# Tagged Questions

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### Find an equation for a moving rod

The two endpoints of a 1-metre long rod have an initial position at $(0,0),(0,1).$ The rod slides continuously to the position $(1,0),(0,0)$ sweeping out a region in the positive quadrant. Determine ...
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### A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
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### Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?

Given two concentric circles with radiuses $r < R$, can we estimate the number of chords in between the circles? With more details, fix a point $P$ on the inner circle. Trace a tangent to the ...
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### Equivalence of the two cosine definitions

There are at least two ways to define the cosine function: You can define it with a right triangle in the unit circle and extend the definition to $\mathbb{R}$. (classic definition) The other ...
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### What does $E^d$ mean?

I was reading the paper "Cutting Hyperplanes for Divide-and-Conquer" by B. Chazelle and in the introduction I came across the following: "Let $H$ be a set of $n$ hyperplanes in $E^d$." What does $E^d$ ...
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### The Least Characteristic of Shapes in $\mathbb{R}^n$

Fix the following notations: "Shape" denotes a closed curve in $\mathbb{R}^2$ or a closed surface in $\mathbb{R}^3$. $P$ denotes the circumference of a shape in $\mathbb{R}^2$. $A$ denotes the area ...
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### Cardinality of the set in $\mathbb{R^2}$

I am trying to understand the following exercise: Let $v_1$ and $v_2$ be non-collinear vectors of $\mathbb{R}^2$. Estimate the cardinality of the set $(m,n) \in \mathbb{Z}^2$ such that ...
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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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The question is: In $\mathbb{R}^2$, with the usual topology, let $S_i$ be a dense set of points that can be partitioned into parallel lines. Let $m_i$ be the slope of these lines. For any $n\in ... 0answers 40 views ### show that$g\circ f :S_1 \to S_3$be also smooth and$d_p(g\circ f)=(d_{f(p)}g)\circ (d_pf)$proposition: Let$S_1$,$S_2$,$S_3$be 3-regular surfaces and$f:S_1 \to S_2$and$g: S_2 \to S_3$be smooth maps. Then show that$g\circ f :S_1 \to S_3$be also smooth and $$d_p(g\circ ... 1answer 53 views ### Show that every open subset of a surface is a Surface. Show that every open subset of a surface is a Surface. I think, first of all, i need to define an atlas \{\sigma_{\alpha}: U_{\alpha}\to \Bbb R^3\} for surface S, Where U is an open set. After ... 1answer 130 views ### The double cone is not a surface. My question is that A double cone ( also named as "circular cone") is not a surface. I know its reason. But I cannot show this mathematically. Suppose \sigma : U \to S\cap W Is a surface ... 1answer 42 views ### Show that a paraboloid is asurface . That I know about paraboloid is all in the picture. I wrote its surface patch. (Hopefully, it is correct) From there, what do I need to do in order show that a paraboloid is a surface. ... 1answer 47 views ### Taylor expansions on manifolds.. can one consider Taylor expansions of functions defined between smooth manifolds? If so, is there a reference for learning more about it? Thanks 2answers 55 views ### Rational translates of the unit circle cover the plane Is it true that the translations of the unit circle by vectors with both coordinates rational cover the plane? This comes to solving$$ x=a+\cos \theta, \ y=b+\sin \theta$$with unknowns$a,b$... 1answer 49 views ### Convex Analysis Question I need to show that$F(x,y,z) = (y - z, z- x, x - y)$is Lipschitz on the closed ball$S = \{(x,y,z): x^2 + y^2 + z^2 \le 1\}$. I get that F should be Lipschitz since S is closed and bounded (I ... 2answers 105 views ### Regular$n$-gon in the plane with vertices on integers? For which$n \geq 3$is it possible to draw a regular$n$-gon in the plane ($\mathbb{R}^2$) such that all vertices have integer coordinates? I figured out that$n=3$is not possible. Is$n=4$the only ... 0answers 89 views ### Line Segments for a Triangle What are the requirements for three line segments to be able to form a triangle? What would be the proof of that? Thanks! 0answers 92 views ### Is there such an example? Is there an example of a sequence of point sets$\left\{ S_{n}\right\} _{n=1}^{\infty}$in which$S_{n}$is a set of$n$points inside the unit triangle, such that the minimum altitude of the triangles ... 1answer 35 views ### What is this total length What is the value of the total length of all the edges connecting the vertices of a regular$k$-gon that is inscribed on a unit circle? 1answer 40 views ### Would this be bounded Let$a_{m}$be supremum of the minimum of the angle between the line segments between any$m$points, in which the supremum is taken over all configurations of$m$points. Is$\sqrt{m}a_{m}$bounded ... 1answer 86 views ### Is this bounded Let$d_{k}$be supremum of the minimum of the pairwise distances between any$k$points in the unit square. Is$kd_{k}$bounded as$k\rightarrow\infty$? 0answers 35 views ### How to show this [duplicate] Given$15$lines in the plane, can anyone show that there are at least$3$of the lines, such that the angle between any two of them is less than$\frac{\pi}{4}$? 1answer 106 views ### Why is this lemma true? In Lemma 2 of this paper, which states that, given$l$lines in the plane, and given$\delta > 0$, there are at least$l/( [\delta^{-1}] +1)$of the lines, such that the angle between any two of ... 1answer 36 views ### A sequence on a line through the origin in$\mathbb{C}$must converge on that line? Does this make sense? Lemma: Consider a sequence$(z_n) \subset \mathbb{C}$. If$z_n \to z$and$Arg(z_n) = Arg(z_m)$for all$n,m \in \mathbb{N}$, then$Arg(z) = Arg(z_n)$. Proof: Fix$n \in ...
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 ...