This homepage is dedicated to my mom, who died at
April 9, 2020 in the age of 93 years. In retrospect, the proposed
solution concepts of different problem areas (the Riemann Hypothesis
& the inconsistent quantum theory with Einstein's gravitation
theory) originate in some few simple common ideas / basic conceptual
changes to current insufficient "solution attemps".
(a) the verification of several Riemann Hypothesis (RH) criteria (b) a truly circle method for the analysis of binary number theory problems. The Kummer function based Zeta function theory is basically about a replacement of the integral exponential function Ei(x) by a corresponding integral Kummer function. It enables the validation of several RH criteria, especially the "Hilbert-Polya conjecture", the "Riemann error function asymptotics" criterion and the „Beurling“ RH criterion. The latter one provides the link to the fractional function and its related periodical L(2) Hilbert space framework, (TiE). Regarding the tertiary Goldbach problem Vinogradov applied the Hardy-Littlewood circle method (with its underlying domain "open unit disk") to derive his famous (currently best known, but not sufficient) estimate. It is derived from two estimate components based on a decomposition of the (Hardy-Littlewood) "nearly"-circle into two parts, the „major arcs“ (also called „basic intervals“) and the „minor arcs“ (also called „supplementary intervals“). The „major arcs“ estimates are sufficient to prove the Goldbach conjecture, unfortunately the „minor arc“ estimate is insufficient to prove the Goldbach conjecture. The latter one is purely based on "Weyl sums" estimates taking not any problem relevant information into account. However, this estimate is optimal in the context of the Weyl sums theory. In other words, the major/minor arcs decomposition is inappropriate to solve the tertiary and the binary Goldbach conjecture. The primary technical challenge regarding number theoretical problems is the fact that only the set of odd integers has Snirelman density ½, while the set of even integers has only Snirelman density zero (because the integer 1 is not part of this set). The additional challenge regarding binary number theoretical problems is
the fact that the problem connects two sets of prime numbers occuring with
different density (probability) during the counting process; regarding the
Goldbach conjecture this concerns the fact, that the number of primes in the
interval (2n-p) is less than the number of primes in the interval (1,p).
Therefore, two different „counting methods“ are required to count the numbers
of primes in the intervals (1,p) and (p,2n-p). In order to overcome both technical challenges above a truly circle method in a Hilbert space framework with underlying domain „boundary of the unit circle“ is proposed. The nonharmonic Fourier series theory in a distributional periodic Hilbert scale framework replaces the power series theory with its underlying domain, the "open unit disk". The proposed nonharmonic Fourier series are built on the (non-integer) zeros of the considered Kummer function, (which are only imaginary whereby for their real parts it holds >1/2) replacing the role of the integers of exp(inx) for harmonic Fourier series. They are accompanied by the zeros of the digamma function (the Gaussian psi function). The set of both sequences are supposed to enable appropriate non-Z based lattices of functions with domain "negative real line & "positive" critical line". This domain is supposed to replace the full critical line in the context of the analysis of the Zeta function, in order to anticipate the full information of the set of zeros of the Zeta function (including the so-called trivial zeros), while omitting the redundant information provided by the critical zeros from the zeros from the "negative" part of the critical line. With respect to the analysis of the Goldbach conjecture it is about a replacement of the concepts of trigonometric (Weyl) sums in a power series framework by Riesz bases, which are "close" (in a certain sense) to the trigonometric system exp(inx). The nonharmonic Fourier series concept of almost periodic functions is basically about the change from integers n to appropriate sequence a(n). Such a change also makes the difference between the Weyl method and the van der Corput method regarding exponential sums with domains (n,n+N), (GrS), (MoH). Selberg‘s proof of the large sieve inequality is based on the fact, that the characteristic functions of an interval (n,n+N) can be estimated by the Beurling entire function of exponential type 2*pi, applying its remarkable extremal property with respect to the sgn(x) function, (GrS). The Riesz based nonharmonic Fourier theory enables the split of number theoretical functions into a sum of two functions dealing with odd and even integers separately, while both domains do have Snirelman density ½. In case of an analysis of the Goldbach conjecture it also enables the definition of two different density functions, „counting“ the numbers of primes in the intervals (1,p) resp. (p,2n-p). The trigometric system exp(inx) is stable under sufficienctly small perturbance, which leads to the Paley-Wiener criterion. Kadec's 1/4-theorem provides the "small perturbance" criterion, which is fulfilled for both sets of zeros, the considered Kummer function and the digamma function. A striking generalization of "Kadec's 1/4-theorem", (YoR) p. 36, with respect to the below is Avdonin's "Theorem "1/4 in the mean", (YoR) p. 178. The Fourier transformed system of the trigonometric system forms an orthogonal basis for the Paley-Wiener Hilbert space (PW space), providing an unique expansion of every function in the PW-space with respect to the system of sinc(z-n)-functions. Therefore, every PW-function f can be recaptured from its values at the integers, which is achieved by the cardinal series representation of that function f (YoR) p. 90. When the integers n are replaced by a sequence a(n)) the correspondingly transformed exponential system builds a related Riesz basis of the PW-space with the reproducing sinc(z-a(n))-kernel functions system. For the link of the nonharmonic Fourier series theory with its underlying concepts of frames and Riesz bases to the wavelet theory and sampling theorems, which is part of the solution concept of part B, we refer to (ChO), (HoM), (ReH).
