10 thoughts on “So many posts to do, so little time.”

1. Market fundamentalist admitting that US needs government restrictions to curb trade and restore the balance! This is apocalypse anyway you look at it.

   Your criticism of Gurumurthy then was invalid and with the latest point with your admission of the precarious US currency situation, it is meaningless. And you got much of the data wrong in the rebuttal
   Indian savings rate is 25% not 30% and it is much lower than 40% in East Asian countries
   The 50 billion v/s 20 billion is not what you say it is, India does not receive that much of FDI (7 billion is closer to reality) that is the total investments made by each country (GSecs plus FII plus FDI).
   And if you think that FDI funds growth, it is well to note that China gets only about 40 billion FDI, as against about 440 billion it invests indegenously (Now don’t put words into my mouth, I am not arguing against FDI or denying its benefits, just saying that it is not a growth engine.)
   World over Financial investments are many times greater than FDI and most of the financial investments flow into US and developed countries as Gurumurthy has pointed out.

2. I could be wrong about every single fact and my criticism of Gurumurthy would still be valid because my major criticism of him is not that he got his facts wrong. Gurumurthy could be right about every one of his facts. He is in fact right in his analysis of the US, as I had mentioned in the post. But what policy prescriptions for India follow from his analysis? As far as I can see, it is, 1) Save 2) Run a trade deficit (at least against the US) 3) Get Foreign Investment in India. Running a trade surplus against the US, or holding lots of dollars (“saving” in Gurumurthy’s lexicon) is suicidal according to Gurumurthy’s own analysis. But that is what his advice seems to be, though he is never clear.

   Look at it another way. Suppose a bank is about to go bankrupt. Would your advice to a customer of that bank be 1) Maintain a savings account with the bank because “it is good to save” or 2) Get your money out of the bank asap. See if you can get a loan while you are at it?

3. http://www.liberalpartyofindia.org/

   Did you know about this?

4. To start off with, Gurumurthy’s article (including the Mahabharata bit) was about US and not about India, I couldn’t notice any suggestion for India, definately not the ones you said -1) Save 2) Run a trade deficit (at least against the US) – The latter one would be a blasphemy for a Swadeshi activist.
   He is plainly refering to the fact that most countries of the world are clamoring for dollar deposits that give dollar the strength to buy without any current export capability or even the savings (=investments) that can generate any future economic power

5. Corrector, so what are we arguing about?

   Lakshmi, I do know about it. As Nitin Pai says (in hindi) such efforts are in the same state the Indian National Congress was in before the arrival of Gandhiji.

6. Summation of the Debate on the New Real Number System and the Resolution of Fermat’s last theorem – by E. E. Escultura

   The debate started in 1997 with my post on the math forum SciMath that says 1 and 0.99… are distinct. This simple post unleashed an avalanche of opposition complete with expletives and name-calls that generated hundreds of threads of discussion and debate on the issue. The debate moved focus when I pointed out the two main defects of Andrew Wiles’ proof of FLT and, further on, the discussion shifted to the new real number system and the rationale for it. Naturally, the debate spilled over to many blogs and websites across the internet except narrow minded ones that accommodate only unanimous opinions, e.g., Widipedia and its family of websites as well as websites that cannot stand contrary opinion like HaloScan and its sister website, Don’t Let Me Stop You. SciMath stands out as the best forum for discussion of various mathematical issues from different perspectives. There was one regular at SciMath who did not debate me online but through e-mail. We debated for about a year and I learned much from him. The few who only had expletives and name-calls to throw at me are nowhere to be heard from.

   There was one unsigned feeble attempt from the UP Mathematics Department to counter my arguments online. But it wilted without a response from the science community because it lacked grasp of what mathematics is all about.

   The most recent credible challenge to my positions on these issues was registered by Bart van Donselaar in the online article, Edgar E. Escultura and the Inequality of 1 and 0.99…, to which I responded with the article, Reply to Bart van Donselaar’s article, Edgar E. Escultura and the inequality of 1 and 0.99…; a website on the Donselaar’s paper has been set up:

   http://www.reddit.com/r/math/comments/93n3i/edgar_e_escultura_and_the_inequality_of_1_and/

