User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:15:34Z http://mathoverflow.net/feeds/user/6137 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56938/what-does-the-adjective-natural-actually-mean What does the adjective "natural" actually mean? unknown (google) 2011-02-28T23:49:35Z 2012-09-13T20:18:25Z <p>Terms like "in the natural way" or "the natural X" are used frequently in mathematical writing. While it is certainly clear most of the time what is meant, on occasion, I have been confounded. The word "natural" seems to be one of the most ambiguous terms used in formal mathematics. I have never seen anyone actually define it. People just use it and expect others to understand it.</p> <p>What exactly is meant by "natural" in mathematical writing?</p> http://mathoverflow.net/questions/57280/numbers-associated-with-boundaries-of-manifolds Numbers associated with boundaries of manifolds unknown (google) 2011-03-03T19:19:58Z 2011-03-31T22:22:20Z <p>I don't know what name if any is attached to the numbers I'm about to describe.</p> <p>For a line segment, [a,b]<br> the number is 1 if for any k in (a,b)<br> and 2 if k=a or k=b. </p> <p>For a square, [a,b] cross [c,d],<br> the number is 1 if k is in the interior<br> the number is 2 if k is on an edge<br> the number is 4 if k is a corner </p> <p>For a cube, [a,b] cross [c,d] cross [e,f],<br> the number is 1 if k is in the interior<br> the number is 2 if k is on an face<br> the number is 4 if k is on an edge<br> the number is 8 if k is a corner </p> <p>The concept I'm interested in might change these numbers if the spaces are non-rectangular. So,</p> <p>For a trapzoid,<br> the number is 1 if k is in the interior<br> the number is 2 for the edges<br> at each corner number the number is inverse of the fraction of the angle of that corner compared to R^2. </p> <p>Does this ring any bell for names that I can use for searching?</p> http://mathoverflow.net/questions/57280/numbers-associated-with-boundaries-of-manifolds/57287#57287 Answer by unknown (google) for Numbers associated with boundaries of manifolds unknown (google) 2011-03-03T21:04:42Z 2011-03-03T21:04:42Z <p>I was calculating the approximate density of a set A (line segment, square, cube, etc) in a ε-neighbourhood of a point x. It's related to the Lebesgue's density theorem. (Actually I was calculating the inverse of these numbers.)</p> http://mathoverflow.net/questions/29285/in-bayesian-statistics-must-i-use-a-marginalized-prior-in-conjunction-with-a-mar In Bayesian statistics, must I use a marginalized prior in conjunction with a marginalized distribution?// unknown (google) 2010-06-23T22:03:29Z 2010-07-08T02:22:16Z <p>Suppose I have some sampling distribution g(x,y,z) which has been marginalized over some variables (say y and z) giving us the marginal distribution which we'll call gx(x).</p> <p>Suppose I now wish to use Bayes Theorem but on the marginalized distribution to obtain the posterior marginal distribution. Suppose I also know the a good prior for all the variables, call it k(x,y,z).... To use Bayes theorem on the marginalized distribution, must I also marginalize the prior? Or does it makes sense to use the full prior, which of course makes, the answer depend on </p> http://mathoverflow.net/questions/60583/how-is-a-n-dimensional-gaussian-distribution-reduced-to-an-n-1-dimensional-distri Comment by 2011-04-05T18:29:56Z 2011-04-05T18:29:56Z The other places to ask (math exchange) gave wrong answers. I think it is pretty clear what I am trying to do but here's another way to explain it. Suppose a have a 2D gaussian and one of the variable's (say Y) variance approaches zero, how can I show that the other variable X is described by a 1D normal distribution? That is, in the limit of sigma_Y goes to zero? – unknown (google) 0 secs ago http://mathoverflow.net/questions/60583/how-is-a-n-dimensional-gaussian-distribution-reduced-to-an-n-1-dimensional-distri/60613#60613 Comment by 2011-04-05T18:27:20Z 2011-04-05T18:27:20Z This is nice response. I still think something is missing though. There's no notion of &quot;when&quot; it's okay to use the lower dimensional Gaussian. http://mathoverflow.net/questions/60583/how-is-a-n-dimensional-gaussian-distribution-reduced-to-an-n-1-dimensional-distri/60593#60593 Comment by 2011-04-04T18:59:09Z 2011-04-04T18:59:09Z