April 10, 2021, change to previous version, p. 18
dead end road“ theories towards a
common gravity and quantum field theory. The physical waymarking labels directing into
those dead end roads may be
read as dead end road label (1): " towards space-time regions with not constant
gravitational potentials governed by a globally constant speed of light",
(UnA)dead end road label (2): " towards Yang-Mills mass gap".
The waymarker labels of the royal road towards a geometric gravity and quantum field theory may be r oyal road label 1: towards mathematical concepts of „potential“, „potential operator“, and „potential barrior“ as intrinsic elements of a geometric mathematical model beyond a metric space (*) royal road label 2: towards a Hilbert space based hyperboloid manifold with hyperbolic and conical regions governed by a „half-odd-integer“ & „half-even integer“ spin concept royal road label 3: towards the Lorentz-invariant, CPT theorem supporting weak Maxwell equations model of „proton potentials“ and „electron potentials“ as intrinsic elements of a geometric mathematical model beyond a metric space : royal road label 4 towards „the understanding of physical
units“, (UnA) p. 78, modelled as „potential barrior" constants, (*),(**), (***), (****), (*****)(*) Einstein quote, (UnA) p. 78: „ The principle of the
constancy of the speed of light only can be maintained by restricting to space-time
regions with a constant gravitational potential.“ (**) The Planck action constant may mark the " potential barrior" between the
measurarable action of an electron and the action of a proton, which "is acting"
beyond the Planck action constant barrior.(***) The „ potential barrior“ for the validity of the Mach
principle determines the fine structure constant and the mass ratio constant of
a proton and an electron: Dirac’s
large number hypothesis is about the fact that for a hydrogen atom with two masses, a proton
and an electron mass, the ratio of corresponding electric and gravitational
force, orbiting one another, coincides to the ratio of the size of a proton and the size of the
universe (as measured by Hubble), (UnA) p. 150. In the proposed geometric model
the hydrogen atom mass is governed by the Mach principle, while the Mach
principle is no longer valid for the electron mass, governed by the CPT spin statistics. (****) The norm quadrat representation of the proposed " potential" definition indicates a representation of the fine structure constant in the form 256/137 ~ (pi*pi) - 8. In (GaB) there is an interesting approach (key words: "Margolus-Levitin theorem", "optimal packaged information in micro quantum world and macro universe") to „decrypt“ the fine
structure constant as the borderline multiplication factor between the range of
the total information volume size (calculated from the quantum energy densities)
of all quantum-electromagnetic effects in the universe (including those in the
absense of real electrodynamic fields in a vacuum; Lamb shift) and the range of the total information volume size of all matter in the four dimensional universe (calculated from the matter density of the universe).