   and the discussion is coming to a close as no new issues are being raised. Needless to say, none of my criticisms of Wiles’ proof of FLT or my critique of the real and complex number systems have been challenged successfully across the internet. In peer reviewed publications there is not even a single attempt to refute my positions on these issues.
   We highlight some of the most contentious issues of the debate.
   1) Consider the equation 1 = 0.99… that almost everyone accepts. There are a number of defects here. Among the decimals only terminating decimals are well-defined. The rest are ill-defined or ambiguous. In this equation the left side is well-defined as the multiplicative identity element while the right side is ill-defined. The equation, therefore, is nonsense.
   2) The second point is: David Hilbert already knew almost a century ago that the concepts of individual thought cannot be the subject matter of mathematics since they are unknown to others and, therefore, cannot be studied collectively, analyzed or axiomatized. Therefore, the subject matter of mathematics must be objects in the real world including symbols that everyone can look at, analyze and study collectively provided they are subject to consistent premises or axioms. Consistency of a mathematical system is important, otherwise, every conclusion drawn from it is contradicted by another. In order words, inconsistency collapses a mathematical system to nonsense. Consider 1 and 0.99…; they are certainly distinct objects like apple and orange and to write apple = orange is simply nonsense.
   3) The field axioms of the real number system are inconsistent. Felix Brouwer and I constructed counterexamples to the trichotomy axiom which means that it is false. Banach-Tarski constructed a contradiction to the axiom of choice of the field axioms, a variant of the completeness axiom. One version says: if a soft ball is sliced into suitably little pieces and rearranged without distortion they can be reconstituted into a ball the size of Earth, a topological contradiction.
   4) Vacuous concept generally yields a contradiction. For example, consider this vacuous concept: the root of the equation x^2 + 1 = 0. That root does not exist and yet it is denoted by i = sqrt(-1). The notation itself is a problem since sqrt is a well-defined operation in the real number system that applies only to perfect square. Certainly, -1 is not a perfect square. Mathematicians extended the operation to non-negative numbers. However, the counterexamples to the trichotomy axiom show at the same time that an irrational number cannot be represented by a sequence of rationals. In fact, a theorem in my paper, The new mathematics and physics, Applied Mathematics and Computation, 138(1), 127 – 149 [4], says that the rationals and irrationals are separated, i.e., the union of disjoint open sets (see also [6].
   At any rate, if one is not convinced of the mischief that vacuous concept can play, consider this:
   i .= sqrt(-1) = sqrt1/sqrt(-1) = 1/i = -i or i = 0. 1 = 0, and both the real and complex number systems collapse. To hide this contradiction some mathematicians invented two supposed roots, the other i = -sqrt(-1) neither of which exists but choose the latter which does not yield a contradiction. There is no logical or mathematical reason for this choice other than to hide the contradiction. Both of them are nonsense anyway since they are both ill-defined.

   5) With respect to Andrew Wiles’ proof of FLT it has two main defects: a) Since FLT is formulated in the inconsistent real number system it is nonsense and, naturally, the proof is also nonsense. The remedy is to first remove the inconsistency of the real number system which I did and reformulate FLT in the consistent number system, the new real number system. b) The use of complex analysis deals another fatal blow to Wiles’ proof. The remedy for complex analysis is in the appendix to the paper, The generalized integral as dual to Schwarz Distribution, in press, Nonlinear Studies [5].

   6) By reconstructing the defective real number system into the contradiction-free new real number system and reformulating FLT in the latter, countably infinite counterexamples to it have been constructed showing the theorem false and Wiles wrong.

   7) In the course of making a critique of the real number system some new results have been found: a) Gauss diagonal method of proving the existence of nondenumerable set only generates a countably infinite set; b) as of this time no nondenumerable set exists; c) only discrete set has cardinality, a continuum has none..

   8) The new real number system is a continuum, countably infinite, non-Hausdorff and Non-Archimedean and the subset of decimals is also countably infinite but discrete, Hausdorff and Archimedean. The g-norm simplifies computation considerably.

   Finally, we note that all the issues about the new real number system, my critique of Wiles’ proof of FLT and my counterexamples to FLT to prove it false have been debated thoroughly in cyberspace during the last 12 years and ALL resolved COMPLETELY in my favor. Not a single hole has been punched on my entire work.

   References

   [1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 – 61.
   [2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
   Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
   [3] Escultura, E. E. (1997) Exact solutions of Fermat’s equation (Definitive resolution of Fermat’s last theorem, Nonlinear Studies 5(2), 227 – 2254.
   [4] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
   [5] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
   [6] Escultura, E. E., The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations 17 (2009), 59 – 84.
   [7] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
   [8] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
   Theory, Methods and Applications; online at Science Direct website
   [9] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
   [10] Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
   [11] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
   [12] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
   [13] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.

   E. E. Escultura
   Research Professor
   V. Lakshmikantham Institute for Advanced Studies
   and Departments of Mathematics and Physics
   GVP College of Engineering, JNT University

7. CLARIFICATION OF THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
   By E. E. Escultura

   Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:

   1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.

