PRECISION OF DETERMINATION USING SINGLE WAVELENGTH APPARATUS INSTEAD OF DUAL-WAVELENGTH APPARATUS Ⅱ. DETERMINATION OF TWO COMPONENT MIXTURES BY DUAL-WAVELENGTH SPECTROPHOTOMETRY

Schiff Bases and Secondary Amines Containing Two Benzo-15-crown-5 Units II. Potassium Ion Selective PVO Membrane Electrodes Based on Bis-Crown Ethers as Carrier

Study of Novel Compounds Containing Both Phosphorus and Pyrethroid Skeleton Ⅱ. Synthesis of Compounds Containing Bicyclic Phosphate and 3-Substituted-2, 2-dimethylcyclopropane Carboxylate and Study of Its Reaction

The present paper provides the Raman spectra of YVO4 crystals doped with Tm3+ , Ho3+ and both of them, measured at room temperature and with the laser projections parallel and perpendicular to axis C.

The addition of penicillin-K increases the viscosity of both theupper phase and down phase of ASTP 1 region and ASTP2 region, and also changesthe distribution of each component in the above two phases.

The results indicated that In(OTf)3 and FeCl3 6H2O were both good catalysts for the synthesis of bis(indolyl)methanes, but the catalytic system of FeCl3 6H2O showed better results than that of In(OTf)3 in the recycle of Lewis acids.

We obtain a criterion for rational smoothness of an algebraic variety with a torus action, with applications to orbit closures in flag varieties, and to closures of double classes in regular group completions.

A Deodhar-type stratification on the double flag variety

We describe a partition of the double flag variety G/B+ × G/B- of a complex semisimple algebraic group G analogous to the Deodhar partition on the flag variety G/B+.

Also, we discuss possible connections to the positive and cluster geometry of G/B+ × G/B-, which would generalize results of Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double Schubert cells.

Hardy Spaces on the Plane and Double Fourier Transforms

Motivated by the physical concept of special geometry, two mathematical constructions are studied which relate real hypersurfaces to tube domains and complex Lagrangian cones, respectively.

The theory is applied to the case of cubic hypersurfaces, which is the one most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special K?hler manifolds.

Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K].

We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system.

There are two well known combinatorial tools in the representation theory ofSLn, the semi-standard Young tableaux and the Gelfand-Tsetlin patterns.

The cohomology algebra of the classifying space of a compact Lie group admits the structure of ann-Hopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also ann-Hopf algebra.

In a first step we prove that the Satake compactification of the modular variety of genus 2 and level 3 is the normalization of the dual of the Burkhardt quartic.

The second part consists in the normalization of the Burkhardt dual.

Finally, we study their reducibility of the action of the Casimirs on the zero-weight spaces of self-dual g-modules and obtain complete classification results for g = sln and g2.

This boils down to a Duistermaat-Heckman exact stationary phase calculation, involving a Poisson structure on the dual symmetric space G0/K discovered by Evens and Lu.

Fr?nsdal [Fr1, Fr2] made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebraUq(g).

We find closed formulas for the 1-point functions in both cases in terms of Jacobi θ-functions.

We characterize irreducible Hermitian symmetric spaces which are not of tube type, both in terms of

This partition is a refinement of the stratification into orbits both for B+ × B- and for the diagonal action of G, just as Deodhar's partition refines the orbits of B+ and B-.

Hardy spaces of analytic functions are studied both on strongly pseudoconvex domains in ?n and on domains of finite type in ?2.

A new approximation method is proposed in this article for the discussion of molecular structures,and this new method includes the two well-known theories,molecular orbital theory and electron-pair bond theory as two special cases.Let a molecule have n bonds and let the ith bond be described by the anti-symmetrical two-electron bond function ψ_i(v_(2i-1),v_(2i)).(If there exist one- electron,three-electron or many-electron bonds,they can be similarly described by the corresponding one-electron,three-electron...

