Nucleosynthesis in Thin Layers E-Book

Otto Ziep

• Herausgeber: epubli
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 332

Veröffentlichungsjahr: 2024

Beschreibung

'Nucleosynthesis in thin layers' develops a unified theory of five interactions within a fractal universe based on information-currents from an algebraic algorithmic point of view. A fractal universe is viewed as a fluctuating hyperelliptic/elliptic singularity. An alternative view to our universe as a stationary state is developed by the author as a specialist in theoretical physics. A stationary universe requires stationary mass generation rate which is suspected in photosynthesis and cosmic rays. The presented theory for the interior of an elementary charge helps to solve the Riemann hypothesis as well, the cosmological constant problem, the magnetic monopole problem and involves the Dyson-Macdonald identity.

Leseprobe

Nucleosynthesis in Thin Layers

A unified theory of five interactions in a fractal, algorithmic- algebraic universe

Otto Ziep

Imprint:

Text: © Copyright by Dr. rer.nat. Dr. sc.nat.(habil.) Otto Ziep

Cover design: © Copyright by Dr. rer.nat. Dr. sc.nat.(habil.) Otto Ziep

1st edition 2024

ISBN: 978-3-7584-9006-4

Publisher:

Otto Ziep

Am Wasserturm 19a

13089 Berlin

[email protected]

E-Book design: Dr. Bernd Floßmann www.ihrtraumvombuch.de

Print: epubli – ein Service der Neopubli GmbH, Berlin

Dedicated to the memory of my dear wife Magdalena

dedicated to Stephan and Tobias

Preface

'Nucleosynthesis in thin layers' concerns the biblical theme of an information current based universe. Current astrophysical data can be interpreted as being consistent with a nonzero value of the cosmological constant implying a generation of matter. The trilogy 'Nucleosynthesis in thin layers', ‘Fractal zeta universeand cosmic-ray-charge- cloud superfluid’, ‘The sensitive balance’ describes a fractal zeta universe as a new unified theory of strong, weak, electromagnetic, gravitational, and dark interaction. ‘Fractal zeta universeand cosmic-ray-charge- cloud superfluid’ relates nontrivial zeros of the e.g. Riemann zeta function to charges, atmospheric clouds and cosmic rays. An oscillating quadrupolar-like potential near zeta function zeros is predicted as a stability region for several phenomena like the quantum Hall conductivity plateau, gravitational waves, sources of creation of matter and a cosmic-ray-charge-cloud superfluid. The large number hypothesis is demonstrated by self-similarity between the Millikan experiment and cosmic-ray cloud seeding. A number-theoretic treatment to the iterated regulator index enables a Lovelock-like expansion into five fields and their coupling constants. 'Nucleosynthesis in thin layers'unifies computer algorithms like a Mandelbrot zoom with algorithms in algebra (iterated hyperelliptic and elliptic theta functions and invariants) and number theory. The presented definition of a bi spinor defines a quantum statistics norm as a bicubic norm. Unified forces are time-thermal rotatory motions of quadrupolar potentials as tidal and dissipation less exchange processes. 'Nucleosynthesis in thin layers'interprets a high-resolution fractal Mandelbrot zoom. The cosmological constant problem is solved by a unified vacuum density being a mean density of period-doublings as additions on elliptic line bundles. The Riemann hypothesis (1859), the Dyson-Macdonald identity (1972) and Dirac’s large numbers hypothesis (1937) are approached by iterated invariants of elliptic line bundles. A magnetic monopole, a charge quantum as well atmospheric clouds and cosmic rays are set in context to nontrivial zeros of iterated zeta functions. Predicted ultra-high masses due to thermal quadrupolar and rotatory temperature gradients offer a new energy production mechanism next to fusion and fission as a‘creatio ex nihilo’. A fractal zeta universe is an open universe connected with a quadrupolar background susceptibility and a thermal-based redshift. Unlike gravitational wave losses, quadrupolar oscillations are interpreted as an energy gain. Conductivity plateaus as coupling constant plateaus are understood by one-dimensional maps. Self-similar thermal global gradients are equally valid for thin electronic layers, atmospheric layers and the whole universe layer, involving step-like discontinuities and differential oscillations.