   2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].

   3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
   x^n + y^n = z^n, (F)

   for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.

   4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.

   The following references include references used in the consolidated paper [8] plus [2] which applies [8]

   References

   [1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 – 61.
   [2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
   Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
   [3] Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
   [4] Escultura, E. E. (1997) Exact solutions of Fermat’s equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
   [5] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
   [6] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
   [7] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
   [8] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
   [9] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
   Theory, Methods and Applications; online at Science Direct website
   [10] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
   [11] Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
   [12] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
   [13] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
   [14] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.

   E. E. Escultura
   Research Professor
   V. Lakshmikantham Institute for Advanced Studies
   GVP College of Engineering, JNT University
   Madurawada, Vishakhapatnam, AP, India
   http://users.tpg.com.au/pidro/

8. Two Fatal Defects in Andrew Wiles’ Proof of FLT
   By E. E. Escultura

   1) The field axioms of the real number system are inconsistent; Felix Brouwer and this blogger provided counterexamples to the trichotomy axiom and Banach-Tarski to the completeness axiom, a variant of the axiom of choice. Therefore, the real number system is ill-defined and FLT being formulated in it is also ill-defined. What it took to resolve this conjecture was to first free the real number system from contradiction by reconstructing it as the new real number system on three simple consistent axioms and reformulating FLT in it. With this rectification of the real number system, FLT is well-defined and resolved by counterexamples proving that it is false. (Main reference: Escultura, E. E., The new real new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 17 (2009), 59 – 84).

   2) The other fatal defect is that the complex number system that Wiles used in the proof being based on the vacuous concept i is also inconsistent. The element i is the vacuous concept: the root of the equation x^2 + 1 = 0 which does not exist and is denoted by the symbol i = sqrt(-1) from which follows that,

   i = sqrt(1/-1) = sqrt 1/sqrt(-1) = 1/i = i/i^2 = -i or

   1 = -1 (division of both sides by i),

   2 = 0, 1 = 0, i = 0, and, for any real number x, x = 0,

   and the entire real and complex number systems collapse. The remedy is in the appendix to the paper, The generalized integral as dual to Schwarz distribution, in press, Nonlinear Studies.

   Another example of a vacuous concept is the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N 1. The first inequality is clearly false. If N > 1, then N^2 > N, contradicting the choice of N. therefore N = 1. This is the original statement of the Perron paradox and it is blamed on the vacuous concept N. In general, any vacuous concept yields a contradiction.

   E. E. Escultura
   Research Professor
   V. Lakshmikantham Institute for Advanced Studies
   GVP College of Engineering, JNT University
   Madurawada, Vishakhapatnam, AP, India
   http://users.tpg.com.au/pidro/

9. THE FASCINATING AND SUCCESSFUL SEARCH FOR THE BASIC CONSTITUENT OF MATTER

   For over half a century physicists have smashed the nucleus of the atom in search of the basic constituent of matter. What is the score so far? They have found only unstable elementary particles with half-life of split second. The basic constituent must not only be stable but nondestructible; otherwise, our universe would have collapsed a long time ago. On the contrary, it has not only existed for 8 billion years but also evolved to higher order. Just look at the biological laws that we now enjoy. They were non-existent a couple of billion years after the Big Bang for there were no biological species then to reveal them.

   Now, where is this atom-smashing leading to? The prospects are not bright. As the energy of the collider is raised more elementary particles are produced alright but they all disappear in split second. In fact, nothing appears nondestructible in the nucleus of the atom. This is the time to pause and change gears. Let us look at the present methodology of physics, quantitative modeling (formerly called mathematical modeling) that describes the appearances of nature mathematically its main tools being computation and measurement. This methodology has limitations for it cannot describe the very small or the unobservable such as latent energy. In fact, this is the reason it has left long standing problems unsolved, e.g., gravitational n-body and turbulence problems, and fundamental questions unanswered, e.g., what the basic constituent of matter and structure of the electron are.

   Our remedy is the introduction of qualitative or non-quantitative modeling that explains nature and its appearances in terms of its laws. It goes much deeper than description of appearances by explaining the internal dynamics and interactions of a physical system. We include under physical systems all motions of matter including cosmic wave which is synchronized vibration of the medium. We shall find out what that medium is. Our strategy is to identify what we consider the most fundamental law of nature and proceed to find others consistent with it and if there is some phenomenon that appears to contradict it we find another natural law that reconciles them. This is always possible under our premise that there is order in our universe.

