8 July 2009
Comprehensive Formulations and their Cognitive Challenge
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Periodic Table: predictive formulations
Equation of Life
Fractal Mandelbrot set
The specific concerns here on global formulations form part of continuing interest summarized in Unexplored Potential of Mathematics and Geometry -- in reframing psycho-social challenges (2008).
As noted above, an extensive summary has recently been produced (D. H. Rouvray, et al., The Mathematics of the Periodic Table, 2005). The three examples considered below are those of Jean-Claude Perez, Jozsef Garai and R. Buckminster Fuller.
Perez: A seemingly controversial proposal was originally made in French in 1997 by Jean-Claude Perez (Mendeleiev Periodic Table Prediction Equation, 1997-2008). Perez had the idea to seek a single mathematical equation which would organize the information of the most heterogeneous table of science -- generating and predicting its structure.
The equation of Jean-Claude Perez (Generic Predictive Equation of Mendeleiev's Periodic Table, 1997-2008) is of the following form:
For Perez [comments and links added]:
With respect to the above equation, he argues that when:
then c(p), from the equqtion, is the number of elements contained in the c(p) layer of order p (by applying the above formula), namely:
Perez gives as examples:
Garai: Jozsef Garai (Mathematical Formulas Describing the Sequences of the Periodic Table, International Journal of Quantum Chemistry, 2008), notes the suggestion by Anton van den Broek that the fundamental organizing principle of the table is not the weight but rather the nuclear charge. The charge distribution of the nucleus affects the electron density distribution of the atoms, thus the sequence of the nuclear charge distribution might show resemblances to the periodicity of the elements. Garai focuses on the periodicity of the nuclear charge occurring in the structural development of a double tetrahedron nucleus as revealing the periodicity of the elements. He derived an analytical solution describing this periodicity, noting that that the number of elements in the period n is known as:
but notes the proposal of a new formula by Eugene S. Kryachko (International Journal of Quantum Chemistry, 2007, 107, 372):
However Garai noted the recognition that no numerical solutions describing the fundamental and atomic number sequences of the periodic table were known. He had himself proposed a double tetrahedron shape with alternately arranged protons and neutrons in face-centered cubic lattice for the structure of the nucleus (The double tetrahedron structure of the nucleus, 2003). This reproduced the symmetry of both quantum mechanics and the periodic system with no discrepancy. On this basis he derived an analytical solution for the number of charges in the shell and the nucleus thereby describing the sequences of the periodic table.
Garai shows the the relationship between the periods (n) and the sequence numbers (m) to be described as:
He then shows how the atomic number sequence of the periodic table can be described as:
or, substituting for m (from above), the atomic number sequence for a given period n, is expressed by:
Garai thereby reproduces the fundamental, periodic, and atomic number sequences of the periodic table.
Buckminster Fuller: In his magnum opus, R. Buckminster Fuller (Synergetics: explorations in the geometry of thinking, 1975-1979), presented as a "co-ordinate system of the Universe", all phenomena reduce to geometric-energetic constructs based on the tetrahedron (4-sided), the octet truss (8-sided) and the coupler (8-faceted with 24 phases). Fuller notes the Possible Relevance to Periodic Table of Elements, arguing, as follows, that the 8-face, 24-phase coupler underlies the 8-fold division of the chemical elements on the Mendeleyev Periodic Table: .
Arguably Garai had effectively provided an analytical solution to the earlier structural intuition of Buckminster Fuller (without recognizing his articulation of it). No reconciliation between that of Perez and Garai has been suggested.
Jean-Claude Perez subsequently integrated his preoccupation with the Periodic Table with his explorations of a possible numerical structure of DNA, genes and genomes, the golden ratio and Fibonacci numbers laws, and has recently proposed an Equation of Life (2008), as summarized in book form (Codex Biogenesis; les 13 codes de l'ADN, 2009). The equation of Jean-Claude Perez (Is there an equation of life?, 2008) is of the following form, explained in detail in Codex Biogenesis; les 13 codes de l'ADN (2009):
In other writings, whose themes are now integrated into Codex Biogenesis; les 13 codes de l'ADN (2009), Perez relates (in documents originally in French):
In mathematics, the Mandelbrot set is a set of points in the complex plane, the boundary of which forms a fractal. It can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2+ c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z0= 0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets. The Mandelbrot set is then defined as the set of all points c such that the above sequence does not escape to infinity.
More formally, if
Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number c either belongs to M or it does not. A picture of the Mandelbrot set can be made by colouring all the points c which belong to M black, and all other points white. The more colourful pictures usually seen are generated by colouring points not in the set according to how quickly or slowly the sequence
diverges to infinity. The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials Pc(z). That is, it is the subset of the complex plane consisting of those parameters c for which the Julia set of Pc is connected.
The potential significance of the Mandelbrot set has been discussed in:
The innovative formula of David X. Li with regard to the Gaussian copula function is of interest since its successful use is alleged to be at the root of the overconfidence of the global financial community in taking the high orders of investment risk which led to the global financial crisis of 2008, and its consequences. It is admirably described by Felix Salmon (Recipe for Disaster: the formula that killed Wall Street, Wired, 17.03, March 2009) -- or on the title page of the issue as The Secret Formula that Destroyed Wall Street. As Li had indicated in 2005 "Very few people understand the essence of the model" (Mark Whitehouse, Slices of Risk, The Wall Street Journal, 12 September 2005). A second description is offered by Kevin Drum (The Gaussian Copula, Mother Jones, 24 February 2009).
Li's original paper (On Default Correlation: A Copula Function Approach, Journal of Fixed Income 9, 2000, pp. 43-54) was the first appearance of the Gaussian copula models for the pricing of collateralized debt obligations (CDO's). This quickly became a tool for financial institutions to correlate associations between multiple securities -- allowing CDOs to be accurately priced for a wide range of investments that were previously too complex to price, such as mortgages. In this respect they were at the core of the subprime crisis.
Expressed succinctly, the formula is:
An articulated expression of the formula is:
The potentially fundamental psychosocial implications of recent symmetry groups work, and the challenge for their comprehension, have been discussed in:
Marcus du Sautoy (Burden of Proof, New Scientist, 26 August 2006)
Michael Forger and Sebastian Sachse (Lie superalgebras and the multiplet structure of the genetic code. I. Codon representations J. Math. Phys. 41, 5407, 2000)
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