Journal of Physics Volume 1| Issue 2 Deuteron States Kalmbach...Nessa Publishers| Page 1 Journal of...

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Journal of Physics

Volume 1| Issue 2

Review Article Open Access

Deuteron States

Gudrun Kalmbach H.E.1*

1 Germany

*Corresponding author: Gudrun Kalmbach H.E., Germany; Email: [email protected]

Citation: Gudrun Kalmbach H.E. (2017) Deuteron States: Nessa Journal of Physics.

Received: 21st November 2017, Accepted: 5th December 2017, Published: 20th December 2017

Copyright: © 2017 Gudrun Kalmbach H.E. et al. This is an open-access article distributed under the terms of the

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original author and source are credited.

1. Introduction

In the MINT-WIGRIS model of [1] basic states of a mathematical deuteron are described in an octonian vector

space. In superposition states can be mixed like frequencies of waves. In this lecture they are arranged entangled in

sequences which often belong to symmetries. To demonstrate the inner deuteron and nucleon dynamics there are 5

videos which use the symmetries of a nucleons quark triangle, the general and special relativistic scaling factors as

Moebius transformations, the projective norming of the strong interactions 5-dimensional sphere to a nucleons and

atomic kernels inner space time as grid in its environment and Fano Gleason frames for measurements.

1. The first sequence 12 roll mill shows 6 force integrations using color charged quark nucleon triangles rgb for

their presentation. In between two integrations are shown the gluon exchanges between quarks.

2. The second sequence rotor presents three rotating color charged force vectors at the quark vertices in pair wise

orthogonal position. This shows the confinement of quarks in a bag: an angular momentum of these rotations

cannot be separated from a charged loop (the boundary of rgb), rotating in an interacting vector field which

crosses the triangle area; the use of the 3-dimensional cross product is known for this as definition of induction

in electromagnetism. Barycentrical coordinates are generated to determine a nucleons barycenter.

3. In the video membrane oscillation it is shown that the vertices of the nucleon triangle are fixed in space by a

rotating blue color charge momentum which lets also the gluon-sides of the triangle move between two vertices

like vibrating strings. Generated are 3 cones (paired 6 half cones) which have the loops as boundary and the

third triangle vertex as tip of the cone.

4. The sequence pulsation shows a gravitational contraction and expansion of the nucleon triangle using rgb-

gravitons and phonons. The scaled 3 basic spin lengths are the triangles Bohr radii. This is an observable

projection in a plane E of the two rgb spherical nucleons from the deuteron figure. The stretching and

squeezing is not in the deuteron, but only in E and can be seen as in a video for the actions of Moebius


Journal of Physics Volume 1| Issue 2

transformations [4] in the internet: the distance between E and the deuteron is changing. Observable is their

stretching and squeezing in E. It is postulated that the weak and the spherical, barycentrical coordinates of the

strong nucleon interaction are in special relativistic motion towards one another.

5. In the video sequence wheel two nucleon (proton, neutron) triangles are located as in the figure deuteron

below. It is shown how isospin exchanges of the two nucleon proton-nucleon states generate local Euclidean

spin coordinates for the weak interaction WI and electromagnetism EM.

No infinite dimensional Hilbert space is necessary for this nor a Schrodinger psi-wave description. Notice that in 3.

The vertices of a nucleon are fixed in space as quarks. This is not a probability distribution. The confinement of the

quarks through the strong gluon exchange is demonstrated in the first and second video which includes the

integrations for a nucleons properties and set barycenters and barycentrical coordinates. The sides of the triangle can

be stretched and squeezed (video 4) in bounds given by the cones loops of video 3. The cones are whirls, not waves,

with a loop oscillation added. The spherical strong coordinates are in special relativistic motion to Euclidean weak

and EM coordinates generated by the wheel in video 5. General relativity as a metrical device is included in the

model by its relativistic scaling factor which arises as an un symmetric distance measure. It provides in the form of a

Moebius transformation of order 6 the time cycles of the strong rotor and weak wheel.

MINT-WIGRIS uses a linear real 8-dimensional, complex4-dimensional octonian vector space E with a base (e0,

e1,..., e7). To simplify the geometry, the vectors are shown in the figure above as the vertices of a cube. They are

enumerated by triples for their location on the cube L left, R right, T top, B bottom, F front, H hidden. LBH is drawn

as an input vector e0. It can be projected as an energy force vector into the E space to any initial point and in any

direction.