The author became a specialist in semiconductor theory through diploma work on electrons and photons in zero-gap semiconductors, through PhD with net recombination rate theory and band structure in lead chalcogenides and through doctoral thesis on electron transport in semiconducting microstructures in the presence of strong electric and magnetic fields. The trilogy of books continues his freelance work on computer algorithms for discrete dynamical systems. As an independent researcher, the author currently publishes on unified fields, algebraic foundations of quantum statistics, one- dimensional chaotic maps and number-theoretic calculations of coupling constants.

Berlin, March 2024

1.Introduction

Nucleosynthesis in thin layers offers a quantitative mathematical unified theory of all known and a hitherto unknown dark interaction as a fractal universe due to quadratic maps. Orbits of quadratic maps as balls, complex planes and spheres are investigated on five-dimensional complex Riemann hypersurfaces. The Hausdorff measure of fractal, generalized Riemann surfaces is given by rational points of Kummer surfaces K(X). An algebraic 5-fluid model unifies quantum statistics, general relativity, and dark matter uniquely by sextic number field 𝕂[∂½] with a pure cubic subfield with ∂=2⅓. The sextic field norm answers the cosmological constant Λ problem by a Born-Oppenheimer parameter 10-53. An interaction w=1,2,3,4,5 contains a w-fold Kronecker product of algebraic units leading to a circulant global background current jw on w different topological paths due to period-doubling bifurcations. Fluctuating period lattices of elliptic theta in line bundles allow to calculate coupling constants Gw of five interactions as field regulators. A finite cosmological constant Λ reflects a stationary but rare particle creation. Densities are norms of a cubic number field Nm(𝕂[∂½]). A norm of a quadratic number in quantum statistics field differs by orders of magnitude from the cubic norm Nm(𝕂[∂]) for w>3. The corrected vacuum energy density jvac is smaller than the zero-point energy. A stationary open fractal universe is an irrotational potential flow j of world points X around point-like torus singularities ln(z-zu). Period-doubling bifurcation and Cayley-Dickson doubling yields fluctuating lattice periods.

1.1.Motivation

A motivation to this book is the net rate for all known direct intra- and interband scattering in semiconductors as a dimensionless quantity summed over periods of the imaginary parts of the dielectric constant ε [1]. This book extends Q to a complex time-thermal rate of a quadratic map in five-dimensional complex space. The quadratic term is present in five interactions as an interference in quantum statistics, a Gauss- Bonnet-term in gravity and a dark (radiation less, Auger) interaction for tidal forces between complex world points. A tight-binding model of the quantized Hall conductivity rises a fundamental question [2]. A Born- Oppenheimer parameter (me/Mw)¼ requires Planck masses M4 to explain conductivity fluctuations with an accuracy of 10-5. The accepted current view excludes Planck energy scales from the largest and smallest observable scales of the cosmos. The present book develops a fractal elliptic theta bag model of coupling constants in which every world point is capable to build a black hole encapsulating massive balls as orbits. An information- universe model of coupling constants is confirmed calculating the regulator of a circulant algebraic fields [3]. A fractal universe with particle creation developed here allows a zoom of the order of |Mg|5 with Mg the Monster group and of a Minkowski bound Mb which compares to a computational zoom ≃102000 for the Mandelbrot map. This model explains strictly two-dimensional Laughlin-and Haldane states as exact conjugated 𝕂[∂½] bi spinor states [4] [5]. Reference made to the connection between quantized Hall plateaus and elementary particles is extended by this book [6] [7] [8].

