Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

The proportiones perfectus law states that σ_x^y=(x+√(x^2+4y))/2 is a proportione perfectus if 1≤y≤x such that for an arbitrary positive integer , there exists an integer sequence defined simultaneously by the quasigeometric relation h_(n+1)=round(σ_x^y h_n ),n≥1 and the arithmetic relation h_(n+2)=〖xh〗_(n+1)+〖yh〗_n,n≥1. When x=y=1 σ_x^y is the golden mean. When x=2,y=1,σ_x^y is the silver mean. In previous works we introduced the theory of number genetics – a framework of logic within which the golden section is studied. In this work we apply the concept to all proportiones perfectus. Let be defined as above. Furthermore, let h1 satisfy {█(round(h_1/(σ_x^y ))=w@round(wσ_x^y)≠h_1 ). ┤Now let H'nbe Hn with h1=1 Again let be defined as A robust universal computing machine is herein developed for the purpose of establishing the relationship h_i=g_(n+i-1)±h_i^',i,n≥1, which relationship is key to the logic protocol. It is clear therefore that this system of logic presents as the building block of every (including itself) within a particular σ_x^y regime. Clearly number genetics takes central place in the proportiones perfectus theory.

A REMARK ON THE TRACE-MAP FOR THE SILVER MEAN SEQUENCE

In this work, we study the Silver mean sequence based on substitution rules by means of a transfer-matrix approach. Using transfer-matrix method, we find a recurrence relation for the traces of general transfer-matrices which characterizes electronic properties of the quasicrystal in question. We also find an invariant of the trace-map.

Transmission properties of multilayered Period Doubling and Silver-Mean graphene structures

ABSTRACTWe present the propagation properties of Dirac-electrons in multilayered Period-Doubling (MPDGS) and Silver-Mean (MSMGS) graphene structures. The multilayered graphene structures are built arranging breaking and non-breaking symmetry substrates such as SiC and SiO2 following a given quasirregular substitution rule locating on them a graphene sheet. We have implemented the Transfer Matrix technique to calculate the transmittance of these multilayered graphene structures. This technique allows us to analyze readily the main differences of the transmission properties between MPDGS and MSMGS.

Conductance Properties of Multilayered Silver-Mean and Period-Doubling Graphene Structures

ABSTRACTIn this work we alternate breaking-symmetry-substrates (BSS) and non-breaking-symmetry-substrates (NBSS) such as SiC and SiO2,following the Silver-Mean (MSMGS) and Period-Doubling (MPDGS) sequences. We implement the Transfer Matrix technique to calculate the transmittance and the linear-regime conductance as a function of the most relevant parameters of the multilayered graphene structures: energy and angle of incidence, widths of BSS and NBSS regions and the generation of the quasi-regular sequence. We analyze the main difference of the transmission and conductance properties between MSMGS and MPDGS.