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This homepage provides solutions to the
following Millennium problems
- the Riemann Hypothesis
- well-posed 3D-nonlinear, non-stationary Navier-Stokes equations
- the mass gap problem of the Yang-Mills equations.
A common distributional
Hilbert space framework provides an answer to Derbyshine's question ((DeJ) p.
295): „ What on earth does the distribution of prime numbers
have to do with the behavior of subatomic particles?". It also enables a quantum
gravity theory based on an only (energy related) Hamiltonian formalism, as the
corresponding (force related) Lagrange formalism is no longer defined due to the
reduced regularity assumptions to the domains of the concerned pseudo differential operators.
The key ingredients of the Zeta function
theory are the Mellin transforms of the Gaussian function and the fractional
part function. To the author´s humble opinion the main handicap to prove the
RH is the not-vanishing constant Fourier term of both functions. The Hilbert
transform of any function has a vanishing constant Fourier term. Replacing the Gaussian function
and
the fractional part function by their corresponding Hilbert transforms enables an alternative
Zeta function theory based on two specific Kummer functions and the cotangens function. The imaginary part of the zeros of one of the Kummer functions play a key role defining alternatively proposed arithmetic functions to solve the binary Goldbach conjecture.
The common distributional
Hilbert space framework goes along with reduced regularity assumptions for the domain of the momentum (or pressure) operator. In the context of the 3-D-NSE problem this enables energy norm estimates "closing" the Serrin gap, while at the same point in time overcoming current "blow-up" effect handicaps.
The classical Yang-Mills theory is the generalization
of the Maxwell theory of electromagnetism where chromo-electromagnetic field
itself carries charges. As a classical field theory it has solutions which
travel at the speed of light so that its quantum version should describe
massless particles (gluons). However, the postulated phenomenon of color
confinement permits only bound states of gluons, forming massive particles.
This is the Yang-Mills mass gap. The variational representation of the time-harmonic Maxwell equations in the proposed "quantum state" Hilbert space framework H(-1/2) builds on truly fermions (with mass) & bosons (w/o mass) quantum states / energies, i.e. a Yang-Mills equations model extention is no longer required.
The common mathematical (geometrical Hilbert space based) model also enables a quantum gravity model based on Bohm's "hidden variables" theory (with the concept of "quantum potentials"), in line with Einstein's ether vision and his Special Relativity theory, Wheeler's gravitation & inertia conception and Schrödinger's "view of the world". At the same point in time Dirac's model of the point mass density of an idealized point mass is replaced by Plemelj's definition of a mass element.
A decomposition of the newly proposed energy Hilbert space H(1/2) provides a model for "complementary" thermodynamic energy and ether (ground state) energy. The first one is governed by Fourier's (one-parameter) waves, Kolmogorow's (statistical) turbulence model, Einstein's Special (Lorentz invariant) Relativity, Klainerman's global nonlinear stability of the Minkowski space, Vainberg's conceptions of second order surfaces in Hilbert spaces (hyperboloid (conical and hyperbolic regions) defined by corresponding potential barriers), Almgren's varifold geometry (in the context of least area problems) and the Heisenberg's uncertainly relation, while the second one is governed by Calderón's (two-parameter) wavelets (to go from scale "a" to scale "a-da"), Bohm's revisited quantum potential and Plemelj's mass element conceptions.
The thermodynamic Hilbert (energy) space H(1) is compactly embedded into the newly proposed Hilbert (energy) space H(1/2). From a statistical point of view it means that the probability to catch a quantum state/"elementary particle", which is able to collide with another one, is zero. This compactly embeddedness enables a new interpretation of the entropy phenomenon as the change process from thermodynamical (kinetic) energy to ether (ground state, "dark", "quantum potential", "Leibniz's living force") energy.
Mathematically speaking the expanded new energy Hilbert space (1/2) (where the Heisenberg uncertainty inequality is valid) enables the Hamiltonian formalism, only. Only for the standard energy Hilbert space H(1) (which is a compactly embedded, separable Hilbert (sub-) space of H(1/2)) the corresponding Lagrange formalism is defined due to a valid Legendre transformation, because of appropriate regularity of the Hilbert space H(1). In other words, Emmy Noether's theorem is valid only in the H(1) framework. It means that if the Lagrange functional is an extremal, and if under corresponding infinitesimal transformation the functional is invariant to a certain definition, then a corresponding conservation law holds true.
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 Braun K., RH, YME, NSE, GUT solutions, overview |
latest update: Dec 4, 2019, pp. 9, 12, 23
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The journey
started in 2010 resulting in the paper
Braun
K., The three Millennium problem solutions (RH, NSE, YME) and a Hilbert
scale based quantum geometrodynamics, July 2019
The
journey is still ongoing: in order to support an appropriate change traceability this paper is split into
PART
I:
Braun
K., A Kummer function based alternative Zeta function theory to solve the Riemann
Hypothesis and the binary Goldbach conjecture
Braun K., RH solutions |
last update : July 31, 2019
PART
II:
Braun
K., 3D-NSE and YME mass gap solutions in a distributional Hilbert scale frame
enabling a quantum gravity theory
Braun K., 3D-NSE, YME, GUT solutions |
last update : July 31, 2019
With respect to the NSE topic we also refer to
http://www.navier-stokes-equations.com/
With respect to the ground state energy and quantum gravity topic we also refer to
https://quantum-gravitation.de/