1.2.Stable floating mass regions as germs and buds

The book shows that charges are floating- mass-regions of the order of the Planck mass within elliptic theta bags of coupling constants. Like black hole the outer fractal shells are screened by dark matter with a coupling of 10-167 in case of Cantor strings. Masses by orders of magnitude higher the proton mass can be generated below the quantum statistical vacuum currents jvac in rare cases, however. To guarantee energy conservation ultra- high-energy radiation should be observed after mass generation on earth. The negative energy term arises from a quadratic equation for masses defining a Kummer surface detail [9] [10]. This equation is the hyperelliptic addition theorem on Kummer surfaces as details of a high genus complex Riemann surface and generalizes the Eddington equation for the determination of the electron-to-proton mass ratio. The book offers an algebraic algorithm to scan period-doubling bifurcation of a quadratic map leading to a polynomial . Bifurcation is associated with algebraic units. Algebraic units are simulated by the Weber-Schlaefli invariant f(√Δ) within a Monte-Carlo like algorithm. Bi spinor states ψ≃(ab) are binary invariants for 24th powers of f(√Δ) for simplest cycles k, k+1,…,k+3 in .Binary forms (ab) leave the differential and the Legendre module λ of lattice fluctuations invariant. This explains the Dyson-Macdonald identities as a ψ- dependence on Dedekind eta function η(ω) via f(√Δ). The ⅓ factor in [11]

(1.1)

passes through the standard model. Nine possible discriminants with class number hΔ=1 ∆ offering regular and not stochastic maps are thought as six flavor species of elementary particles and three bosons (W, Z and Higgs). Period-doubling and Cayley-Dickson doubling in complex Riemann surfaces are bulbs and cardioids which generate susceptibility/conductivity plateau and phase transitions. As Hausdorff measure of rational points the Kummer surface K(X) declares an area 2πX(u)jX(v)≃0. A differential form of the K(X) is with partial derivatives where and ϑε(u) are hyperelliptic theta with 16 characteristics [12]. Using (1.5) one gets the Eddington equation.

(1.2)

with a mass ratio

being for (α, β, γ)=(1,-136,10) like a proton-to-electron mass ratio me/mp and for a huge mass [13] [10]. Combining 26 steps as orthogonal substitutions of the area Xt(u)jX(v)=0 one gets a multiplicity m± =26 of lines dv =m+du++mu- in and a mass comparable to the Planck mass. Within w-ball charge model this is explained as a dense texture as required background to define coordinates.

1.3.w-ball orbits

A complex space ℂw with w=1,2,3,4,5 balls as orbits yield a total potential as a sum ΣκBOAw with κBO=(Mw’/Mw)¼. For w=4 theBorn- Oppenheimer parameter κBO=(me/M4)¼ reaches Planck masses M4. So at least 4- balls are necessary for accuracy 10-5 in a w-dimensional complex space ℂw. A strictly two-dimensional state is given by 8-component bi spinors as sextic number field 𝕂[∂½] field having five complex conjugates if both axes are the real and imaginary part of 𝕂[∂½]. Charges and fine structure constant αf are a Hausdorff measure within Hieb’s suggestion [14]. The Eddington quadratic equation for the electron-proton mass ration with coefficients (αβγ) turns out to be the hyperelliptic addition theorem on Kummer surface K(X) giving values (αβγ)≈ of hyperelliptic characteristics ε on 120 syzygetic (Rosenhain) quadruple of hyperelliptic characteristics exist [15]. These Xt(u)jX(v)=0 is assigned to possible atoms of the periodic table. But details for high-genus surfaces predict high masses M4 and rare ultra-high energy radiation up to M4c2. Masses of the bag model are exactly screened by Auger (dark force) susceptibilities with coupling constant G5≃ 10-167 which is small but dominates due to elliptic invariances . A layer structure of X(u)jXt(v)=0 on K(X) favors mass generation, e.g. in organic leafs and thin semi-insolating (semiconducting) layers. A possible cause of origin of ultra-high-energy rays are thin layers on earth perturbed by small oscillations, e.g. solar radiation, high magnetic fields accompanied by small electric oscillations.

