Moment problem determined

A necessary and sufficient condition for the existence of a solution of the Hamburger moment problem (5.25) is that the Hankel determinants D > 0,... 

In the case > 0 for n = 0,1,2,..., the solution of the Hamburger moment problem may be unique, in which case we speak of a determined moment problem-, or there may be infinitely many solutions and the moment problem is called undetermined. Notice that the possibility of an inde-... 

In Section V, we have formally provided simple expressions [Eqs. (5.14), (5.15), and (5.16)] that allow passing from the moments to the parameters of the continued fractions. From a purely algebraic point of view the situation is satisfactory, but not from an operative point of view, an aspect which has often been overlooked in the literatiu e. Indeed, formulas bt ed on Hankel determinants could hardly be used for steps up to it == 10, because of numerical instabilities inherent in the moment problem. On the other hand, in a variety of physical problems (typical are those encountered in solid state physics ), the number of moments practically accessible may be several tens up to 100 or so the same happens in a number of simulated models of remarkable interest in determining the asymptotic behavior of continued fractions. In these cases, more convenient algorithms for the economical evaluation of Hankel determinants must be considered. But the point to be stressed is that in any case one must know the moments with a... 

Pq( ) = 1]. The expression of D can be recognized as the standard expression of Hankel determinants, wtiich are known to be essentially positive quantities (in the classical moment problem). 

The modified moments are useful whenever it is possible to guess an auxiliary distribution n E) closely simulating the actual one. However, in the construction of the N matrix, the principal ingredients are just the modified moments, whose determination from the power moment is still a notoriously ill-conditioned problem. 

By the moment method we mean the technique of directly using power moments to determine the Green s function and to reconstruct the spectral density. From a mathematical point of view, this problem goes back to the last century (see Chapter III), but the applications in several branches of physics have more recent origin. 

NiO (56) (Section 4.1). The only way to avoid such a calibration apart from extrapolation using a particular shape for f(>e) is to estimate the total spin by integration of the moment density determined by Fourier transformation of single crystal data, which carries its own problems (Section 3.7) as well as being a vastly more time-consuming experiment. This procedure has been carried out (73) however for octahedral and tetrahedral Fe + in yttrium iron garnet (YIG). 

Measurements of relaxation times fall broadly into two classes, those which monitor the populations of some chosen states, and those which measure in some way the impedance of the system to the propagation of a thermal disturbance many laser experiments fall into the first class, whereas ultrasonic dispersion or shock-tube measurements fall into the second. Although artefacts can occur if unsuitable population v. time profiles are used [76.P3], there is, in general, no real difficulty in using equation (2.14) to obtain the vibrational relaxation rate we need not discuss this point further at the moment. Problems may well arise, though, in the determination of rotational relaxation rates in this way, as I will show. 

Sample Problem 8.5 shows how to use bond lengths and dipole moments to determine the magnitude of the partial charges in a polar diatomic molecule. 

Problem Determine the three principal moments of inertia for a water molecule. 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

The Compton scattering cannot be neglected, but it is independent of molecular structure. Then, fitting experimental data to formulas from gas phase theory, the concentration of excited molecules can be determined. Another problem is that the undulator X-ray spectrum is not strictly monochromatic, but has a slightly asymmetric lineshape extending toward lower energies. This problem may be handled in different ways, for example, by approximating its spectral distribution by its first spectral moment [12]. 

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