OVERVIEW
LOOKING BACK
RIEMANN HYPOTHESIS
QUANTUM GRAVITY
NAVIER-STOKES EQUATIONS
WHO I AM
LITERATURE


This homepage is dedicated to my mom, who died at April 9, 2020 in the age of 93 years.

This homepage considers multiple research areas. Looking back the proposed solution concepts originate in some few simple ideas / basic conceptual changes to current insufficient "solutions": 

(A) a Riemann Hypothesis (RH) solution in line with the Hilbert-Polya conjecture (which is about the existence of a proper self-adjoint integral operator going along with the concept of convolution operators, (CaD)) needs to overcome the mathematical problem of the not vanishing constant Fourier term of the Jacobian theta function. Every Hilbert transformed function has a vanishing constant Fourier term. The Hilbert transform of the Gaussian function is the Dawson function with its relationship to a specific Kummer function

(A1) the proposed solution concept, replacing the Gaussian function by its Hilbert transform, supports the proof of several RH criteria:

The Euler product formula connects the Riemann function z(s) (the analytic continuation of the power function representation beyond the halfplane Re(s)>1) with primes. Taking the log of both sides of the Euler product formula results into a Stieltjes integral representation of the log(z(s)) function with a (prime) density dJ(x) ((EdH) 1.11).

The entire Zeta function Z(s), fullfilling the Riemann duality equation Z(s) = Z(1-s), is basically a product of the three functions z(s), (s-1) and G(1+s/2), whereby G(s) denotes the Gamma function ((EdH) 1.13). The method for deriving the formula for J(x) is then basically about the calculation of the Fourier inverses of log(Z(s)), -log(s-1) and -log(G(1+s/2)). The Fourier inverse of the principle term -log(s-1) becomes the logarithmic integral represention of the Li(x) function. The Fourier inverse of the term -log(G(1+s/2)) leads to the famous Riemann approximation error function between the prime density function J(x) and the Li(x) function ((EdH) 1.16).

An appropriate convergence behavior of the Riemann error function is one of several RH criteria.

Replacing the Gaussian function by its Hilbert transform leads to an alternative entire Zeta function definition with same zeros in the critical stripe as Z(s). It basically replaces the term (s/2) of the term G(1+s/2) = (s/2)*G(s/2) by the term tan((p/2)s) = cot((p/2)(1-s)), whereby p denote the circle number „pi“. The method for deriving the formula for J(x) is then about the calculation of the Fourier inverses of log(Z(s)), -log(s-1), -log(cot(p/2)(1-s)) and -log(G(s/2)). We note that the Zeta function on the critical line and the divergent Fourier series representation of the cot(x) function (BeB), which is Cesàro summable, are both elements of H(-1). The term -log(tan(p/2)s) is L(2) integrable and can be represented as a convergent series (ElL). It results into a correspondingly modified Riemann error function or a correspondingly modified definition of the Li(x) function. In both cases the approximation behavior between the „Li(x)“ function and the prime density function gets improved.

The entire Zeta function is the baseline for all RH criteria, i.e. the alternative entire Zeta function provides new opportunities for all existing RH criteria

(A2) Bagchi's "Hilbert space based reformulation of the Nyman-Beurling RH criterion" (BaB) is based on the periodical distributional Hilbert space H'(-1). It goes along with the definition of the zeta function z(s) based on the periodical fractional part function r(x). The first derivative of the Hilbert transform of r(x), the log(sin)-function with its related Fourier series representation, is given by the divergent Fourier series representation of the cot(x) function. It is an element of the Hilbert space H'(-1). The fractional part function and its Hilbert transform are elements of H'(0).  Therefore, trying to verify the Bagchi-Nyman convergence criterion (i.e. verifying its underlying density proposition) in a variational (weak topology) framework puts the spot on the H'(-1/2) Hilbert space,: the Fourier series representation of the cot function or the constant sequence g:=(1,1,1,1,....) are distributional "function" in the sense that they are elements of H'(-1). The latter can be interpreted as the counterpart "function" of the zeta function on the critical line (BaB). Therefore, it can be representated as a singular, symmetric, bounded convolution integral operator B from H(0) --> H(-1)) with Bv = g. A corresponding variational formulation is about test function elements of L'(2)=H'(0) functions, enabling a corresponding selfadjoint Friedrichs extension operator supporting the verification of Bagchi criterion. For an analysis of convolution operators in the context of the zeros of entire functions we refer to (CaD)