1.4.Spin, flavor, color, isospin as rational points

Singular world points X on Kummer surfaces are a singular interchanged 2·2 ⇿ 4·4 matrices generating a self-similar process within quadrifocal configuration. A 24∙24matrix a⊗b of general world point shift δX= aqq’Xq’ - bqq’Xq’ requires 256 components. Matrix b measures an arbitrary coordinate change. A quadrifocal tensor Q has 34focal within a linear east squares algorithm (LLS) [17]. The present calculation shows that arbitrary changes δX yield a systematic focal error εfocal which serves as a basis for our understanding of atomism and quantum statistics. The error εfocalis related to algebraic units of elliptic curves and is limited by the partition function estimate ½ sh-1π√Δ<10-5 [18]. A systematic εfocalis caused by quadrifocal values Q which are equivalent to a singular Kummer surface detK(X)=0. 16 singularities of K(X) generate a geodesic polytope texture with high-dimensional vertex configuration. Spin indices s=1,2,3,4 are equivalent to simplest cycle number k of four images in quadrifocal configuration, color indices c=1,2,3 are equivalent to image coordinate indices xc, flavor indices f=1,…,9 correspond to period class Δ in image frequencies and isospin indices I=0,1 are indices of the two-component fundamental tensor . The minimum of F(x,x’) estimates periods of the Riemann surface. Usually, elementary particles are imaged via one camera in torus configuration in nuclear experiments. A quadrifocal configuration with four cameras of imaging world points is equivalent to a singular 4∙4 matrix of a Kummer K(X) and a Weddle surface W(Y).

1.5.Information-theoretic universe vs. numerology

In distinction to numerology in this book quadratic maps are designed algebraically by theta functions of arbitrary genus subjected to invariant theory. Predicted period lattices are viewed as an evaluation tool not only for a Mandelbrot zoom, but also for the standard model of particle physics. Kummer surfaces K(X) subjected to invariant theory have binary invariants (ab)2(ac)(bc) [19]. Fractional substitutions of one-dimensional roots a,b,c of a cubic polynomial are used to describe a bifurcating chaotic one dimensional map. The Tschirnhausen- Hermite map or shortly Hermite map γ(ϕ3) of an integer invariant cubic polynomial ϕ3 is extended to complex algebraic units in ℂ which leave (ab)2(ac)(bc)=0 invariant. Fixed point of the Hermite map γ(ϕ3) are viewed as a Hermitian map with unit determinant, Dirac matrix commutator σμν and zero-trace nonabelian skew matrix where c=1,2,3 and I=0,1. A Fatou set of simplest cycles q={k, k+1,k+2,k+3} with complex poles of the geometric zeta function of intervals reflects γ(ϕ3)- fixed points with Lorentz coordinates q⇿xμ. Discrete indices get coordinates describing dynamics. Here γ(ϕ3) is defined as matrix for binary values . K(X)-iterates alternate between following one-dimensional values.

(1.3)

1.6.Mean world points

The image (1.3) between x and X is not 1:1. A world point X has a dark component which is not accessible even in four-focal configuration. A world point change δX imaged by elliptic curves reflects a mean point on torus Xmean/toruswhich is like a planet suffering tidal forces.