The three vectors on top are for setting a weight like mass at a barycenter RTF and a color charge energy turquoise

(rbar) inside a volume (e5); a frequency, linear momentum and blue color charge energy LTF (e6); a central rotation

about an axis LTH (e3) as angular momentum energy L = r x p, r radius, p linear momentum and color charge

magenta (green bar).The initial points of these vectors are the vertices of the grey triangle which can be a nucleon



triangle with three quarks of color charge red, blue, green. An rgb-graviton sets the neutral color charge of quarks in

a nucleon and the vectors (e3, e5, e6) generate a dynamical SI rotor of the strong interaction SI. Their orientation is

negative.

The 3-dimensional Euclidean space (x, y, z)-coordinates of spin are drawn at LBF (e1, color charge red) and have

positive (e1,e2,e3) orientation (see the video wheel). For heat and inner entropy (e2, color charge green) in a volume

a 2-dimensional ball surface is drawn at RBF. At RBH is drawn a helix line (e7) on a cylinder for an angular

frequency which expands in time along the cylinders surface. This is also for periodic functions like the exponential

ψ-waves. The scaled time coordinate ct, t time, and c constant speed of light, at RTH (e4, color charge b bar, yellow)

carries also the magnetic momentum vector μ which occurs in the gyro magnetic relation as multiple of spin. As field

quantum, μ is a whirl. Also the rgb-gravitons are whirls. Since the mathematical description of whirls through the

operators’ div, rot is dual to waves; the particle-wave character of energies is extended to the triple energy characters

particle-wave-whirl.

particle

In the Fano figure below, it is shown that the octonians allow seven measuring triples like the Pauli spin 123-spin

coordinate and Pauli matrices for space (at 1 red in the cube). Deleting the letter e in the octonian coordinate list, the

octonian triples in the Fano figure are 123, 145, 167, 246, 257, 347, and 356. The space 356 is for the SI rotor with

barycentrical coordinates of the triangle. 167 is for periodic functions and light with cylindrical coordinates (at 7).

Other GF lines are 145 (at 4, cone, magnetism, color charge y), 246 (at 2, sphere S², entropy, color charge g), 257 (at

5, barycenter, mass and gravity, color charge t), 347 (at 3, rotation axis with an orthogonal orbit plane, wheel, color

charge magenta), 356 (at 6, kinetic frequency, rotor, color charge b.

The triples are taken as 3-dimensional frames for Gleason measures. They arise through the real cross product,

known for the Pauli spin matrix multiplication. It is not commutative like most operator compositions. In [1] chapter

6 from them is constructed real Gleason measures which can extend to complex measures on the complex 3-

dimensional space C³. They allow three real weights for their coordinates. If the complex version is drawn as the

vectorial caps on the hedgehog sphere S² (the figure after Fano) the weights can be extended to 6. This is also the

order of the matrix G arising from the general relativistic scaling factor related to the Schwarzschild radius Rs. As

sixth roots of unity the action of G is like a discrete compass whose needle can set in π/3 rotation six numerical

weights, but it also sets color charges on the circular segments between them or 6 electrical charges. The rgb-



graviton figure below shows 3 such segments for the quark triangle whirls in a nucleon. The whirls stack like

magnetic field quantum whirls (used for developing quantum computers). The rgb frame 126 is a cross product of

the Gell-Mann matrices λ1, λ2, λ3. As nucleons neutral color charge of quarks 126 allows the SI rotor whirl rotation

for integrations.

For the energy characters in the above figure take for EM the Fano triples containing 1: 123 is for a rotating point

charge 1 as pole on a loop (see the EM charge loop in first left figure S2 leptons); 145 is for a conic whirl rotation of

an energy vector 4 (see magnetic field quantums); 167 is for a frequency energy as exponential wave function 7

expanding on a cylinder (see photons). Other Fano triples containing 2 are for particles 257 with a mass barycenter

inside a volume 246 containing heat, adding rgb-graviton whirls 126 (the Gell-Mann matrices λj, j=1,2,3). The

triples containing 3 are for the nucleon rotor strong integrations 356, for angular frequency, rotations 347 where for

weak boson with mass (Heegard decays) a Gleason isospin triple 235 (the G powers of the matrix Gj, j=2, 4, 6),

replacing the 123 spin triple, can be added.

