Rahul
Reputation
22,733
79/100 score
 Mar9 comment Finding potential function for a vector field If $\nabla f=\mathbf F$ then $f(\mathbf b)-f(\mathbf a)=\int_{\mathbf a}^{\mathbf b} \mathbf F(\mathbf r)\cdot\mathrm d\mathbf r$ over any path from $\mathbf a$ to $\mathbf b$. Let $\mathbf a=(0,0,0)$, $\mathbf b=(x,y,z)$, and pick any path you like from one to the other. Mar9 comment Euler-Lagrange Equation and “Eigen Value ” You can just use **two asterisks** for bold. Mar9 comment Is It Always Possible to Draw A Connected Compact Set in $\mathbb R^2$? This probably doesn't help answer the question, but your $f_\epsilon\circ F_\epsilon$ is known as closing in image processing. Mar8 comment Spaces that satisfy closure with respect to addition and scalar multiplication but aren't vector spaces? Just define any two functions $V\times V\to V$ and $\mathbb K\times V\to V$ and call them addition and scalar multiplication. Odds are, they won't satisfy the axioms. For example, take $V=\mathbb R^n$, $\mathbb K=\mathbb R$, $(x_1,\ldots,x_n)+(y_1,\ldots,y_n)=(x_1-y_1,\ldots,x_n-y_n)$, and $a\cdot(x_1,\ldots,x_n)=(a+x_1,\ldots,a+x_n)$. Mar8 comment Given the projection, can we determine the object? What object is a circle in plan, front and side views? (Besides a sphere) Mar7 comment Four holes in a sphere Do the holes stop at the center or go all the way to the other side? If the latter, then they form the space diagonals of a cube. Mar6 comment Olbers' Paradox in an Euclidean universe with randomly located stars Consider two disjoint measurable sets $S_1$ and $S_2$. Is the distribution of the number of points in $S_1$ independent of the distribution of the number of points in $S_2$? If so, then $P$ is a Poisson process and the question becomes easy. If not, then I don't know. Mar4 comment Picking the shortest path without square root (from non-linear paths) Sorry, you have to use the square root. It's not that expensive. Mar4 comment An alien comes to Earth and says $7\times7=41$. How many fingers does he have? I didn't know the Babylonians had 60 fingers Mar3 comment Unique circle through two points perpendicular to a given line? Perform an inversion with respect to one of the points? Mar3 comment Singular value decomposition for matrices that are not square? Your matrix has 3 rows and 2 columns, so just to make the dimensions match you must have $U$ with 3 rows and $V^T$ with 2 columns. Is that what you mean by the "missing last component"? Mar3 comment How to calculate the critical density estimation for “continuum” percolation model in “3D space” when we have “spatial correlation”? This looks more like a percolation theory problem to me. Feb28 comment Insertion into an optimal route – is it still optimal? For a concrete example, consider points SABCDE lying clockwise on a circle, with distance SA < SE; then the optimal path is SABCDE. Now insert X between S and E so that SX = XE < SA; now the optimal path is SXEDCBA. Feb28 comment Insertion into an optimal route – is it still optimal? Unfortunately, the new optimal route cannot always be the old optimal route with the new point inserted, otherwise you could solve the travelling salesman problem in polynomial time by inserting one point at a time. Feb25 comment Intersection of 8 spheres: find the volume I'm pretty sure that by symmetry the intersection looks like a rounded octahedron. It's the union of 8 disjoint congruent pieces, one of which is the portion of the unit sphere centered at $(0,0,0)$ that lies in the region $x,y,z\ge1/2$. Feb23 comment How is it possible to change the pitch and the tempo of an audio track independently of each other? Relevant Wikipedia article: en.wikipedia.org/wiki/Audio_time-scale/pitch_modification. Possibly relevant StackExchange site: dsp.stackexchange.com. Feb23 awarded Enlightened Feb23 awarded Nice Answer Feb20 comment Irregular Implicit Plot in Mathematica What Mathematica is actually doing is trying to plot the regions where $y-x\tan(\cdots)$ changes sign. Feb20 comment Are there real solutions to $\exp(X)=-I$? Hint: $-I$ is also a rotation by $\pi$.