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".
(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: (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 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 whereby G = T (denotes the
Einstein tensor and G 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)
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".
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