User tobias hagge - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:42:03Z http://mathoverflow.net/feeds/user/4313 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8846/proofs-without-words/17320#17320 Answer by Tobias Hagge for Proofs without words Tobias Hagge 2010-03-06T21:55:58Z 2012-10-23T15:24:00Z <p>Also elementary, but here is a proof that</p> <p>$C_n = \binom{2n}{n} - \binom{2n}{n+1} = \frac{\binom{2n}{n}}{n+1},$</p> <p>where $C_n$ is the $n$th Catalan number.</p> <p><a href="http://utdallas.edu/~hagge/images/Catalan.pdf" rel="nofollow">http://utdallas.edu/~hagge/images/Catalan.pdf</a></p> <p>Sorry for the link; new users may not use image tags.</p> <p>Here's the image:</p> <p><img src="http://i.imgur.com/3fJi1.png" alt="alt text"></p> http://mathoverflow.net/questions/89069/should-the-formula-for-the-inverse-of-a-2x2-matrix-be-obvious/89088#89088 Answer by Tobias Hagge for Should the formula for the inverse of a 2x2 matrix be obvious? Tobias Hagge 2012-02-21T07:31:10Z 2012-02-21T07:37:56Z <p>Mnemonic: make the product diagonals the determinant, then scale.</p> <p>The off diagonals are zero because the area of a parallelogram with planar edge vectors $c_1,c_2$ is the length of the scaled projection $|c_1 \cdot i c_2| = |c_2 \cdot i c_1|$, and the mnemonic sets row $r_k$ in the inverse to $(ic_{3 - k})^T$.</p> http://mathoverflow.net/questions/41210/does-every-frobenius-algebra-in-a-monoidal-category-give-a-q-system/44852#44852 Answer by Tobias Hagge for Does every Frobenius algebra in a monoidal *-category give a Q-system? Tobias Hagge 2010-11-04T18:00:58Z 2010-11-04T18:00:58Z <p>This is a comment on Noah's answer, posted as an answer due to lack of reputation. The semion MTC is inequivalent to Vec(Z/2) as a fusion category; it is the other rank two fusion category. Confusingly, there is a change in sign in one of the F-matrices AND a change in sign in the pivotal structure which gives unitarity; the two occur simultaneously in most diagrams.</p> http://mathoverflow.net/questions/21881/how-should-one-present-curl-and-divergence-in-an-undergraduate-multivariable-calc/43222#43222 Answer by Tobias Hagge for How should one present curl and divergence in an undergraduate multivariable calculus class? Tobias Hagge 2010-10-22T19:02:53Z 2010-10-26T05:03:41Z <p>Here as another way to define curl:</p> <p>1) Curl is a vector describing "instantaneous rotation". The line integral over a gradient vector field is zero on any closed curve, so whatever instantaneous rotation is, it should have the property $curl(\nabla f) = 0$.</p> <p>2) Every symmetric $3\times 3$ matrix is $D^2f$ for some homogeneous polynomial $f$ in three variables. Thus a vector field $F:\mathbb{R}^3 \to \mathbb{R}^3$ is locally well-approximated by an irrotational gradient vector field when $DF$ is symmetric.</p> <p>3) The function $g(\vec v) = \vec v \times \vec p$, where $\vec p$ is fixed but arbitrary, is both the general rotational velocity field about an axis $\vec p$ through the origin, with angular velocity $||\vec p ||$, as well as the linear function described by an arbitrary antisymmetric matrix. When $DF$ is antisymmetric, $F$ is locally well-approximated by a rotational velocity field.</p> <p>4) $DF = \frac{(DF + DF^T) + (DF - DF^T)}{2}$. The first summand is irrotational, the second, a rotational velocity field. Define the instantaneous rotation field for F to be the linear function given by $(DF - DF^T)$ (ignoring the factor of $2$). The vector $\vec p$ which gives us the corresponding function $g$ is easily determined from the coefficients of $(DF - DF^T)$, and is precisely $curl(F)$.</p> http://mathoverflow.net/questions/8846/proofs-without-words/17320#17320 Comment by Tobias Hagge Tobias Hagge 2010-10-26T05:46:50Z 2010-10-26T05:46:50Z Sorry for not noticing your question (much) earlier. The differences between adjacent terms in Pascal's triangle form another triangle which obeys the same generation rules. In my picture of that triangle, the yellow squares count some of the downward paths on a square grid which has been rotated $45^\circ$, namely those that never fall to the left of the top square. One definition of $C_n$ is that it is the number of such paths which terminate at the bottom corner of an $n \times n$ grid. http://mathoverflow.net/questions/21881/how-should-one-present-curl-and-divergence-in-an-undergraduate-multivariable-calc/43222#43222 Comment by Tobias Hagge Tobias Hagge 2010-10-26T05:07:49Z 2010-10-26T05:07:49Z Edited to $D^2f$.