A quadrifocal configuration images world point changes δX of ultra elliptic surfaces. Minkowski coordinates xM are mean-field mass points on a real surface as field norm Nm𝕂[∂]. Images as exact elliptic theta describe a precession of a spinning top on a sphere rationalized by A= (1, -θ, θ2). Hyperelliptic theta functions on K(X) are exact solutions in fluid dynamics and mechanics. Then image lines are a precession of a spinning classical top with kinetic energy . Here cameras are rotated by a Cayley-Klein parameter γ(ei) with Euler parameter ei consisting of Jacobi theta functions ϑ1(u) [21]. A stationary bifurcation of simplest cycles of K(X) yields times u± of two hyperelliptic theta X, X’. If period-doubling is governed by a Hermite map the images on four cameras xμ∈𝕂[∂] are correlated. This correlation can be represented in terms of a Euclidean vierbein eMμ as Euler parameter multiplied by twiddle factor 1¼ of a radix-4 discrete Fourier transform (DFT). Minkowski metric is set in context to a squared tensor of twiddle factors of DFT-4. Similarly, relativistic effects are frequencies in two two-eye images. Regular period-doubling also explains relativistic coordinates if an image texture is DFT-4 periodic as a four-point fluid motion . By means of a metric tensor and Ricci scalar the apparent relativistic kinetic energy depends on the differentiated square of a Jacobi theta function eMμ=vecγ. The set of squares of theta functions is one definition of a Kummer surface as subsequent squared theta function . But stationary and periodic quadratic maps of the interval detect a mean mass. Iterated variables z in a Mandelbrot zoom are generation rates being a complex Ricci scalar. If curvature changes sign or gets complex, the system gets singular . A summed-up Ricci scalar generalized to two points x and y as a convolution integral reduces to the Einstein-Hilbert action for y→0. The complex Fourier transform Q(q) of Q(y) is a complex time- thermal-thermal rate Q with a d dimensional DFT for d periodic iterates.

1.7.Relation to real Riemann surfaces

Einsteins field equations correspond to a real Riemann surface which can be mapped to a one-dimensional complex surface with Lorent-invariant Hermitian matrix σμxμ without branch points where σμ are Pauli matrices. A field equation is replaced by an iteration process of a complex quadratic map. A Minkowski model maps the world to a complex plane (hyperbolic plane). Masses in real Riemann surfaces belong to regions with positive/negative curvature κ as spheres/saddle points. Here the geodesic texture as a two-dimensional layer determines an open or closed universe via the sign of curvature κ as sign of stress-energy as a concave or convex manifold.

Open manifolds with κ<0 are negative energy regions which allow particle creation as shown for curvature κ of a metric lens near a massive point. The question whether the universe is open or closed is answered by a fractal texture of geodesics which differs from the existing models. Then real Riemann surface is viewed as a set of mean-field points of a complex manifold. The complex manifold is a generalized Riemann surface as an underdetermined system of equations which has more functions than equations, i.e. a genus p≤5 surface. A universe is described mathematically by geodesics as curves of zero curvature. An open fractal universe has branches of inflectional lines of simplest cycles of line bundles of elliptic curves near zero curvature.

A period zu results from Sharkovsky’s ordering and consecutive periodic combinatorial dynamics of a real interval for bifurcation diagram of the logistic equation for{2.4<t0=r<4}. [image from Wikimedia Commons]. A summation over lattice positions in Weierstrass zeta functions ζ(u,ω) leading to in lnf(√Δ) is identical to coexisting cycles. Here ordering of periods of cycles is explained by modular units of elliptic curves g(aω) a∈ℚ2.

1.8.k- test series of a quadratic map

(1.4)

SE (3) steps hold only for rational X. Multiplication by g is understood as composition of characteristics ε of world points . It is assumed that a solution of (1.4) exists only in 4-focal geometry for X- projections to images xμ =PμX ∈ℙ2 of camera μ=1,2,3,4 with Dirac equation . Here an invariant differential d on K(X)-W(Y) is expanded into a four-component basis ψs =(ab) for a four -point-cycle a,b,c,d.

1.9.Ginzburg – Landau-Higgs- mechanism as a cubic field

A singular matrix satisfies the simple SE(3) equation with four 3∙3 skew rotation matrices with color components Aμcc’ where μ=1,2,3,4 indexes Kummer surfaces Xμ [20]. Around rational SE(3) steps with curvature κ=1 a spinor is defined as a quadrifocal eigenstate in a simple real field 𝕂[∂½] with complex SU(3) rotations in the vicinity of rational points and the S4(A,a) term equivalent to Higgs mechanism. Therefore a simple dynamical interpretation of the Higgs term consists in rotational invariance of a massive spherical shell as a fourth power S4(A,a). SE(3) invariance of X and X-Y yields equivalences

(1.5)

An orthogonal tensor aij quadratic in Euler parameter eμ=ϑ(u) is as well a hyperelliptic theta function and a Jacobi function where det(aij)=e2. Euler- Rodrigues rotation demands a renormalization where e2=e’2=1. A Cayley transform between orthogonal and skew symmetric tensor exists if e2=1 which requires an additional iteration procedure. Then composition is as well interpretable as a discrete SE(3) step in space. The Cartan-like relation expresses the relationship between SO (3) and SO(4) rotations. [22].

