User keenan pepper - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:19:48Z http://mathoverflow.net/feeds/user/9021 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130890/can-distinct-open-knots-correspond-to-the-same-closed-knot Can distinct open knots correspond to the same closed knot? Keenan Pepper 2013-05-17T01:46:25Z 2013-05-20T19:53:30Z <p>A topological ("closed") knot is an embedding of a circle in $\mathbb{R}^3$. It's possible for a knot to be distinct from the unknot because there are no free ends to move around and untie the knot. An alternative notion of a knot is an embedding of a closed line segment in $\mathbb{R}^2 \times [0,1]$ such that the two boundary points of the line segment get mapped to the two boundary planes of $\mathbb{R}^2 \times [0,1]$ respectively (one end on each plane), and all other points get mapped to the interior of $\mathbb{R}^2 \times [0,1]$. I'll call this an "open knot". It's possible for an open knot to be distinct from the unknot because, although it has two ends that can move around, they are stuck on the boundary planes, and no part of the knot can get around them, so it can't get untied.</p> <p>My question is whether these two kinds of knot are fundamentally equivalent. <s>Clearly there are some differences related to chirality/reversibility issues, for example left- and right-handed "overhand" open knots are distinct even though the closed trefoil knot is equivalent to its mirror image (just turn it over). I want to know if that's the full extent of the difference.</s> Whoops, don't know what I was thinking here. If you turn a right-handed trefoil over it remains right-handed.</p> <p>There's an obvious function mapping from open knots to closed knots - just connect the two ends together. But if you try to invert this function you have to make a choice of where to cut the knot, and you might get different results depending on where you cut it.</p> <p>Do all choices of where to cut a closed knot result in the same open knot? If not, what's an example?</p> http://mathoverflow.net/questions/91817/in-the-classical-construction-of-conic-sections-where-does-the-axis-of-the-cone In the classical construction of conic sections, where does the axis of the cone intersect the plane? Keenan Pepper 2012-03-21T12:22:19Z 2012-03-23T12:19:47Z <p>Everybody knows that if I take the intersection of a right circular cone with a plane, I get a conic section. My question is, where does the symmetry axis of the cone intersect the plane? Does this point relative to the conic have a name, or a simple description? For example, for an ellipse I first guessed that it was one focus of the ellipse, but that is false.</p> http://mathoverflow.net/questions/76832/minimally-6-connected-3d-discrete-lines-that-are-convex-lattice-sets Minimally 6-connected 3D discrete lines that are convex lattice sets Keenan Pepper 2011-09-30T08:08:35Z 2011-09-30T09:19:58Z <p>There are several definitions of 3D discrete lines, e.g. <a href="http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html" rel="nofollow">http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html</a> , <a href="http://dx.doi.org/10.1007/978-3-642-19867-0_4" rel="nofollow">http://dx.doi.org/10.1007/978-3-642-19867-0_4</a> . However, I know of none that has all the following properties:</p> <ul> <li>Minimal 6-connectedness: The discrete line is 6-connected. Also, assuming WLOG that the direction vector is in the first octant, for any integer N there is exactly one voxel $(x,y,z)$ of the line such that $x+y+z=N$ (the "minimal" part).</li> <li>Well-behaved projections: The projections onto the $xy$, $xz$, and $yz$ planes are all 4-connected 2D discrete lines, such as would be produced by a modified Bresenham's algorithm.</li> <li>Convexity: The voxels of the discrete line form a convex lattice set in $\mathbb Z^3$; that is, it is equal to the intersection of $\mathbb Z^3$ with its own convex hull.</li> </ul> <p>Is there a kind of discrete line that has all these properties?</p> <p>It's surprising and intersting to me that discrete lines are so easy to define in 2D, but so hard to pin down in 3D. I guess being codimension-1 makes it easy.</p> http://mathoverflow.net/questions/75908/minimum-cardinality-of-a-difference-set-in-rn Minimum cardinality of a difference set in $R^n$ Keenan Pepper 2011-09-20T04:59:50Z 2011-09-20T07:23:27Z <p>Cross-posted from <a href="http://math.stackexchange.com/questions/65195/minimum-cardinality-of-a-difference-set-in-mathbb-rn" rel="nofollow">http://math.stackexchange.com/questions/65195/minimum-cardinality-of-a-difference-set-in-mathbb-rn</a>.</p> <p>Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the same $(n-1)$-dimensional hyperplane, consider the set of difference vectors:</p> <p><code>$\{x-y \, | \, x,y \in S\}$</code></p> <p>What is the minimum cardinality of this set, as a function of $m$ and $n$?</p> <p>(The sets that minimize this should be "small" subsets of a lattice, but I don't know what specific shapes minimize it.)</p> <p>What is the status of exact results for this problem for small $n$ (say $n = 2$ or $3$)?</p> http://mathoverflow.net/questions/73621/maximal-number-of-edges-and-triangular-cells-for-n-points-in-a-triangular-lattice Maximal number of edges and triangular cells for n points in a triangular lattice Keenan Pepper 2011-08-25T01:39:08Z 2011-08-28T11:43:38Z <p>Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.</p> <p>What is the maximum, over all such subsets, of the number of edges? This sequence appears to start 0, 1, 3, 5, 7, 9, 12, 14, 16...