<HTML><BODY><p style="margin-top: 0px;" dir="ltr">Dear Krassimir, <br> Thanks a lot for your question, now the discussion will become hotter!</p>
<p dir="ltr">First of all, we never stated that consciousness lies either on a n-sphere or on an Euclidean n-space.<br> Indeed, in our framework, consciousness IS the continuous function. <br> Such function stands for a gauge field that restores the brain symmetries, broken by sensations. <br> Concerning brain and gauge fields, see my PLOS biology paper: <br> http://journals.plos.org/plosbiology/article?id=10.1371%2Fjournal.pbio.1002400</p>
<p dir="ltr">When consciousness lacks, the inter-dimensional projections are broken, and the nervous higher functions temporarily disappear. </p>
<p dir="ltr">Concerning the question about which are the manifolds where brain functions lie, it does not matter whether they are spheres, or circles, or concave, or flat structures: we demonstrated that the BUT is valid not just for convex manifolds, but for all the kinds of manifolds. <br> See our: <br> http://onlinelibrary.wiley.com/doi/10.1002/jnr.23720/abstract?userIsAuthenticated=false&deniedAccessCustomisedMessage=<br></p>
<p dir="ltr">Therefore, even if you think that brain and biological functions are trajectories moving on concave structures towards lesser energetic levels, as suggested by, e.g., Fokker-Planck equations, it does not matter: you may always find the antipodal points with matching description predicted by BUT. </p>
<p dir="ltr">Ciao!</p>
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<p dir="ltr">--<br> Inviato da Libero Mail per Android</p>
</div>sabato, 26 novembre 2016, 06:23PM +01:00 da Krassimir Markov <a href="mailto:markov@foibg.com">markov@foibg.com</a>:<br><br><blockquote id="mail-app-auto-quote" style="border-left:1px solid #85AF31; margin:0px 0px 0px 10px; padding:0px 0px 0px 10px;" cite="14801811490000025350">
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<p><font size="4" face="Times New Roman">Dear FIS colleagues,</font></p>
<p><font size="4" face="Times New Roman">I think, it is needed to put discussion
on mathematical foundation. Let me remember that:</font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font face="Times New Roman"><font size="4">The <b>Borsuk–Ulam theorem</b>
(BUT), states that every </font><a title="Continuous function" style='href: "https://en.wikipedia.org/wiki/Continuous_function"'><strong><font size="4">continuous function</font></strong></a><font size="4"> from an </font><a title="N-sphere" style='href: "https://en.wikipedia.org/wiki/N-sphere"'><font size="4"><i>n</i>-sphere</font></a><font size="4"> into </font><a title="Euclidean space" style='href: "https://en.wikipedia.org/wiki/Euclidean_space"'><strong><font size="4">Euclidean <i>n</i>-space</font></strong></a><font size="4"> maps some pair
of </font><a title="Antipodal point" style='href: "https://en.wikipedia.org/wiki/Antipodal_point"'><font size="4">antipodal points</font></a><font size="4"> to the same point.
</font></font></p>
<p><font size="4" face="Times New Roman">Here, two points on a sphere are called
antipodal if they are in exactly opposite directions from the sphere's
center.</font></p>
<p><font size="4" face="Times New Roman">Formally: <strong>if</strong> <span><span style="display: none">f : <em>S<sup> n</sup> → R</em> <sup>n</sup> </span></span> <strong>is</strong>
<strong>continuous</strong> then there exists an <span><span style="display: none">x ∈ S n </span></span> such that: <span><span style="display: none">f ( −
x ) = f ( x )</span></span>.</font></p>
<p><font size="4" face="Times New Roman">[ </font><a title="https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem" href="https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem" target="_blank" ><font size="4" face="Times New Roman">https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem</font></a><font size="4" face="Times New Roman"> ] </font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font face="Times New Roman"><font size="4">Who may proof that consciousness is
a </font><a title="Continuous function" style='href: "https://en.wikipedia.org/wiki/Continuous_function"'><font size="4" ><strong>continuous
function</strong></font></a><font size="4" > from
reflected reality ???</font></font></p>
<p><font face="Times New Roman"><font size="4">Who may proof that consciousness is
an </font><a title="Euclidean space" style='href: "https://en.wikipedia.org/wiki/Euclidean_space"'><font size="4" ><strong>Euclidean
<i>n</i>-space</strong></font></a><font size="4" >
???</font></font></p>
<p><font size="4" face="Times New Roman" >After
proving these statements we may think further.</font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font size="4" face="Times New Roman">Yes, discussion is interesting but, I am
afraid, it is not so scientific.</font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font size="4" face="Times New Roman">Friendly regards</font></p>
<p><font size="4" face="Times New Roman">Krassimir</font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font size="4" face="Times New Roman"></font></p></div></div>
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