User some_random_guy - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:57:17Z http://mathoverflow.net/feeds/user/3071 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique Is the direction of the longest line of a polytope unique? some_random_guy 2010-01-08T18:06:39Z 2010-05-29T01:11:56Z <p>The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions. </p> <p>The affine subspace is given by:</p> <p>$X \mbox{ u} = y$</p> <p>where </p> <p>$u$ &isin; $\mathbb{R}^p$.</p> <p>$X$ is a $m$ &times; $p$ matrix with $m$ &lt; $p$ and </p> <p>$y$ is a $m$ dimensional vector.</p> <p>The hypercube is given by:</p> <p>$0$ &le; $u$ &le; $u_{max}$ </p> <p>The longest line of this polytope is clearly one of the lines that join the vertices of the polytope. My question is:</p> <p>Is the direction of the longest line independent of $u_{max} \mbox{&amp;nbsp}$ for some $u_{max}$ &ge; u* ?</p> <p>Some simulations I did in Matlab indicates that the answer to the above question is yes but I am not sure if this will hold in general. I am assuming of course that the polytope actually resides in the hypercube defined by $0$ &le; $u$ &le; $u_{max}$. </p> <p>Any pointers to relevant literature in applied math or some approaches to answer the question would be very helpful.</p> <p>Thanks</p> http://mathoverflow.net/questions/16177/can-t-test-be-used-for-non-inferiority-hypothesis-testing/16454#16454 Answer by some_random_guy for Can t-test be used for non-inferiority hypothesis testing? some_random_guy 2010-02-25T21:41:48Z 2010-02-25T21:41:48Z <p>Perhaps, a slight re-formulation may help to deal with the issue pointed our by Sheldon response. State your null as follows:</p> <p>$H_0: A - B = \delta$</p> <p>$H_a: A - B >= \delta$</p> <p>Then you can do a standard one-tailed t-test as the issue raised by Sheldon is no longer applicable. If you have a small sample size then perhaps bootstrap testing will be useful. See <a href="http://en.wikipedia.org/wiki/Bootstrapping_(statistics" rel="nofollow">http://en.wikipedia.org/wiki/Bootstrapping_(statistics</a>) for details of this approach.</p> <p>Hope that helps.</p> http://mathoverflow.net/questions/15028/do-you-find-your-students-are-less-competent-in-basic-algebra-and-arithmetic-and/15037#15037 Answer by some_random_guy for Do you find your students are less competent in basic algebra and arithmetic, and, if so, do you believe that this is due to overuse of calculators at an early level? some_random_guy 2010-02-11T21:53:58Z 2010-02-11T21:53:58Z <p>I have the same issue with business students in my class so I guess the problem is more widespread than just math undergrads. In order to combat the issue, I re-designed my course so that repetition is the key theme. In other words, the same concept/formula is emphasized via in-class examples that are solved by me, via out-of-class graded assignments, via in-class ungraded assignments and sample exams. Students are allowed to work with each other on assignments and sample exams but the exams are individual exams.</p> <p>My hope is that repeated use of the same formula/concept in different contexts and allowing them to talk to each other for assignments/sample exams will help them internalize the ability to answer questions involving basic algebra/arithmetic on the exam as well. I am not sure to what extent my answer generalizes to math or if it gives you any ideas for your own course. Their performance on the first exam (scheduled for next week) will probably tell me if my approach is working or not.</p> http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique/11179#11179 Comment by some_random_guy some_random_guy 2010-01-11T14:39:48Z 2010-01-11T14:39:48Z ok I understand. Thanks http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique Comment by some_random_guy some_random_guy 2010-01-11T14:38:54Z 2010-01-11T14:38:54Z I edited the question to fix the terminology. http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique Comment by some_random_guy some_random_guy 2010-01-09T13:49:13Z 2010-01-09T13:49:13Z The way I have defined u_{max} above - it is a vector. However, all elements of u_{max} have the same value and hence we have a hypercube in p dimensions. X can have positive or negative values. The original context in which the above issue arises would constrain X to have only +1, 0 or -1 values but I did not mention this constraint as I felt that the general problem would have a positive answer. http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique/11179#11179 Comment by some_random_guy some_random_guy 2010-01-09T02:41:44Z 2010-01-09T02:41:44Z Never mind my comment reg $P_v$. The argument does not work. However, I am confused about why you think a rectangle that is perpendicular to the top and the bottom faces of a cube becomes skinnier as $v$ increases. Such a plane will always have the same shape irrespective of the value of $v$. As $v$ increases, the plane shifts towards the origin but retains its orientation. (Note: The longest line is not unique as either diagonal is a longest line but that is a different issue.) http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique/11179#11179 Comment by some_random_guy some_random_guy 2010-01-08T21:29:40Z 2010-01-08T21:29:40Z How about this: Define $S$ as the set of points where $L$ intersects the half-planes defined by $u'_i = 0$ and u'_j &gt;= 0 for all $j$ not equal to $i$. I guess we are done if we can show that for some $v &gt; v^*$ all the points in the set $S$ are less than $1$. The above would suggest that $P_v$ lies entirely in the hypercube and hence 'retains' its shape. Does that make sense? ps; I am unable to edit my comments after posting. I hope the above is clear. http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique/11179#11179 Comment by some_random_guy some_random_guy 2010-01-08T20:33:33Z 2010-01-08T20:33:33Z A clarification: Do you mean the transformation $u'\rightarrow u/v$ instead of $u \rightarrow vu$? http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique Comment by some_random_guy some_random_guy 2010-01-08T18:37:51Z 2010-01-08T18:37:51Z @Qiaochu: I am afraid I do not understand your question. My knowledge of math is fairly limited. http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique Comment by some_random_guy some_random_guy 2010-01-08T18:20:04Z 2010-01-08T18:20:04Z @Deane: Cleaned up notation to indicate that the question relates to real spaces. http://mathoverflow.net/questions/11169/is-the-direction-of-the-longest-line-of-a-polytope-unique Comment by some_random_guy some_random_guy 2010-01-08T18:15:18Z 2010-01-08T18:15:18Z Sorry for the confusion. I do not usually use LaTeX. I cleaned up the LaTeX glitches.