A new approximation method is proposed in this article for the discussion of molecular structures,and this new method includes the two well-known theories,molecular orbital theory and electron-pair bond theory as two special cases.Let a molecule have n bonds and let the ith bond be described by the anti-symmetrical two-electron bond function ψ_i(v_(2i-1),v_(2i)).(If there exist one- electron,three-electron or many-electron bonds,they can be similarly described by the corresponding one-electron,three-electron or many-electron bond func- tions.) Then the stationary state of the molecule is represented by the follow- ing wave function Ψ, where the summation is over all permutations of 1,2,……,2n except those within the interior of the functions,since each ψ_i is already anti-symmetrical.Obviously (2~n/((2n)/!))~(1/2) is the normalization factor. By quantum mechanics the energy of the molecule equals (1) here H_i,T_(ij) and S_(11)' are respectively the following three kinds of operators, (2) (3) (4) The third term of equation (1) is the exchange integral of electrons 1 and 1', while (1,2') is that of electrons 1 and 2'.According to the definition of bond functions,ψ_i may be written as (5) Substituting equation (5) into equation (1) and carrying out the integration over spin coordinates,we obtain (6) It can be easily seen from equation (6) that the combining energy of a mole- cule consists of two parts,one being the binding energy of the bonds represent- ed by the first term of equation (6),and the other being the interaction energy of the bonds denoted by the second term of that equation. If we choose certain functions φ_i~('s) involving several parameters and substi- tute them into equation (6),we may determine the values of those parameters by means of the variation principle. For the discussion of bond interaction energies,we develop a new method for the evaluation of certain types of three-center and four-center integrals.The interaction energy of a unit positive charge and an electron cloud of cylindrical- symmetry distribution may be written as (7) where (8) and R_0~2=a~2+b~2+c~2 The interaction energy of two electron clouds both of cylindrical-symmetry distributions with respect to their own respective axes is evaluated to be (9) (10) where is to sum over j from zero to the lesser value of n-2i and m, is to sum over i from zero to the integral one of n/2 and (n-1)/2,and is to sum over all cases satisfying the relation =m-j,while b_(n,n-2i) represents the coefficient of x~(n-2i) in the n th Legendre polynomial.

Although James and Coolidge (1933) solved the molecular hydrogen problem in almost complete agreement with experiment by using a 13-term 2-electron eigenfunction, his method can hardly be applied to more complex molecules. For this and other reasons (Coulson, 1938), it is still desirable to obtain a good one-electron eigenfunction, i.e., molecular orbital, for the hydrogen molecule. The best molecular orbital treatment available in the literature was given by Coulson (1938), who used a trial eigenfunction in...

Although James and Coolidge (1933) solved the molecular hydrogen problem in almost complete agreement with experiment by using a 13-term 2-electron eigenfunction, his method can hardly be applied to more complex molecules. For this and other reasons (Coulson, 1938), it is still desirable to obtain a good one-electron eigenfunction, i.e., molecular orbital, for the hydrogen molecule. The best molecular orbital treatment available in the literature was given by Coulson (1938), who used a trial eigenfunction in elliptical coordinates involving 5 parameters and obtained 3.603 eV for the binding energy of H_2, which is to be compared with the ex- perimental value of 4.72 eV. In the present investigation we have proposed a new type of trial eigenfunction for the molecular orbital: (1) with p = centers a, b, g, c, d,…… i = electron 1 or 2 (2) where the p's are centers along the bond axis a-b (Fig. 1). In this simple problem both the Fock and Hartree methods yield the same result. The molecular orbital ψ must satisfy the following integral equation: (3) where ε is the energy of the molecular orbital, F is the Fock operator which is equal to H+G(1), while H is the one-electron Hamiltonian operator: H = -1/2▽~2-1/r_a-1/r_b (4) and G(1) is the interaction potential (5) Substituting (1) into (3), we obtain the linear combination coefficients c_p, which must satisfy the following secular equation: (6) where is the solution of the secular determinant and The F_(pq)'s are not at first known, but depend upon the c_p's. A method of successive approximation must therefore be adopted. A set of c_p values may be assumed, the F_(pq)'s calculated, the secular determinant (7) solved, and a new set of c_p values found. This process is repeated until a "self-consistent" set of c_p values is obtained. The above procedure was first proposed by Roothaan (1951), not for H_2 but for more complex molecules. It was called by him the "LCAO SCF (linear combination of atomic orbitals self-consistent field) method". The new feature of the present investigation is that we not only use LCAO but also LCNAO (linear combination of non-atomic orbitals, such as x_g, x_c, x_d, …). The order of secular determinant (7) may be reduced to half if we replace the eigen- functions x_a, x_b .... by their symmetrical and anti-symmetrical linear combinations x_a + x_b and x_a-x_b. Numerical calculations have been carried out both for the three- and the two-centered molecular orbitals. The three-centered molecular orbital is (10) (11) where S_(ab) and S(ag) are the overlapping integrals between x_a and x_b, and between x_a and x_g respectively. The parameters a and g have 'been obtained to give minimum energy by the method described above. They are a=l.190, g=0.22, and the binding energy is 3.598 eV, which is almost as good as that obtained by Coulson (3.603 eV) using a trial function of 5 parameters. The two-centered molecular orbital is (12) (13) which gives a maximum binding energy of 3.630 eV for a=1.190 and R~(ac)=R(bd)=0.105 (Fig. 1). This result is 'better than Coulson's. If we allow different values for the ex-ponent α in x_a and x_g in equation (11), or if we use a four-centered molecular orbital, such as ψ=a(x_a + x_b) + b(x_c + x_d) with four parameters, namely α_a=α_b, α_c=α_d, R_(ac)=R_(bd) and the ratio b/a, it is possible to obtain a still better result. Extension of the present method to the treatment of more complex molecules is now under investigation.

