A Geometric Approach to Differential Forms

“…Physically, the neglect of SOC in RP systems composed of light atoms comes from the fact that the electron orbital angular momentum of the unpaired electron is practically quenched. Second, the B vector is highly dependent on the nucleotide-nucleotide space curvature-gauge field [20] (or gauge potential A, using the language of the gauge field theory [21]). Third, the presence of A creates a geometric phase on a curved manifold (twist structures, nested structures etc.)…”

Section: Resultsmentioning

confidence: 99%

Spin nature of genetic code

Tulub¹,

Стефанов²

2013

JBPC

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Nature has developed codon as a tool to manipulate a two-electron spin symmetry (short-living electrons, forming a radical pair, arise from the Mg-bound nucleosidetriphosphate cleavage at the triplet/singlet (T/S) crossing), which permits or forbids further nucleotide synthesis (DNA/RNA) and the synthesis of proteins. The thesis is confirmed by conducting DFT:B3LYP (6-311G^** basis set) computations (T/S potential energy surfaces) with the model system composed of the template (C-G-C-G-A nucleotide sequence) and the growing chain (G-C-G nucleotide sequence, DNA or RNA). The origin of codon is in hyperfine interaction between a single electron, transferred onto the template, and three ³¹P nuclei built into the phosphorus fragments of nucleotides. The nuclei, together with the polynucleotide structure, form a spiral twist that is homeomorphic to a triangle patch on the Poincare sphere. Each triangle has unique angle values depending on the nucleotide nature and their position in the codon. The patch tracing produces the Berry phase changing the electron spin orientation from “up” to “down”. The Berry phase accumulation proceeds around the (T/S) conical intersections (CIs). The CIs are a result of complementary recognition between nucleotide bases at distances exceeding the commonly accepted Watson-Crick pairing by 0.17 A. Upon changing spin symmetry, the DNA or RNA chain is allowed to elongate by attaching a newly coming nucleotide. Without complementary recognition between the bases, the chain stops its elongation. The Berry phase accumulation along the patch tracing explains the effect of Crick’s wobbling when the second nucleotide plays a primary role in recognition. The data is directly linked to creation of a quantum computing device.

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“…Thus, the scattering power is accounted for in the joint delay Doppler pdf for the first time. The application of differential forms [33] is necessary to derive the differential areas and to carry out the two-dimensional integration in the prolate spheroidal coordinate system (PSCS). Eventually, our generalized M2M channel model is capable of describing any time-variant, single-bounce M2M scenario with arbitrarily oriented scattering planes, e.g., V2V, A2A, drone-to-drone, or drone-to-vehicle scenarios.…”

Section: Introductionmentioning

confidence: 99%

Geometric Analysis of the Doppler Frequency for General Non-Stationary 3D Mobile-to-Mobile Channels Based on Prolate Spheroidal Coordinates

Walter

Shutin

Schmidhammer

et al. 2020

IEEE Trans. Veh. Technol.

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Mobile-to-mobile channels often exhibit timevariant Doppler frequency shifts due to the movement of transmitter and receiver. An accurate description of the Doppler frequency turns out to be very difficult in Cartesian coordinates, and any subsequent algebraic analysis of the Doppler frequency is intractable. In contrast to other approaches, we base our investigation on a geometric description of the Doppler frequency with the following three mathematical pillars: prolate spheroidal coordinate system, algebraic curve theory, and differential forms. The prolate spheroidal coordinate system is more appropriate to algebraically investigate the problem. After the transformation into the new coordinate system, the theory of algebraic curves is needed to resolve the ambiguities. Finally, the differential forms are required to derive the joint delay Doppler probability density function. This function is normalized by the equivalent ellipsoidal area of the scattering plane bounded by the delay ellipsoid. The results generalize in a natural way our previous model to a complete 3D description. Our solutions enable insight into the geometry of the Doppler frequency and we were able to derive a Doppler frequency that is dependent on the delay and the scattering plane. The presented theory allows describing any time-variant, single-bounce, mobile-to-mobile scattering channel.

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A Geometric Approach to Differential Forms

Cited by 23 publications

References 0 publications

Differential Forms in , I

Differential Forms in , I

Spin nature of genetic code

Geometric Analysis of the Doppler Frequency for General Non-Stationary 3D Mobile-to-Mobile Channels Based on Prolate Spheroidal Coordinates

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