</p> <p>What is the maximum number of triangular lattice cells? (Not the number of all triangles, just the number of smallest possible equilateral triangles in the lattice.) This sequence appears to start 0, 0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 13...</p> <p><a href="http://oeis.org/A047932" rel="nofollow">http://oeis.org/A047932</a> is related to the first sequence but I have no proof it's the same. (There might be some other way of arranging the pennies that yields a higher number of contacts. A047932 is a lower bound on my sequence.) I can't find any OEIS sequences relevant to the second one.</p> http://mathoverflow.net/questions/70035/searching-for-an-inhomogeneous-diophantine-approximation-algorithm Searching for an inhomogeneous diophantine approximation algorithm Keenan Pepper 2011-07-11T17:57:13Z 2011-07-16T03:35:19Z <p>Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ and $b$ such that $|ax + by - z| &lt; \epsilon$?</p> <p>Note that if the restriction that $a$ and $b$ be coprime is lifted, the problem becomes very simple. One possible algorithm is:</p> <ul> <li>Find $a_1$ and $b_1$ such that $0 &lt; a_1 x + b_1 y &lt; \epsilon$ using the extended Euclidean algorithm.</li> <li>Let $\displaystyle a = a_1 \left[ \frac{z}{a_1 x + b_1 y} \right]$ and $\displaystyle b = b_1 \left[ \frac{z}{a_1 x + b_1 y} \right],\,$ where $[\cdot]$ is the nearest integer function.</li> </ul> <p>However, the integers $a$ and $b$ provided by this algorithm are usually not coprime. I'm looking for an algorithm that produces the same kind of approximation but guarantees that $a$ and $b$ are coprime.</p> http://mathoverflow.net/questions/130890/can-distinct-open-knots-correspond-to-the-same-closed-knot/130902#130902 Comment by Keenan Pepper Keenan Pepper 2013-05-20T19:53:53Z 2013-05-20T19:53:53Z Thanks, that's exactly what I was looking for! http://mathoverflow.net/questions/91817/in-the-classical-construction-of-conic-sections-where-does-the-axis-of-the-cone Comment by Keenan Pepper Keenan Pepper 2012-03-23T01:58:32Z 2012-03-23T01:58:32Z Okay, so the answer is there is no special point - there is a continuum of different points that are on the axes of different cones, all of which have that same ellipse as a section. Both Pietro Majer's and alvarespaiva's comments sound more like answers. http://mathoverflow.net/questions/91817/in-the-classical-construction-of-conic-sections-where-does-the-axis-of-the-cone Comment by Keenan Pepper Keenan Pepper 2012-03-21T14:07:35Z 2012-03-21T14:07:35Z Right, for that one specific kind of hyperbola the point goes off to infinity. http://mathoverflow.net/questions/75908/minimum-cardinality-of-a-difference-set-in-rn/75918#75918 Comment by Keenan Pepper Keenan Pepper 2011-09-20T21:43:56Z 2011-09-20T21:43:56Z I have no idea why I didn't come across this paper on my own, because I was using all those search terms. Now it should be easier for others to find. http://mathoverflow.net/questions/75906/sampling-from-a-partition-of-a-hypercube-by-convex-polytopes Comment by Keenan Pepper Keenan Pepper 2011-09-20T05:12:02Z 2011-09-20T05:12:02Z Can't you just sample points uniformly from the entire hypercube, discarding points in polytopes which already have enough points? The only reason this would be inefficient is if your polytopes have widely differing (hyper)volumes. Do they? http://mathoverflow.net/questions/75908/minimum-cardinality-of-a-difference-set-in-rn Comment by Keenan Pepper Keenan Pepper 2011-09-20T05:02:17Z 2011-09-20T05:02:17Z I added the &quot;arithmetic-progression&quot; tag because the solution for n=1 is any arithmetic progression, giving a difference set with cardinality $2m-1$. So in some sense the higher-dimensional solutions generalize arithmetic progressions. http://mathoverflow.net/questions/1083/do-good-math-jokes-exist/7072#7072 Comment by Keenan Pepper Keenan Pepper 2011-08-25T23:12:05Z 2011-08-25T23:12:05Z I don't get it. Does she intentionally say &quot;under a dollar&quot; instead of &quot;under a buck&quot;, or accidentally? Or does she not get it? http://mathoverflow.net/questions/73621/maximal-number-of-edges-and-triangular-cells-for-n-points-in-a-triangular-lattice/73625#73625 Comment by Keenan Pepper Keenan Pepper 2011-08-25T06:01:52Z 2011-08-25T06:01:52Z I had already seen and dismissed <a href="http://oeis.org/A186705" rel="nofollow">oeis.org/A186705</a> , and it took me a while to figure out the difference between that and your definition. &quot;Maximum times the <i>same</i> distance can occur&quot; is different from &quot;maximum times the <i>minimum</i> distance can occur&quot;. First element where the sequences actually differ is for n=9 points. http://mathoverflow.net/questions/70035/searching-for-an-inhomogeneous-diophantine-approximation-algorithm/70150#70150 Comment by Keenan Pepper Keenan Pepper 2011-07-12T17:42:45Z 2011-07-12T17:42:45Z Yes, that's what I was looking for! Not sure why you changed all the variable names, but the last part is what I was missing: it proves that once you've found a good enough continued fraction convergent p/q you can just stick with that; it's not necessary to try different p,q values in order to find a solution that's both coprime and accurate enough (which I had feared might be the case). http://mathoverflow.net/questions/70035/searching-for-an-inhomogeneous-diophantine-approximation-algorithm Comment by Keenan Pepper Keenan Pepper 2011-07-12T05:09:04Z 2011-07-12T05:09:04Z I don't see how it's specifically related to math.stackexchange.com/questions/46100 . There's nothing in there about the coefficients being coprime.