Karrer and Enslin reported that the structure of alstyrine (S_(19)H_(22)N_2), a selenium- dehydrogenation-degradation product obtained by Sharp from the alkaloid alstonine, is identical with that of corynanthyrine, α-[2-(4, 5 diethylpyridyl)]-β-ethylindole (Ⅱ), despite the fact that the melting points of these two substances have a 5°difference. In his investigation, the author of the present paper has synthesized two homologues of (II), α-(2-pyridyl)-β- methylindole (Ⅲ) and α-(2-pyridyl)-β-ethylindole (Ⅳ),...

Karrer and Enslin reported that the structure of alstyrine (S_(19)H_(22)N_2), a selenium- dehydrogenation-degradation product obtained by Sharp from the alkaloid alstonine, is identical with that of corynanthyrine, α-[2-(4, 5 diethylpyridyl)]-β-ethylindole (Ⅱ), despite the fact that the melting points of these two substances have a 5°difference. In his investigation, the author of the present paper has synthesized two homologues of (II), α-(2-pyridyl)-β- methylindole (Ⅲ) and α-(2-pyridyl)-β-ethylindole (Ⅳ), finding that the former has almost the same ultraviolet absorption spectrum as that of alsyrine and that, like alstyrine, the metho- sulphate of the compounds (Ⅲ) and (Ⅳ) gives the same red colour reaction upon treatment with dilute sodium-hydroxide solution. The Fischer indole synthesis to cyclize the phenylhydrazone of the corresponding alkyl- 2-pyridylketone in the presence of mineral acid was employed by the author in the preparation of compounds (Ⅲ) and (Ⅳ). Besides, three a-carboline derivatives, namely, 2-(cyclohexylmethyl)-β-carboline (Ⅻ), 2-[(4'-methylcyclohexyl)-methyl]-β-carboline (XIII) and 2-[(2'-methylcyclohexyl)-methyl]- β-carboline (XIV), have also been synthesized; none of them is identical with alstyrine in physical and chemical properties. The preparation of compounds (Ⅻ), ((XIII)) and ((XIV)) was carried out according to the method originated by Bischler and Napieralski. This involved the condensation of tryptamine with a suitable acid, or acid chloride, first to form an amide, which was then cyclized with phosphorus pentoxide, and finally dehydrogenated .partially with selenium. The melting points of the two α-(2-pyridyl)-β-alkylindoles, three β-carbolines, their inter- mediates, and a few derivatives of theirs are as follows: ■