User vinayak pathak - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:33:18Z http://mathoverflow.net/feeds/user/6645 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40100/example-of-a-function-that-behaves-like-another-function Example of a function that behaves like another function Vinayak Pathak 2010-09-27T07:10:33Z 2010-09-30T09:30:14Z <p>I need a function $f(x)$ with the following properties -</p> <ol> <li>It should be monotonically non-decreasing.</li> <li>For $x \geq 1$, $x + \frac{1}{x} - f(x) &lt; \epsilon$ where $\epsilon$ is an extremely small positive real number.</li> <li>It should look simple. I know this sounds like a very vague requirement, but I hope the meaning of "simple" is clear to some extent. For example, the function should definitely not have a piecewise definition. One should be able to write it by using not more than 15 characters, etc. etc.</li> </ol> <p>Can anyone think of such a function? It's fine even if the above requirements are satisfied only for positive values of x.</p> http://mathoverflow.net/questions/28092/feasibility-of-linear-programs/28146#28146 Answer by Vinayak Pathak for Feasibility of linear programs Vinayak Pathak 2010-06-14T17:45:13Z 2010-06-14T17:45:13Z <p>It seems this can be done in linear time. Algorithms that solve linear programs are also capable of deciding whether the LP is feasible or not and 2-d linear programs can be solved in linear time (linear in terms of the number of constraints). So to decide whether a set of n halfplanes is non-empty or not, just solve the LP that has those halfplanes as its constraints with any objective function.</p> http://mathoverflow.net/questions/28092/feasibility-of-linear-programs Feasibility of linear programs Vinayak Pathak 2010-06-14T04:31:51Z 2010-06-14T17:45:13Z <p>It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to <em>deciding</em> whether the intersection is non-empty?</p> http://mathoverflow.net/questions/76623/growing-random-trees-on-a-lattice-rightarrow-voronoi-diagrams Comment by Vinayak Pathak Vinayak Pathak 2011-09-28T12:01:30Z 2011-09-28T12:01:30Z Two questions. 1. Do the unit length edges you add have to be horizontal or vertical? 2. How is the first edge added for a given seed? Should the seed be one of its endpoints or should it just lie on it somewhere? http://mathoverflow.net/questions/40100/example-of-a-function-that-behaves-like-another-function/40114#40114 Comment by Vinayak Pathak Vinayak Pathak 2010-09-27T17:45:03Z 2010-09-27T17:45:03Z Awesome. This works, thanks! Can you give me some idea about how you came up with this by the way? http://mathoverflow.net/questions/39371/maximum-number-of-points-in-two-disks Comment by Vinayak Pathak Vinayak Pathak 2010-09-20T13:22:38Z 2010-09-20T13:22:38Z I am confused. Doesn't the second figure show that we <i>can</i> do 10 points? http://mathoverflow.net/questions/9255/can-you-determine-whether-a-graph-is-the-1-skeleton-of-a-polytope/23102#23102 Comment by Vinayak Pathak Vinayak Pathak 2010-07-25T02:57:52Z 2010-07-25T02:57:52Z I am talking about the generalization of the Czaszar polyhedron. It's a polyhedron whose skeleton is $K_7$. For n = 6, 7, 8, 9, 10, 11, there cannot be a polyhedron whose skeleton is $K_n$, which can be shown using the Euler characteristic. So the next candidate is $K_12$, and that's still open. I had assumed that 3-polytopes are the same as polyhedra. But it seems they are the same as &quot;convex&quot; polyhedra? http://mathoverflow.net/questions/9255/can-you-determine-whether-a-graph-is-the-1-skeleton-of-a-polytope/23102#23102 Comment by Vinayak Pathak Vinayak Pathak 2010-07-24T17:53:12Z 2010-07-24T17:53:12Z I meant $K_{12}$. http://mathoverflow.net/questions/9255/can-you-determine-whether-a-graph-is-the-1-skeleton-of-a-polytope/23102#23102 Comment by Vinayak Pathak Vinayak Pathak 2010-07-24T17:52:40Z 2010-07-24T17:52:40Z I have heard that it's still not known if the complete graph on 12 vertices can be realized as the 1-skeleton of a 3-polytope. But according to what you have said, it should have been decidable. So can't we just use Tarski's algorithm once on $K_12$ and check? Or is the input size really really big for this case? http://mathoverflow.net/questions/9255/can-you-determine-whether-a-graph-is-the-1-skeleton-of-a-polytope/9262#9262 Comment by Vinayak Pathak Vinayak Pathak 2010-07-24T14:46:58Z 2010-07-24T14:46:58Z Then, do we know of a theorem that characterizes the 1-skeleton of some other kind of polytopes (i.e., not necessarily convex)? http://mathoverflow.net/questions/9255/can-you-determine-whether-a-graph-is-the-1-skeleton-of-a-polytope/9262#9262 Comment by Vinayak Pathak Vinayak Pathak 2010-07-24T04:18:08Z 2010-07-24T04:18:08Z I wonder what he meant by &quot;similar theorem&quot;. Did he mean that a theorem characterizing the 1-skeleton of convex polytopes is not known or that a theorem characterizing the 1-skeleton of general polytopes is not known? http://mathoverflow.net/questions/27418/reading-blog-archives Comment by Vinayak Pathak Vinayak Pathak 2010-06-08T03:30:59Z 2010-06-08T03:30:59Z Kevin, sorry about that. But there are two reasons why I thought this would be a good place to ask - 1. The solutions may not be limited to Google Reader... all I want is some tool that can be used for reading all posts of a blog, that allows bookmarks and where one can 'star' items. 2. People have asked similar questions on the Google support website, but there isn't any satisfactory answer. Suresh, it doesn't show all posts. It only shows posts that are at most three months old.