(*****) The vacuum is a homogeneous, dielectric medium, where no charge distributions and no external currents exist. It is governed by the dielectric and the permeability constants, which together build the speed of light; the fine structure constant can be interpreted as the ratio of the circulation speed of the electron of a hydrogen atom in its ground state and the speed of light. This puts the spot on the Maxwell equations and the " still missing underlying laws governing the "currents" and "charges" of electromagnetic particles. ...The energetical factors are unknown, which determine the arrangement of electricity in bodies of a given size and charge", (EiA), p. 52:
- the kinematical energy Hilbert space of the fermions is the standard weak Hilbert space H(1), equipped with the "Dirichlet integral" inner product - the Hilbert space H(1) allows the split into two complementary spaces, providing a model for the physical concepts of "half-odd-integer" and "even-odd-integer" spins with related repulsive and attractive effects, mathematically governed by Riesz bases coming along with the non-harmonic Fourier series theory - the Friedrichs extension of the Laplacian operator is a selfadjoint, bounded operator B with domain H(1). Thus, also the operator B induces a decomposition of H(1) into the direct sum of two subspaces, enabling the definition of a „potential“ and a corresponding „grad“ potential operator. Then a potential criterion defines a manifold, which represents a hyperboloid in the Hilbert space H(1) with corresponding hyperbolic and conical regions ((VaM) 11.2) - the kinematical Hilbert space H(1) is compactly (coarse-grained) embedded into an overall energy Hilbert space H(1/2), whereby the closed complementary subspace of H(1) governs all non-kinematical energies, i.e. the closed sub-space H(1,ortho) replaces all current bosons-type elementary particles - theoretical physics models are defined as weak (distributional Hilbert scale based) variational (PDE) representations of the considered mechanical system, which is compactly (coarse-grained) embedded into an overall dynamical system. A first proof of concept is given by the Millennium problem of a well-posed initial-boundary value problem of the 3D non-linear, non-stationary Navier -Stokes Equations, where the corresponding variational representation accompanied with an extended H(1/2) energy Hilbert space norm is well-posed - the physical concept of " potential"
is an intrinsic element of the geometric quantum field model, and is no
longer an appropriately interpreted physical phenomenon of the
considered physical situation; mathematically speaking, the concept of
"potential" is an intrinsic element of the geometric Hilbert scale
framework, and do not need to be chosen properly as a given functions of
the considered PDE- the " electron potential" (:=
electric potential energy/unit charge) in the geometric quantum field
(H(1/2) energy related Hilbert space) model corresponds to the electric
potential as defined by the weak representation of the Maxwell
equations; the hyperboloid manifold
in the Hilbert space H(1) with corresponding hyperbolic and conical
regions enables a corresponding "proton potential" governed by
the related "spin type"; in other words, also the concept of repulsive
electrons and attractive positrons are intrinsic parts of the geometric
model; the differentiator from a physical perspective is about the fact,
that the Planck action constant marks the barrior between the
measurarable action of an electron and the action of a proton,which is
beyond the Planck action constant- Fourier transforms do not allow localization in the phase space, leading to the concept of windowed Fourier transforms. From a group theoretical perspective windowed Fourier transforms are identical to the wavelet transforms. The wavelet admissibility condition puts the spot on the Hilbert space H(1/2), (LoA). From a „ mathematization“ perspective „wavelet analysis may be considered
as a mathematical microscope to look by arbitrary (optics) functions over the
real line R on different length scales at the details at position b that are
added if one goes from scale „a“ to scale „a-da“ with da>0 but infinitesimal
small“, (HoM) 1.2- the one-dimensional counterpart of the below considered S(3) unit sphere is the unit circle. For the continuous wavelet transform of a function over the unit circle with respect to a wavelet g we recall from (HoM): „ Geometrically, the wavelet transform
of a function over the circle T is a function over the half-cylinder
R(+) x T. … The Poisson summation formula shows a vanishing constant
Fourier term and the positive and negative frequencies do not mix,
enabling a corresponding (+/-) split of the L(2) space. … the wavelet
transform over the circle conserves energy. .. The wavelet transform
with respect to a progressive, admissible wavelet is an isometry. …. In
case where g=h it is now an orthogonal projector on the image of the
wavelet transform.