Hedgehog

The geometry of the strong interaction has as factor the 5-dimensional unit sphere S5 in a complex 3-dimensional

subspace of the octonians with octonian coordinates (e1,...,e6). S5 projects down to an inner 4-dimensional space

time V of deuteron and nucleons with the hedgehog 2-dimensional boundary S² (Riemannian sphere) in the

following figure. V is a Pauli grid in an environment. There are 6 complex poles on S² where force vectors are

located for the energy exchange between V and its environment. The 6 polar caps present the 6 color charges of

quarks whose mass sits at the initial points of the vectors in S². A red-green-blue rgb-graviton marks a nucleon and

magenta-turquoise-yellow m,t,y a dual second nucleon. Its dual colors are not observable but change in a

measurement to r, g, b. Isospin turns the nucleons in the wheel dynamics as described later on. The nucleon states are



described through a dynamical model without using the quantum mechanical Hilbert space model and its infinite

dimensional ψ vectors.

hedgehog

The hedgehog figure shows that the energy vectors are input/output vectors for the energy exchange of a nucleon

with its environment. The Heisenberg uncertainty relations are drawn on one line as oppositely oriented vectors.

They are discrete and act as poles on the boundary S² of a nucleon. S² has as new symmetry for poles, added to those

of the standard model of physics, the Moebius transformations MT. As well the three Pauli matrices as those of the

quark triangle. Symmetry, the special and general relativistic scaling matrices M, G are MTs. beside sink or source

poles in fluids arise dipoles, for instance known as magnets S, N poles. The MTs for dipoles a, b, are complex

fractions (z-a)/(z-b), zεC.

In the next figure a 2-dimensional potential flow about the hedgehog caps is marked in a projection. This model, in

[3] as 6 roll mill, is described through an elliptic umbilic and has 3 pairs of driving motors (figures at left). It runs

technical constructed. The rotations of the motors POTm (potentials), SIm (strong interaction) run with speed v1 and

WIm (weak interaction) y, b with a different speed v2. The two speeds relate also to the two cosmic speeds of

gravity. The motors names POTm, SIm, WIm are not from [3]. They are chosen for the Heisenberg uncertainties as

color charge coupling of octonian coordinates. POTm, SIm and WIm are presented in the hedgehog as r,t on the x-

axis, g,m on the y-axis, b,y on the z-axis, respectively.

WIm

POTm SIm



3. Integrations

Integration of force vectors or fields to potentials (radial integration), speeds (time integration) or through differential

or difference equations are shown as changes of color in the video 12 roll mill. A vector or vectorial field crosses in a

suitable angle (often orthogonal) a nucleon triangle. The example from physics is a rotating electrical currents loop

crossed by a magnetic field, used for the definition of electrical induction as angular momentum. Complex contour

integrations can also be used. Rotations of the color charge loops are not marked in these figures.

On the first page of the 12 roll mill video is shown a location of three quarks in a nucleon, each having 2 poles or

handles for its two electrical and mass/color charge poles. A potential inner flow as 2 roll mill is added. Some more

illustrations follow. Then the integration cycle starts. In between two of them is marked a gluon exchange between

paired quarks. For the cycles time the general relativistic matrix G of order 6 can be used, The symmetry of the

nucleon triangle of order 6 describes the strong integration cycle.

In the video rotor the three reflections of the nucleons triangle symmetry are demonstrated by a rotation about these

axes. Barycentrical coordinates are generated and have as common point the barycenter of the nucleon. The

rotations are alternating clockwise cw and mathematical positive mpo oriented. If the three quarks are named 1, 2, 3

then their rotor 6 cycles are the permutations of these three numbers.

4. Oscillation

The blue vector is repeated after the former video rotor in its motion for an oscillation of the nucleon membrane. The

triangles sides bend and follow the motion of the b initial point on the side. The shape of the bended membrane is

then half of a cone.



5. Weak Cycle and Gravity

It is assumed that the weak Euclidean coordinate system of deuteron is in special relativistic motion against the

spherical, barycentrical coordinate system of SI and GR. As mentioned in the introduction, the flat projection of a

nucleon is in Euclidean co-ordinates spiral rotating for the stretching and squeezing with three Bohr radii. For the

barcentrical stretching and squeezing there is no rotation involved. The three radii for equilibrium states arise

through the Moebius transformations of the triangle symmetry. One of its degenerate numerical orbit has the values

1/z = -1/2 for the t,m integrations, the value -z-1 = 1 for the y, b integrations and the value z = -2 for the r,g

integration. Their abolute value presents also the scaled three basic spin length for fermions, bosons and gravitons.

