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This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues.


We apply this criterion of algebraic integrability to two examples: finitezone potentials and the elliptic CalogeroMoser system.


In the second example, we obtain a proof of the ChalyhVeselov conjecture that the CalogeroMoser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.


The least upper bound for the degrees of elements in a system of generators turns out to be independent of the number of vector variables.


We consider 3parametric polynomialsPμ*(x; q, t, s) which replace theAnseries interpolation Macdonald polynomialsPμ*(x; q, t) for theBCntype root system.

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 A waterextracted polycaprolactam sample was fractionated from a 2% solution in 85% formic acid at 25℃, water being used as precipitant. Carboxyl endgroup titration and viscosity measurements in 40% H_2SO_4 at 25℃ were carried out for the fractions obtained. The experimental data fit either of the following equations: [η] = 5.92 × 10~(4) M~(0.686) [η] = 2.44 × 10~(5)M + 0.080 in the molecular weight range of 300013000, concentrations being in g/dl. Viscosity data were treated according to the empirical... A waterextracted polycaprolactam sample was fractionated from a 2% solution in 85% formic acid at 25℃, water being used as precipitant. Carboxyl endgroup titration and viscosity measurements in 40% H_2SO_4 at 25℃ were carried out for the fractions obtained. The experimental data fit either of the following equations: [η] = 5.92 × 10~(4) M~(0.686) [η] = 2.44 × 10~(5)M + 0.080 in the molecular weight range of 300013000, concentrations being in g/dl. Viscosity data were treated according to the empirical equatibns lnη_r / c = [η]  β[η]~2c (1) η_(sp) / c = [η] + k＇[η]~2c (2) log(η_(sp) / c) = log[η] + k[η]c (3) The intrinsic viscosities obtained from (1) and (3) are identical, while those obtained from (2) are smaller by 12%. The values of β and k' vary with molecular weight. They increase appreciably with decreasing molecular weight. This anormalous behavior indicates that the value of k' is in no way connected with the solventpower of the solvent for the polymer, when strong solvation is of prime importance for the dissolution of polymer. In 40% H_2SO_4 solution, polycaprolactam shows no appreciable degradation and no polyeletrolyte 'behavior. We have also acertained that the solution is Newtonian by deter mining the efflux times in a capillary viscometer under various external hydrostatic heads. So we concluded that 40% H_2SO_4 is a suitable solvent for the viscometric determination of molecular weight of polycaprolactam. From the intrinsic viscositymolecular weight relation obtained, the Stokes radii of the macromolecules in solution have been calculated according to the theory of Debye and Bueche. The resuk shows that polycaprolactam molecules in 40% H_2SO_4 solution are quite coiled and can be regarded as random coils.  (1)聚己內醯胺試樣在85％甲酸溶液中加水分級沉澱,得到分子量不同的級份,經羧基滴定,並於40％硫酸溶液中,在25°時测定粘度,得到下面的特性粘數分子量關係式: [η]=5.92×10~(4) M~(0.686)或 [η]=2.44×10~(5) M+0.080濃度單位是克/分升,分子量範圍是300013000。 (2)聚己內醯胺的40％硫酸溶液的粘度數據,試用了三種外推公式: lnη_r/c=[η]β[η]~(2)c (1) η_(sp)/c=[η]+k′[η]~(2)c (2) logη_(sp)/c=log[η]+k[η]c (3) 用式(1)和式(3)得到的[η]值相同,式(2)得到的略小12％。β和k′值隨分子量的减小而顯著地增大,這和一般的高聚物——溶劑體系的行為相反。當高分子與溶劑分子間的氫鍵作用是高聚物溶解的主要因素時,用k′值來做溶劑能力的估計,是完全沒有意義的。 (3) 聚己內醯胺在40％硫酸裏,溶液粘度的切變速度依賴性是可以忽略的。我們認為40％硫酸是測定聚己內醯胺的粘均分子量的最合適溶劑。 (4) 從粘度數據依照Debye和Bueche的特性粘數理論,算出聚己內醯胺分子在40... (1)聚己內醯胺試樣在85％甲酸溶液中加水分級沉澱,得到分子量不同的級份,經羧基滴定,並於40％硫酸溶液中,在25°時测定粘度,得到下面的特性粘數分子量關係式: [η]=5.92×10~(4) M~(0.686)或 [η]=2.44×10~(5) M+0.080濃度單位是克/分升,分子量範圍是300013000。 (2)聚己內醯胺的40％硫酸溶液的粘度數據,試用了三種外推公式: lnη_r/c=[η]β[η]~(2)c (1) η_(sp)/c=[η]+k′[η]~(2)c (2) logη_(sp)/c=log[η]+k[η]c (3) 用式(1)和式(3)得到的[η]值相同,式(2)得到的略小12％。β和k′值隨分子量的减小而顯著地增大,這和一般的高聚物——溶劑體系的行為相反。當高分子與溶劑分子間的氫鍵作用是高聚物溶解的主要因素時,用k′值來做溶劑能力的估計,是完全沒有意義的。 (3) 聚己內醯胺在40％硫酸裏,溶液粘度的切變速度依賴性是可以忽略的。我們認為40％硫酸是測定聚己內醯胺的粘均分子量的最合適溶劑。 (4) 從粘度數據依照Debye和Bueche的特性粘數理論,算出聚己內醯胺分子在40％硫酸裏的等效Stokes半徑,說明聚己內醯胺分子在40％硫酸溶液裏的形態,可以看作是無規則的線團。  By heat treatment study, extraction experiments and powder photography, the differential thermal effects occuring below 250°of three amorphous sulfides of antimony are inter preted in this paper. At about 220°, all the three amorphous sulfides produce crystalline Sb_2S_3. In the neighborhood of 160°, the pentasulfide and tetrasulfide segregate into easily extractable sulfur and amorphous Sb_2S_3. The latter change has been identifid in the main with the following reaction: 1/xS_x→1/8S_8+~3 kcal. On the basis... By heat treatment study, extraction experiments and powder photography, the differential thermal effects occuring below 250°of three amorphous sulfides of antimony are inter preted in this paper. At about 220°, all the three amorphous sulfides produce crystalline Sb_2S_3. In the neighborhood of 160°, the pentasulfide and tetrasulfide segregate into easily extractable sulfur and amorphous Sb_2S_3. The latter change has been identifid in the main with the following reaction: 1/xS_x→1/8S_8+~3 kcal. On the basis of the above identification, the pentasulfide and tetrasulfide prepared from Na_3SbS_4·9H_2O are probably complex systems of [Sb_2S_3+S_μ+S_λ] and of [Sb_2S_3+S_μ] respectively. The fact that a synthetic sample containing 91% amorphous Sb_2S_3 and 9% Sμ gives essentially the same differential thermal diagram as the pentasulfide and tetrasulfide strongly supports the proposed model. According to Currie, Bunsen's pentasulfide contains no easily extractable sulfur. However, Bunsen's pentasulfide may not be a chemical individual either, inasmuch as it produces easily extractable sulfur at about 160°. In the light of the proposed theory, Bunsen's pentasulfide is probably a complex system of [Sb_2S_3+2S_μ]. The differential thermal diagram of the pentasulfide is probably a pertinent reference to its quality.  銻的三種無定形硫化物在250°以內的差熱效應,曾於熱處理實驗、硫的提取實驗與粉末法的結果的配合下取得了鑑定與解釋。五硫化二銻與四硫化二銻在160°附近進行一放熱的、析出容易脫去的硫的析硫作用,並在220°附近進行無定形Sb_2S_3的晶化作用。析硫作用主要是下列變化: 1/xS_x→1/8S_8+～3千卡而從硫代銻酸鈉製得的、含有～8％容易脫去的硫的五硫化二銻樣品基本上是下列複雜的無定形體系 [Sb_2S_3+S_μ+S_λ]Currie按照Bunsen方法製得的、不含容易脫去的硫的五硫化二銻樣品可能是下列複雜的無定形體系 [Sb_2S_3+2S_μ] 合成樣品[Sb_2S_3+S_μ]的差熱曲線與所謂五硫化二銻和四硫化二銻者並無二致。文中最後還曾指出了從差熱曲線來檢驗五硫化二銻質量的可能性等。  Even though in the application of adsorption to industry or research one almost never deals with solutions containing only a single solute, yet the number of investigations on the adsorption from mixed solutions is very limited. Following the lead of Freundlich the generally held opinion is that the effect of one adsorbate on the adsorption of another is simple displacement. While this may be true for some special cases, it is undoubtedly too simple to be of general validity, because the various interactions,... Even though in the application of adsorption to industry or research one almost never deals with solutions containing only a single solute, yet the number of investigations on the adsorption from mixed solutions is very limited. Following the lead of Freundlich the generally held opinion is that the effect of one adsorbate on the adsorption of another is simple displacement. While this may be true for some special cases, it is undoubtedly too simple to be of general validity, because the various interactions, such as solidsolvent, solutesolute, solutesolvent, have been neglected in this theory. In the theoretical study of chromatography it is necessary to know the equation for the adsorption