(A3) Literature (RH) Polya‘ s theorem“: "Attempts have been made to apply Polya’s general theorem about the zeros of the Fourier transform of a real function to the Zeta function (PoG). After a change of variable t --> log and approximating the integral over the half-line (positive x-axis) by integrals over finite intervals (which are the essential restriction/handicap of Polya’s theorem to be applied to the Zeta function), one can obtain a theorem about zeros of Mellin transforms. For the Zeta function, the Müntz formula has been used, but the Polya theorem only applies in the interval (0,1), and no information is obtained about the zeros of the Zeta function in the critical open stripe of validity of Müntz’s formula".

We note that in the context of Tauberian Theorems for Generalized Functions ((VlV) p. 57), the crucial condition of Polya's theorem is equivalent to one of the characterization criteria of an automodel "function".


(B)  A quantum gravity model requires some goodbyes from current postulates of quantum mechanics/dynamics models and Einstein’s field model as per definition both theories are not compatible:

(KaM) p. 12: „Because general relativity and quantum mechanics can be derived from a small set of postulates, one or more of these postulates must be wrong. The key must be to drop one of our commonsense assumptions about Nature (with respect to the underlying physical models, which are (1) continuity, (2) causality, (3) unitarity, (4) locality, (5) point particles), on which we have constructued general relativity and quantum mechanics.“

The approach of this homepage is about challenging the postulates of both theories with respect to the underlying mathematical postulated concepts. The key ingredient of quantum mechanics is the L(2) = H(0) Hilbert space to model quantum states with correspondingly related quantum momenta as elements of the Hilbert space H(1). The key ingredients of Einstein’s field equations are Riemann‘s differentiable manifolds (whereby the differentiability condition is w/o any physical meaning) in combination with the concept of affine connexion (enabled by the differentiability condition) to build the metric g based (Riemann manifold) metric space (M,g).

The proposed quantum gravity model is based on the following changes:

(1) the Dirac „function“ to model the charge of a point particle (going along with a Hilbert space H(-n/2-e), where n denotes the space dimension, and e>0), is replaced by elements of the Hilbert space H(-1/2)

(2) Dirac’s concept of a spin of an electron is replaced by quanta elements of the (dual) Hilbert space pair H(-1/2) (position) and H(1/2) (momentum/energy)

(3) the solution of classical theoretical physics PDE is interpreted as an approximation solution to the solution of the underlying (weak) variational PDE representation of the PDE and not the other way around; from a mathematical point of view this allows reduced regularity requirements of the concerned PDE solution(s)

(4) all „Nature forces“ are based on the same concept of underlying „living (bosons) and kinetic (fermions) energies“; the (dual) Hilbert space pair H(-1/2) and H(1/2) ensures a valid Hamiltonian formalism, while the applied Lagrange formalism of the SMEP is not valid due to reduced regularity assumptions of variational solution in the above Hilbert space pair framework; however, the Lagrange formalism keeps valid for the classical approximation solutions with its underlying notions of "Nature forces". This is due to the fact that the Langrange and Hamiltonian formalisms are equivalent, if the Legendre transformation is valid, which is the case for the classical approximation solutions

With respect to (3) concerning "scalability" we quote from Smolin L., Einstein's Unfinished Revolution, The search what lies beyond the quantum, xvii:

In these chapters I hope to convince you that the conceptual problems and raging disagreements that have been bedeviled quantum mechanics since its inception are unsolved and unsolvable, for the simple reason that the theory is wrong. It is highly successful, but incomplete. Our task - … - must be to go beyond quantum mechanics to a desription of the world on an atomic scale that makes sense“.