- a coarse-grained mechanical Hilbert space L(2) embedded into the Hilbert space H(-1/2) allows a re-interpretation of the notion "entropy" in the context of the "discrete" Shannon entropy vs. the log(x)-function based "continuous entropy"; we note that the (still best known) relationship between the notions "volume V", "number of particles N", temperature T", "pressure P", "entropy S" and the related total energy E, as a function of the parameters S, V, N, is given by the total differential dE=TdS-PdV+c*dN- the coarse-grained mechanical (statistical) Hilbert space L(2) embedded into the overall Hilbert space H(-1/2) puts the spot on the Hawking-Hartle (probability theory related) " Interpretation of "The Wave Function of the Universe"" in (DrW): "the ground state is the amplitude for the Universe to appear from nothing"; ... "Physical probabilities, as exemplified by radioactive decay,
start with something, a first situation (particle in space and time) becoming
another situation (other particles in space and time). The probability is the
chance that the transition from situation one to situation two happens during a
certain interval of time, or that a particle is found in a certain volume of
space, or something like that". In this context we note that the embeddedness of
L(2) into H(-1/2) relates to the embeddedness of the set of rational numbers
into the set of real numbers, whereby the countable set Q is a „zero quantity set“ with
respect to the L(2) inner product (i.e. the probability to pick rational numbers ot of the set real numbers is zero). We note that the set of the (infinite) solutions of the field equations has the same cardinality as the set of real numbers- the model supports also Einstein's composition model of radiation (Physikalische Zeitschrift 10 (1909), 185–193), while at the same point in time indicates to revisit the related discourse note with W. Ritz (Physikalische Zeitschrift, 10, (1909), 323-324) - the extended H(1/2) (energy) Hilbert space framework replacing the Dirichlet integral based inner product space H(1) (derived from the classical Newton potential equation accompanied with Green's identities) leads to a well-posed 3D-non-linear, non-stationary Navier-Stokes boundary-initial value problem accompanied with Plemelj's extensions of Green's identities based on intrinsic surface notions " mass element" and "flux"). At the same time, it omits the Yang-Mills system (the expansion of the Maxwell equations in the context of the SMEP) accompanied with the corresponding (physical) Yang-Mills mass gap problem- the SMEP (with its underlying separation into the categories fermions (EPs with spin 1/2 (electron, neutrino, quarks) and bosons ("EPs" with spin 0 (Higgs), spin 1 (gluons, W/Z-boson, spin 2 (graviton)) is replaced by a purely fermion based EP model defined in the coarse-grained Hilbert scale pair ((H(0),H(1)), governed by the complementary closed sub-space with respect to the overall ((H(-1/2),H(1/2)) Hilbert scale framework. As a consequence of the (" color") confinement problem of quarks (the phenomenon that "color-charged" particles (such as quarks and gluons) have not been isolated until today, i.e. they have not been observed until today), the "quark" concept is omitted by the original proton- fermions are governed the Fermi statistics, whereby two identical fermions (EP with mass) cannot occupy the same quantum state (in a corresponding mathematical world this means that a mathematical „fermion“ object behaves like a rational number). The Fermi oscillator is „ a particularly simple system. It is a thing capable only for
two levels, zero and „epsilon““, (ScE) p.20. The Fermi oscillator is
basically the mathematical model for his famous „spin-concept“. „The so-called „spin
statistics theorem“ is one of the few theorems in theoretical physics, which
are proven based on very very little
assumptions. .. it states that EPs with integer spin are representated by symmetric wave functions, while EPs with half integer spin (k+1/2, k=0,1,2,...) are representated as anti-symmetric wave functions“ (HeW) p. 103. All bosons are governed by the Bose-Einstein
statistics, which is concerned with “photon gases”. A characteristic of the
Bose-Einstein „photon gases“ statistics is, that the concerned particles do not
restrict the number of them that occupy the same (continuous) quantum state . All
„photon gas“ particles can be brought into the energetically lowest quantum
state (below the critical temperature of “normal gas”), where they show the
same “collective” behavior. They occupy a single (continuous) quantum
state of zero momentum, while „normal gas“ particles all have finite momentum. The statistical thermodynamics model for "normal gases" is the Planck oscillator; in the proposed model the related Planck-Schrödinger statistics of the n-particle problem starts with k=1, while to case k=0, the ground state energy case, is interpreted as physical condition for an model adequate ground state energy value approximation, from which the reduced Planck action variable can be derived, (ScE)- Pauli’s spin concept and its related spin statistics theorem is about amathematical object with the 2-value attributes, "half integer" and "integer" multipliers of the reduced Planck action quantum. The "connection" model of those spins is mathematically given by „complex number multiplication“. The proposed fermions model is about the three observable