   Our choice for the fundamental law is the first law of thermodynamics which has a long history of applications and, therefore, verification. However, this is partial for it does not take latent energy into account. Therefore, we enrich and modify it as follows:

   Energy Conservation. In any physical system and its interactions, the sum of kinetic and latent energy is constant, gain of energy is maximal and loss of energy minimal.

   Then we notice from our experience that there are universal configurations and motions of matter. They are universal because they are optimizing with respect to the accumulation or disposition of energy. Therefore, they are expressions of energy conservation. We capture it in the next natural law.

   Energy Conservation Equivalence. Energy conservation has other forms: order, symmetry, economy, least action, optimality, efficiency, stability, replicative-self-similarity (fractal), coherence, resonance, quantization, smoothness, uniformity, motion-symmetry balance, non-redundancy, non-extravagance, evolution to infinitesimal configuration, helical and related configuration, e.g., circular, helical, spiral and sinusoidal and, in biology, genetic encoding, reproduction and order in diversity and complexity of function, configuration and capability.

   Non-redundancy means that nature does not create another physical system with the same functions. The so-called third quark in the nucleus of an atom outside the proton discovered in 2004 joins two positive quarks, one from each of two protons; the negative quark joins the two positive quarks from the same proton. They do the same functions; therefore, they must be the same negative quark but in different places in the nucleus. This is an example of how qualitative mathematics solves physical problems without computation using natural laws alone. This is how we shall determine the structure of the superstring, primum (elementary particle), etc.

   Observation through the Hubble says that matter steadily forms in the Cosmos first as cosmic dust that gets entangled in cosmological vortices then it collects around their eyes at the rate of one star per minute. In fact, there are “star nests” in the Cosmos that release stars rapidly [26,27]. Recently, Sky Cable BBC reported a baby galaxy (at its nascent phase). By energy conservation what appears as empty vacuum in the Cosmos is filled with matter that is not observable with present technology. Thus, we have this natural law.

   Existence of Two Fundamental States of Matter. There exist two fundamental states of matter: visible and dark; the former is directly observable, the latter is not.

   Our medium for observation is light and the reason for our failure to observe dark matter is it consists of pieces of matter of sizes finer than the finest wavelength of visible light which is of order of magnitude 10^(-14) meters. The appearance of matter in the Cosmos is due to conversion of dark to visible matter. With the existence of dark matter verified we can now ask this legitimate question: what does dark matter consist of? The answer: the superstring. For now it is just a name but we shall endow it with structure and properies as soon as we discover enough natural laws. First, we introduce physical concepts some of which tentative pending the discovery of the appropriate natural laws that well define them.

   ENERGY is motion of matter; therefore, matter and energy are inseparable and neither pure energy nor pure matter exists. FLUX is motion of matter with identifiable direction at every point, e.g., water current. CHAOS is mixture of order none of which is identifiable, e.g., the onset of tropical cyclone on tropical depression. The immensity of colliding atoms makes it impossible to monitor or predict the trajectory of every atom but each atom is subject to the laws of natur (order). Every physical system vibrates due to the impact of cosmic waves coming from all directions in the Cosmos. Basic cosmic or electromagnetic waves are generated by the normal vibration of atomic nuclei and propagated in all directions across dark matter. We introduce the next natural law needed for the discovery of the superstring but also crucial for the study of earthly turbulence like typhoon and tornado.

   Flux-Low-Pressure Complementarity. Low pressure sucks matter around it and the initial chaotic rush of matter towards low pressure stabilizes into coherent (stable) flux; conversely, coherent flux induces low pressure around it.

   The only possible force in the Cosmos that can possibly destroy a superstring is the energetic cosmic wave because it resonates with it, i.e., its size has the same order of magnitude as the wave length. Now, what configuration does the superstring have to make it non-destructible? It must have a generalized nested fractal sequence configuration [25]. This means that the superstring has a closed circular helical configuration (like s lady’s spring bracelet), by energy conservation and energy conservation equivalnce (explained below) containing a superstring traveling through its helical cycles (its motion as matter) called toroidal flux or flux torus at the speed of 7 x 10^22 cm/sec [1]. Then the latter superstring contains a superstring with the same motion and properties as the former, etc., ad infinitum. Since cosmic waves interacts only with the first term of the fractal sequence that alone can possibly be destroyed by it leaving the tail sequence still a generalized nested fractal sequence of superstrings, i.e., a superstring. Thus, the superstring survives each time the first term is destroyed by cosmic wave.

   We define the superstring to be non-agitated if its cycle length (CL) is less than 10^(-16) meters, semi-agitated if 10^-16 < CL < 10^-14 meters and agitated or primum, unit of visible matter if 10^-14 meters < CL.