The notion "force" becomes ("only") an intrinsic part on each of the considered physical situations, mathematically represented as classical PDE, which are governed by mathematical notions like "continuity" or "differentiability". The scale-up capability from weak/quantum (Hilbert space based) variational representations to e.g. continuous or differentiable function spaces is given by the Sobolev embedding theorems.

With respect to (4) we note that the proposed model is in line with the concepts of the "thermal time hypothesis" and a "kinematical state space" equipped with an scalar product, (RoC) 3.4, 6.2. However, "defining the coupled gravity + matter system by adding the terms defining the matter dynamics to the gravitational relativistic hamiltonian", ((RoC) (7.32)), as the baseline concept for the loop quantum gravity is avoided, (RoC) 7.3. As a consequence, the Yang-Mills mass gap problem "just" disappears, (RoC) 7.2.1.

With respect to the consequences of the proposed model for well-posed non-linear, non-stationary Navier-Stokes equations with correspondingly reduced regularity assumptions to the initial and boundary value functions see (B16) below.

(B1) the concept of differentiable manifolds required for properly defined classical Einstein field equations needs to be avoided:

(a) Weyl’s world metric to build a „Purely infinitesimal geometry (excerpt)“ is still only based on the metric space (M,g). From a mathematical point of view in order to define a geometric framework a metric space is not sufficient (the field of real numbers equipped with the distance metric is a metric space; everyone would agree that this field does not show a geometric structure). The concept of an inner product is required leading to the concept of a Hilbert space. As the related norm of an inner product is a metric, each Hilbert space is also a metric space. Our proposed Hilbert space model provides an alternative approach for a "purely infinitesimal (truly) geometry"

(b) the "differentiability" requirement is without any physical meaning and even continuous manifolds would be hard to be united with a Hilbert space based quantum theory, (KaM) 1.2

(B2) functional analysis, Hilbert spaces and operators build the foundation of quantum mechanics. One famous conclusion out of it, is the Heisenberg uncertainty relationship. When applying an operator in physical models it is not all the time correctly defined as its underlying domain, which is beside the mapping the second essential part of the definition of an operator, is not specified. The standard unspoken domain assumption in quantum mechanics seems to be, that, what ever it is, it needs to fit to the "quantum state" Hilbert space model: this is the Lebesgue integral based L(2) Hilbert space, which is used e.g. in mathematical statistics and physical (Kolmogorow) turbulence and thermodynamics theory; however, the quantum mechanics model requires a Hilbert space, only

(B3) the Dirac „function“ concept with its underlying space-time depending (distribution) regularity needs to be avoided just from a mathematical perspective, as well as from its sophisticated physical interpretation as a "mathematical point" particle charge; we note that when picking a real number out of the x-axis the probability that it is an irrational or even a transcendental number is 100%; this is quite an unusual measure of a physical quantity; with respect to (B1) we note that the completeness axiom required to define irrational numbers is also essential for the definition of the notion "continuity"

(B4) replacing the Dirac "function" concept by H(-1/2) distributions goes along with the definition of an inner product for differentials (BrK); the replacement can be compared with a replacement of the Archimedean ordered field of "real" numbers by the non-Archimedean ordered field of hyperreal numbers. The latter ones are also called ideal numbers, which goes back to the monadology concept of Leibniz. The term "real" is somehow missleading: every irrational number "is" its own universe, i.e. it is defined as an infinite limit of rational numbers. We note that both fields do have the same cardinality and that the Archimedean axiom basically states, that each positive real number "x" can be "measured" as product of an integer "n" times another real (standard length) number "y". Another non-Archimedean field is the Levi-Civita field

(B5) Hawking S. W., „A Brief History of Time“, chapter "Elementary Particles and the Forces of Nature“:

All known particles in the universe can be divided into two groups: particles of spin ½, which make up the matter in the universe, and particles of spin 0, 1, and 2, which give rise to forces between matter particles“.

