most recent 30 from http://mathoverflow.net 2013-05-21T20:20:26Z http://mathoverflow.net/feeds/user/5566 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22473/request-intermediate-level-proof-every-2-homology-class-of-a-4-manifold-is-gen Herb 2010-04-25T01:12:29Z 2012-11-10T21:28:58Z

<p>Hi, everyone:</p> <p>For the sake of context, I am a graduate student, and I have taken classes in algebraic topology and differential geometry. Still, the 2 proofs I have found are a little too terse for me; they are both around 10 lines long, and each line seems to pack around 10 pages of results. Of course, I am considering cases for "reasonable" spaces, being the beginner I am at this point.</p> <p>It would also be great if someone knew of similar results for H_1 (equiv. H_3).</p> <p>Thanks in Advance.</p>

http://mathoverflow.net/questions/22490/correlation-and-causation-when-can-we-believe-correlation-reasonably-at-least Herb 2010-04-25T06:58:15Z 2010-07-19T23:54:01Z

<p>We always hear, when reading on correlation, that "correlation does not imply causation."</p> <p>Still, I have never seen any source that tries to answer the question of when can we reasonably conclude a causal relation between variables X, Y from a correlation.</p> <p>After talking about it with some friends, we conclude these factors are at least necessary, if not sufficient to conclude causation:</p> <ol> <li><p>A high value of r², with r the correlation coefficient. Therefore, the value of r itself is also close to &plusmn;1. </p></li> <li><p>Data on X, Y was obtained under controlled/experimental conditions, and data for X, Y, taken under similar experimental conditions produces similar values for r². </p></li> <li><p>The time interval t between the independent X and the independent Y in many trials is short (if not immediate), and the values of t from different trials fall in a narrow interval.</p></li> </ol> <p>Anyone have comments?</p>

http://mathoverflow.net/questions/22905/when-is-there-a-derham-duality-relation-between-the-fundamental-class-and-a-top-f Herb 2010-04-28T23:04:09Z 2010-06-24T00:01:52Z

<p>Hi, everyone: I am reading a small expository paper on properties of CP², in which the intersection form is defined as an integral of the wedge of two forms $w_1$, $w_2$, and these forms $w_1$, $w_2$ (no problem with compact support, since CP² is compact) seem to have been obtained from the fundamental class [z] of H₂(CP²)-- a copy of CP¹ (embedded in CP²), after which we integrate $w:=w1\wedge w2$ to get the intersection number.</p> <p>I am curious on whether I am reading the above correctly, i.e., that the volume form in CP² is obtained by using the fund. class [z] in H₂. If not, would someone explain; if this is correct, if we are we using some form of deRham's theorem to turn a purely topological object like [z] into an object like $w$, for which we must have a differentiable structure defined)?</p> <p>Thanks in Advance.</p>

http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23545#23545 Herb 2010-05-05T04:25:01Z 2010-05-05T04:25:01Z

<p>I don't know if this is what you are looking for, but I keep hearing that "a differentiable function is one that is locally linear", not one whose local variation can be approximated linearly. No one stops to think about e.g, x², and the fact that its graph does not look like a line at any value of x.</p>

http://mathoverflow.net/questions/22480/triangulations-of-torus-general-and-euler-number-hopefully-more-interesting-r Herb 2010-04-25T03:52:32Z 2010-04-26T17:28:48Z

<p>Hi, everyone:</p> <p>I have been going over some simplicial homology recently, hoping to get<br> some geometric insight that I don't know how to get from the algebraic machinery alone.</p> <p>I have been trying to find the homology of the torus this way, i.e., by triangulating it ( i.e., finding a carrier for the torus), but the smallest triangulation I have been able to do , has 18 triangles/faces --I checked it works; there are 8 vertices and 26 edges. Still: does anyone know of a simpler triangulation, ie., one with a smaller total number of triangles (and, of course, fewer vertices and edges resp.). ?</p> <p>I had tried the long shot of solving the very simple equation:</p> <p>V-E+F =0 </p> <p>in positive integers.</p> <p>but this alone does not seem to help . Any ideas.?. Any ideas for finding minimal triangulations of surfaces, or higher-dimensional manifolds.?</p> <p>Thanks.</p>

http://mathoverflow.net/questions/22238/working-with-intersection-forms-in-homology-computation/22282#22282 Herb 2010-04-23T00:35:51Z 2010-04-23T02:00:40Z

<p>Thanks , David, both for the formatting and the ref. Unfortunately, I think my question may be much simpler than your refs: I have a matrix representation of a form in cohomology , which can be dualized (and I give this dualized form) to homology. This form/matrix is supposed to output an integer value; this value is the number of points of intersection of two submanifolds of a 4-manifold, with the sign having to see with the orientation of the two submanifolds.</p> <p>I am just not clear on how I can get this integer value from the matrix, i.e., how I can get the intersection number using this form; I know I need to evaluate this matrix on some 2x1 vector, I just have no idea of what this vector would be.</p> <p>Herb.</p>

http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23545#23545 Herb 2010-05-25T23:59:00Z 2010-05-25T23:59:00Z