1.10.Bifurcating world line elements

The exact line integral can be deformed on complex plane around ζ(z, ω) zeros without changing its value and the exact expression (1.1) remains valid during iteration and depends on a fluctuating discriminant Δ.

The differential of the uniformization parameter v expanded on K(X) travels in n-polytopes. Four lines yield an invariant bifurcation line

where . This Ansatz is equivalent to . The geometric zeta function ζ(l,m,z)∈ℚ introduces a rational factor though z∈ℂ is complex [23]. The Ansatz should be reducible to the exact elliptic lnf(√∆). Cycles of q quadruples k, k+1, k+2, k+3 as simplest cycles of an interval correspond to a quartic polynomial reduced to cubic invariant polynomial ϕ3. The Hermite map is periodic for transformations of quartic roots. First a scalar map γ(ϕ3(f)) generates X[f] =(1,-f,f2,1) and Y[f]=(1,-f,f2,-f3) as world points of invariant (ab)2(ac)(bc). Iterates f(√∆) yield a precession of world point X as a classical spinning top X’=m[γ(ϕ(f))]X with eigenvalues with phase where is quadratic in f(√∆k). Thus, iterates are compatible with SE(3) steps and define the rotation matrix A. Replacing ζ(u) by geodesic dF(X,Y) one gets

compatible with quantum statistics. SE(3) steps of Xi yield a complicated functional A[ψ] but a simple equation with singular det Sμ=0. An infinite fast zoomed fractal point Xi in (1.4) implies detS=0. The theory embeds field equations of five interactions into singular matrices detK(X)=detW(Y)=detS(A,a) =0. Quadratic maps have a metrical condition det . Here the Cayley-Menger metric determinant CM(a,b,c) is a singular 4∙4 matrix of triangles if three points a,b,c ∈ℙ1 are rational

det

1.11.Riemann hypothesis

As shown in [24] nontrivial zeros znt of the Riemann zeta function ζ(z) are associated with poles of the quantum statistical scattering amplitude A. Section 7 shows that znt≃λmin with a 4-component Legendre module λ at plateaus of the topological entropy ht. Thus, masses connected with znt are defined by a compound spinor ψ as a quadratic equation in f12(√∆) as an eigenfunction of supposing that f12(√∆) describes a cycle. The mass generated from a zero state is related to non-trivial zeros znt of the Riemann zeta function ζ(z). An S-matrix step of the Weber invariant f(ω) is related to poles of quantum-statistical amplitudes which confirms [24]. In addition, the book shows that zeroes znt correspond to quadruples of Legendre modules λq of elliptic line bundles 𝕃. Masses μn in znn result from an S-matrix expansion.

1.12.References

[1] O. Ziep und M. Mocker, „A new approach to Auger recombination. Application to lead chalcogenides, “ physica status solidi (b), Bd. 98, p. 133–142, 1980.

[2] R. Keiper und O. Ziep, „Theory of quantum Hall effect,“ physica status solidi (b), Bd. 133, pp. 769-773, 1986.

[3] B. Ganter, „Deciphering the fine structure constant, Germany: 2020; 96 p; epubli; Berlin (Germany); ISBN 978-3-750296-12-1.

[4] F. D. M. Haldane, „A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus,“ Journal of Mathematical Physics, Bd. 59, p. 071901, 2018.

[5] R. B. Laughlin, „Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations,“ Physical Review Letters, Bd. 50, p. 1395, 1983.

[6] O. Ziep, „On Fractional Quantization and Triplet Correlations in Quantum Hall Theory,“ physica status solidi (b), Bd. 168, p. K9–K14, November 1991.

