Tagged Questions

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Crosscap function in $\mathbb{R}^4$ - and how to show it is proper?

I found the Cross-cap function in $\mathbb{R}^3$ as follows: $$f(x,y,z)=(yz,2xy,x^2-y^2),$$ My questions are (I couldn't show any progress for Q1,2.I have thought hard but had no clue): Q1: Is ...
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Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
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Find conditions on a1,a2,b1,b2 so that the intersections of two second-order curves are perpendicular

$a_1x^2+b_1y^2=1;$ $a_2 x^2+b_2y^2=1;$ I'm tried to use derivatives in points of crossing (that it crossed at right angle too), but nothing has helped. Thanks!
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What would this set look like

Let $S\subseteq\mathbb{R}^{3}$ be the set of $\left(x,y,z\right)$, $x\ge y\ge z$ , which are the three eigenvalues of $diag\left(1,2,3\right)+Udiag\left(-1,-2,-4\right)U^{T}$, where $U$ is an ...
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Angle between two centered noisy vectors

Let $\mathcal{H} = \lbrace u\in \mathbb{R}^n \mid \langle x, (1, 1, ...., 1) \rangle = 0 \rbrace$, the hyperplain where the avarage is zero i.e. $\frac{1}{n}\sum\limits_{i=1}^n x_i = 0$. Given two ...
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I want to classify all these values by using an equivalence relation

We apply the Mean value theorem to a real analytic function $f$ (defined on $\mathbb R$) in the interval $(u,a)$ such that $u<a$ and $f(u)=0$ to find a $c\in(u,a)$ such that: the expression ...
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The unsolved mathematical light beam problem

I have the following problem: Imagine that you have a sphere sitting at the interface of two media(like water and oil). And the position(the heigth) of the interface to the center of the sphere is ...
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Why is there this contradiction or what is wrong

In the second paragraph on Page 30 of this published paper, it says that the intersection of the convex hull of points $(\alpha_{1}+\beta_{P_{1}},\alpha_{2}+\beta_{P_{2}})$ with the convex hull of ...
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Occurrence of $e$ in intersecting circles.

Consider two identical circles that share a radius such that they intersect. The radii of the circles are $\pi\over 2$. If this new shape sits such that its major axis is horizontal and the shortest ...
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Shape Analysis with Morphometric Parameters of Binary Digital Images

I am doing shape analysis on some binary digital images. My target is to find the blobs in binary image that has shape like in this figure. I recently encountered this site that talks surfacely about ...