In SRM, for changing rate of modified global trajectory equiangular spiral extension {Δ^{2}S_{Tm} ->0_{+}}, and for all other MTCs (modified time constructs), such as changing rate of Hubble expansion – allegedly all are exponentially changing. Such always exponentially changing (likewise for unending derivatives) modified global variables, such as MTCs, MGS (Modified Global Simultaneity), might also be referred to as cosmic time t_{c}, which would seem very smooth; essentially analytical like?

All analytical functions are smooth, but not conversely. Also a function can be described utilizing a power series, such as for exponential function exp x=1+ x+ x^{2}/2! + x^{3}/3! + x^{4}/4! + … And extension of modified global trajectory is allegedly always exponentially changing; likewise for first, second, n^{th} derivative, giving again the exponential function. Hence essentially as smooth as power series expansion describing an analytical function?

So are most all functions that can be expressed as a power series – analytical? Then can MTCs’ sets and set of MGSs be utilized as part of a function? For example, is Hubble expansion always exponentially changing, and so too it’s derivatives; thus essentially analytical like? MTCs, abstracted and generalized as modified 1-manifold M^{1}_{m} , allegedly (together with it’s derivatives) are always exponentially changing; hence modified analytical? Such as for positive definite S_{Tm} ->0_{+ }element of {S_{Tm} ->0_{+}} trajectory of 3-manifold, and in general for 1-manifold M^{1}_{m} ->0_{+} member of {M^{1}_{m} ->0_{+}} for all MTCs?

Since integers can be mapped 1:1 etc.; thus defining a function) to MTCs {S_{Tm} ->0_{+}} and to change in rate of 3-volume expansion {Δ^{2} V^{3}_{L} } (i.e. domain), then can such modeling indicate that integers (herein considered co-domain) might be considered, in this model, as analytical like? Thus would power series expansion for such function [mapping of MTC (MGSs) domain to integer co-domain] give the latter i.e. integers, an analytical like description? If an integer(s) can be considered as a mathematical object, then might one consider the neighborhood of such 1-manifold object(s) as exploring not only ‘*inbetweeness’ concept of continuity, but also smoothness, even if no sea of irrationality context?
*

*J. Wallis: π/2 = 2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 …*

Thus an irrational number can be constructed from an infinite multiplicative series of rationals i.e. integer ratios. Then the rationals, together with the irrational gaps, can form a continuity of real numbers. However continuity, and thus manifold, can be described by mapping, not just with real number set, but also with rationals or just integers. Thus does it not seem so unreasonable, such as based on above series, for integers or rationals to also be considered analytical like?

From perspective of overall divergent set M^{1}_{m}≡{Δ^{2}S_{Tm} ->0_{+}}, an element of such set would have an infinitesimal description. Also from an elemental perspective, one could describe a neighborhood i.e. continuity, of such positive definite element, and for a function (mapping of sets), a power series expansion – giving an analytical like description? From such overall divergent conceptual perspective, and more so specifically from a neighborhood perspective, might one then seem to have an analytical like description for such function, and thus for a set of modified global time-like separations (i.e. instants) {ΔS_{Tm} ->0_{+}} along equiangular spiral trajectory; thus effectively likewise for mapping set of integers?

So for set of elements for MGSs, respective MTCs, including changing rate of Hubble expansion (i.e. one rendition of common cosmic time), rather than just a chronometer beating out integers, could one also consider the System as an endless evolving analytical engine? Hence again the importance of conceptual perspective, even for a description of integers, which appear to have an analytical like description, since mapped (i.e. constructing a function) to set of positive definite MTC elements, domain for such function in this model. Integers – analytical like, in context of this model?

In a contrasting model, for assumed divergent clustering of primes, wherein if each cluster were considered as an element, then there would be no relevant neighborhood, nor constructed function (i.e. mapping); thus no power series expansion. Hence differing from any alleged contrasting analytical – like description for set of integers, relating to a function i.e. mapping of sets. Thus might the *nature of time* construct, in regards to our evolving 3-manifold and alleged divergent set, System SΞ{U_{T}}, a set of total universes wherein U_{T}Ξ{{V^{3}_{R}}_{p}}, be considered as both always exponentially changing, and also analytical like; then would integers appear analytical like, in context of model? So then would *‘flow of time’* would seem to be very smooth and analytical like?

Likewise our finite appearing (coarser than Planck scale) 3-manifold *high perch* would seem rather smooth.

* ‘The number of ways in which a manifold can be smoothed is greatest in 3 or fewer dimensions.’ *

Consistent with our residing in an evolving smooth 3-manifold, component of a set of 3-manifolds of a smooth dynamic endlessly evolving System i.e. an analytical engine?

Reiterating, the equiangular spiral trajectory of our 3-manifold can seemingly be rendered as a set of successive outward spiraling positive definite elements {ΔS_{Tm} ->0_{+}} mapped to integers, conveniently with positive integers for future positive definite moments, and negative integers for past moments, and zero for present modified global instant, which would seem smooth. Thus for any positive definite modified global instant i.e. common *‘now’*, one has MGS ->0_{+} as an element of set of concentric positive definite thick spherical shells, for all patches for all 3-manifolds.

Also always exponentially changing modified global tangent velocity Vv_{R }MTC, member of {Vv_{R}}, for regional volume’s i.e. changing LSS (large scale streaming) tangent vector to equiangular spiral; likewise for other respective MTCs, concomitantly changing exponentially. This would seem a consistent description, that maps MGSs and respective MTCs to integers; giving an analytical like function relating to such sets? See also c/p/c and SRM abbreviations. TMM

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Thus in such modeling, would one have two concomitant renditions of an overall common cosmic time; the Hubble Expansion, intrinsic to changing 3-volume, and MGS, intrinsic to an evolving dynamic System, described as an analytical engine, consisting of an entangled array of manifolds, endlessly evolving? apeiron

Comment by zankaon — January 21, 2016 @ 2:15 pm