"A particle of spin 0 is like a dot: it looks the same from every direction. A particle of spin 1 is like an arrow: it looks different from different directions. Only if one turns it round a complete revolution (360 degrees) does the particle look the same. A particle of spin 2 is like a double-headed arrow: it looks the same if one turns it round half a revolution (180 degrees). Similarly, higher spin particles look the same if one turns them through smaller fractions of a complete revolution. ... there are particles that do not look the same if one turns them through just one revolution: one has to turn them through two revolutions! Such particles are said to have spin 1/2."

„The matter particles obey what is called Pauli’s exclusion principle. … It says that two similar particles cannot exist in the same state; that is, they cannot have both the same position and the same velocity, within the limits given by the uncertainty principle. The exclusion principle is crucial because it explains why matter particles do not collapse to a state of very high density under the influence of the forces produced by the particles of spin 0, 1, and 2; if the matter particles have very nearly the same positions, they must hve different velocities, which means that they will not stay in the same position any longer. If the world had been created without the exclusion principle, quarks would not form separate, well-defined protons and neutrons. Nor would these, to gether with electrons, form separate, well-defined atoms. They would all collapse to form a roughly uniform, dense „soup““.

Mathematically speaking, the uncertainty principle is caused by different domains of the quantum position and momentum operators. (We note that an operator is only well-defined by both criteria, the mapping rule of the operator and its domain). In other words, putting both physical parameters, position and momentum, as one („Nature forces“ type specific) "spin-" attribute of a corresponding particle type violates the prerequisites for well-defined position and momentum operators.

In our model Dirac’s  spin(1/2)-concept and its related SMEP interaction particles with spin 0, 1, and 2 are no longer required. All (energy/mass) fermions are modelled as elements of the Hilbert space H(1); the corresponding fermion states are modelled as elements of the corresponding Hilbert space H(0). The complementary sub-space H(1,ortho) of H(1) in H(1/2) provides a (closed sub-space) bosons model of „energy/momentum interaction elements" between fermions, replacing the three SMEP „interaction particles" model with spin 0, 1, and 2. The corresponding fermions state Hilbert space is given by H(0), while the corresponding bosons state Hilbert space is given by H(0,ortho), which is a closed sub-space of H(-1/2). Pauli’s exclusion principle is still valid and is given implicitly, as the separable Hilbert space H(1) (the "actors") is compactly embedded into H(1/2) (the "stage"), resp. the separable Hilbert space H(0) (the "actors") is compactly embedded into H(-1/2) (the "stage"); see also (PeR) 1.3, "Phase space, and Boltzmann's defintion of entropy".

Therefore, in our model the "Nature forces" phenomena become "just" implicit part of the considered (Hamiltonian formalism based) variational representations of the considered (classical) Partial Differential Equations. We mention that the concept of a compactly embedded, sparable Hilbert space follows the same building principles and related properties, as the field of rational numbers is compactly embedded into the field of real numbers.

(B6) The discrete Shannon entropy is derived from a set of axioms showing a bunch of nice properties that it exhibite. The formally defined related "continuous" entropy based on the Riemann integral concept in (MaC) (Marsh C., Introduction to Continuous Entropy) shows several weaknesses; it "is highly problematic to the point that, on its own, it may not be an entirely useful mathematical quantity".

The current phase space concept can be easily adapted to the Hilbert space pairs H(0) := (H(0),H(1)) resp. H(-1/2) := (H(-1/2),H(1/2)) coming along with the Lebesgue integral resp. the Plemelj/Stieltjes integral concepts (BrK).