Now I'm not so sure. This ALWAYS produces a 1D gaussian from a 2D. Requires not other condition. http://mathoverflow.net/questions/60583/how-is-a-n-dimensional-gaussian-distribution-reduced-to-an-n-1-dimensional-distri Comment by 2011-04-04T18:53:32Z 2011-04-04T18:53:32Z What I'd like to do is say something along the lines that if sigma_Y -&gt; 0, then the distribution for X approaches a gaussian. http://mathoverflow.net/questions/57280/numbers-associated-with-boundaries-of-manifolds Comment by 2011-03-03T20:53:25Z 2011-03-03T20:53:25Z How is it consistent with all my examples? Three out of the four examples aren't even in R^3 so solid angle doesn't even apply. In the lone R^3 example, solid angle only applies for the corners. http://mathoverflow.net/questions/57280/numbers-associated-with-boundaries-of-manifolds Comment by 2011-03-03T20:03:41Z 2011-03-03T20:03:41Z I'm only measuring solid angles at the smallest non-empty boundary, not in general. http://mathoverflow.net/questions/57280/numbers-associated-with-boundaries-of-manifolds Comment by 2011-03-03T19:28:00Z 2011-03-03T19:28:00Z The think my question is related to the Lebesgue's density theorem. http://mathoverflow.net/questions/56938/what-does-the-adjective-natural-actually-mean/56956#56956 Comment by 2011-03-03T16:12:57Z 2011-03-03T16:12:57Z Hi, Sandor. Your answer and John Sidles' answer seem to be competing for the #1 spot. I know very little about category theory. I asked Sandor to try to explain your answer in terms of a simple example using a two-dimensional vector space and the &quot;natural&quot; inner product. Another thing that is still confusing me, is that even if there are &quot;natural&quot; choices under category theory, that still requires the reader take category theory as an assumption. If they do no such thing, I suppose it invalidates any formal notion and renders the word having only its &quot;ordinary speech&quot; definition. http://mathoverflow.net/questions/56938/what-does-the-adjective-natural-actually-mean/56948#56948 Comment by 2011-03-01T17:38:22Z 2011-03-01T17:38:22Z Interesting reply. A follow-up: suppose we were talking about a vector space of 2-tuples, $V$. When people say something like &quot;let $V$ have the natural inner product&quot;, we all know it means $a_x b_x+a_y b_y$ but one can go very far in the mathematics and physics literature just assuming natural means &quot;the most common, most Euclidean thing&quot;. Could you briefly explain why is the inner product above &quot;natural&quot; in the sense of category theory, expanding on your &quot;arbitrary choice of coordinates should make no difference&quot; comment. http://mathoverflow.net/questions/55245/canceling-positive-definite-functions-from-an-integral-that-equals-zero Comment by 2011-02-14T01:41:13Z 2011-02-14T01:41:13Z $f = -3x + 100$ and $g= 1x+1$ from -10 to 10 is even nicer, with integer coefficents. http://mathoverflow.net/questions/55245/canceling-positive-definite-functions-from-an-integral-that-equals-zero Comment by 2011-02-13T23:22:25Z 2011-02-13T23:22:25Z Here's a solution using only lines: $f(x) = -(3/100) x + 1$, $g(x) = 4 x + 4$ has $\int_{-10}^{10} f(x) g(x) dx = 0$ but $\int_{-10}^{10} g(x) dx = 80$ and $\int_{-10}^{10} f(x) dx = 20$. http://mathoverflow.net/questions/55245/canceling-positive-definite-functions-from-an-integral-that-equals-zero Comment by 2011-02-13T21:34:05Z 2011-02-13T21:34:05Z I realized shortly before my last update, that this wasn't even the question I wanted to ask, I had intended to write if <i>both</i> f(x) and g(x) are greater than zero except on boundary, a question which if the answer is false would require a bizarre solution that stretches the imagination. It was a long day yesterday, I had that case in mind even while writing exactly the opposite above. http://mathoverflow.net/questions/55245/canceling-positive-definite-functions-from-an-integral-that-equals-zero Comment by 2011-02-13T00:13:32Z 2011-02-13T00:13:32Z It's the second mean value theorem that is most relevant so far. Unfortunately, f seems to require certain monotonicity properties even in one-dimension. A counter example is f(x) = 1 + 16 Delta(x+9) and g(x)=1 for x&gt;-8 and g(x)=-1 for x&lt;-8. A construction can be done without the delta function, or even discontinuous functions. Yuck.