾向於縮小亦因之較雌性为甚。 (四)兩性交配,与色型之差異,全無關係。 (五)大部份個數皆呈現中間色型。其趨向於兩極端者(深色或淡色特别發達者)乃屬少数。因此分類學者於記述变異極大之種類時,應发中間色型者為代表。  The main difficulty of the oneelectron theory of optical activity lies in the calculation of the socalled vicinal actions. This may be overcome, as suggested in the present investiga tion by assuming that the vicinal actions are due to the interactions of various chemical bonds in the molecule with the chromophoric electron. For the calculation of these interactions the charge distribution of a single covalent bond is assumed to consist of two nuclei, each of unit positive charge, separated by the bond distance... The main difficulty of the oneelectron theory of optical activity lies in the calculation of the socalled vicinal actions. This may be overcome, as suggested in the present investiga tion by assuming that the vicinal actions are due to the interactions of various chemical bonds in the molecule with the chromophoric electron. For the calculation of these interactions the charge distribution of a single covalent bond is assumed to consist of two nuclei, each of unit positive charge, separated by the bond distance and two electrons with opposite spins and a spherically symmetrical orbital eigenfunction represented by (?) where r is the distance measured from a certain point on the bond axis and near the center of the bond. The location of this point is so determined that the model will give the proper dipole moment of the bond. The exponent α, which determines the extent of diffusivity of the electron cloud, is obtained by an approximate variational treatment. The quadruple moment of this model, calculated for the CH bond, agrees very well with that estimated by Lassettre and Dean from a study of the potential barrier hindering the internal rotation of the ethane molecule. To test the theory and also to illustrate the procedure which is followed in calculating optical rotations using the new model of vicinal actions, the optical rotations of several methyl derivatives of cyclopentanone have been calculated and the following conclusions have been reached: (1) The proposed model gives rise to optical rotations in agreement with the observed values. (2) The effect of the internal rotation of the methyl group on the molecular rotation [M]_D of 3methylcyclopentanone is large, [M]_D being +44° and 130° for the cis transconformations, respectively. (3) The observed [M]_D may be regarded as composed of two parts, namely, the contribution from the multipole terms and that from the orbital overlapping. The latter corresponds to the "incomplete screening of atomic nuclei", and has been regarded as the most important vicinal action in earlier calculations. In the present work, it is shown, however, that this effect contributes only a few percent of the total [M]_D, unless the perturbing group is very near to the chromophoric group as it is in the case of 2methyl cyclopentanone. The failure of the previous calculations is, therefore, due mainly to the omission of the multipole (especially the quadrupole) terms. (4) The absolute configuration of 3methylcyclopentanone, indicated by the present calculation, is the opposite of the one suggested by Eyring, since he based on the overlapping contribution alone, which bears a sign opposite to that of the total [M]_D (5) That the observed [M]_D of 2,4,5,5tetramethylcyclopentanone is considerably smaller than that of 2,4dimethylcyclopentanone may be due to one or both of the following possibilities: (a) that the conformation of the 2 and 4methyl groups slightly deviates from the transposition in the tetramethyl compound, due to the presence of the two additional methyl groups; (b) that the two additional methyl groups in 5,5positions are not exactly symmetrical with respect to the ring so that they make contributions to [M]_D.  本文在量子力學的單電子旋光理論的鄰近作用問題上,作了如下的貢獻: 1.指出旋光度應由分子中各化學鍵,而不是分子中各原子(如像前人所假定的)對於生色團電子的微擾作用來計算,兩者的主要不同點在於是否考慮鍵的多極矩。 2.建議在旋光度的計算中,共價單鍵可以看作是由兩個處於鍵端的正電荷和一個以單中心狀態函數,表示出來的電子雲所組成。根據這個假定計算了環戊酮的甲基衍生物的旋光度,其結果與實驗值甚爲一致(詳見結果討論)。 3.計算結果證明甲基的內旋轉對於旋光度的影響很大,例如順式和反式構型的3甲基環戊酮的旋光度,應分別為+44°和30°。 4.指定了3甲基環戊酮的絕對構型,其結果Eyring所指定者相反。   << 更多相关文摘 
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