*Articoli di Fisica Teorica con connessioni inerenti la Teoria dei Numeri elaborati da Febbraio 2011 ad oggi***On some equations concerning the Casimir Effect Between World-Branes in Heterotic M-Theory and the Casimir effect in spaces with nontrivial topology. Mathematical connections with some sectors of Number Theory.**(Pubblicato in Febbraio 2012)

Abstract

The present paper is a review, a thesis of some very important contributes of P. Horava, M. Fabinger, M. Bordag, U. Mohideen, V.M. Mostepanenko, Trang T. Nguyen et al. regarding various applications concerning the Casimir Effect. In this paper in the Section 1 we have showed some equations concerning the Casimir Effect between two ends of the world in M-Theory, the Casimir force between the boundaries, the Casimir effect on the open membrane, the Casimir form and the Casimir correction to the string tension that is finite and negative. In the Section 2, we have described some equations concerning the Casimir effect in spaces with nontrivial topology, i.e. in spaces with non-Euclidean topology, the Casimir energy density of a scalar field in a closed Friedmann model, the Casimir energy density of a massless field, the Casimir contribution and the total vacuum energy density, the Casimir energy density of a massless spinor field and the Casimir stress-energy tensor in the multi-dimensional Einstein equations with regard the Kaluza–Klein compactification of extra dimensions. Further, in the Section 1 and 2 we have described some mathematical connections concerning some sectors of Number Theory, i.e. the Palumbo-Nardelli model, the Ramanujan modular equations concerning the physical vibrations of the bosonic strings and the superstrings and the connections of some values contained in the equations with some values concerning the new universal music system based on fractional powers of Phi and Pigreco. In the Section 3, we have described some mathematical connections concerning the Riemann zeta function and the zeta-strings. In conclusion, in Section 4, we have described some mathematical connections concerning some equations regarding the Casimir effect and vacuum fluctuations. In conclusion (Appendix A), we have described some mathematical connections between the equation of the energy negative of the Casimir effect, the Casimir operators and some sectors of the Number Theory, i.e. the triangular numbers, the Fibonacci’s numbers, Phi, Pigreco and the partition of numbers.

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/nardelli2012a.pdf

**On some equations concerning Fivebranes and Knots, Wilson Loops in Chern-Simons Theory, cusp anomaly and integrability from String theory. Mathematical connections with some sectors of Number Theory.**(Pubblicato in Settembre 2011)

Abstract

The present paper is a review, a thesis of some very important contributes of E. Witten, C. Beasley,R. Ricci, B. Basso et al. regarding various applications concerning the Jones polynomials, the Wilson loops and the cusp anomaly and integrability from string theory. In this work, in the Section 1, we have described some equations concerning the knot polynomials, the Chern-Simons from four dimensions, the D3-NS5 system with a theta-angle, the Wick rotation, the comparison to topological field theory, the Wilson loops, the localization and the boundary formula. We have described also some equations concerning electric-magnetic duality to N = 4 super Yang-Mills theory, the gravitational coupling and the framing anomaly for knots. Furthermore, we have described some equations concerning the gauge theory description, relation to Morse theory and the action.

In the Section 2, we have described some equations concerning the applications of non-abelian localization to analyze the Chern-Simons path integral including Wilson loop insertions. In the Section 3, we have described some equations concerning the cusp anomaly and integrability from String theory and some equations concerning the cusp anomalous dimension in the transition regime from strong to weak coupling. In the Section 4, we have described also some equations concerning the “fractal” behaviour of the partition function.

Also here, we have described some mathematical connections between various equation described in the paper and (i) the Ramanujan’s modular equations regarding the physical vibrations of the bosonic strings and the superstrings, thence the relationship with the Palumbo-Nardelli model, (ii) the mathematical connections with the Ramanujan’s equations concerning π and, in conclusion, (iii) the mathematical connections with the aurea ratio and with 1,375 that is the mean real value for the number of partitions p(n).

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/nardelli2011b.pdf

**On some applications of the Eisenstein series in String Theory. Mathematical connections with some sectors of Number Theory and with Φ and π.**(Pubblicato in Febbraio 2011)

Abstract

In this paper in the Section 1, we have described some equations concerning the duality and higher derivative terms in M-theory. In the Section 2, we have described some equations concerning the moduli-dependent coefficients of higher derivative interactions that appear in the low energy expansion of the four-supergraviton amplitude of maximally supersymmetric string theory compactified on a d-torus. Thence, some equations regarding the automorphic properties of low energy string amplitudes in various dimensions. In the Section 3, we have described some equations concerning the Eisenstein series for higher-rank groups, string theory amplitudes and string perturbation theory. In the Section 4, we have described some equations concerning U-duality invariant modular form for the D^6R^4 interaction in the effective action of type IIB string theory compactified on T^2 . Furthermore, in the Section 5, we have described various possible mathematical connections between the arguments above mentioned and some sectors of Number Theory, principally the Aurea Ratio Phi, some equations concerning the Ramanujan’s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan’s identities concerning π and the zeta strings. In conclusion, in the Appendix A, we have analyzed some pure numbers concerning various equations described in the present paper. Thence, we have obtained some useful mathematical connections with some sectors of Number Theory. In the Appendix B, we have showed the column “system” concerning the universal music system based on Phi and the table where we have showed the difference between the values of Phi^(n/7) and the values of the column “system”.

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/nardelli2011a.pdf

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