EPs (electron, neutrino, positron) considered in a Hilbert space framework. The underlying field is the three-dimensional unit sphere S(3) enjoying the following properties i) S(3) is isometric to "rotations in the three dimensional space" ii) S(3) relates to the multiplication of triplets, Hamilton’s concept of quarternions, (UnA1) p. 148 iii) quaternionic multiplication of a spatiotemporal derivative with „electromagnetic potential“ leads to two terms precisely matching the known expressions for the electric and the magnetic fields of the Maxwell equations, (UnA1) p.152 iv) the governing mathematical concept of the Maxwell equations is the Lorentz transformation, which is a linear transformation mapping space-time onto space-time, preserving the scalar product - the S(3) model in combination with a complex Lorentz
group puts the spot on the famous Spin-Statistics-PCT theorem, (StR), pp. 9, 13: "The PCT
theorem is a fundamental symmetry of physical laws. It is the only
combination of C, P, and T transformations (see below) that is observed to be an exact symmetry
of nature at the fundamental level. In terms of Einstein's SRT, "the CPT
theorem says that CPT symmetry holds for all physical phenomena, or more
precisely, that any Lorentz invariant local quantum field theory with a
Hermitian Hamiltonian must have CPT symmetry""- regarding general irreducible spinor fields and assumed "wrong" connections of spins with statistics of integer and half-odd integer spins, in which all fields either commute or anti-commute, we refer to the Spin-Statistics Theorem for General Spin, (StR) Theorem 4-10- wave-mechanical vibrations correspond to the motion of particles of a gas resp. the eigenvalues and eigen-functions of the harmonic (Planck) quantum oscillator modelled in the Hilbert scales L(2)/H(1)). The alternatively proposed energy space (H(1/2) = H(1)+H(1,ortho) indicates to revisit Schrödinger's " purely quantum wave" vision, which is about
half-odd integer wave numbers, rather than integer quantum numbers. "The wave point of view in both cases (Bose-Einstein and Fermi-Dirac statistics) or at least in the Bose case (mathematically a simple oscillator of the Planck type) raises another interesting question: we may ask whether we ought not to adopt for the quantum numbers half-odd integers. .... On the other side, if we adopt straightaway, we get into erious trouble, especially on contemplating changes of the volume (e.g. adiabatic compression of a given volume of black body radiation), because in this process the (infinite) zero-point energy seems to change by infinite amounts!" (ScE) p. 50. We emphasis that the proposed model overcomes this challenge, as there is no n-particle problem anymore in the purely continous spectrum world of the considered complementary sub-space of the statistical Hilbert space L(2)- „ Hubble’s observations of a shift to the red in the
spectra of the spiral nebulae—the farthest parts of the universe—indicated that
they are receding from us with velocities proportional to their distances from
us“ (DiP). In the context of the spherical wave conjecture regarding
observable the space-time hyperbolic world) Dirac’s „new basis for cosmology“
((DiP) 1937), suggests „a model of the (physical) universe in which there is a
natural velocity (requiring the concept of time) for the matter at any point, varying continuously from one
point to a neighbouring point. Referred to a four-dimensional space-time
picture, this natural velocity provides us with a preferred time-axis at each
point, namely, the time-axis with respect to which the matter in the
neighbourhood of the point is at rest. By measuring along this preferred time-axis
we get an absolute measure of time, called the epoch.“ - An elliptic mathematical „reality“ model of the universe governs two complementary subsets. There is the parabolic/hyperbolic "physical reality world“, (which is a very „small“ (in the sense of a zero quantity set) countable subset of the overall mathematical „reality“ model), accompanied with physical-kinematical notions like „time“, „velocity“, „action“ and e.g. the second law of thermodynamics. Then there is the dominant closed, complementary subset of the "physical reality“ model, accompanied with purely mathematical-not measurable notions like „ground state energy“. In this mathematical reality, (UnA1), concepts like Dirac’s „ epoch“
or Penrose’s „cycles of time“ (for an alternative steady state model to describe, what came before the big bang), may be
revisited to describe how elementary particles with mass are generated out of
the purely mathematical, non-kinematical „world“ and how a kind of "annihilation force" occurs measurable by the second law of thermodynamics. The guiding principle may be, that "time" (resp. only "time duration") as perceived by human consciousness aggregates and reflects the "actions" (accompanied with related "action variables", (HeW)) of observed physical systems, which are detectable and understandable (e.g. the scattering theory based methods or Einstein's composition of radiation model) only by statistical ((L(2)-Hilbert space based) measurement methods. Regarding the prospects success measuring the generation of an elementary particles with mass we note that the sum of countable zero quantity sets is again a zero quantity setThe spectrum of a Hermitian, positive definite operator H in a complex valued Hilbert space H with domain