8 July 2009 | Draft

Comprehensive Formulations and their Cognitive Challenge

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Annex 2 of Periodic Pattern of Human Knowing: implication of the Periodic Table as metaphor of elementary order (2009)


Introduction
Periodic Table: predictive formulations
Equation of Life
Fractal Mandelbrot set
Gaussian copula
Symmetry group

Introduction

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).

Periodic Table: predictive formulations

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Perez: A seemingly controversial proposal was originally made in French in 1997 by Jean-Claude Perez (Mendeleiev Periodic Table Prediction Equation, 1997-2008).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:

Mendeleev Periodic Table Equation -- Jean-Claude Perez

For Perez [comments and links added]:

  • this equation predicts the number of elements of any layer of period "p" in the table according to the only value of this period "p".
  • beyond this mathematical modeling of the periodic table of the elements, this equation underlines, in its formulation, the "trace" of the 4 fundamental quantum numbers. [principal, azimuthal, magnetic, spin projection; see Quantum Numbers and Electron Configurations]
  • this modeling predicts the structure of the hypothetical extensions of the table of Mendeleev towards possible eléments (real) unknown which would be located beyond the last known radioactive elements.
  • this modeling also makes it possible to imagine an infinity of other elements (virtual) which one could however predict positioning towards the low layers of the table, like their quantum properties.

With respect to the above equation, he argues that when:

  • c(p) is a horizontal layer of elements of the table of Mendeleev [divided into groups],
  • "p" is the period associated with this c(p) layer such as p = [ 1 2 3 4 5 6 7 … ],
  • Int(v) is the whole part of the numerical value "v" (so that, for example, if v=2.35, then Int(2.35)=2)

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:

c(p) = 2 [Int ((p+2) /2 )]*2

Perez gives as examples:

p = Period c(p) = Number of elements
1 2
2 8
3 8
4 18
5 18
6 32
7 32

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: .

955.20 Modular Development of Omnisymmetric, Spherical Growth Rate Around One Nuclear Sphere of Closest-Packed, Uniradius Spheres: The subtraction of the 144 modules of the nuclear sphere set from the 480-module inventory of the vector equilibrium at initial frequency, leaves 336 additional modules, which can only compound as sphere fractions. Since there are 12 equal fractional spheres around each corner, with 336 modules we have 336/12ths. 336/12ths = 28 modules at each corner out of the 144 modules needed at each corner to complete the first shell of nuclear self-embracement by additional closest-packed spheres and their space-sharing domains.

955.21 The above produces 28/144ths = 7/36ths present, and 1l6/144ths = 29/36ths per each needed.

955.30 Possible Relevance to Periodic Table of the Elements: These are interesting numbers because the 28/l44ths and the 116/144ths, reduced to their least common denominator, disclose two prime numbers, i.e., seven and twenty-nine, which, together with the prime numbers 1, 2, 3, 5, and 13, are already manifest in the rational structural evolvement with the modules' discovered relationships of unique nuclear events. This rational emergence of the prime numbers 1, 3, 5, 7, 13, and 29 by whole structural increments of whole unit volume modules has interesting synergetic relevance to the rational interaccommodation of all the interrelationship permutation possibilities involved in the periodic table of the 92 regenerative chemical elements, as well as in all the number evolvements of all the spherical trigonometric function intercalculations necessary to define rationally all the unique nuclear vector-equilibrium intertransformabilities and their intersymmetric-phase maximum aberration and asymmetric pulsations.

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.

Equation of Life

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):

For Perez:

Not only does this law distinguish between CONHSP bio-atoms -- the main building 'blocks of life' -- and the remaining population of all other atoms, but also by providing an uniform and regular numerical scale of integer numbers common to these 6 bio-atoms, to the UTCAG nucleotides of RNA and DNA, and to the 20 amino acids, then this law will allow the UNIFICATION of the 3 "languages" of biology that are DNA, RNA and amino acids.

Then this "common" numerical scale, this "Numerical CODE" will bring unknown new levels of organizations and structures, "meta-levels", kinds of "meta-codes" or "meta-languages" organizing at a large scale all biological structures, genes and whole genomes.

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):

Fractal Mandelbrot set

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

denotes the nth iterate of Pc(z) (i.e. Pc(z) composed with itself n times), the Mandelbrot set is the subset of the complex plane given by:

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:

Gaussian copula

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:

Gaussian copula
Pr =
probability that any two potential sources of risk
(A and B)
in a proposed investment will both default
Φ2 =
couples the individual probabilities
associated with A and B
=
TA = amount of time between now and when A may be expected to default Φ-1 (FA(1)) = probability of how long A is likely to survive before defaulting
TB = amount of time between now and when B may be expected to default Φ-1 (FB(1)) = probability of how long B is likely to survive before defaulting
. γ = correlation parameter, reducing correlation to a single constant

Symmetry group

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)

The proof of the classification of finite simple groups, a kind of periodic table of mathematical symmetry, for example, was announced in 1982. Stretching to over 10,000 pages, it was authored by hundreds of mathematicians - and it turned out to be incomplete. In the early 1990s mathematicians trying to master the argument in its entirety discovered that a portion of the proof was missing. After battling for some years the gap was finally plugged in 2004, but it took a paper whose proof was more than 1200 pages long

Michael Forger and Sebastian Sachse (Lie superalgebras and the multiplet structure of the genetic code. I. Codon representations J. Math. Phys. 41, 5407, 2000)

It has been proposed by Hornos and Hornos [Phys. Rev. Lett. 71, 4401-4404 (1993)] that the degeneracy of the genetic code, i.e., the phenomenon that different codons (base triplets) of DNA are transcribed into the same amino acid, may be interpreted as the result of a symmetry breaking process. In their work, this picture was developed in the framework of simple Lie algebras. Here, we explore the possibility of explaining the degeneracy of the genetic code using basic classical Lie superalgebras, whose representation theory is sufficiently well understood, at least as far as typical representations are concerned. In the present paper, we give the complete list of all typical codon representations (typical 64-dimensional irreducible representations), whereas in the second part, we shall present the corresponding branching rules and discuss which of them reproduce the multiplet structure of the genetic code.

See also: Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication (2008)


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