Power moments

The expression (94) can easily be established in an inductive manner, as pointed out in Ref. [56]. Hereafter, we use the notation Q ,s(u) for the sth derivative of Qn(u) with respect to u instead of the more standard notation Q (u). The reason for this is that the notation Q (u) is customarily employed in the theory and applications of the so-called delayed orthogonal polynomials constructed from power moments, autocorrection functions, or signal points // +s = C +s = c +s, where the first s < N points /a, = C, = cr (0 < r < s - 1) are either skipped or simply missing from the full set of N elements. Delayed time signals c +s are thoroughly studied in Refs. [2, 25]. If the expression (84) is differentiated m times and Eq. (94) is used, the following recursion is obtained inductively for the derivatives lQn,m(U) [47] ... 

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,... 

The solution (299) is the continued fraction of the order n. The system (293) is encountered in statistical mechanics [69] when applying the method of power moments to obtain a sequence of approximations to the partition function Q(/3) defined as the Stieltjes integral /0°° exp (-pE)dcp(E), where fi is a parameter proportional to the reciprocal temperature of the investigated system, d[Pg.218]

When H is Hermitian, the power moments ju are real quantities, and even moments are positive both properties are lost in the case of a general relaxation operator. 

It is well known that the determination of abscissas and weights of the Gaussian quadrature from power moments is an exponentially ill-conditioned problem due to the presence of rounding errors. 

By construction, the first row of M contains the power moments, and the first column the modified moments. 

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. 

In solid state theory the power moments of order n are related with closed paths of length n and depend thus on the topology and composition of the lattice. They can be evaluated by a walk-counting technique or by manageable computational procedures. 

Insights into people, places, and times give us clues to our own lives— what we want and what we don t want from our lives. Insights are the shared moments between writer and viewer, the point at which we are closest. In script writing, they are the most powerful moments in the act of telling a dramatic story. 

PB It s as if the play itself and the characters in that dramatic location are themselves the knock 1 felt this when I watched the Lyric production. The characters onstage are themselves on a site that is a kind of outside. They are on an outside dimension of another inside site. There s that powerful moment when Sara enters in that amazing costume It s in scene 3 and the stage direction describes Sara s costume as She wears a ground-length loose coat of stiff sky-blue silk. It is covered with metal spoons. They are stitched to the silk so they cannot swing loosely but can knock against each other when the coat moves. ... 

The general problem of reconstructing a function from its power moments (i.e. the Moment Problem),... 

In addition to the Hankel-Hadamard moment quantization method for the bosonic ground state energy (as well as other states, provided their nodes axe known), two other, powerful, moment quantization methods have been recently developed and used to generate the energy and wavefunction for arbitrary states. Both use the same basis representation, that corresponding to the Multiscale Reference Function (MRF) formalism. 

The concepts described in Section 4.1 apply equally well to particle size distributions and to other quantities commonly characterized by number statistics— for example, dollars, populations, or the dimensions of machined parts. What makes particle size statistics more complicated is that we frequently measure or need to know some quantity that is proportional to particle size raised to a power (moment), such as surface area, which is proportional to d, or mass, which is proportional to d. There is no counterpart of this in number statistics, for what is the meaning of or (people) The need to use moment averages for particle statistics arises because aerosol size is frequently measured indirectly. For example, if you have a basket of apples of different sizes, you could determine the average size by measuring each apple with calipers, summing the results, and dividing by the total mun-ber of apples. This procedure is direct measurement. If, however, each apple were... 

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