   We summarize the conversion of the superstring to its three phases: non-agitated, semi-agitated and agitated (visible) and determine its structure and behavior using the natural laws. All we know initially is that dark matter consists of superstrings. We knew nothing about it at the start except that it was a piece of matter.

   Cosmic waves traverse dark matter from all directions. When non-agitated superstring is hit by cosmic wave it gets thrown off and bounces against the other superstrings. When it loses imparted energy, it grinds to a halt as non-agitated superstring. But when it gets near its previous path it is sucked by it, by flux-low-pressure complementarity, and forms a loop. By energy conservation and energy conservation equivalence it evolves to helical, semi-agitated superstring, with the original superstring its toroidal flux traveling at staggering speed along its helical cycles, 7 × 1022 cm/sec [1]. As a superstring, this toroidal flux has toroidal flux, a superstring, etc. Thus, we have a generalized nested fractal sequence of superstrings.

   Another possibility is when a non-agitated superstring is hit by cosmic wave its first term as nested fractal sequence expands to a semi-agitated superstring. We summarize the structure of the superstring by the following natural law.

   Seemi-superstring Formation. When suitable cosmic wave hits (agitates) a non-agitated superstring one of the following occurs: (a) its first term as nested fractal sequence expands and becomes a semi-agitated superstring with the rest of the sequence its toroidal flux; (b) it is projected into the first term of a new superstring with itself the toroidal flux or loses the energy imparted by the cosmic wave and remains dark.
   The next law governs conversion of dark matter to visible matter.

   Dark-to-Visible-Matter Conversion. When suitable shock wave hits a semi-agitated superstring one of these occurs: (a) the outer superstring breaks, its flux torus remaining non-agitated superstring; (b) a segment bulges into a primum, unit of visible matter.

   While a semi or non-agitated superstring interacts neither with other superstrings nor with visible matter, the primum, being visible, does. Subjected to bombardment by cosmic waves from all directions its toroidal flux is thrust into erratic motion called spike within its neighborhood along the helical cycles [22] and pulls the superstrings around the primum into a vortex flux around its cylindrical eye along its axis at linear speed of 7 x 10^22 cm/sec, much greater than that of light. The vortex flux of a simple primum is measured as charge, its visible or kinetic energy. This makes the primum a magnet, and using the right-hand-rule with the index finger pointing in the direction of the toroidal flux, the thumb points to the north or N-pole of a positive primum like the positron, the anti-matter of the electron and it mirror image with respect to a plane between their equators normal to their common equatorial plane. Viewed from its N-pole, the toroidal and induced flux of the positron are counterclockwise, its charge positive. Naturally, its mirror image or anti-matter, the electron, has clockwise vortex rotation that provides it with negative charge, the unit of charge by convention, -1 or 1.6 × 10^-19 coulombs [22]. Another primum, the positive quark, has charge +2/3 and still another, the negative quark, charge -1/3 [24].

   Since the primum and its induced flux are visible matter (the latter detected by its charge) they are affected by centrifugal force. Thus, there is greater concentration of it along the equatorial plane. Then the primum’s profile is pointed sinusoidal arc of high even power of the sinusoidal curve, corresponding to the exponent m of our qualitative-computational model of a primum in flight below, so that the induced vortex flux is also discular like that of a cosmological vortex. This is only one form of duality between macro and quantum gravity.

   It is clear that simple primum is charged and since the neutrino is neutral it is a coupled pair of prima of opposite but numerically equal charge, say -q and +q, so that -q +q = 0 and they neutralize each other’s flux and charge. The proton consists of two positive quarks joined by a negative quark at their equators on account of flux compatibility [22] (graphics in [5]). (Energy conservation requires that their axis be coplanar [22]). Thus, the charge of the proton is: 2/3 – 1/3 + 1/3 = +1. This means that there is a net coherent vortex flux around the cluster with the individual primal fluxes eddies in it. By flux compatibility the electron can attach itself to a positive quark of the proton at any point but energy conservation and energy conservation equivalence require it attaches itself between them beside the negative quark as the most stable position but pushes the negative quark a bit by flux compatibility so that their centers viewed from the N-pole form the vertices of a quadrilateral. In its interior are the coherent vortex fluxes of the positive quarks, negative quark and electron that make it a region of low pressure or depression. By flux low pressure complementarity this interior sucks neutral prima around it since charged prima are repelled by charged prima already in the cluster. Therefore, only suitably light neutral primum fits in and that is the neutrino. Thus, we have just composed the neutron consisting of a proton, electron and neutrino. Its charge is: +2/3 – 1/3 +2/3 – 1 + 0 = 0, i.e., neutral, and there is no net coherent vortex flux around it. The vortex flux of a coupled primum is also discular for the same reason as the simple primum’s is. The electron and positive and negative quarks are the basic prima because they are constituent of every atom. They are converted from dark matter to visible matter at enormous rate in the Cosmos and cellular membranes of every living thing, plant or animal [23,26,27].