The Boltzmann (statistics) entropy formula in the context of the physical phase space can be interpreted as a coarse graining entropy in the H(0) framework. The Hilbert space pair H(0) comes along with Dirac's mass density concept. It is dense in H(-1/2) (with respect to the H(-1/2) norm), coming along with Plemelj's mass element concept; the decomposition of H(-1/2) into H(0) and its complementary pair of two closed sub-spaces enables the definition of a H(-1/2)-based entropy definition, which can be derived from a set of axioms formulated in the separable H(0) framework.

The mathematical analysis tool of the fermion Hilbert space H(1) is the Fourier transform governed by the (one-parameter) Fourier waves; the corresponding analysis tool for the complementary closed subspace of H(1) in the H(1/2) framework is the continuous (two-parameter) wavelet transform, going back to Calderón's reproducing formula for radial L(1)-functions with vanishing constant Fourier term (LoA).

(B7) The geometry of the granular fermions Hilbert space H(1) (in the sense of its compactly embeddedness into H(1/2)) in combination with specific properties of the Friedrichs extension of the Laplacian operator (whereby the latter defines the Newton potential) allows to distinguish between repulsive and attractive fermions:

the Friedrichs extension of the Laplacian operator is a selfadjoint, bounded operator B with domain H(1). Thus, 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). This direct sum of two subspaces of H(1) is proposed as a model to distinguish between repulsive and attractive fermions

(B8) the regularity of the distribution Hilbert space H(-a) containing the Dirac function is given by the condition a=n/2+e (e>0), where n denotes the space dimension of the underling R(n) field; the Sobolev embedding theorem in the form that H(a) is continuously embedded into C(0), denoting the Banach space of continuous functions, provides the linkage of the Dirac point charge concept the concept of continuity, where both notions a purely mathematical concepts (without any physical meanings on elementary quantum level) even defined resp. demanded by axioms, only; at the same point in time both concepts are used in all classical theoretical physics (Ordinary or Partial Differential Equation, ODE or PDE) model

(B9) The classical Maxwell Equations are PDE with respect to the space parameter „x“ and ODE with respect to the time parameter „t“. They build the foundation of Lorentz’s theory of electrons. Its underlying Lorentz transformation builds the foundation of Einstein’s SRT. The electric and magnetic fields in „(source) free regions“, i.e. regions without charges and magnetic fields (i.e. even a Dirac point particle charge is not allowed), can travel with any shape, and will propagate at a single speed, which turned out to be light velocity c. Mathematically, the underlying hyperbolic wave equations are derived by applying the curl operator to the electric and magnetic field equations (going along with additional regularity requirements to both fields) in source free regions. Then both equations reduce to the identical vector wave equation with the single parameter c. Therefore, applying the (hyperbolic, time-symmetric) wave equation as model for gravitation waves and corresponding ODEs to „calculate back“ to early universe states already anticipates that „one of the assumed nicest properties of the universe“ is based on the assumption that every vacuum is source free

(B10) A variational representation of the Maxwell equations in an extended Hilbert quantum state framework H(-1/2) with source free regions in H(0) resp. H(1), only would still allow classical Maxwell and wave equation models as approximations to the “truly” quantum gravity model. However, the concepts of space, time, cause and action are only defined and valid as part of the classical PDE approximation models; the required non zero vaccuum (energy) states are element of the complementary sub space to the classical variational Hilbert spaces H(0) resp H(1). The model then would allow a correspondingly extended modified SRT including energy “quanta” into Lorentz’s theory of electrons, which is claimed to overcome Einstein’s mathematical problem to include gravitation forces into his (mathematically well defined) SRT 

(B11) (WeH)  p. 171: „On the basis of rather convincing general considerations, G. Mie in 1912 pointed out a way of modifying the Maxwell equations in such a manner that they might possibly solve the problem of matter, by explaining why the field possesses a granular structure and why the knots of energy remain intact in spite of the back-and-forth flux of energy and momentum“. This concept is in line with our proposed compactly embedded „fermions“ energy Hilbert space H(1) into H(1/2), where a H(1/2)-based (energy) field possesses a H(1)-based granular (matter) structure