The u-quarks mass defect is used for the relative speed similar to the mass defect of electrons in solving the

Schrodinger equation. It is needed for the generation of a common group speed v<c of the nucleon parts. Added is

here that the mass defect of neutrons in atomic kernels has a similar use and can contribute energy for the isospin

exchange.



The weak isospin exchange between paired quarks u,d in the deuteron figure is demonstrated as rotations. In a

machine constructed macroscopically the opposite located shovels in a plane are driven by water in the mpo sense:

yz rotation generates the spin x-axis, xz rotation the spin y-axis and x y rotation the spin z-axis. The local Euclidean

coordinates of deuteron for the weak interaction, electromagnetism and space are generated.

Repeating some items and explanations of the figures above:

According to a physics time table, postulating a big bang, first quarks appear, also leptons. Then the gluon

confinement of quarks in nucleons arises through the generation of the strong interaction SI. After that the weak

interaction WI splits from gravity GR and SI.

Hence it is postulated that GR is the first basic force. Since quarks carry two potential forces, electrical charge EM

and GR mass, it seems that the first basic force is a unified version POTm of EM+GR. This view is adopted in a

book of Schmutzer [4] where a 5-dimensional potential Pot is constructed for this. He then gets a splitting of his 5-

dimensional version into three four-dimensional potentials and spaces.

The isospin wheel:



Gravity

radius 2 radius1 radius 1/2

central projection

POTm is the first motor splitting of Pot for quarks. Two frequencies hit orthogonal in proportion 1:2 and generate a

1-dimensional central lemniscates for quarks (Lissajous figure) with two poles for EM 1 and GR 5 charge. The EM

charge is +2/3 or -1/3 the GR one is a color charge red, green, blue (or their dual colors).

Consider the Einstein general relativity factor (r-Rs)/r, r radius, Rs Schwarzschild radius of a central mass system.

Normed as a complex Moebius transformation on a Riemannian sphere it is the matrix G of order 6 with first row 1

(-1) and second row 1 0. This matrix is for a compass generating the sixth roots of unity: for EM six charges + or –

as sign of 1/3, 2/3, 1.Similarly the compass (see Higgs bosons) generates the six color charges.

There are six energies generated as splittings of POTm potentials. POTm is mathematically integrated by the radius

differential dr, the other four ones either by the time differential dt or a difference or mixed differential equation.

They are for the magnetic force 4, heat 2, rotations 3 and kinetic energy 6. The strong motor SIm carries two poles

heat and rotation and the weak motor WIm carries magnetism and linear momentum. The polar structure is shown in

the figure hedgehog as six caps 1, 2, 3, 4, 5, 6 where a nucleon can exchange energies with its environment. The

coupling of energies on the three coordinate lines of space is according to the Heisenberg uncertainties.

The action of the scaling factor of Schwarzschild metric is demonstrated in this model as a Moebius transformation.

It arises from a central projection as used in computer graphics. In homogeneous 5 coordinates instead of 4 it allows

an un symmetric distance measure between a central mass of a system Q and a smaller mass system P as r = |QP| and

(r-Rs) = |PQ|. The homogeneous quotient of the central projection is the general relativistic factor.

The reason for three quarks confined in a nucleon is that two time cycles SI, WI can be generated for the inner

exchange of forces between quarks.



The model is the integration or definition of the EM induction through a loop of an EM current, rotating in an

orthogonal (the loop crossing) magnetic field. This needs the real 3-dimensional cross product, not only a 2-

dimensional Riemannian sphere. The nucleon triangle is crossed in rotation by the six vectorial fields generating the

six integrated energies 1, 4, 5, 3, 6, 2 in the SI time cycle. The cross products are listed in the Fano figure as six lines.

