Auxiliary polynomial

If all the elements of a partieular row are zero, then they are replaeed by the derivatives of an auxiliary polynomial, formed from the elements of the previous row. 

Remark 7.30 (Existence of zero-knowledge proof schemes). Zero-knowledge proof schemes exist for all languages in NP under certain computational assumptions, but of course, those schemes have not been constructed and proved with respect to the definition made here. [GoMW91] works with arbitrarily powerful provers and in the non-uniform model, but makes a remark that the proof should also work for provers with arbitrary auxiliary inputs. [Gold93] is for polynomial-time provers and uniform, but, as mentioned, only proves the existence... 

Although it is easy to write down the explicit solution of the system (127), here we shall provide only a qualitative discussion of the solution. The main features are then best demonstrated with the help of a figure. Eliminating idler and auxiliary mode variables from Eq. (127) we get a differential equation of the third order for the annihilation operator of the signal mode. Its characteristic polynomial (on substitution as (t)=as (0) exp (iXt))... 

This identity in the auxiliary variable s means that the function hjyg property that, if it is expanded in a power series in 8, the coefficients of successive powers of s are just the Hermite polynomials H ( ), multiplied by 1/n . To show the equivalence of the two definitions 11-13 and 11-14, we differentiate S n times with respect to s and then let s tend to zero, using first one and then the other expression for S the terms with v < n vanish on differentiation, and those with v > n vanish for s —> 0, leaving only the term with v = n ... 

Let us make the observation that Eqs. (23) and (24) represent an alternative partial wave expansion of the SAPF in which the individual terms are defined by the degrees I of the Legendre polynomials. To distinguish this expansion from the PW/m expansion (15), where the individual terms are defined by pairs of orbital momentum quantum numbers of one-electron wave functions employed in the Cl representation of the pair function, we shall refer to it as auxiliary PW expansion and denote it by the acronym PW/a. This expansion turned out to be well suited for representing the first-order pair functions at the interelectronic cusp. Unlike the PW/m expansion is it not directly related to the Cl approach. Let us stress the important fact that for pairs defined by other than s-electrons the PW/m and PW/a expansions of the second-order energies need not be the same. 

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