(B12) the notion "symmetry" with all its mathematical (group theory, Lie groups) and physical (gauge symmetry, Higgs' symmetry break down, hidden symmetry) flavors should be replaced by the notion "self-adjointness", which is the central property of a linear operator of the Hilbert space based spectral theory in the context of the Friedrichs (self-adjoint) extension of a linear symmetric operator; a self-adjoint operator allows the definition of a related "energy" inner product /norm, (VeW)

(B13) (ChD1) pp. 1, 10-13: „Einstein’s field equations is about an unified theory of space-time and gravitations; the space-time (M,g) is the unknown, where M denotes a 4-dimensional manifold; one has to find an Einstein metric g, fulfilling the Einstein field equations. This is basically the equality G = T, whereby G denotes the Einstein tensor and T denotes the energy momentum tensor (e.g. the Maxwell equations). The Einstein-Vacuum equations (in the absense of matter, i.e. T = 0) are given by R = 0, whereby R denotes the Ricci tensor. Its simplest solution is the Minkowski space-time with its canonical coordinate system. Apart from Minkowski space-time it is not known, if there are any smooth, geodesically complete solution, which becomes flat at the infinity on any given spacelike direction. The main difficulties one encounters in the proof for the Cauchy Einstein-Vacuum equations with given initial data are: (1) the problem of coordinates (2) the strongly nonlinear hyperbolic features of the Einstein equations. The problem of coordinates comes along with the concept of manifolds. To write the equations in a meaningful way, one seems forced to introduce coordinates. Such coordinates seem to be necessary even to allow the formulation of well-posed Cauchy problems and a proof of a local in time existence result. Nevertheless, as the particular case of wave coordinates illustrates, the coordinates may lead, in the large, to problems of their own.

The concept of manifolds was introduced by Riemann to model the physical phenomenon „force“ as a consequence of a hyperbolic geometry, replacing Newton’s concept of a „far distance force" by a „near distance force" concept. The alternative approach of this homepage is about keeping the „Riemann's formula“ „force“ = „geometry“ ((WeH3) III, 15), but introducing a truly geometric Hilbert space framework coming along with an inner product (whereby the related Hilbert space norm defines a metric), alternatively to the current affine connected manifold framework (based on the concepts "affine connexion", "covariant derivative" and "geodesics of an affine connexion"; Schrödinger E., Space-Time Structure) to enable the definitions of a metric and an (at least) exterior product. We emphasize that the affine connexion concept is not suitable to overcome open contact body problems in the context of interaction of elementary particles

(B14) The Newton gravitation model is about the potential equation. The counterpart of the underlying Laplace operator of the potential equation in the Einstein gravitation model G = T (whereby G denotes the Einstein tensor and T denotes the energy momentum tensor) is the Einstein tensor G. The weak variational formulation of the potential equation leads to the energy Hilbert space H(1). Its norm is equivalent to the L(2)-norm of the gradient of the considered field. If the Newton (L(2)-based variational) gravitation model is interpreted as an approximation on a more accurate H(-1/2)-based variational potential equation model th corresponding potential solution can be intepreted as a compact disturbance of the Newton potential solution, which could cover all strongly nonlinear hyperbolic features of the Einstein equations enabled by "Convex Analysis in General Vector Spaces" (Zalinescu C.)

(B15) the overall physical principle is the minimum action principle (in the context of the compactly embeddedness of the today's standard energy Hilbert space H(1) into the newly proposed "energy" Hilbert space H(1/2)) with its mathematical counterpart, the variational calculus, where a self-adjoint operator enables the definition of a corresponding energy norm based minimization problem, (VeW), (NiJ1); the counterpart of non-linear problems is given by the Garding inequality, which can be interpreted as a decomposition of the non-linear operator into the sum of a linear, self-adjoint operator and a compact disturbance operator, e.g. (LiP1)

