<div dir="auto">My pain was unbearable until I saw my neurologist who prescribed Gabapentin and then my primary care physician added Cymbalta. Both relieve nerve pain. My left arm is partially paralyzed. I can't raise it above my chest. All of this is because of an impingement of a nerve on C6, left side. It was amazing how the neurologist diagnosed that. It involved tiny needles and mild shocks.<div dir="auto"><br><div dir="auto"><div dir="auto"><div dir="auto"><br><br><div data-smartmail="gmail_signature" dir="auto">-----------------------------------<br>Frank Wimberly<br><br>My memoir:<br><a href="https://www.amazon.com/author/frankwimberly">https://www.amazon.com/author/frankwimberly</a><br><br>My scientific publications:<br><a href="https://www.researchgate.net/profile/Frank_Wimberly2">https://www.researchgate.net/profile/Frank_Wimberly2</a><br><br>Phone (505) 670-9918</div></div></div></div></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sat, Mar 9, 2019, 4:28 PM Steven A Smith <<a href="mailto:sasmyth@swcp.com">sasmyth@swcp.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div text="#000000" bgcolor="#FFFFFF">
<p>Nick -</p>
<p>I do know that reading my missives *can be* excruciatingly
painful but I do trust those without such masochistic tendencies
to use their <delete> or <next> buttons freely.</p>
<p>Frank -</p>
<p>Sorry I can't commiserate better with your physical pain... but
in an ironic reversal of roles, my pain is entirely abstract
(existential angst) while yours sounds to be entirely embodied!</p>
<p>- Steve<br>
</p>
<div class="m_8799912283011008667moz-cite-prefix">On 3/9/19 4:23 PM, Nick Thompson wrote:<br>
</div>
<blockquote type="cite">
<div class="m_8799912283011008667WordSection1">
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366">Sorry,
everybody,<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366"><u></u> <u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366">I
am experiencing phantom pain in Steve’s body. <u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366"><u></u> <u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366">Gotta
read these threads more carefully. <u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366"><u></u> <u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366">Nick
<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366"><u></u> <u></u></span></p>
<div>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366">Nicholas
S. Thompson<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366">Emeritus
Professor of Psychology and Biology<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366">Clark
University<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366"><a href="http://home.earthlink.net/~nickthompson/naturaldesigns/" target="_blank" rel="noreferrer"><span style="color:#0563c1">http://home.earthlink.net/~nickthompson/naturaldesigns/</span></a><u></u><u></u></span></p>
</div>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#993366"><u></u> <u></u></span></p>
<div>
<div style="border:none;border-top:solid #e1e1e1 1.0pt;padding:3.0pt 0in 0in 0in">
<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext">From:</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:windowtext">
Friam [<a class="m_8799912283011008667moz-txt-link-freetext" href="mailto:friam-bounces@redfish.com" target="_blank" rel="noreferrer">mailto:friam-bounces@redfish.com</a>] <b>On Behalf
Of </b>Steven A Smith<br>
<b>Sent:</b> Saturday, March 09, 2019 4:17 PM<br>
<b>To:</b> The Friday Morning Applied Complexity Coffee
Group <a class="m_8799912283011008667moz-txt-link-rfc2396E" href="mailto:friam@redfish.com" target="_blank" rel="noreferrer"><friam@redfish.com></a><br>
<b>Subject:</b> Re: [FRIAM] Manifold Enthusiasts<u></u><u></u></span></p>
</div>
</div>
<p class="MsoNormal"><u></u> <u></u></p>
<p>Nick -<u></u><u></u></p>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1f497d">All
I can say is, for a man in excruciating pain, you sure
write good. Your response was just what I needed. </span><u></u><u></u></p>
</blockquote>
<p>Something got crossed in the e-mails. *I'*m not in
excruciating pain... that would be (only/mainly/specifically)
Frank, I think. But thanks for the thought!<u></u><u></u></p>
<p>Any excruciating pain I might be in would be more like
existential angst or something... but even that I have dulled
with a Saturday afternoon Spring sunshine, an a cocktail of
loud rock music, cynicism, anecdotal nostalgia, and
over-intellectualism. Oh and the paint fumes (latex only)
I've been huffing while doing some touch-up/finish work in my
sunroom on a sunny day is also a good dulling agent.<u></u><u></u></p>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1f497d">Now,
when I think of a manifold, my leetle former-english-major
brain thinks shroud, and the major thing about a shroud is
that it <i>covers</i> something. Now I suspect that this
is an example of irrelevant surplus meaning to a
mathematician, right? A mathematician doesn’t give a fig
for the corpse, only for the properties of the shroud.
