scholarly journals The occurrence and properties of molecular vibrations with V ( x ) = ax 4

1945 ◽  
Vol 183 (994) ◽  
pp. 328-337 ◽  
Keyword(s):  
Harmonic Oscillator ◽  
Energy Levels ◽  
Wave Functions ◽  
Successive Approximations ◽  
Fourth Power ◽  
Method Of Successive Approximations ◽  
Group Iii ◽  
Modes Of Vibration ◽  
The Difference ◽  
Four Levels

It is shown that in certain modes of vibration of plane rings the potential energy for small displacements is proportional to the fourth power of the displacement, provided that there is free rotation about the bonds of the ring. This type of vibration is termed a ‘fourth-power vibration’. It is likely to occur in cyclobutane and its derivatives, in a number of halides having the formula X 2 Y 6 , and in the hydrides of group III elements. The energies and wave functions of the first four levels of a one-dimensional oscillator with V ( x ) = ax 4 have been derived by a method of successive approximations, and asymptotic formulae are given for the higher levels. The wave functions are qualitatively similar to those of a harmonic oscillator, but the energy levels differ considerably. A comparison is made between energy levels for oscillators with V ( x ) = a q | x q | and different values of q . The selection rule for dipole radiation from a fourth-pow er vibration is discussed. Overtones will be more numerous than in the spectrum of a harmonic oscillator. Estimates are made of the spectrum frequencies of fourth-power vibrations in actual molecules, with special reference to cyclobutane and diborane. For these two molecules there are observed infra-red frequencies of approximately the expected value. The isotope effect should provide a means of discriminating experimentally between harmonic and fourth-power vibrations. The contribution of a fourth-power vibration to any thermodynamic function will differ from that of a harmonic vibration with the same fundamental spectrum frequency. Figures are given for the specific heat, where the difference should be detectable experimentally. In the general case V ( x ) = a q | x q | the energy levels derived from the quantum theory lead to expressions for the thermodynamic functions which agree with the predictions of classical theory at high temperatures.

EXACT SOLUTIONS OF A MANY-ANYON PROBLEM

1992 ◽  
Vol 07 (32) ◽  
pp. 7931-7942 ◽  
Author(s):  
STEFAN V. MASHKEVICH
Keyword(s):  
Exact Solutions ◽  
Stationary State ◽  
Arbitrary Number ◽  
Statistical Parameter ◽  
Analytic Solutions ◽  
Wave Functions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Exact Analytic Solutions ◽  
State Functions

A scheme for obtaining exact analytic solutions of the problem of an arbitrary number of anyons in a harmonic well is developed. Its essence consists in establishing a set of wave functions with the demanded interchange properties followed by finding stationary state functions within this set by the method of successive approximations. The energy of the corresponding states depends linearly on the statistical parameter. The discussion of the obtained states is carried out.

scholarly journals Methods of calculating the deflection of an orthotropic inhomogeneous plate on an elastic basis

2019 ◽  
pp. 106-109
Author(s):  
M. V. Lavrenyuk
Keyword(s):  
Iterative Process ◽  
Poisson Ratio ◽  
Elastic Equilibrium ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Shear Stresses ◽  
Orthotropic Plates ◽  
Winkler Model ◽  
The Difference ◽  
Elastic Basis

The problem of elastic equilibrium of an orthotropic nonhomogeneous rectangular plate on an elastic basis (one-parameter Winkler model) is considered, hingedly fixed from all sides. We use the Navier method for finding the deflection function at each step of the iterative process and perturbation methods and successive approximations as iterative methods for solving the problem. The suitability of the method of successive approximations and the method of perturbations for the numerical solution of the problem of determining the stress-strain state of such a plate, the limits of the applicability of these methods, their accuracy and convergence of the iterative process in solving the deformation problems of heterogeneous orthotropic plates have been analyzed. The dependence of the deflection on the mechanical and geometric parameters of the plate and the base is established. It was found that the Poisson ratio practically does not affect the stress state of the plate (when the Poisson ratio is changed two times, the difference between the intensities of the shear stresses does not exceed 10%), it is possible to consider it as a constant using the methods of successive approximations and disturbances. It is also established that the method of successive approximations and the method of perturbations has a limit on the nature of inhomogeneity, the convergence essentially depends on the nature of the heterogeneity.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

scholarly journals The role of L/W in the solar radiation for Saharian cities

2019 ◽  
Vol 91 ◽  
pp. 05006
Author(s):  
Rami Qaoud ◽  
Alkama Djamal
Keyword(s):  
Solar Radiation ◽  
Solar Energy ◽  
Energy Levels ◽  
Direct Solar Radiation ◽  
Test Field ◽  
Natural Lighting ◽  
The Difference ◽  
The Impact ◽  
The Relationship

The urban fabric of the desert cities is based on the principle of reducing the impact of urban canyons on direct solar radiation. Here comes this research, which is based on a comparative study of the periods of direct solarisation and values of the solar energy of urban canyons via two urban fabrics that have different building densities, where the ratio between L/W is different. In order to obtain the real values of the solar energy (thermal, lighting), the test field was examined every two hours, each three consecutive days. The measurement stations are positioned by the three types of the relationship between L/W, (L≥2w, L=w, L≤0.5w). According to the results, we noticed and recorded the difference in the periods of direct solarization between the types of urban engineering canyons, reaching 6 hours a day, the difference in thermal values of air, reaching 4 °C, and the difference in periods of direct natural lighting, reaching 6 hours. It should be noted that the role of the relationship between L/W is to protect the urban canyons by reducing the impact of direct solar radiation on urban canyons, providing longer hours of shading, and reducing solar energy levels (thermal, lighting) at the urban canyons. This research is classified under the research axis (the studies of external spaces in the urban environment according to the bioclimatic approach and geographic approach). But this research aims to focus on the tracking and studying the distribution of the solar radiation - thermal radiation and lighting radiation - in different types of street canyons by comparing the study of the direct solarization periods of each type and the quantity of solar energy collected during the solarization periods.

Inhibition of cardiac vagal component of baroreflex by group III and IV afferents

1991 ◽  
Vol 260 (3) ◽  
pp. H730-H734 ◽  
Author(s):  
P. N. McWilliam ◽  
T. Yang
Keyword(s):  
Peroneal Nerve ◽  
Baroreceptor Reflex ◽  
Interval Prolongation ◽  
Group Iii ◽  
Control Value ◽  
Pressure Elevation ◽  
Group I ◽  
Afferent Fibers ◽  
The Difference ◽  
The Right

The action of electrically evoked activity in somatic afferent fibers on the sensitivity of the baroreceptor reflex was examined in decerebrate cats. The sensitivity of the reflex was expressed as the difference between the maximum prolongation of R-R interval in response to carotid sinus pressure elevation and the mean of 10 R-R intervals immediately before pressure elevation. The control value of R-R interval prolongation was 192 +/- 50 ms. Stimulation (10 Hz) of group I and II fibers of the right peroneal nerve (evoked volleys recorded from the sciatic nerve) had no effect on R-R interval prolongation (171 +/- 45 ms). Recruitment of group III fibers (10 Hz) conducting at 23.6 +/- 0.65 m/s reduced the prolongation of R-R interval to 52 +/- 14 ms. Recruitment of group IV fibers (10 Hz) conducting less than 2.5 m/s further reduced the prolongation of R-R interval to 1.0 +/- 8.0 ms. It is concluded that the inhibition of the cardiac vagal component of the baroreceptor reflex produced by electrical stimulation of the peroneal nerve is mediated by afferent fibers of groups III and IV.

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