Proof of P Vs. NP Millenium Prize Problem with Application to New

Proof of P Vs. NP Millenium Prize Problem with Application to New ECommerce Field Theory with self energy and Artificial Intelligence with SO(32) Higgs-Englert-Bosonic Mean Field Mechanism

Soumitra K. Mallick, IISWBM & Calcutta University and www.repec.org, Management House, College Square West, Kolkata 700 073, West Bengal, India [email protected] Nick Hamburger, HN Consulting & Megatron Consulting, Hollinger Avenue, Fairfax, Virginia, USA

Sandipan Mallick, Undergraduate student and researcher, NSHM College of Pharmacy & MAKAUT, BL Saha Road, Kolkata 700 053, West Bengal, India September 18, 2017

1. Statement of the P vs. NP problem : The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time (http://www.claymath.org/sites/default/files/pvsnp.pdf (Cook & Levin (1971)). 2. Description of the Problem : If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution ( http://www.claymath.org/millennium-problems/p-vs-np-problem) 3. Proof : Since the Millenium Prizes are the latest phenomenon in scientific research I am submitting hereby for your possible consideration the solution to the P=NP Millenium Problem for the scientific fields of Economics and Financial Economics combined and separately and Mathematics and Statistics combined. The paper titled Implementing Pareto Optimality in India's Asset Markets (Archived Journal of Mathematical Finance and Quantitative Finance (variation) in 2011 (please see my Research Gate site under "Soumitra K. Mallick").

The Proof solves the problem in the Edgeworth Box in its most generality for the above fields and Proves that the solution space of problems in these fields viz. that of existence of computable General Equilibrium and computable Pareto Optimality do not have the same solution space necessarily because of the above fields governing Law of Demand which can be resolved into the Information Law of Demand and the Financial Law of Demand (vide my paper on Political Environment and Information Fields in Financial Markets (please see my Research Gate page "Soumitra K. Mallick") for Systems Classification and Systems Integration Laws (also the same set of papers) which make them mutually exclusive for Space and Time Closure which can have D functions with slope ={0,infinity} and in such cases the P=indeterminate even outside the set of Real Numbers which

were the original solution spaces. Hence, the problem of computing Pareto Optimality from the set of General Equilibria and the set of verifying Pareto Optimality from the set of General Equilibria of the two equivalent fields from the basic definition of the problems spanning the spaces and hence the fields, which are nontrivial scientific fields, depends on the field of statistics and cannot be solved mathematically alone. Hence the "P=NP problem is not transitive".

The proof uses Category Theory more particularly Cayley's Theorem for sub permutation groups and draws homologous correspondence with Differential Topology solutions (Nummela, Eric (1980), "Cayley's Theorem for Topological Groups", American Mathematical Monthly (Mathematical Association of America) 87 (3): 202–203, doi:10.2307/2321608, JSTOR 2321608) thereby establishing uniform closure in surjective subgroups of Demand and Prices by negation. The Yoneda Lemma (Nobuo, Yoneda (1954). "On the homology theory of modules". J. Fac. Sci. Univ. Tokyo. Sect. I 7: 193–227.) proves the correspondence between Cayley's Theorem for Group Homology and Category Theory with Functor correspondence. Thereby the above negation is preserved. Category Theory extends the logic to truth tables with Fields by Completeness Arguments of Commutativity. Hence Definition 1 and Definition 2 in Mallick (2011) violates Group Homology, Category correspondence and Truth Tables. In fact since it is not factorisable violates the Systems Classification Tgeorem and Systems Integration Theorem in Mallick (2014) hence cannot be solved by society (Markets) either as P=NP problem or P=/=NP problem. Failure of Transitivity in this lower dimensional non trivial case.

This extension is obvious. Proved.

4. Application to E-Commerce Dbranes String Artificial Genetic Gravitational Network with self energy and artificial intelligence :

I would like to draw your attention to our research for the Millenium Prize P vs. NP problem which develops the Dbranes String Functor Algebra Calculus which is also a method for Physical Chemical Technology Analysis and we derive a Modern Physics E Commerce Field Theory with self energy (Dbranes) and self information (Quantum Information, protocols and simulation) Quantum Mechanical Information and Energy Archimedean Stock and Flow in AGNN Networks and is similar to the Higgs-Englert-Bosonic Flow mechanism.

The Relevant Renormalisation Group (SO(32) Heterotic String Group Computer System) for dimensional cooling and conducting properties Field Theory is as follows. Renormalisation Group (SO(32) Heterotic String Group Computer System) for dimensional cooling and conducting properties :

The infinity cancellation and compactification of the system by the objectives of Physical (Systems) Technological Efficiency*{Economic Market Efficiency*Financial Market} Efficiency coupled e-commerce network (closure in the consumption action) has a equilibrium equation system of 23 (Mallick (2014)) + 6 algorithm steps + 3 BLUE Steps of Gauss-Markov Econometric Error (System Noise) Minimisation to be classified in the String Theoretic SO(32) Heterotic String with Higgs-Englert-Bosonic Mean Field electron flow.

