Potential constants and mean amplitudes of vibrations of N-methyl acetamide

A normal co-ordinate treatment based on general quadratic forcefield has been applied to N-methyl acetamide molecule. In these calculations, the structure parameters of N-methyl acetamide, as given recently by Katzet al., and the Raman and infra-red data obtained by the authors are used and the potential constants have been obtained. The potential energy distribution among the various symmetry co-ordinates of each normal vibration and the mean amplitudes of vibrations have been calculated.     

Vibrational spectra and normal vibrations of acetanilide and N-deuterated acetanilide

The Raman and infra-red spectra of acetanilide and N-methylacetamide and of their N-deuterated compounds have been recorded. The vibrational frequencies of acetanilide have been assigned and the nature of the Amide I, II and III bands in acetanilide and N-methylacetamide has been investigated by the study of the changes in the frequencies of these bands in the spectra of their deuterated compounds. Acetanilide and deuterated acetanilide molecules have been subjected to normal co-ordinate treatment and the mixing up of the skeletal frequencies arising out of the in-plane vibrations has been determined. These results indicate that as in secondary amides, the amide II and amide III bands in acetanilide are due to the combined contribution ofδ (NH) andν (C-N) vibrations, but the contribution ofν (C-N) to the amide II band is less in acetanilide and deuterated acetanilide than in cases of N-methylacetamide and deuterated N-methylacetamide.     

Potential constants,mean amplitudes of vibration,and thermodynamic properties of ethylene oxide,ethylene oxide-d4 and ethylene sulphide

A normal co-ordinate analysis of ethylene oxide, ethylene oxide-d₄ and ethylene sulphide is carried out and the potential energy constants are determined using both the most general quadratic potential function and Urey-Bradley force field. Using the potential energy constants obtained by the former method the mean amplitudes of vibration of these molecules are calculated by the secular equation method. The thermodynamic properties of these molecules are reported for the ideal gaseous state at one atmosphere pressure for eleven temperatures from 100° K. to 1000° K.     

Force constants and mean amplitudes for some anions of MX6 type (X=F,Cl, Br)

Force constants of the Urey Bradley type for twelve MX₆ ^n? ions ofOh symmetry have been reported. Generalized mean square amplitudes and mean amplitudes of vibration for bonded and non-bonded distances have also been calculated. Thermodynamic quantities for these molecules have been calculated at various temperatures from 50° K to 1600° K using a rigid rotator, harmonic oscillator approximation.     

Force constants,generalised meansquare amplitudes of vibration,shrinkage constants and coriolis coupling coefficients of GeH3CCH and GeD3CCH

General Valence force constants have been evaluated for GeH₃CCH and GeD₃CCH. Generalised mean-square amplitudes of vibration have been obtained and the shrinkage constants have been calculated using the perpendicular mean-square amplitudes of vibration. Coriolis coupling constants have also been determined.     

Infrared and Raman spectra and normal vibrations of N,N-dimethylthioformamide and N,N-dimethylthioacetamide

The Raman and infrared spectra of N, N-dimethylthioformamide and N, N-dimethylthioacetamide were recorded and the vibrational frequencies are assigned. The normal co-ordinate treatment of these molecules has been carried out and the potential energy distributions calculated in order to clarify the nature of normal vibrations and investigate the magnitude of mixing up of various skeletal frequencies. The results in regard to the nature of the C-N and C=S stretching frequencies in these thioamides are compared with those of primary and secondary thioamides on the one hand and those of the corresponding ordinary tertiary amides on the other. It has been shown that the force constants of the general quadratic force field type are transferable in the series of molecules of tertiary amides and tertiary thioamides. The lowest frequency in the region of 150 cm.^?1 to 160 cm.^?1 has been recorded by the authors in the Raman spectra of the two tertiary thioamides under consideration. In the normal vibrational analysis of related molecules, the earlier workers only calculated this frequency, but did not record it.     

Microwave spectra and centrifugal distortion constants of orthofluoropyridine

The microwave spectra of orthofluoropyridine due to high J transitions has been analysed in order to determine its centrifugal distortion constants. The final values of rotational and centrifugal distortion constants are: $$\begin{gathered} A = 5870 \cdot 856 \pm 0 \cdot 011 MHz \tau _{aaaa} = - 2 \cdot 46_5 \pm 0 \cdot 016 KHz \hfill \\ B = 2699 \cdot 971 \pm 0 \cdot 005 MHz \tau _{bbbb} = - 0 \cdot 42_2 \pm 0 \cdot 003 KHz \hfill \\ C = 1849 \cdot 244 \pm 0 \cdot 004 MHz \tau _{cccc} = - 0 \cdot 0 (fixed) KHz \hfill \\ \tau _{abab} = - 0 \cdot 61_8 \pm 0 \cdot 015 KHz \hfill \\ \end{gathered} $$ Using these constants a number of transitions have been assigned upto J = 60.     

