Rahul
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 21h comment Sane solution for an ODE with physical interpretation Ah, for Newtonian drag you should have $mv'=f_0-f_1v-f_2v|v|$ instead with $f_1\ge0,f_2\ge0$, so that the drag forces always oppose the instantaneous velocity. Perhaps this clarifies things: when $v(0)$ and $f_0$ have opposite signs, the solution will appear to have a finite-time singularity, but as soon as it crosses zero it will switch to another analytical form where it asymptotically approaches a terminal velocity. As for your complex-valued solution though, I don't know; you might have to follow Julián's advice and consider the cases carefully. 1d comment I call them squares. They called them arrays. What do they mean? Can you just contact the teacher and ask? May25 comment Visualize $z+\frac{1}{z} \ge 2$ That's not true for $z<0$. Maybe you mean $z\in\mathbb R^+$. Anyway, for a geometrical interpretation, one can compare the graph of $y=1/x$ and $x+y=2$: wolframalpha.com/input/?i=plot%20y%3D1%2Fx%2C%20x%2By%3D2 May23 comment Ideal shape for underwater habitat I'm not sure I understand your analogy. A drop of liquid typically does not change its volume -- do you mean its surface area? Also, if the external air pressure is uniform it does not influence the shape of an incompressible drop, because the work done against a uniform pressure field depends only on the change in volume. May23 comment Sane solution for an ODE with physical interpretation You can't avoid singularities, because any solution to $v'=v^2$ with $v$ initially positive goes to infinity in finite time. Your "physical process" is not physically realizable. May23 comment Are closed simple curves with that property necessarily circles? Consider any region with four-fold rotational symmetry, such as a square... May23 comment Is 'difference of two squares' in this limit proof acceptable? I'm afraid you can't plug in the "last element" of an infinite sequence like that. The moment you started talking about $a^{k/\infty}$ your proof was invalid. May23 comment Origin of the term dual space? Typically, the term dual is applied when the dual of the dual is the original object. Wikipedia has a host of examples. May23 comment Explicit Bezier Curves: Lerping between curves same as lerping control points? This is a consequence of the fact that $f$ is linear in $A$, $B$, and $C$, and similarly for $g$. May23 comment Distribution formed by taking two random points on an open disc and graphing their midpoint The probability density at any point $p$ is proportional to the area of the set of points $q$ such that both $q$ and its reflection about $p$ lie inside the disk. This is definitely higher in the middle than near the boundary. May22 comment Given two points, how to find a circle through them that's also tangent to the $x$-axis? A Euclidean geometry construction: cut-the-knot.org/Curriculum/Geometry/GeoGebra/PPL.shtml May21 comment Does a fluid with $0$ divergence have $0$ density? If $\mathrm d\rho/\mathrm dt=0$, it doesn't mean that $\rho=0$. It just means that $\rho$ is constant. (Everyone forgets the constant of integration, grumble grumble...) May21 comment Way to measure the similarity/difference of 2D point clouds If the curves are noisy, that could still throw the arc length parametrizations out of sync (imagine one curve is noisier near the beginning and the other is noisier near the end, and you get the same situation as my first comment). Which is why I suggested the Hausdorff distance, which is independent of parametrization. May21 comment Way to measure the similarity/difference of 2D point clouds This is halfway to a good idea. But if the first point set has a lot of points near the beginning of the curve and the second has lots of points near the end of the same curve, your metric will still give a large dissimilarity. What you should do, once you have a notion of the curves represented by each point set, is to measure the geometrical dissimilarity between the curves themselves, for example using the Hausdorff distance. May21 comment Total area for a natural nested set of convex polygons. Whoops, that's not quite true. When $n$ is large and odd, $P^*$ forms two simple polygons of area close to $A$, so $A^*\approx A$, or a single "doubly-wrapped" polygon whose central area is double-counted, and again $A^*\approx A$. I'll edit my answer. May20 comment Why $(A\vec{x})'A \vec{x} = \vec{0}$ implies that $A\vec{x} = \vec{0}$ You still have $A^TAx=x^TA^TAx$, which is also incorrect. May20 comment Why $(A\vec{x})'A \vec{x} = \vec{0}$ implies that $A\vec{x} = \vec{0}$ The argument as written is incorrect. It is true that if $x^TA^TAx=0$ then $Ax=0$, but it is not true that $x^TA^TAx=Ax$. (Nor that $A^TAx=x^TA^TAx$, as written in the immediately previous line.) May19 comment Identification of a quadrilateral as a trapezoid, rectangle, or square @Daniel: Even if that were the case, do we expect students taking the test to answer the question as given, or do we expect them to somehow divine the intent of the question setters? ...On second thought, given the quality of standardized testing in many places, don't answer that. May19 comment mathematical symbol for vector appending This is a natural application of block matrix notation. If we can write $$\begin{bmatrix}\mathbf A&\mathbf b\\\mathbf b^T&c\end{bmatrix}$$ where $\mathbf A$, $\mathbf b$, and $c$ are a matrix, a column vector, and a scalar respectively, then surely we can write $$\begin{bmatrix}\mathbf v\\w\end{bmatrix}$$ to denote $$\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\\w\end{bmatrix}$$ (and we do). May19 comment The difference between a basis of vectors vs functions Your space of functions is not a subspace of $\mathbb R^2$, because none of the functions in it are elements of $\mathbb R^2$. The $xy$-plane is not a subspace of $\mathbb R^2$ either -- maybe that was a poorly chosen example, consider the subspace $x\begin{bmatrix}1\\2\\0\end{bmatrix} + y\begin{bmatrix}3\\-1\\-2\end{bmatrix}$ instead. The point is that you should stop thinking of two-dimensional vector spaces as being inherently tied to $\mathbb R^2$.