which is about a Hilbert space of differential forms with appropriately defined inner product replacing an exterior algebra (with exterior derivatives)
(RoC) xi, " The characteristic of an affine geometry is the fact, that only parallel distances can be measured against each other, i.e. other kinds of infinitesimal small "actions" between not parallel "objects" are not considered in this kind of "continuum". Vectors are the mathematical model of such translations (resp. parallel displacements) and the underlying (affine) geometry is mathematically described by the group properties of vectors (WeH). An affine geometry with space dimension n is the “same” as its related (n-1)-dimensional projective group. The “enrichment” of the today's n-dimensional space-time affine geometry (manifolds and affine connexions, (ScE), (WeH1) and quotes § 18 below) goes along with the concept of The Sobolev H(1/2) space on the circle plays a key role of universal period mapping universal Teichmüller parameter space for all Riemann surfaces via quantum calculus: Biswas I., Nag S., Jacobians of Riemann Surfaces and the Sobolev Space H(1 2) on the Circle Nag S., Sullivan D., Teichmüller theory and the universal period mapping via quantum calculus and the H(1 2) space on the circle.
- enabling an infinitesimal small geometry model with an inner product defined by "rotating differential forms", alternatively to based onexterior derivatives differentiable (!) manifolds- enabling a truly infinitesimal geometry, alternatively to the affine connexions (affine, parallel infinitesimal displacements, only)- not changing the way, "how to measure distances" ( Archimedean axiom), but changing the "what to be measured", i.e. the structure of the underlying field from an ordered to a non-ordered field- not increasing the "degree" of transcendence "complexity" (knowing that this is a question of yes/no, of course), if this is measured by Cantor's definition of cardinality (as the field of Non-Standard numbers *R does have the same cardinality than the field of real numbers R) - applying the Riesz and Caldéron-Zygmund Pseudo Differential Operators (PDO) with domains in Hilbert spaces H(-a), a>0 enabling convergent (!) quantum oscillator energy series in a Hilbert space H(-a), for appropriate a>0.Here we are:
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