The velocity for the Phosphorus -Ceramic Dielectric Material Group has been computed after implementing the E-Commerce Network and successful periodic equilibrium system in Lagrangian commercial market actions using the experimental equilibrium prices for that time period in the Indian markets. This is a lower dimensional space than the E8xE8 Heterotic String Group and by Newtons Law of Cooling can be cooled for superconducting properties of the E-Commerce (also genetic i.e.growing within the Edgeworth Box by collective Lagrangian genetic market actions) satellite system (see also Sullivan, Kolokythas, Raychaudhury, Vrtilek, Kantharia (2013) for radiative cooling in Astrophysics for lowering of dimensions)). It satisfies the Boolean Field property because of the linearity of the Gauss Markov Equilibrium solution space. Hence satisfies systems closure for information and energy coupled networking and Econophysics Arrow of Time,Symmetric and normalized Equilibrium Actions.

5. The Local Laws using Boolean Logical Artificial Genetic Gravitational Neural National Network Circuit design : (see Mallick (2014), Sullivan, Kolokythos, Raychaudhuri, Vrtilek & Kantharia (2013)) :

(1)

Information Law of Demand

(2)

Financial Law of Demand

(3)

Systems Classification Theorem

(4)

Systems Integration Theorem

This implies =>

The Systems Classification (Systems Factorisation) : SO(32)~SO(23)*SO(6)*SU(3) The Systems Integration : 6 Algorithm Steps  The calculation is (Millenium Prize P vs. NP problem solution (Mallick, Hamburger, Mallick (2016 b)) [23 equations of Brownian Motion Classical Quantum Stock Market Systems Genetic Quality Arrow of Time Equilibrium + 1 Money Price (Walrasian Renormalisation)] *1/ 4 dimensional systems Closure of Edgeworth Box Schroedinger Physical Chemical (Mallick et.al. (2016 a & b), “jump quantum” (Hamburger switches design) Nanowires Potential String Matching Haag Theorem Bypass (Mallick (1993, 2014, 2015, 2016)) Field=6, which can be analysed by other Nanomethods as we have found. We have used the method of Sullivan et. al. (2013) in our computed dimensions by using chiral symmetry properties of satellites and our Archimedean HiggsEnglert-Bosonic Stock & Flow mechanism in lower dimensional network fields constructed above. The 3 transverse equations arise from the Gauss-Markov Theorem for Econometric Systems noise minimisation which are equivalent to gauge field equations (Judge, Griffiths, Hill, Lutkepohl, Lee (1985)).

References :

1.Cook, Steven and Leonid Levin , The P vs. NP problem, http://www.claymath.org/millennium-problems/p-vs-np-problem, 1971. 2. Judge, G., W.E. Griffiths, R.C.Hill, H. Lutkepohl, T.C. Lee, 1985, The Theory and Practice of Econometrics, 2nd. Edition, Wiley & Sons., New York. 2. Mallick, S.K.: Bounded Rationality and Arrow Debreu Economies, unpublished PhD dissertation, Dept. of Economics, New York University, 1993. 3. _______, A set theoretic forecasting model for Investment in the Pharmaceutical Sector of Emerging Markets, Mathematical Finance, Archived, 2007 . 4.______, Implementing Pareto Optimal Asset Prices, Mathematical Finance, Archived, 2011. 5._______, The String Theory of Indian Stock Market nanostructure system, Journal of Physics A, Archived, 2014. 6. ______, Developing Stock Market Nanotechnology Systems and Geometric Fractal representation of data using Econophysics Haag’s Theorem, European Journal of Physics, Archived, 2015. 7.______, How can we build Stock Markets like NSEIL with BSE SENSEX using String Theory (developing String Theoretic Reservoir Engineering), Journal of Physics D, Archived, 2016. 8. ______, Nick Hamburger & Sandipan Mallick : Design of gated switches using HAAG Theorem bypass technology, Nick Hamburger website, January 10, 2016 a. 9.______, ______, & ________, : Proof of P vs. NP problem with E Commerce Field Theory,https://www.researchgate.net/publication/316602343_Proof_of_P_Vs_NP_Millenium _Prize_Problem_3_Clay_Mathematics_Institute_with_Electronic_Commerce_Field_Theory, August 2016 b. 10.Nummela, Eric , Cayley's Theorem for Topological Groups, American Mathematical Monthly (Mathematical Association of America) 87 (3): 202–203, 1980

11.Nobuo, Yoneda , On the homology theory of modules. J. Fac. Sci. Univ. Tokyo. Sect. I 7: 193–227, 1954. 12. Sorli, A. , Communications on www.researchgate.net, 2017. 13.Sullivan, E.O., K. Kolokythas, S. Raychaudhury, J. Vrtilek, & N. Kantharia, First results from the complete local-volume groups sample, Bulletin of Astronomical Society of India, 14, 2013.