The seventh line 176 is reserved for EMI. The nucleon neutral color charge is postulated as rgb-graviton whirl, a

superposition of three color charges. Gluons are a superposition of 2 or six color charges. The rgb-graviton acts at

the same time as the SI cycle and generates by the lever law and barycentrical coordinates the barycenters between

two or three quarks. There are three equilibrium states for the nucleon Bohr radii, similar to the Bohr radii of

electrons in an atoms shell. In case the two rows of the matrix G are interchanged, the special relativistic factor M is

generated. In case the G fraction (z-1)/z is numerical solved with the value 2 for rgb-gravitons spin length, the M

value is (-z+1) = -1. Deleting its sign it is the bosons spin length 1 for instance of weak bosons. Weak bosons

generate by the Hopf splitting two fermions, leptons or quarks. The matrices (Moebius transformations) belonging to

the SI cycle, allow the third numerical value ½ of fermions as a degenerate cycle with z/(z-1) = ½.

The special relativistic factor M is needed for the mass rescaling - for u-quarks in nucleons, for neutrons in atomic

kernels and for electrons in an atoms shell. The WIm drives with a different speed than the combined POTm

GR+SIm motors. The polymer solution of the hedgehog is macroscopic constructed in catastrophe theory and

includes the elliptic umbilic catastrophe. The special relativistic metric includes the Minkowski cone MC in the form

r²-c²t² = 0 with a singularity at (0, 0). The universes mass carrying systems have a speed v<c, light speed v = c and

dark energy DE inside a light cone can have speed v>c. This cannot be measured by the Minkoski metric with the M

factor. If MC is homogeneous closed by a circle at infinity, a Horn torus is obtained in side DE frequencies can move

with a greater speed than c. For dark matter DM with a high inverted mass at Rs it is postulated that it is located

inside a volume of radius Rs. Also DM cannot be measured in the universes Schwarzschild metric, scaled by the G

factor.

The two cosmic speeds v1, v2 of GR are obtained as follows: v1 is essentially the GR potential, adding to v1 the

v2=v1√2, this is obtained by setting sin² β = Rs/r = v2²/c². The matrix G belongs to cos² β.

Particle Zoo

Observable series, called particles, are in this model described, geometrically and with symmetries added:

Fermions, two 12 series of spin ½.



Leptons with the SU (2) symmetry and Hopf map presentation, blown down 2-dimensional as sphere S2,

Description:

S2: electrical charged loop rotation (red) S2: neutrino oscillation,

Magnetic momentum and spin (yellow) as Gleason frame weights

Blown up through the Heegard decompositions of the 3-dimensional sphere S3:

S3: electrical charge loop rotation on torus S3: neutral leptons spindle torus helicity, spin and momentum aligned

Heegard decomposition of S3 into 2 leptons of genus 1, circle

The weak Heegard decompositions allow splitting of energies into two parts. Drawn is it above for leptons. The

frequency of the electrical and mass fields energy states hit orthogonal and form a circle as Lissajous figure which is

3-dimensional weak blown up by a circular orthogonal Hopf fibre to a torus.



Quarks with the color charge symmetry D3xZ2 of order 12. In case of the quark series, the frequencies proportion is

1:2 which generates a Lissajous central lemniscate for their 3-dimensional brezel of genus 2. For leptons the

frequency proportion is 1:1. A nucleon Gleason frame triple red-green-blue 126 as

An rgb-graviton Gleason frame is for a vectorial triple like the three spin coordinates, in a dual rotor: a vector charge

rotation on a loop [a], a vectorial field [c] crossing the loops area [ab] and a rotational momentum [b = axc] as cross

product of these two (see the next figure).

Proton with an inner flow of a quark at lower right (cross sections)

Field quantums and pseudo particles



2 color charge gluons with the SU (3) symmetry (8 gluons) at left; at right:

Higgs bosons with the general relativistic factors Moebius transformation G oforder 6 as discrete compass,

generating an energy bifurcation and 6 charges:

Next figure:

Octonian energy bifurcation from the e0 Higgs energy to six energy vectors ej of index j=1,...,6, as described in the

book MINT-WIGRIS and with output of atoms as light e7 coordinate; the Gleason frames of the Fano figure are

added as 3-dimensional coordinate extension, spin/space coordinates to 1as x, entropy volume to 2 as y, angular

momen-tum with rotation axis to 3 as z, magnetic whirls to 4 as ict (time), barycenter (mass) in a volume to 5, sinus

or cosine projection to 6 for frequency, helix line on a cylinder to 7 light, also for psi-waves; physics space time is