Yes, I did not read the question very carefully. I realize it is not a good comment, and, yes, it is more of a abd heuristic than anything else.

http://mathoverflow.net/questions/23665/why-is-the-intersection-of-complex-submanifolds-always-positive Herb 2010-05-06T23:14:32Z 2010-05-06T23:14:32Z

Scott:why can't we argue the same for Gl(n,R) , using Gaussian elimination.?

http://mathoverflow.net/questions/23369/powers-of-quotient-maps Herb 2010-05-04T06:11:07Z 2010-05-04T06:11:07Z

I think it is true: If q:X--&gt;Y is a quotient map, then U in Y is open iff q&lt;sup&gt;-1&lt;/sup&gt; is open in X. If q&lt;sup&gt;2&lt;/sup&gt;:X&lt;sup&gt;2&lt;/sup&gt;-Y&lt;sup&gt;2&lt;/sup&gt; is a quotient map too, then UxU' is open in Y&lt;sup&gt;2&lt;/sup&gt; iff (q&lt;sup&gt;-1&lt;/sup&gt;(U),q&lt;sup&gt;-1&lt;/sup&gt;(U') is open in X&lt;sup&gt;2&lt;/sup&gt;. In general : q^n:X^n--&gt;Y^n , then q^n-1(U1xU2x...Un)= q^-1(U1)xq^-1(U2)x...xq^-1(Un) is open in X^n , as the product of open sets. I think the other side follows. in

http://mathoverflow.net/questions/22490/correlation-and-causation-when-can-we-believe-correlation-reasonably-at-least/22542#22542 Herb 2010-05-02T01:40:56Z 2010-05-02T01:40:56Z

O.K., Andrew, you are right, I did misread. Given that I chose not to use my full name, I should be more measured in my response. My apologies to F.Dorais for flying of the handle and assuming condescencion without enough evidence. Sorry, Francois.

http://mathoverflow.net/questions/22238/working-with-intersection-forms-in-homology-computation Herb 2010-04-28T05:41:25Z 2010-04-28T05:41:25Z

Thanks, Tim. I know that somehow CP^1 is a basis, but I am trying to understand why. I think it is because CP^1 is the characteristic class for H_2, but I am not sure. Would you please suggest.?

http://mathoverflow.net/questions/22473/request-intermediate-level-proof-every-2-homology-class-of-a-4-manifold-is-gen/22509#22509 Herb 2010-04-28T05:34:27Z 2010-04-28T05:34:27Z

Thank you both. Dylan's answer is nice , in that one can see the actual geometry. Torsten's answer is a bit out-of-my-league at this point, but I will still try to understand it.

http://mathoverflow.net/questions/22480/triangulations-of-torus-general-and-euler-number-hopefully-more-interesting-r Herb 2010-04-28T04:32:56Z 2010-04-28T04:32:56Z

Sorry. I meant that after having gained no real insight into the geometry behind spaces by using the big algebraic machinery, I am trying to do some computations of simplicial homology by hand. This is why I asked this question.

http://mathoverflow.net/questions/22490/correlation-and-causation-when-can-we-believe-correlation-reasonably-at-least/22542#22542 Herb 2010-04-28T04:27:57Z 2010-04-28T04:27:57Z

F.Dorais: The tone of my response had to see with what I perceived was a condescending attitude on your part; phrases like &quot;little question&quot; do seem to indicate condescencion. And my point (which I could have explained better) was that if/when conditions 1,2 and 3 were met, then it may be worthwhile to conduct experiments to detect possible causality between 2 variables. I never claimed that conditions 1,2, and 3 were indicative of the existense of causality.

http://mathoverflow.net/questions/22480/triangulations-of-torus-general-and-euler-number-hopefully-more-interesting-r Herb 2010-04-28T04:11:17Z 2010-04-28T04:11:17Z

To Charlie Frohman: This is not a homework question. I am computing the actual simplicial homology of spaces to get insights I do not know how to get by using the algebraic machinery alone (e.g., with simplicial homology) As to not stating my name, I have to admit I feel somewhat intimidated in this forum, being a first-year student at a school other than one of the top 10, specially after having read the resumes/CV's of many here. If this is against MO policy, I apologize, and I will drop out.

http://mathoverflow.net/questions/22480/triangulations-of-torus-general-and-euler-number-hopefully-more-interesting-r Herb 2010-04-25T03:53:58Z 2010-04-25T03:53:58Z

My apology. I mistakenly, and carelessly, entered triangulated categories as tags. Sorry.

http://mathoverflow.net/questions/22473/request-intermediate-level-proof-every-2-homology-class-of-a-4-manifold-is-gen Herb 2010-04-25T01:35:08Z 2010-04-25T01:35:08Z

Thanks, but this is a different question from the one you are referring/linking to. The only thing common to both is that both questions are on, or relate-to, 4-manifolds. AFAIK, the intersection form assumes the existence of representative surfaces, but does not prove their existence. An etiquette question: I was able to answer my own previous question, the one you linked to. But since I saw it was of such low interest in this site, I thought I would not comment on this. Is that O.K in this site.?