What?? You right brain people are driving me nuts! 😁 --- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505
505 670-9918 Santa Fe, NM On Mon, Oct 25, 2021, 12:22 AM Stephen Guerin <stephen.gue...@simtable.com> wrote: > On Sun, Oct 24, 2021 at 5:00 PM Frank Wimberly <wimber...@gmail.com> >> wrote: >> I want to clarify what a dual space is. > > > The domain of dual.space can be mapped to any other domain :-) > > _______________________________________________________________________ > stephen.gue...@simtable.com <stephen.gue...@simtable.com> > CEO, Simtable http://www.simtable.com > 1600 Lena St #D1, Santa Fe, NM 87505 > office: (505)995-0206 mobile: (505)577-5828 > twitter: @simtable > z <http://zoom.com/j/5055775828>oom.simtable.com > > > On Sun, Oct 24, 2021 at 5:00 PM Frank Wimberly <wimber...@gmail.com> > wrote: > >> I want to clarify what a dual space is. I think it is much more specific >> than Jon thinks it is. A vector space is a linear space which consists of >> vectors for which addition and scalar multiplication are defined. Scalars >> are usually real numbers but may elements of other fields such a complex >> numbers. >> >> An important example of a (finite dimensional) vector space is the set of >> n-tuples of real numbers. A linear functional on a vector space is a >> function defined on the set of vectors the into the set of scalars. This >> is a (0,1) tensor on the space. The set of linear functionals is also a >> vector space called the dual space. The vectors of the original space >> define linear functionals on the dual space as follows: v(f) = f(v). >> >> The only other dual space of which I am aware of is the dual topological >> space which Google tells me is >> >> In functional analysis and related areas of mathematics a dual topology >> is a *locally convex topology on a dual* pair, two vector spaces with a >> bilinear form defined on them, so that one vector space becomes the >> continuous dual of the other space. >> >> I am being very succinct and may not remember these definitions >> correctly. Jon may be speaking "metaphorically" when he talks about >> pheromone trails being duals of ants. >> >> Anyway... >> >> Frank >> >> --- >> Frank C. Wimberly >> 140 Calle Ojo Feliz, >> Santa Fe, NM 87505 >> >> 505 670-9918 >> Santa Fe, NM >> >> On Sun, Oct 24, 2021, 1:05 PM Jon Zingale <jonzing...@gmail.com> wrote: >> >>> """ >>> The problem for me with this view is that I don't understand how seeing >>> pheromone as 'organizing itself in space' is intuitively useful. >>> """ >>> >>> I suppose that even if I didn't find this view *useful*, which I do and >>> will attempt to explain momentarily, I continue to find that it offers >>> a theoretical completeness that I find aesthetically compelling. Much >>> like magnetism and electricity can appear as distinct phenomena or as >>> two aspects of an integrated whole, stigmergy points to a similar duality >>> between agent and environment, another integrated whole. Lifting to such >>> a perspective offers insights into a class of possible implementations, >>> all preserving the underlying dynamics. >>> >>> For instance, when attempting to reason about the ant-pheromone system, >>> I find it useful to view the ants as inefficient *raster-like* update >>> to the state, but of course one could also choose a less brownian, >>> less resource-limited or less discrete approach. For instance, I believe >>> it makes the analysis more clear if we instead picture a continuum of >>> ants acting on the space and begin with pheromone of very little effect. >>> Then slowly turning up the potency, we begin to see the pheromone >>> organize and as a side-effect (and as an epiphenomenon from my view) >>> the ants follow suit. To view the ants as an implementation detail, for >>> me, yields clarity into the problem, while the pheromone takes the role >>> of first class citizens in the ABM. >>> >>> """ >>> Toward the end, you wrote, "I only meant to emphasize that stigmergy >>> appears to me as a local concept." I'm not sure what that means. >>> """ >>> >>> By local, I mean local as it often manifests in mathematics, but I >>> gather that you would prefer a different tack. Here I am referring to a >>> pair of related concepts for me: >>> >>> 1. Excision of glider's from Conway's game. >>> >>> 2. Characterization of subjectivities (one's subjective experience, say) >>> relative to objectivity. >>> >>> When one watches Conway's game unfold, it is challenging to maintain the >>> view that gliders are not agents but simply a local patch of board state >>> in the process of updating itself as a whole. The ease with which we >>> perceive gliders as agents facilitated the discovery of "glider guns", >>> and ultimately the construction of assemblages of "glider guns" to make >>> logic gates. Further, there is a smallest possible toroidal board such >>> that one can have a glider (and only a glider) walk along the surface >>> forever. The relation of this smallest board to any other board state is >>> (in my view) an analogy, an inclusion relation. >>> >>> Localization, here for me, is a way of bracketing the baby from the >>> bathwater while continuing to acknowledge that the meaning of both is >>> in relation to a whole. There have been a number of discussions on this >>> forum (and quite a few papers by its participants) where work is done >>> to emphasize the complications associated with non-trivial mereological >>> systems, systems of parts with non-trivial structural constraints. Two >>> recent papers that come to mind are: >>> >>> 1. https://www.biorxiv.org/content/10.1101/2021.02.09.430402v1.full >>> 2. https://arxiv.org/pdf/1811.00420.pdf >>> >>> .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - >>> . >>> FRIAM Applied Complexity Group listserv >>> Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam >>> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com >>> FRIAM-COMIC http://friam-comic.blogspot.com/ >>> archives: >>> 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ >>> 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >>> >> >> .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . >> FRIAM Applied Complexity Group listserv >> Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam >> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com >> FRIAM-COMIC http://friam-comic.blogspot.com/ >> archives: >> 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ >> 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >> > > .-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn UTC-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ > archives: > 5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/ > 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ >
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