[7] O. Ziep, „Quantum transport and breaking of space-time symmetries,“ in Mathematical Physics X: Proceedings of the Xth Congress on Mathematical Physics, Leipzig, 1991.

[8] O. Ziep, „Hadron Production in the Quantum Hall Effect,“ unpublished, 1991.

[9] O. Ziep, „Linear map and spin I. n-focal tensor and partition function,“ Journal of Modern and Applied Physics, 2023.

[10] O. Ziep, „Linear map and spin II. Kummer surface and focal error,“ Journal of Modern and Applied Physics, to be submitted.

[11] H. Weber, „Lehrbuch der Algebra, Band III,“ Elliptische Funktionen und Algebraische Zahlen, F. Vieweg und Sohn. Braunschweig, 1908.

[12] H. Weber, „Ueber die Kummersche Fläche vierter Ordnung mit sechzehn Knotenpunkten und ihre Beziehung zu den Thetafunctionen mit zwei Veränderlichen,“ Journal für die reine und angewandte Mathematik (Crelles Journal), Bd. 1878, p. 332–354, 1878.

[13] A. S. Eddington, Relativity theory of protons and electrons, CUP Archive, 1936.

[14] M. Hieb, Feigenbaum’s Constant and the Sommerfeld Fine-Structure Constant, 1995.

[15] A. Krazer, Lehrbuch der Thetafunktionen, Bd. 12, BG Teubner, 1903.

[16] Triggs, Bill. "Matching Constraints and the Joint Image." 5th International Conference on Computer Vision (ICCV'95). IEEE Computer Society, 1995.

[17] R. I. Hartley, „Computation of the quadrifocal tensor,“ in European Conference on Computer Vision, 1998.

[18] C. Meyer, „Bemerkungen zum Satz von Heegner-Stark über die imaginär-quadratischen Zahlkörper mit der Klassenzahl Eins.,“ Journal für die reine und angewandte Mathematik, Bd. 244, p. 179–214, 1970.

[19] H. F. Baker, An introduction to the theory of multiply periodic functions, University Press, 1907.

[20] J. M. Selig, Geometric fundamentals of robotics, Bd. 128, Springer, 2005.

[21] F. Klein und A. Sommerfeld, „Ueber die Theorie des Kreisels,“ Leipzig: Teubner, 1897.

[22] E. Jahnke, „Über orthogonale Substitutionen und die Differentialrelationen zwischen den Thetafunktionen von zwei Argumenten.,“ Journal für die reine und angewandte Mathematik (Crelles Journal), Bd. 1908, p. 243–283, 1908.

[23] H. Herichi und M. L. Lapidus, „Riemann zeros and phase transitions via the spectral operator on fractal strings,“ Journal of Physics A: Mathematical and Theoretical, Bd. 45, p. 374005, 2012.

[24] G. N. Remmen, „Amplitudes and the Riemann zeta function,“ Physical Review Letters, Bd. 127.24, p. 241602 (2021).

[25] G. Tiozzo, „Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set,“ Advances in Mathematics, Bd. 273, p. 651–715, 2015.

Coverfoto: https://unsplash.com/de/@saadchdhry?utm_content=creditCopyText&utm_medium=referral&utm_source=unsplash

2.Elliptic theta bag model of coupling constants

‘Nucleosynthesis in thin layers’ describes a unified theory of five interactions via discrete one-dimensional dynamics. Mass and charge can be generation in a fractal stationary universe with finite cosmological constant Λ which is inherent as a constant c in a Mandelbrot map . A quadratic map is viewed as an iterated time-thermal rate. A physical state is defined as a minimal field regulator which enables lots of units as information densities. Five interactions result from cycles of period-doubling bifurcations in a discrete dynamical one-dimensional map represented in five-dimensional complex space of balls of orbits. Riemann surfaces of the iterated function yield n-polytopes details of which consists of chaotic quadruples q={ k,k+1,k+2,k+3} where folded into 3-cubes of 𝕂[∂] folded into Moebius triangles of a 2-cube [1]. A map γ(ϕ3)z→z can be represented by complex SE(3) steps of world point X=(1,-z,z2,1). Addition on an elliptic curve Eλ is closely related to a map . The n-polytope, e.g. the hyperelliptic 4-cube decays into a 3-cube of 𝕂[∂], 𝕂’[∂], 𝕂‘‘[∂] of conjugates of a pure cubic and sextic field. Regular maps are expected if a normal field ℕ[√Δ]=𝕂[∂]𝕂’[∂]𝕂‘‘[∂] are iterates of a cubic polynomial ϕ3(f(√Δ)) with its conjugates of 𝕂[∂].