1234, 2356 a 4-dimensional octonian subspace is for a nucleon rotor and 1456 for the special relativistic light cone

Horn torus, opening to a light cone by a circular cut at homogeneous infinity,



drawn in a 2-dimensional figure with a planar cross section for a Kepler ellipse as orbit.

rgb - gravitons as super position of three conic color charge whirls. The rgb- graviton shows the 3 segments for its

quark triangle whirls in a nucleon. The whirls stack like magnetic field quantum whirls, used for developing quantum

computers. The rgb frame 126 is a cross product of the Gell- Mann matrices λ1, λ2, λ3. As nucleons neutral color

charge of quarks they allow the SI rotor whirl rotation for integrations. In the 12 plane they generate the orbits for a

small mass system in GR interaction with a huge mass system as free fall, rosette or ellipse rotations and escape

(parabola/hyperbola), using cosmic speeds.

Magnetic field quantum whirls

Phonons, pseudo particle without spin, transferring energy and momentum



Photons with the U(1) symmetry and as helix in cylindrical coordinates (light wave)

Three weak bosons, SU (2) symmetry and S3 geometry, see the above Heegard decompositions of S3

Atomic Kernels

Deuteron from weak isospin

The decay products of Higgs bosons show that this generates the 3-dimensional real cross product for energies, but

not only the three spin coordinates S = (Sx,Sy,Sz) using the three Pauli matrices. Also the rgb -graviton for quarks

confinement (using gravity and the strong interaction SI with the lambda (1, 2, 3) Gell-Mann matrices) in a nucleon

is such a cross product. The seven octonian cross products are the lines in the homogeneous Fano Figure.

Their use is described in the SI rotor and pulsation. After the generation of protons p+, the fusion process allows

through p+ decays the generation of paired nucleons in atomic kernels. This is shown in the deuteron figure for the

two isospin states of nucleons, proton and neutron. Their six u,d -quarks sit in the spherical drawn triangles of the



deuteron figure in triples as endpoints of the x,y,z-coordinate axes an rgb-graviton proton u1,2u,6d and dual drawn as

n neutron 5d,d3,4u. In every isospin exchange the locations of p+, n are interchanged. This is not shown in the

deuteron figure itself. The earlier video shown for this is the wheel: a pair of shovels driven by water sets them in a

plane (here yz, xz, xy) in motion and generates beside the p+, n exchange as orthogonal rotation axes the Sx, Sy, Sz

coordinates of deuteron and atomic kernels, using the real cross product.

Deuteron: quarks1u,2u,6d, 3d,4u,5d, gluon exchanges 12,16,26,34,35,45, weak isospin exchanges 15,23,46,126,345

nucleon rgb-graviton locations, observed stretching/squeezing through a central projection (see [5]), 6 roll mill

driven by 3 motors (see [3]) gravity plus electrical potentials (see [4]) POTm (rolls color charges red redbar as

Heisenberg uncertainties couplings), strong motor SIm (rolls green green bar) and weak motor WIm (rolls blue

yellow).

The location of an electron in a deuterons shell is well understood and not geometrical demonstrated here. The

symmetry of light emitted from atoms is a circle U(1) which generates cylindrical coordinates for the

electromagnetic interaction. Also for periodic wave functions this kind of coordinates is recommended. The 8

octonian coordinates are then: e7 for light as a rolled circle, e1,e2,e3 as linear, Euclidean space coordinates to which

e4 time is added and the Minkowski special relativistic metric. e0 is for setting vectors or vector fields and e5,e6 are

a mass, frequency plane with the line mc² = hf. The scaled general relativistic factor is presented as the Moebius

transformation G of order 6 which generates the inner strong and weak cycles of gluon, membran and isospin

exchanges for nucleons and deuteron.



References

1. G. Kalmbach H.E., MINT-WIGRIS, MINT Verlag, Bad Woerishofen, 2017

2. G. Kalmbach H.E. (Chef-Hrsg.), MINT (Mathematik, Informatik,Naturwissenschaften, Technik), MINT

Verlag, Bad Woerishofen, 1997-2017, vol.1-36

3. T. Poston and I. Stewart, Catastrophe theory and its applications, Pitman, London, 1978

4. Internet video under YouTube: Moebius Transformations Revealed 2014

MINT, PF 1533, D-86818

Bad Woerishofen, Germany

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