But is there a mathematics of the relation between the
shroud and the corpse? And what is THAT called. </span><u></u><u></u></p>
</blockquote>
<p>Hmm... I don't know if I can answer this fully/properly but
as usual, I'll give it a go:<u></u><u></u></p>
<p>I think the Baez paper Carl linked to has some help for this
in that. I just tripped over an elaboration of a topological
boundary/graph duality which might have been in that paper.
But to be as direct as I can for you, I think the two
properties of <i>shroud</i> that *are* relevant is
*continuity* with a surplus but not always irrelevant meaning
of *smooth*. In another (sub?)thread about <i>Convex Hulls</i>,
we encounter inferring a continuous surface *from* a finite
point-set. A physical analogy for algorithmically building
that <i>Convex Hull</i> from a point set would be to create a
physical model of the points and then drape or pull or shrink
a continuous surface (shroud) over it. Manifolds needn't be
smooth (differentiable) at every point, but the ones we
usually think of generally are. <u></u><u></u></p>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1f497d"> So,
imagine the coast of Maine with all its bays, rivers and
fjords. Imagine now a map of infinite resolution of that
coastline, etched in ink. I assume that this is a
manifold of sorts. </span><u></u><u></u></p>
</blockquote>
<p class="MsoNormal">In the abstract, I think that coastline
(projected onto a plane) IS a 1d fractal surface (line). To
become a manifold, it needs to be *closed* which would imply
continuing on around the entire mainland of the western
hemisphere (unless we artificially use the non-ocean political
boundaries of Maine to "close" it).<br>
<br>
<u></u><u></u></p>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1f497d"> Now
gradually back off the resolution of the map until you get
the kind of coastline map you would get if you stopped at
the Maine Turnpike booth on your way into the state and
picked a tourist brochure. Now that also is a manifold of
sorts, right? In my example, both are representations of
the coastline, but I take it that in the mathematical
conception the potential representational function of a
“manifold” is not of interest? </span><u></u><u></u></p>
</blockquote>
<p>I think the "smoothing" caused by rendering the coastline in
ink the width of the nib on your pen (or the 300dpi printer
you are using?) yields a continuous (1d) surface (line) which
is also smooth (differentiable at all points)... if you
*close* it (say, take the coastline of an island or the entire
continental western hemisphere (ignoring the penetration of
the panama canal and excluding all of the other canals between
bodies of water, etc. then you DO have a 1D (and smooth!)
manifold.<u></u><u></u></p>
<p>If you zoom out and take the surface of the earth (crust,
bodies of liquid water, etc), then you have another manifold
which is topologically a "sphere" until you include any and
all natural bridges, arches, caves with multiple openings. If
you "shrink wrap" it (cuz I know you want to) it becomes
smooth down to the dimension of say "a neutrino". To a
neutrino, however, the earth is just a dense "vapor" that it
can pass right through with very little chance of
intersection... though a "neutrino proof" shroud (made of
neutrino-onium?) would not allow it I suppose.<u></u><u></u></p>
<p>This may be one of the many places Frank (and Plato) and I
(and Aristotle) might diverge... while I enjoy thinking
about manifolds in the abstract, I don't think they have any
"reality" beyond being a useful archetype/abstraction for the
myriad physically instantiated objects I can interact with.
Of course, the earth is too large for me to apprehend directly
except maybe by standing way back and seeing how it reflects
the sunlight or maybe dropping into such a deep and perceptive
meditative state that I can experience directly the
gravitational pull on every one of the molecules in my body by
every molecule in the earth (though that is probably not only
absurd, but also physically out of scale... meaning that
body-as-collection-of-atoms might not represent my own body
and that of the earth and I think the Schroedinger equation
for the system circumscribing my body and the earth is a tad
too complex to begin to solve any other way than just
"exisiting" as I do at this location at this time on this
earth.) <u></u><u></u></p>
<p>If you haven't fallen far enough down a (fractal
dimensioned?) rabbit hole then I offer you:<u></u><u></u></p>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<p><a href="https://math.stackexchange.com/questions/1340973/can-a-fractal-be-a-manifold-if-so-will-its-boundary-if-exists-be-strictly-on" target="_blank" rel="noreferrer">https://math.stackexchange.com/questions/1340973/can-a-fractal-be-a-manifold-if-so-will-its-boundary-if-exists-be-strictly-on</a><u></u><u></u></p>
</blockquote>
<p>Which to my reading does not answer the question, but kicks
the (imperfectly formed, partially corroded, etc.) can on down
the (not quite perfectly straight/smooth) road, but DOES
provide some more arcane verbage to decorate any attempt to
explain it more deeply?<u></u><u></u></p>
<p>- Steve<u></u><u></u></p>
<p>PS. To Frank or anyone else here with a more acutely
mathematical mind/practice, I may have fumbled some details
here... feel free to correct them if it helps.<u></u><u></u></p>
<p><u></u> <u></u></p>
</div>
<br>
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