   We conclude this article by sharing the amusement of the superstring. Physicists are still looking for it in the atomic nucleus when it has been staring at us since 1811, the year Ernest Rutherford discovered the electron. As we have seen the electron is a primum, an agitated superstring.

   [1] Atsukovsky, V.A., General ether-dynamics. Simulation of the matter structures and fields on the basis of the ideas about the gas-like ether. Energoatomizdat, Moscow, 1990 (in Russian).
   [2] Escultura, E. E., The solution of the gravitational n-body problem, Nonlinear Analysis, Series A: Theory, Methods and Applications, 30(8), Dec. 1997, 521 – 532.
   [3] Escultura, E. E. (1999) Superstring loop dynamics and applications to astronomy and biology, J. Nonlinear Analysis, 35(8), 259 – 285.
   [4] Escultura, E. E. (1999) Recent verification and applications, Proc. 2rd International Conf.: Tools for Mathematical Modeling, St. Petersburg, vol. 4, 74 – 89.
   [5] Escultura, E. E. (2001) From macro to quantum gravity, J. Problems of Nonlinear Analysis in Engineering Systems, 7(1), 56 – 78.
   [6] Escultura, E. E. (2001) Quantum gravity, Proc. 3rd International Conference on Dynamic Systems and Applications, Atlanta, 201 – 208.
   [7] Escultura, E. E. (2001) Turbulence: theory, verification and applications, J. Nonlinear Analysis, 47(2001), 5955 – 5966.
   [8] Escultura, E. E. (2001) Vortex Interactions, J. Problems of Nonlinear Analysis in Engineering Systems, Vol. 7(2), 30 – 44.
   [9] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
   [10] Escultura, E. E. (2003) Macro and quantum gravity and the dynamics of cosmic waves, J. Applied Mathematics and Computation, 139(1), 23 – 36.
   [11] Escultura, E. E., (2005) Dynamic Modeling of Chaos and Turbulence, Proc. 4th World Congress o f Nonlinear Analysts, Orlando, June 30 – July 7, 2004; Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, e519-e532.
   [12] Escultura, E. E., (2005). The theory of everything, Nonlinear Analysis and Phenomena, II(2), 1 – 45.
   [13] Escultura, E. E., The Pillars of the new physics and some updates, Nonlinear Studies, 14(3), 2007, 241 – 260.
   [14] Escultura, E. E., The physics of the mind, accepted, The Journal of the Science of Healing Outcome.
   [15] Escultura, E. E., The cosmology of our universe, submitted, Problems of Nonlinear Analysis in Engineering Systems.
   [16] Escultura, E. E., (2007) Dynamic Modeling and the new mathematics and physics, Neural, Parallel and Scientific Computations, 15(4), 2007, 527 – 538.
   [17] Escultura, E. E., The grand unified theory, contribution to the Felicitation Volume on the occasion of the 85th birth anniversary of Prof. V. Lakshmikantham: Nonlinear Analysis: TMA, 69(3), 2008, 823 – 831.
   [18] Escultura, E. E. The mathematics of the grand unified theory, in Nonlinear Analysis, C-Series: Theory, Mthods and Applications, 71 (2009) e420 – e431.
   [19] Escultura, E. E. Dynamic and mathematical models in physic, Proc. 5th International Conference on Dynamic Systems and Applications, June 30 – July 5, 2007, Atlanta, 164 – 169.
   [20] Escultura, E. E. (2004) Dynamic Modeling of Chaos and Turbulence, NA, TBA, 63(5-7), e519 – e532.
   [21] Escultura, E. E. The basic concepts and dynamics of quantum gravity with applications, in press, Nonlinear Studies
   [22] Escultura, E. E., Qualitative model of the atom, its components and origin in the early universe, Nonlinear Analysis: B-Series: Real World Applications, 11 (2010), 29 – 38.
   [23] Escultura, E. E., Genetic alteration, modification and sterilization with applications to the treatment of genetic diseases, accepted, The Journal of the Science of Healing Outcomes.
   [24] Gerlovin, I. L. (1990) The Foundations of United Theory of Interactions in a Substance, Leningrad: Energoattomizdat
   [25] Lakshmikantham, V., Escultura, E. E. and Leela, S. The Hybrid Grand Unified Theory, Atlantis
   (Elsevier Science), 2009, Paris.
   [26] Science, Glow reveals early star nurseries, 281(5375), July 1998, 332 – 333.
   [27] Science, (a) Science, Starbirth, gamma blast hint at active early Universe, 282(5395), December, 1998, 1806; (b) Gamma burst promises celestial reprise, 283(5402), January 1999, 616; (c) Powerful cosmic rays tied to far off galaxies, 282(5391), Nov. 1998, 1969 – 1971 (100 million times reached in largest particle accelerators), 1023.