isotherm when more than one solute are present in the solution. The most widely adopted equation for this purpose is (x/m)_a=k'_aC_a/(1+k_iC_i) (1) This equation can be easily derived from that for a single adsorbate: x/m = k'C/ (1+kC) (2) by assuming that the presence of an additional solute merely reduces the available surface area of the solid. Formally, this equation is analogous to that of Langmuir for the adsorp tion of gaseous mixtures, replacing pressures by concentrations, but it is actually an empirical equation, because the numerical values of the k's can be obtained only from experimental data and their physical significance is not at all clear. This equation predicts the depression of the adsorption of one solute by another. While it is in line with the current idea, there is no experimental proof of its validity. It is the dual purpose of the present investigation to substantiate or disprove the dis placement theory and to test the applicability of equation (1). We have studied the adsorption from binary solutions of hydrochloric, acetic, and oxalic acids by sugar char. The adsorbates are chosen because of their widely different strength. Sugar char is chosen because it has been shown from previous studies that with this adsorbent the experimental data for single adsorbate follow equation (2). The experimental results are given below: HClCH_3COOH and CH_3COOHH_2C_2O_4 systems: (1) The adsorption of any acid is less than when it is present alone; (2) When the corresponding (x/m)'s are plotted, straight lines with negative slopes are obtained; (3) The slopes of these straight lines vary with the concentration of the acid being displaced; (4) The order of the displacing power, measured by the slope, is HCl> CH_3COOH > H_2C_2O_4, while the order of adsorption is just the reverse; (5) Equation (1) is not valid. HClH_2C_2O_4 system: (6) The adsorption of HCl is decreased; (7) When its concentration is higher than about 0.005 N, the adsorption of oxalic acid is increased; at lower concentrations the adsorption is decreased; (8) The higher the concentration of HCl is, the more pronounced will be the increase of adsorption of oxalic acid; (9) Corresponding (x/m)'s give straight lines whose slopes change sign as C_ox increases beyond 0.005 N. From (3) and (4) it is concluded that the observed decrease of adsorption is not due to simple displacement. Based on the hypothesis that it is primarily the solvent which is displaced with the subsequent shift of equilibrium, a mechanism is proposed to account for the effect of one solute on the adsorption of another, which seems to agree with all the observed results. The increase of the adsorption of oxalic acid disproves conclusively the displacement theory. The fact that there is a critical concentration below which the adsorption of oxalic acid is decreased indicates that there are two opposite influences at work. It is suggested that three factors contribute to the increase of adsorption of oxalic acid: (a) the decrease of dissociation; (b) the lowering of solubility; and (c) electrostatic effect. To determine the relative importance of these factors requires further investigation. Finally, an empirical equation involving three constants has been derived to represent the adsorption of oxalic acid in the presence of hydrochloric acid.  在此工作中我們研究了三種混合酸的吸附。在鹽酸酸的體系中,鹽酸增加糖炭對於草酸的吸附量,倘若草酸的濃度超過0.005N。這個事實否定了在混合酸中各溶質間的作用是彼此頂替的說法。在醋酸草酸及醋酸鹽酸的混合溶液中,各酸的吸附量皆較其單獨存在時低。根據實驗的數據我們指出了各家說法的可疑之點。我們認為在混合溶液中吸附量之減少主要是因為溶質甲頂替了表面上的溶劑,因而使溶質乙的吸附量降低。若是溶質甲能使溶質乙的電離度或溶度降低,或增加溶質乙的Gegenion,則能使乙的吸附量增加。因為Langmuir式的混合吸附公式不能代表吸附之增加,我們提出了一個可以此較滿意的代表實驗結果的三常數經驗公式。各溶質的吸附量皆成直線關係。對此結果我們不能作定量的解釋,雖然此種現象與我們的假設並不矛盾。   << 更多相关文摘 
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