> > 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/
.-- .- -. - / .- -.-. - .. --- -. ..--.. / -.-. --- -. .--- ..- --. .- - . 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/
