0
votes
1answer
22 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
2
votes
1answer
37 views

An ellipse bigger than a circle

Suppose you have a unit ball in $B^2\subseteq \mathbb{R}^2$ and a point $A=(a,0)$ where $a>\sqrt{2}$. I would like to show there is an ellipse $E\subseteq\mathrm{conv}(B^2\cup\{A,-A\})$ such that ...
11
votes
1answer
281 views

A variation of the isoperimetric problem in the plane

The isoperimetric problem in the plane: « The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed ...
1
vote
0answers
15 views

Retanguloids and volumes

I am currently going back through all the "Challenge" questions in preparation for exams, and have come across this, any hints on how to approach?
3
votes
1answer
34 views

Count PI with analytical methods

Is there a differential equation which can be used to count the value of pi? I was able to describe pi only with sequences based on polygons with infinite corners. I think I'll need a continuous ...
1
vote
0answers
19 views

Derivatives of the Green's Function

Consider the function $ g(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ defined as $$ g(x,y) = \log( x^2 + y^2) $$ We know that $ (\partial_x^2 + \partial_y^2) g(x,y) = -2 \pi \delta(x)\delta(y) $. ...
6
votes
0answers
50 views

Geometric interpretation of Euler's identity for homogeneous functions

A function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is called homogeneous of degree $d \geq 0$ if $$f(\lambda x_1, \ldots, \lambda x_n ) = \lambda^d f(x_1, \ldots, x_n)$$ Differentiating both sides ...
1
vote
1answer
32 views

How do I convert OS coordinates (X and Y) to longitude and latitude coordinates?

How do I convert OS coordinates (X and Y) - Eastings and Northings to longitude and latitude coordinates? For example X and Y below X (Eastings): 347904 Y (Northings): 287484
1
vote
0answers
16 views

How do I figure out the grid references necessary to create a circular 1 mile radius around one location?

How do I figure out the grid references necessary to create a circular 1 mile radius around one location? For instance: Grid Reference: SO 47904 87484 X (Eastings): 347904 Y (Northings): 287484 ...
1
vote
0answers
16 views

Berkovich analytifications and non-archimedean geometry of Transseries

In Transseries and Real Differential Algebra by Joris van der Hoeven it is said that Transseries admit a rich non-Archimedean geometry (somewhere on page 13), but since the book isn't about that, ...
1
vote
1answer
61 views

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location?

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location? I don't mind how many coordinates that takes. For instance: Latitude = ...
0
votes
0answers
12 views

Countable product of metric spaces

Let $X=\prod X_i$ of countably many metric spaces $(X_i,d_i)$. Prove that the function which associates to $x=(x_i)$,$y=(y_i) \in \prod X_i$ the number $d(x,y)\in [0,\infty]$ defined by ...
1
vote
2answers
42 views

Is there a family of lines having two (or more) distinct envelopes?

The question: It's on the title. It just needs some clarification: $i)$ The problem takes place in the real plane. $ii)$ The "envelope" has its definition of the $E_2$ in this Wikipedia page: ...
0
votes
1answer
60 views

countable dense subsets

Let $I$ be some uncountable index set. For $\iota\in I$ and $j\in\mathbb N$ let $A_{\iota,j}\subseteq\mathbb R^d.$ Then there exists a countable dense subset in $A:=\overline{\bigcup_{\iota\in ...
3
votes
0answers
50 views

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 ...
1
vote
1answer
34 views

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 ...
1
vote
1answer
43 views

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!
2
votes
0answers
68 views

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 ...
1
vote
0answers
39 views

A question on the minimums

For any given $a_{i},\ a_{i}',\ c\in\mathbb{R},a_{i}\le\ a_{i}',\ i=1,2,\ 3$, let $$ S:=\left\{ \left(x_{1},x_{2},x_{3}\right)\in\mathbb{R}^{3}:\ x_{i}\in[a_{i},a_{i}'],\ i=1,2,3,\ ...
1
vote
1answer
46 views

A surface patch $\tilde{\sigma}(\tilde u,\tilde v)$ is a reparametrization of a surface patch $\sigma (u,v)$

Suppose that a surface patch $\tilde{\sigma}(\tilde u,\tilde v)$ is a reparametrization of a surface patch $\sigma (u,v)$. Let $$\tilde E d\tilde u^2+ 2\tilde F d\tilde ud\tilde v+\tilde G d\tilde ...
0
votes
1answer
30 views

Intersection of Dense Sets of Parallel Lines

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 ...
1
vote
0answers
39 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 ...
1
vote
1answer
34 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 ...
-1
votes
1answer
102 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 ...
0
votes
1answer
39 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. ...
1
vote
1answer
43 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
2
votes
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 $a,b$ ...
1
vote
1answer
45 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 ...
9
votes
2answers
78 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 ...
2
votes
0answers
85 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!
1
vote
0answers
84 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 ...
0
votes
1answer
32 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?
2
votes
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 ...
5
votes
1answer
83 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 $k\rightarrow\infty$ ?
0
votes
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 $\frac{\pi}{4}$?
0
votes
1answer
105 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 ...
2
votes
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 ...
1
vote
0answers
19 views

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 ...
0
votes
1answer
39 views

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 ...
6
votes
3answers
392 views

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 ...
3
votes
1answer
123 views

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 ...
0
votes
1answer
82 views

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 ...
0
votes
0answers
33 views

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 ...
2
votes
1answer
171 views

annihilator of an intersection in infinite dimension

Given two subspaces of an infinite dimensional Banach space, is the sum of their annihilators dense in the annihilator of their intersection?
0
votes
0answers
10 views

$\left( {M_i^n,{q_i}} \right) \to \left( {{R^k},{q_\infty }} \right)$,then there is a rescaling also converging to${R^k}$?

The arrow represent Gromov-Hausdorff convergence.Suppose $\left( {M_i^n,{q_i}} \right) \to \left( {{R^k},{q_\infty }} \right)$,where $Ri{c_{{M_i}}} \ge - 1/i$.Then $\left( {{\lambda _i}{M_i},{q_i}} ...
2
votes
2answers
399 views

How to measure the distance between two cities in the map by knowing latitude point and longitude point of them?

I want to measure the distance between two points in a map. For example between London and Moscow by knowing that the latitude point and longitude point of them. ...
4
votes
1answer
119 views

How can I measure the distance between two cities in the map?

Well i know that the distance between Moscow and London by km it's about 2,519 km and the distance between Moscow and London in my map by cm it's about ...
1
vote
1answer
41 views

When is gradient flow an isometry?

$M$ is a Riemannian manifold,$f$ is a function on $M$. Under what conditions is the gradient flow $F(t)$ of $f$ an isometry for $t>0$?
2
votes
1answer
51 views

Gradient flow of harmonic function is measure preserving?

Let $M$ be a manifold with $Ric \ge - \left( {n - 1} \right)$ and $f:{B_R}\left( p \right) \to R$ is a lipschitz and harmonic function. In a paper, it says "As the gradient flow ${\Phi _t}$ of $f$ is ...
2
votes
1answer
75 views

minimum value of a trigonometric equation is given. the problem is when the minimum value attains

Suppose the minimum value of $\cos^{2}(\theta_{1}-\theta_{2})+\cos^{2}(\theta_{2}-\theta_{3})+\cos^{2}(\theta_{3}-\theta_{1})$ is $\frac{3}{4}$. Also the following equations are given ...