<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">In an affine space (vector space + point), an ant’s position is a point and a path or trail can be described by vectors.<div class=""><br class=""></div><div class=""><div class="">
<div>__________<br class=""><br class="">Ed Angel<br class=""><br class="">Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)<br class="">Professor Emeritus of Computer Science, University of New Mexico<br class=""><br class="">1017 Sierra Pinon<br class="">Santa Fe, NM 87501<br class="">505-984-0136 (home)<span class="Apple-tab-span" style="white-space: pre;"> </span> <span class="Apple-tab-span" style="white-space: pre;"> </span><a href="mailto:edward.angel@gmail.com" class="">edward.angel@gmail.com</a><br class="">505-453-4944 (cell) <span class="Apple-tab-span" style="white-space: pre;"> </span><span class="Apple-tab-span" style="white-space: pre;"> </span>http://www.cs.unm.edu/~angel<br class=""></div>
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<div><br class=""><blockquote type="cite" class=""><div class="">On Oct 24, 2021, at 5:27 PM, Frank Wimberly <<a href="mailto:wimberly3@gmail.com" class="">wimberly3@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="auto" class="">Focusing the application of ant foraging, what is the scalar product of an ant? What is the sum of two ants. Same questions for pheromone trails. If the answer is that scalar product and sum are not general enough. What are the homomorphisms of ants and pheromone trails?<br class=""><br class=""><div data-smartmail="gmail_signature" class="">---<br class="">Frank C. Wimberly<br class="">140 Calle Ojo Feliz, <br class="">Santa Fe, NM 87505<br class=""><br class="">505 670-9918<br class="">Santa Fe, NM</div></div><br class=""><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Sun, Oct 24, 2021, 5:12 PM Jon Zingale <<a href="mailto:jonzingale@gmail.com" class="">jonzingale@gmail.com</a>> wrote:<br class=""></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr" class=""><div class="gmail_default" style="font-size:small;color:rgb(51,51,51)"><pre style="white-space: pre-wrap;" class=""><font face="verdana, sans-serif" class="">I want to clarify what a dual space is. I think it is much more general
than Frank thinks it is: </font><a href="https://en.wikipedia.org/wiki/Adjoint_functors" target="_blank" rel="noreferrer" class="">https://en.wikipedia.org/wiki/Adjoint_functors</a></pre></div></div>
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