Hankel determinant

This finding, which was mentioned in Section 4, exhibits a remarkably regular factorability of the general result for H (c0) as a direct consequence of a judicious combination of symmetry of the Hankel determinant (46) and the Lanczos orthogonal polynomials. Thus, given either the set /3 or qVi0, the Hankel determinant H (c0) or equivalently the overlap determinant detS in the Schrodinger or Krylov basis rj can be constructed at once from Eq. (175). This is very useful in practical computations. 

Here, j32 is given exclusively in terms of the determinant of the Schrodinger overlap matrix Hm(c0) = detSm. Recursive numerical computations of the Hankel determinants Hm(c0) and from Eq. (278) can be carried out by,... 

It is convenient to define the Hankel determinants formed with the sequence of complex numbers Mo> Mi> M2> ... 

The elements of are obtained from the elements of D by advancing the subscripts by unity. In the following we will need also the modified Hankel determinants of the type... 

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,... 

All the mathematical apparatus of Hankel determinants and continued fractions expansion apply also to Hermitian or relaxation superoperators. 

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... 

A Hankel determinant D is a function of 2n +1 independent parameters (the moments) yet when constructed explicitly it requires a matrix with (ra -fl) elements. The problem of finding efficient algorithms, which take into account the peculiar persymmetric structure of the Hankel matrices [left diagonals of (S.13) are formed with the same element], has been considered in the literature by several authors. We discuss here in detail a recent satisfactory solution of this problem, obtained within the memory function formalism, and then compare it with other algorithms. 

With respect to the use of Hankel determinants, the memory function PD algorithm is very convenient because it needs only two arrays of 2n storage locations, instead of = storage locations, to construct n steps of the continued fraction. It is thus possible to use multiple-precision arithmetic, when necessary, and to overcome round-ofif errors and numerical instabilities. 

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). 

In dealing with orthogonal polynomials we have encountered Hankel determinants and modified Hankel determinants, which were an important tool in discussing power series and continued fractions. Thus we may expect a close analogy between the theory of orthogonal polynomials and that of continued fractions. This expectation is corroborated by the following theorem. 

The parameters a and can be obtained from the moments by evaluating Hankel determinants or working out product-difference algorithms. The projected density of states n( ) is then given by... 

The functions PJT(cos 9) are associated Legendre functions of the first kind of degree n and order m, and z (kr) denotes any of four spherical Bessel functions. The choice of the spherical Bessel function depends on the domain of interest, that is, on whether we are looking for the solution inside the sphere (r < a) or outside the sphere (r > a). For the internal field we choose z (kr) = j (kr), where j (kr) is the spherical Bessel function of the first kind of order n. The solution for the external field can be written in terms of spherical Bessel functions j kr) and y kr), where the latter is the spherical Bessel function of the second kind, but it is more convenient to introduce the spherical Hankel function /i / (kr) to determine tj/ for the outer field. 

I. is the identity matrix and z is defined by z (k-v)AT), is determined. In the second step, this model is transformed into a discrete-time state space model. This is achieved by making an approximate realization of the markov parameters (the impulse responses) of the ARX model ( ). The order of the state space model is determined by an evaluation of the singular values of the Hankel matrix (12.). 

Various refinements to the basic LPSVD method include using the noise singular values to correct the signal singular values,100, l01,109 restoration of the Hankel structure of X after truncation of the noise singular values,110,111 and regularization of Eq. (103) to determine K.111,112... 

Theorem 3.4 A moment set is said to be realizable if the Hankel-Hadamard determinants (Gautschi, 2004 Shohat G Tamarkin, 1943) are all non-negative ... 

It is now sufficient to modify the second-order moment to 25 (instead of 26, corresponding to a difference of only 4%) to make the moment set unrealizable. The Hankel-Hadamard determinants are now equal to -179352 and -12 362 344, respectively, whereas the difference table (see Table 3.2) presents negative elements in the column containing the second-order differences (namely A2). If this moment set is fed to the PD algorithm, the resulting quadrature is unable to reproduce the moment set. 

Other anisometric viruses have rod-like helical or cylindrical structures, such as tobacco mosaic virus [495,496,509,533] or alfalfa mosaic virus [551,561,562]. Thus cross-sectional parameters can be determined using / xs Q) Q q->o addition to Rq d I 0) data [537,550]. Stuhrmann plots of the / xs data lead to information on the cross-sectional distribution of protein and RNA. Shell models for the cross-section can likewise be made by analogy with the isometric viruses [550,561,562]. The radial scattering density of the cross-section can be calculated by applying the Hankel transformation to the scattering curve [509]. 

To determine the threshold behavior of the cross-sections for scattering in collision channels with m > 0, we express the Hankel functions in Equation 4.20 in terms of the Bessel and Neuman functions. Using the asymptotic expansions of the Bessel and Neuman functions, we find thata kf as ks 0. This yields the following energy dependence of the cross-section near threshold ... 

Hankel, R. F., Gunther, A., Wirth, K.-E., Leipertz, A., Braeuer, A. (2014). Liquid phase temperature determination in dense water sprays using linear Raman scattering. Optics Express. 22(7), 7962-7971. 

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