Iterates zk≃f(√Δ) and conjugates f(√Δ) ∊ℂ and SE (3) steps of X are calculated within a pure cubic field 𝕂[∂] and sextic field 𝕂[∂½]with cubic irrationality ∂ =2⅓. Poncelet involution in space is(u), a simplest cycle quadruple q and the folding of a band of complex lattice periods into triangles are equivalent identity maps as Moebius triangles with different ωk∊ℕ[√Δk], ωk+1∊

ℕ[√Δk+1]and ωk+2∊ℕ[√Δk+2] multiplied by units εk ,εk+1 and εk+2 in 𝕂’[∂],𝕂‘‘[∂] [1]. Then a 4-layer strip folded at points 1 and 4 into a Moebius triangle (spin triangle) has points 1-4 as quadruples q. Cycles of iterated zk as invariant Hermite maps of a cubic polynomial ϕ3(f(ω)) constitute again half-periods εω, quarter periods K, K’ in torus geometry where only a fundamental domain is visible as an abstract set of representatives of the orbits. The dependence of ω on ℕ[√Δ] units ε[η(ω)] with Dedekind eta function η(ω) reflects a zoom within the Kronecker- Weber (KWT) theorem. Lattice periods ω∊𝕂[∂½] are represented by a stack of triangles in a sextic number field. A minimal band of 4 layers is given by a singular field ℚ[√-3]. A bi- spinor is a sextic field 𝕂[∂½] with a pure cubic field 𝕂[∂] which has 8 parameters [2]. Cycles of period-doubling bifurcations are attributed to inflection points 3u=0. A local view consists in a normalized ω=1 as in [3]. One-periodic cycles of two-periodic elliptic functions are expected near 3u=0 where curvature vanishes. Rational points on K(X) are connected by quaternary continued fractions and steps of the special Euclidean group SE(3) in X and subsets of ternary and binary continued fractions. The Hermite problem of expressing cubic irrationalities ∂ by ternary continued fractions yields non-unique bifurcating periodicities. The path-independent σ- exponent is normalized to unit interval v∊[0,1] but ∫dvζ(v,𝕃) depends on δCk endpoints and therefore, on a fractal string of lines. Twofold differences δk(δkvζ) are invariant which creates a rational factor Qk relating ζ(v,𝕃k) at different n-polytope positions. Fractal δCk differences yield an infinite invariant rational sum Qζ(l, m, Q) which converges if the string multiplicity m is smaller than string length l, m<l. Cycles yield one-periodic modular units g(u=aω) ∊Γ[N] due to the Kronecker- Weber theorem where N=3, 5,7,…. Here a∊ℚ2 is constant during lattice fluctuations. Iterating f(√Δ) means working one-periodic with real quarter period K→0. The infinity of folding possibilities reduced to a spin Moebius triangle represents inert Nth order σ-functions Πσ(ui-uj) which are strictly two-dimensional for K→0 on the complex plane of the sextic field 𝕂[∂½] [4] [5]. As a stable breathing and fluctuating torus, the uniformization parameter u=a2ω2 represents a fractional current (QHE) for K→0 as a thermal background current for a homogeneous system in the whole universe [6]. Iterations of dimensionless complex time-thermal rate Qk yield a time derivative . The rate Qk depends on orthogonal 4x4 substitutions g(α(u),β(u)) on as