Moment problem

N. I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965. 

This demonstrates the equivalence between the harmonic inversion and the moment problem [2],... 

Here, we shall develop a recurrence algorithm for solving a general tridiagonal inhomogeneous system of n linear equations. This will be illustrated for two important classes of problems, such as the power moment problem and spectral analysis, as frequently encountered in physics and chemistry, as well as in linear algebra. 

In the power moment problem [69], there is only one nonzero element of the inhomogeneous column vector D = D, from Eq. (284), that is, D a 5 i, where is the Kronecker 5-symbol. Thus,... 

We have not yet specified if the operator to be handled is Hermitian (real eigenvalues) or whether it is a relaxation operator (eigenvalues either real or in the lower half of the complex plane). Uie moment problem related to a Hermitian operator is addressed as the classical moment problem, while by relaxation moment problem we mean the treatment of relaxation operators. 

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

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-... 

It should be noticed that the original Stieltjes formulation of the moment problem considered the less general case in which the distribution function n(E) could be different from zero only for positive values of E. 

It is not possible to extend right away the results of the classical moment problem to the relaxation moment problem. However, our survey of Section V.B has been done in such a way that it is possible to select which relations maintain their validity in the relaxation moment problem and which are to be disregarded. Thus little remains to be said except for a few comments. 

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). 

Before closing this section, we wish briefly to comment on how to deal with the relaxation moment problem in this case the usual definition of moments,... 

These are momentous problems, and currently many research efforts to elucidate them are being made, whichmay also change our views on evolution. In this context, we have to offer some explanation of the origin of species (see below) that differs from the idea of a systematic development of the major features of evolution of chemotypes. 

Taguani, a. 1999 Hausdorff moment problem and maximum entropy a unified approach. Applied Mathematics and Computation 105, 291-305. 

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. 

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