User david bernier - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:49:10Z http://mathoverflow.net/feeds/user/13292 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110017/injective-morphisms-of-cd-the-c-algebra-of-continuous-functions-on-the-clos injective *morphisms of C(D), the C^* algebra of continuous functions on the closed unit disk D David Bernier 2012-10-18T14:45:08Z 2012-10-18T15:34:40Z <p>This question relates to one on topology and C^*-algebras that was asked two days ago, namely at the link: <a href="http://mathoverflow.net/questions/109772/c-algebras-with-no-nontrivial-endomorphisms" rel="nofollow">http://mathoverflow.net/questions/109772/c-algebras-with-no-nontrivial-endomorphisms</a> <a href="http://mathoverflow.net/questions/109772/c-algebras-with-no-nontrivial-endomorphisms" rel="nofollow">link text</a></p> <p>Let D be the closed unit disk in the plane. Let C(D) be the unital ring of continuous complex-valued functions on D. Then, C(D) is naturally a Banach algebra with pointwise addition and multiplication as the ring operations. Furthermore, the "star-operation" on elements of C(D) can be defined by: $g*(x)$ to be the complex conjugate (pointwise) of g(x), any x in D, and for any function g in C(D).</p> <p>The question in reference 1 above was related to injective star-endomorphisms of some $C*$ -algebras. Without saying so, I think the famous commutative Gelfand-Naimark theorem and the Gelfand representation figured "in the background", because of the interplay of commutative $C*$-algebras and topology on compact spaces ... If $\alpha$ is an injective star-morphism of C(D) to itself, is it possible for $\alpha$(C(D)) to be a proper (i.e. a strict') star-sub-algebra of the $C*$-algebra C(D) ?</p> <p>If so, I cannot find such a special *-morphism $\alpha$, hence my question.</p> http://mathoverflow.net/questions/109635/automorphisms-of-subgroup-of-hamming-cube-under-distance-constraint/109898#109898 Answer by David Bernier for Automorphisms of subgroup of hamming cube under distance constraint David Bernier 2012-10-17T11:49:25Z 2012-10-17T11:49:25Z <p>For the additive group {0, 1}^n , it seems to me that every non-singular binary nxn matrix provides one F_2 linear bijective map from {0, 1}^n to itself . As I recall, asymptotically the number of these bijective linear maps is at least C 2^(n^2), for some C>0 . In other words, a strictly positive proportion of random nxn matrices over F_2 has non-zero determinant, and an nxn matrix over F_2 has nxn entries, each being 0 or 1. When you say automorphism, are you referring to automorphisms of the additive group (F_2)^n ? Thanks. David Bernier</p> http://mathoverflow.net/questions/109772/c-algebras-with-no-nontrivial-endomorphisms/109853#109853 Answer by David Bernier for C*-algebras with no nontrivial endomorphisms David Bernier 2012-10-16T21:41:07Z 2012-10-16T21:41:07Z <p>Trying to add information to Bill Johnson's Answer, I'd just like to say that the paper he mentioned by Cook is available for free on-line. Hopefully, below is a working link to the Cook paper in PDF format: matwbn.icm.edu.pl/ksiazki/fm/fm60/fm60123.pdf</p> <p>David Bernier</p> http://mathoverflow.net/questions/50245/the-hardy-z-function-and-failure-of-the-riemann-hypothesis/56882#56882 Answer by David Bernier for The Hardy Z-function and failure of the Riemann hypothesis David Bernier 2011-02-28T06:49:02Z 2011-02-28T06:54:54Z <p>I'd have another suggestion to replace your $Z(t/\log(t))$ : there's the Riemann-Siegel Theta function described in Harold Edwards' book, $\theta(t)$. The Gram points satisfy: $\theta(g_n) = n\pi$, $n$ = 1, 2, 3, ... So the idea is to look at $W(\alpha) = Z(\theta^{-1}(\alpha))$ . That way, if $g_n$ is the $n$th Gram point, $\theta^{-1}(n\pi) = g_n$ and<br> $W(n\pi) = Z(\theta^{-1}(n\pi)) = Z(g_n) = (-1)^n \zeta(1/2 + i g_n)$ .</p> <p>Cf. Gram's Law' at MathWorld: .</p> <p>Perhaps there's a way to rescale $W(.)$ vertically from $Z(.)$ to get identical square-integrals over corresponding intervals say $[g_n, g_{n+1}]$ for Z and $[n\pi, (n+1)\pi]$ for $W(.)$ ...</p> http://mathoverflow.net/questions/109635/automorphisms-of-subgroup-of-hamming-cube-under-distance-constraint/109898#109898 Comment by David Bernier David Bernier 2012-10-20T17:47:31Z 2012-10-20T17:47:31Z I might be mistaken. But perhaps the autmorphism group is the same as the hroups that preserve the Hamming distance on the metric space {0,1}^n with the Hamming distance metric? Noam Elkies wrote a survey article on linear error-correcting codes that was published in the Notices of the AMS: Elkies, Noam; &quot;Lattices, Linear Codes, and Invariants, Part II&quot;, Notices of the AMS, Volume 47 number 11, December 2000. <a href="http://www.ams.org/notices/200011/index.html" rel="nofollow">ams.org/notices/200011/index.html</a> http://mathoverflow.net/questions/110017/injective-morphisms-of-cd-the-c-algebra-of-continuous-functions-on-the-clos/110021#110021 Comment by David Bernier David Bernier 2012-10-19T07:40:25Z 2012-10-19T07:40:25Z Thanks. $f*$ is obtained by &quot;pre-composing&quot; through the map $f: D -&gt; D$. It's clear to me now. http://mathoverflow.net/questions/110017/injective-morphisms-of-cd-the-c-algebra-of-continuous-functions-on-the-clos/110021#110021 Comment by David Bernier David Bernier 2012-10-18T21:24:57Z 2012-10-18T21:24:57Z Thanks, Vahid Shirbisheh. I'm beginning to understand your answer. I'm not familiar with the definition or naming of $f*$ , given $f$ . I have heard of pull-back maps, but I don't know what they are. The same goes for so-called &quot;push-forward&quot; maps. David Bernier http://mathoverflow.net/questions/110017/injective-morphisms-of-cd-the-c-algebra-of-continuous-functions-on-the-clos Comment by David Bernier David Bernier 2012-10-18T21:09:23Z 2012-10-18T21:09:23Z Regarding having the question in community wiki, I now think it probably wasn't a wise decision. In the usual, non-community wiki mode, the question could have up-votes and down-votes; also, the answer(s) could have up-votes and down-votes. In my opinion, non-community wiki mode would have been preferable. It's an unfortunate decision for which I must take the responsibility. I hope to improve over time :) . http://mathoverflow.net/questions/50245/the-hardy-z-function-and-failure-of-the-riemann-hypothesis/56882#56882 Comment by David Bernier David Bernier 2011-02-28T10:22:46Z 2011-02-28T10:22:46Z Thanks for the help in fixing the LaTeX. What I find intriguing about your quasiperiodicity Conjecture is how (using Terry Tao's projective notion, say) one might have $t_n &gt; exp(exp(exp(exp(n))))$ , i.e. $t_n$ is allowed to grow extremely fast with n . Am I right that the sequence $t_n \element \R$ is allowed to depend on the interval $I$ and also on $\delta$ ?