   E. E. Escultura
   Research Professor
   Lakshmikantham Institute for Advanced Studies and Departments of Mathematics and Physics
   GVP College of Engineering, JNT University, Visakhapatnam, AP, India
   E-mail: escultur36@gmail.com * URL: http://users.tpg.com.au/pidro/

10. BACKGROUNDER ON THE GRAND UNIFIED THEORY (GUT); SOME BASIC INFORMATION by E. E. Escultura

    Why do problems in mathematics and physics defy solution or resolution for a long time? In mathematics the most famous unsolved problem was the 360-year-old Fermat’s conjecture known as Fermat’s last theorem (FLT) and in physics it was the 200-year-old Laplace or gravitational n-body problem. The author posed this question in 1988 after a 17-year absence from his mathematical career. Given that both problems appear to be very clearly stated he came to the conclusion that the difficulty lies in the inadequacy and other defects of their underlying fields. He then proceeded to first make a thorough critique of the underlying fields of FLT, namely, foundations, number theory and the real number system and he found, among others, that the real number system is inconsistent and is, therefore, ill-defined or ambiguous. Consequently, FLT being formulated in it is also ambiguous and cannot be resolved. He proceeded to construct the consistent new real number system on three simple axioms and reformulate FLT in it to make it clear and open to resolution. Indeed, FLT has countably infinite counterexamples in the new real number system.

    What has FLT to do with GUT? The first major theorem in its resolution was the characterization of undecidable (unprovable) propositions that says, essentially, that a proposition is unprovable if it is ambiguous, i.e., involves ambiguous or ill-defined concepts. Being “ill-defined” is the negation of “well-defined” and a concept in a mathematical system is well-defined if its existence, properties or behavior and relationship with other concepts are specified by its axioms. To avoid ambiguity and contradiction (the latter often hides in the former) every concept in a mathematical space must be well-defined and in its construction the choice of the axioms is not complete until this requirement is achieved. When we have two distinct mathematical spaces every concept in one is ill-defined in the other since each mathematical space is well defined only by its axioms. A physical theory is a mathematical space whose axioms are laws of nature. In a mathematical space the axioms are man-made and have nothing to do with the laws of nature.

    In the present methodology of physics called quantitative modeling (formerly called mathematical modeling) natural phenomena are described mathematically and a physical problem is modeled by a mathematical problem so that the solution of the latter is attributed to the solution of the physical problem. Reasoning is purely by analogy since there is no causal relation between the physical and mathematical spaces concerned. This is the reason for the existence of long-standing unsolved problems and unanswered fundamental questions of physics like what the basic constituent of matter and the structure of the electron are.

    The remedy for this inadequacy of methodology is qualitative or non-quantitative modeling (formerly called dynamic modeling) that explains nature or natural phenomena in terms of the laws of nature. While quantitative modeling describes the appearances of nature mathematically, qualitative modeling explains its internal dynamics and interactions including its appearances in terms of its laws. The former is based on computation, measurement and intuition, the latter on qualitative mathematics, rational thought and analysis. Qualitative mathematics includes the following routine activity of the mathematician or scientist:

    Making conclusions, visualizing, abstracting, thought experimenting, engaging in creative activity, intuition, imagination and trial and error to sift out what is more appropriate, negating what is known to gain some insights into the unknown, altering premises to draw out new conclusions, thinking backwards and all other techniques that yield results.

    Qualitative modeling alters the task of the scientist from computation and measurement to the search for the laws of nature. It was used for the first time to solve the gravitational n-body problem in 1997. The solution required the discovery of the basic constituent of matter, the superstring. It required 11 laws of nature to accomplish both. They where the initial laws of nature of GUT known as the flux theory of gravitation then.

    At present particle physicists are still smashing the nucleus of the atom in search of the basic constituent of matter, the superstring, which has been going on for over half a century. Actually, the superstring has been staring at us since 1811 when Ernest Rutherford discovered the electron. The electron is an agitated superstring. A non-agitated superstring is dark, i.e., its size is less than 10^(-10) meters. It is the basic constituent of dark matter, one of the two fundamental states of matter, the other being visible or ordinary matter. Dark matter is not observable with present technology and is known only by its impact on visible matter. When suitably agitated by cosmic waves the superstring expands to a primum, unit of visible matter such as the electron or positive or negative quark. These three prima are called basic prima because they are constituents of every atom. They are converted from dark matter at staggering rate in the Cosmos and in the cells of living things – plants or animals. In the Cosmos alone the prima form cosmic dust that get entangled into cosmological vortices and collect at their cores at the rate of one star per minute.

