Looking for references for an implicit differentiation formula

Question

In a paper which I submitted to a peer-reviewed math journal in April 2010, I proved a formula for the n-th derivative $\frac{d^n z}{dw^n}$ in terms of (as a polynomial over the integers) of the partial derivatives of a given implicit function, $G(z,w)=0$, with respect to $z$ and $w$ (and negative integer powers of the "separant", $G_z$, the first partial derivative of $G$ with respect to $z$).

This is classic first-semester calculus homework exercise: to compute $\frac{d^n z}{dw^n}$ for n=1 and 2, namely,

$\frac{dz}{dw} = - \frac{G_w}{G_z}$

$\frac{d^2 z}{dw^2} = - \frac{G_{zz}G_{ww}}{G_{zzz}} + 2\frac{G_{zw}G_w}{G_{zz}} - \frac{G_{ww}}{G_z}$

I did so not knowing whether any one had proved the general formula first, because I am busy building on, generalizing, and using this result for other things, including chemical processing.

I have since proved the partial differential generalization of this implicit differentiation formula: i.e. given $G(z,w_1,...,w_N)=0$, compute

$\frac{d^{({u_1}+...+u_N})}{{dw_1}^{u_1}\cdots {dw_N}^{u_N}}z$ as a Laurent polynomial over the integers of the partial derivatives of $G$ with respect to $w_1,...,w_N$, and $z$

I do not have access to most peer-reviewed journals. I have had to make do with Google searches, Wolfram Research's MathWorld, online searches through my county library, and the help from one mathematician friend who has sent me related papers.

Most of the papers my friend sent me concern the Faa da Bruno formula (FdBF) and its generalizations, and the Lagrange Inversion Formula (LIF). Both the FdBF and LIF are very closely related to what I am doing, but they can not be trivially applied to get (my) general formula. (I tried... for about 8 months.) I have studied G.P. Egorychev's book: "Integral Representations of Combinatorial Sums" intensely, especially the back, with the multivariable generalizations of the LIF.

No, I am not in school. This is not a homework problem. This is "free-lance" research. I am not asking for a solution to the problem (as I already solved it independently).

I simply want to know yes or no whether someone has already done this. And, if so, where.

Thank you for the responses, in particular to go to arxiv.org, which I forgot, since I had submitted 2 papers there myself.

Answer 1

Zentralblatt has a sample service where you get 3 responses. I think you might want to look at the first item:

Zbl 1186.92069 Pongor, Gábor; Eöri, János; Rohonczy, János; Kolos, Zsuzsanna Direct inversion in the spectral subspace: a novel method for quantitative and qualitative analysis of chemical mixtures. (English) J. Math. Chem. 47, No. 3, 1085-1105 (2010). MSC2000: *92E99 65F15 92E10

Zbl pre05669072 Wang, Weiping Generalized higher order Bernoulli number pairs and generalized Stirling number pairs. (English) J. Math. Anal. Appl. 364, No. 1, 255-274 (2010). MSC2000: *11-99 05-99

Zbl 1183.74370 Wang, Meng-Fu; Au, F.T.K. Precise integration methods based on Lagrange piecewise interpolation polynomials. (English) Int. J. Numer. Methods Eng. 77, No. 7, 998-1014 (2009). MSC2000: *74S30 65D30 $$ $$

$$ $$ Zbl 1186.92069 Pongor, Gábor; Eöri, János; Rohonczy, János; Kolos, Zsuzsanna Direct inversion in the spectral subspace: a novel method for quantitative and qualitative analysis of chemical mixtures. (English) [J] J. Math. Chem. 47, No. 3, 1085-1105 (2010). ISSN 0259-9791; ISSN 1572-8897

Summary: A novel method, called Direct Inversion in the Spectral Subspace (DISS), has been developed for the quantitative (and partly qualitative) analysis of chemical mixtures. The method belongs to the broad group of supervised classification'' methods: its use necessitates the components'pure'' spectra, either experimental or computed. On the basis of three simple conditions, an elegant, linearized system of equations has been deduced, taking into account a sole restriction via the Lagrange multiplier method. This restriction is seemingly redundant but it has been shown that with its use the unknown normalization constant of the components' descriptive weighted average (CDWA) spectrum can be taken into consideration. The system of linearized equations can be solved repeatedly until convergence. Any kind of spectra can be used; the method does not require the non-negativity of spectral data points. Two versions of the new method have been developed: the normalized and the non-normalized versions regarding the components' spectra. In ideal cases, the non-normalized version of the DISS method provides a mixture's accurate composition due to the iteration for getting the correct norm of the CDWA spectrum. Realistically, the normalized version of the DISS method identifies a mixture's composition within a few molar percentage points accuracy, according to the test results in IR and $^{1}$H-NMR spectroscopy. The normalized method functions without any calibration measurements and needs only a control of accuracy; it is hoped that it will be a useful tool for chemical and biochemical analysis as well as for spectral databases. The DISS method is also useful for qualitative analyses in a limited sense: in the case of computed spectra of the components the set of the de facto components determined could be somewhat wider than those existing in the real system.

MSC 2000: *92E99 Appl. of mathematics to chemistry 65F15 Eigenvalues (numerical linear algebra) 92E10 Molecular structures

Keywords: decomposition of molecular spectra; Lagrange multiplier; quantitative analysis; qualitative analysis; IR; NMR; EPR; UV/Vis; Raman; CD; VCD; hexa-chloro-buta-1,3-diene; dioxane; D-Camphor; L-Menthol; supervised classification; spectral databases