(B16) the chaotic inflation state of the early universe does not match to the second law of thermodynamics. The latter requires a permanent increase of the entropy of the universe over time, i.e. the cosmos started with an incredible low probability, but also with an incredible high ordered state, "at the same point in time" ((PeR) 2.6, "Understanding the way the Big Bang was special"). The energy/action minimizing principle is equivalent to a corresponding orthogonal projection onto a compactly embedded sub-space. This orthogonal projection can be interpreted as an extended model (symmetry ---> selfadjointness) of the Higgs "spontaneous symmetry break down" mass generation model. Therefore, this orthogonal projection becomes a "mass generation" operator in the sense that "mass is essentially the manifestation of the vacuum energy". In other words, there is a Hilbert space model for a perfect ordered (only vaccuum energy) system until a very unlikely first event of such a projection occured; this is because the "fermions" Hilbert (sub-) space is compactly embedded into the overall energy Hilbert space. Therefore, from a probability/statistics theory perspective the probability of this first event is zero with respct to the underlying Lebesgue measure. It might sound sophicated or even strange, but it is just the same probability, when picking a rational number out of the field of real (including irrational and transcendental) numbers on the x-axis (which is the domain framework required to define continuous functions)

(B17) the gauge (symmetry) groups S(3)xSU(2)xU(1) of the SMEP (and the still missing graviton gauge group, (KaM)) could be replaced by certain self-adjoint properties of related linear operators; Fourier waves could be replaced by Calderon wavelets, while from a group theoretical perspective Calderon's wavelet and Gabor's windowed Fourier transformations are the same. They are both built by the same construction principle based on the affine-linear group resp. based on the Weyl-Heisenberg group (LoA)

(B18) when changing the variational framework from H(0) to H(-1/2) the non-linear, non-stationary Navier-Stokes equations with correspondingly reduced regularity assumptions to the initial and boundary value functions become well posed, while at the same time the Serrin gap problem disappears; from a physical modelling perspective the extended H(1/2) norm based energy measure of the non-linear term does not vanishes, in opposite to the current H(1) energy norm; at the same point in time the potential incompatibility of the initial boundary values of the NSE with the Neumann problem based prescription of the pressure at the bounding walls dissappears.

(C) Schopenhauer's "theory of explaining" (which he called "about the fourfold root of sufficient reason") is about the different categories explaining the (his four) different root causes & actions of the world's representations, answering the "why?" question, based on the concept "something is, because something else has been before"; in today's world this would go along with the scope of all theoretical physics & neuroscience phenomena/representations, but not including the only suspected cause of a "big bang" "event". Schopenhauer's "(the) world as will and representation" (written about 200 years ago) also addresses the "what?" question, which he answered with the concept of "will", which is a kind of "vital principle" or "living energy" (or "living force" according to Leibniz) affecting both, ("dead") matter and creatures; in the context of this homepage this concept "will" might be interpreted as analogy to the enlarged scope of the mathematical ("dark energy", Einstein's "ether" energy) model as proposed in this homepage

(C1) for some first touchpoints between philosophical "views of the world" and the proposed quantum gravity model we refer to the small books of the „views of (their) world“ from A. Einstein and E. Schrödinger, as well as to Einstein's "ether and the theory of relativity" and Schrödinger's "statistical thermodynamics" and "mind and matter". From the latter we quote (chapter 5): „The great thing (of Kant’s statement) was to form the idea that this one thing – mind or world – may well be capable of other forms of appearence that we cannot grasp and that do not imply the notions of space and time. This means an imposing liberation from our inveterate prejudice. There probably are other orders of appearence than the space-time-like. It was, so I believe, Schopenhauer who first read this from Kant“. ....
"To my view the 'statistical theory of time' has an even stronger bearing on the philosophy of time than the theory of relativity. The latter, however revolutionary, leaves untouched the undirectional flow of time, which is presupposes, while the statistical theory constructs it from the order of the events. This means a liberation from the tyranny of old Chronos.