    References

    [1] Escultura, E. E., The solution of the gravitational n-body problem, Nonlinear Analysis, Series A: Theory, Methods and Applications, 30(8), Dec. 1997, 521 – 532.
    [2] Escultura, E. E. (1997) Exact solutions of Fermat’s equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
    [3] Escultura, E. E. (1999) Superstring loop dynamics and applications to astronomy and biology, J. Nonlinear Analysis, 35(8), 259 – 285.
    [4] Escultura, E. E. (1999) Recent verification and applications, Proc. 2rd International Conf.: Tools for Mathematical Modeling, St. Petersburg, vol. 4, 74 – 89.
    [5] Escultura, E. E. (2001) From macro to quantum gravity, J. Problems of Nonlinear Analysis in Engineering Systems, 7(1), 56 – 78.
    [6] Escultura, E. E. (2001) Quantum gravity, Proc. 3rd International Conference on Dynamic Systems and Applications, Atlanta, 201 – 208.
    [7] Escultura, E. E. (2001) Turbulence: theory, verification and applications, J. Nonlinear Analysis, 47(2001), 5955 – 5966.
    [8] Escultura, E. E. (2001) Vortex Interactions, J. Problems of Nonlinear Analysis in Engineering Systems, Vol. 7(2), 30 – 44.
    [9] Escultura, E. E. (2001) Chaos, turbulence and fractal, Indian J. Pure and Applied Mathematics, 32(10), 1539 – 1551.
    [10] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
    [11] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
    [12] Escultura, E. E. (2003) Macro and quantum gravity and the dynamics of cosmic waves, J. Applied Mathematics and Computation, 139(1), 23 – 36.
    [13] Escultura, E. E., (2003) Dynamic Modeling and Applications, Proc. 3rd International Conference on Tools for Mathematical Modeling, State Technical University of St. Petersburg, St. Petersburg.
    [14] Escultura, E. E., (2004) Problems and Unanswered Questions of physics and their resolution, Nonlinear Analysis and Phenomena, I(1), 1 – 26.
    [15] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
    [16] Escultura, E. E., (2005) Dynamic Modeling of Chaos and Turbulence, Proc. 4th World Congress of Nonlinear Analysts, Orlando, June 30 – July 7, 2004; Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, e519-e532.
    [17] Escultura, E. E., (2005). The theory of everything, Nonlinear Analysis and Phenomena, II(2), 1 – 45.
    [18] Escultura, E. E., (2006) Foundations of Analysis and the New Arithmetic, Nonlinear Analysis and Phenomena, January 2006.
    [19] Escultura, E. E., The Pillars of the new physics and some updates, Nonlinear Studies, 14(3), 2007, 241 – 260.
    [20] Escultura, E. E., The physics of the mind, accepted, The Journal of the Science of Healing Outcome.
    [21] Escultura, E. E., The cosmology of our universe, submitted, Problems of Nonlinear Analysis in Engineering Systems.
    [22] Escultura, E. E., (2007) Dynamic Modeling and the new mathematics and physics, Neural, Parallel and Scientific Computations, 15(4), 2007, 527 – 538.
    [23] Escultura, E. E., The grand unified theory, contribution to the Felicitation Volume on the occasion of the 85th birth anniversary of Prof. V. Lakshmikantham: Nonlinear Analysis: TMA, 69(3), 2008, 823 – 831.
    [24] Escultura, E. E. The mathematics of the grand unified theory, Nonlinear Analysis, A-Series: Theory, Methods and Applications, 71 (2009) e420 – e431.
    [25] Escultura, E. E. Dynamic and mathematical models in physic, Proc. 5th International Conference on Dynamic Systems and Applications, June 30 – July 5, 2007, Atlanta, 164 – 169.
    [26] Escultura, E. E. (2004) Dynamic Modeling of Chaos and Turbulence, NA, TBA, 63(5-7), e519 – e532.
    [27] Escultura, E. E. The basic concepts and dynamics of quantum gravity with applications, in press, Nonlinear Studies
    [28] Escultura, E. E., Qualitative model of the atom, its components and origin in the early universe, Nonlinear Analysis: C-Series: Real World Applications, 11 (2010), 29 – 38.

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