Lipschitz constants for iterates of mean lipschitzian mappings

We present a sharp evaluation of the Lipschitz constant for every iterate for mean lipschitzian mappings, solving a problem posed by K. Goebel and M. A. Japón Pineda.     

Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation

The purpose of this paper is to present efficient and accurate analytical expressions for large amplitude free vibration and post-buckling analysis of unsymmetrically laminated composite beams on elastic foundation. Geometric nonlinearity is considered using Von Karman’s strain–displacement relations. Besides, the elastic foundation has cubic nonlinearity with shearing layer. The nonlinear governing equation is solved by employing the variational iteration method (VIM). This study shows that the third-order approximation of the VIM leads to highly accurate solutions which are valid for a wide range of vibration amplitudes. The effects of different parameters on the ratio of nonlinear to linear natural frequency of beams and the post-buckling load–deflection relation are studied.     

Mean amplitudes of vibration and coriolis coupling constants of some XY3Z type silicon compounds

A theoretical treatment of the generalised mean-square amplitudes of vibration of some silicon compounds of XY₃Z type molecules is presented. The numerical values evaluated for the parallel and perpendicular mean-square amplitudes for all the bonded and non-bonded atom pairs are tabulated. The L matrix elements and Coriolis coefficients have also been evaluated for these molecules.     

The Lebesgue constants for Fourier-Bessel series,mean convergence,and the order of singularity of a Sturm-Liouville equation

Questions of mean convergence of classical orthogonal expansions and rates of divergence of their Lebesgue constants are dealt with, under two aspects. The first aspect is that the known results for Jacobi, Laguerre, and Fourier-Bessel series can be seen to be closely related to each other with respect to the kind of singularities in their Sturm-Liouville equations. The second, and the main aspect is to show that the rate of divergence of the Lebesgue constants for Fourier-Bessel series, which were unknown so far, fits well into this interpretation. For the latter purpose we use the Hankel translation in order to reduce the kernel of the Fourier-Bessel partial sum to a function of one variable, a representation of which is derived by the residue calculus. This method of proof is also discussed in connection with the methods used for the other orthogonal systems and with possible generalizations to more general eigenfunction expansions.     

Raman and infrared spectra and normal vibrations of N-methylpropionamide and N-deuterated N-methyl propionamide

Raman and infrared spectra of N-methyl-propionamide and N-deuterated N-methyl-propionamide were recorded in various states of aggregation—pure amide, in solution and vapour state—and the frequency changes of the functional groups were studied. The implane skeletal vibrations of N-methyl-propionamide and its deuterated compound were calculated treating these molecules as six-body problems. The distribution of potential energy of each normal mode among various symmetry co-ordinates was calculated and the nature of the amide I, II and III bands and the mixing up of the inplane vibrational frequencies of N-methyl-propionamide and the deuterated amide are discussed on a quantitative basis.     

On proportional constants of the mean value of class numbers of quadratic extensions

In this article, we give a refinement of the mean value theorem for the class number of quadratic extensions obtained by Goldfeld-Hoffstein and Datskovsky. More specifically, we determine the proportional constants of the mean value for fields that satisfy any local conditions including wild ramification at places dividing .

     

Optimal boundary control of string vibrations by a force under elastic fixing

We study a boundary control problem based on a mixed problem with an inhomogeneous condition of the second kind at the left end of a string with elastically fixed right end. The difficulty in the solution of that problem is that the fixing condition is absent. Therefore, in addition to a constraint that is an equality of functions in the class L ₂, we need one more condition, to which V.A. Il’in refers as a condition of coordination of the initial and terminal displacements. We develop a new optimization method based on the extension of the terminal conditions to the interval [−T,T]. This permits one to minimize the integral of the squared boundary control. A control minimizing this energy integral is written out in closed form.     

Projection of diffusions on submanifolds: Application to mean force computation

We consider the problem of sampling a Boltzmann‐Gibbs probability distribution when this distribution is restricted (in some suitable sense) on a submanifold Σ of ?ⁿ implicitly defined by N constraints q₁(x) = ? = q_N(x) = 0 (N < n). This problem arises, for example, in systems subject to hard constraints or in the context of free energy calculations. We prove that the constrained stochastic differential equations (i.e., diffusions) proposed in [7, 13] are ergodic with respect to this restricted distribution. We also construct numerical schemes for the integration of the constrained diffusions. Finally, we show how these schemes can be used to compute the gradient of the free energy associated with the constraints. © 2007 Wiley Periodicals, Inc.