With respect to (B) above we note that the "time variable" can be introduced via the "action variable", defined as the solution of a corresponding ODE (HeW)

(C2) overall, it might be said, that while Schopenhauer's concept overcomes the "dialectic" concept of Fichte/Hegel (which is about the "practical ethics" dualism problem of the German idealism between "be" and "should be"), the proposed mathematical model overcomes the Copenhagen "dualism" interpretation (going back to Bohr/Born/Heisenberg) to "explain" the contractions between the apparently "parallel existing explanations" of wave (energy) and particle (matter) behaviors, which both have been verified experimentally by two different experiment

(C3) there is no longer an energy concept, which is somehow interwovened with concepts like forces, matter and causality, but which not includes the 99% "dark" energy / matter of the universe and its non zero vacuum energy. There is an extended energy concept proposed, which distinguishes between those two kinds of energy "classes" modelled as a decomposition of the Hilbert space H(1/2) = H(1,ortho) + H(1)), while the (matter based) "bright" energy Hilbert (sub-) space H(1) is "only" compactly emdedded into H(1/2)

(C4) Schopenhauer's and Schrödinger's views of the world were very much influenced from the Upanishades as presented in the Vedas. The above decomposition concept might be interpreted as analogy to the notion "Brahm", the universal, all flowing power, and the notion "Maya", the world of imaginations. In this case both notions become defined and part of a system with consistently defined related notions, i.e. they become part of the existing as a whole ("das Seiende im Ganzen"). In terms of Schopenhauer's conception of will & representation it corresponds to an aimless, cosmic, universal energy as the reason for the universe (will), and its appearance as representation

(C5) regarding the perspectives of Schopenhauer's philosophy on phenomenology, existentialist philosophy and hermeneutics and the corresponding impact on scientific and metaphysical research we refer to

(ReT) Regehly T., Schubbe D., Schopenhauer und die Deutung der Existenz, J. B. Metzler Verlag GmbH, Stuttgart, 2016

(C6) for a quick overview with incredible insights to latest findings into a neuroscience view of the world and its relationship to chemistry (and therefore also to theoretical physics) we refer to

           (KlS) Klein S., The Science of Happiness, Scribe UK, 2015

(C7) as a kind of bridge to Buddhist philosophy between the personal dedication and some touchpoints to the above we would consider

(KoJ) Kornfield J., The Wise Heart, Bantam Books, Random House Inc., New York, 2009.


Officially accepted solutions of the considered research areas would be honored by several prizes. For hopefully understandable reasons none of the papers of this homepage are appropriately designed to go there. Therefore, after a 10 years long journey accompanied by four main ingredients "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“.

For this purpose this page providing the MS-Word based source documents of some key papers.

A small, closed building area to start with could be to go for "a truly proof of the observed non-linear Landau damping phenomenon based on a variational representation of the Boltzmann-Landau equations".



             

1_Braun K., RH, YME, NSE, GUT solutions, overview


             

2_Braun K., RH solutions


             

3_Braun K., A Kummer function based Zeta function theory to prove the Riemann Hypothesis and the Goldbach conjecture


             

4_Braun K., 3D-NSE, YME, GUT solutions


             

5_Braun K., Global existence and uniqueness of 3D Navier-Stokes equations

      
             

6_Braun K., A new ground state energy model


             

7_Braun K., An alternative Schrödinger momentum operator enabling a quantum gravity model


             

8_Braun K., Comparison table, math. modelling frameworks for SMEP and GUT


             

9_Braun K., An integrated electro-magnetic plasma field model


            

10_Braun K., Unusual Hilbert or Hoelder space frames for the elementary particles transport (Vlasov) equation


            

11_Braun K., A distributional Hilbert space framework to prove the Landau damping phenomenon


Disclaimer: None of the papers of this homepage have been reviewed by other people; therefore there must be typos, but also errors for sure. Nevertheless the fun part should prevail and if someone will become famous at the end, it would be nice if there could be a reference found to this homepage somewhere.