D(A), where its inverse operator is compact, is discrete (measurable/stable). Without the latter property the concept of a continuous spectrum is required (unstable meaurement results). The link back to part A is given by the Berry-Keating conjecture with its mathematical counterpart, the Hilbert-Polya conjecture. It is about the suggestion that the E(n) of the non-trivial zeros ½+iE(n) of the Zeta function are all real, because they are eigenvalues of some Hermitian operator H. Berry’s extensions are that if H is regarded as the Hamiltonian of a quantum-mechanical system then, (BeM) i) the operator H has classical limits ii) the classical orbits are all chaotic (unstable) iii) the classical orbits do not possess time-reversal symmetry. While the full H(-1/2) based quantum-mechanical system behaves chaotic from an observer perspective (measured in the statistical Hilbert (sub) space L(2)), the underlying complementary sub-systems allow a differentiation between those observable thermostatistical effects and the related, not statistically measurable mathematical model parameters, which are physically governed by physical Nature constants, reflecting the border line to those complementary sub-systems (with purely continuous spectrum). The proposed circle method above (part A) comes along with a split of the integer domain N (of number theoretical functions) being bijectively mapped onto two sets of integers („odd“ and „even“) on the boundary of the unit circle, both with Snirelman density ½. The two sets may be used to revisit the elementary „particle“ state numbers, n=2n/2 (n=1,2,3, ...) exdended by n=1/2, and Schrödinger’s half integer state numbers n+1/2=(2n+1)/2 (n=0,1,2,...), (corresponding to the levels of the Planck oscillator, see above resp. (ScE) pp. 20, 50)) to re-organize the chaotic model behavior of the current thermodynamics based purely kinematical quantum-mechanical systems. Further referring to part A we note that the zeta function on the critical line is an element of the sequences Hilbert space l(-1). The Shannon sampling operator is the linear interpolation operator mapping the standard sequences Hilbert space l(2) isomorphically onto the Paley-Wiener space with bandwidth „pi“ (i.e. a signals f represented as the inverse Fourier transform of a L(2) function g can be bijectively mapped to its related sequences f(k) and vice versa). We note that the generalized distributional (polynomial and and exponential decay) Hilbert scales l(a) and l(t,a) allow the definition of corresponding generalized Paley-Wiener scales enabling a (wavelet basis based) convolution integral representation of the zeta function, fulfilling appropriate admissibility conditions for wave functions to support general quantumâmechanical principles.We note that the Fourier transform does not allow localization in the phase space. In order to overcome this handicap D. Gabor introduced the concept of windowed Fourier transforms. From a group theoretical perspective the wavelet concept and the windowed Fourier transform are identical. With respect to the above Hilbert scale framework we note that the admissibility condition defining wavelets (e.g. (LoA)) puts the spot on the proposed H(1/2) (energy) Hilbert space. From a physical perspective the wavelet transform may be looked upon as a „ time-frequency analysis“ with constant relative bandwidth. From a „mathematization“
perspective „wavelet analysis may be considered as a mathematical microscope
to look by arbitrary (optics) functions over the real line R on different length
scales at the details at
position b that are added if one goes from scale „a“ to scale „a-da“ with
da>0 but infinitesimal small“, (HoM) 1.2. Technically speaking, „the wavelet transform allows
us to unfold a function over the one-dimensional space R into a function over
the two-dimensional half-plane “.
H of positions and details (where is
which details generated?). … Therefore, the parameter space H of the
wavelet analysis may also be called the position-scale half-plane since
if g is localized around zero with width „Delta“ then g(b,a) is localized
around the position b with width a*DeltaThe related wavelet duality relationship provides an additional degree of freedom to apply wavelet analysis with appropriately (problem specific) defined wavelets in a Hilbert scale framework where the " microscope observations" of two wavelet
(optics) functions g and h can be compared with each other by a "reproducing"
("duality") formula. The (only) prize to be paid is about additional
efforts, when re-building the reconstruction wavelet.
fun, fun, fun and learning", it looks like a good point in time to share resp. enable more
fun to the readers‘ side, who showed their interest by more than
1 GB downloads per day (on average) during the last years. From (KoJ) p. 148 we quote: „ find a skillful motivation.
Then do the math and enjoy the creativity of the mind“and, with the words of master Yoda: " may the Force be with you", ...:) .In order to support this some MS-Word based source documents of key papers are provided below. For people, who are familar with the german language and who want to get some guidance to autonomous thinking in current grazy times we recommend
the latest book from A. Unzicker: „Wenn man weiß, wo der Verstand ist, hat
der Tag Struktur“.
Braun K., Looking back, part B, (B1)-(B17), Dezember 2, 2020
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