2
votes
2answers
51 views

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 ...
1
vote
1answer
29 views

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?
1
vote
1answer
40 views

Converge of an inversion to a mirrorring

I want to ask something about a mirroring and a inversion in $\mathbb{R}^n$. An inversion in a sphere with center $m$ and radius $\rho$ can be written as $$ v \ \longmapsto \ ...
0
votes
0answers
40 views

Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
1
vote
1answer
33 views

surface area of the graph of a convex function

I started out with the following question: Say $\Omega$ is a nice bounded domain in $\mathbb{R}^{n-1}$. (One can imagine it being a unit ball in $\mathbb{R}^{n-1}$.) Let $f:\Omega\rightarrow ...
0
votes
0answers
38 views

Relationship between n-dimensional ellipsoid's surface area and semi-axes

I'm trying to prove (or falsify) the following claim relevant to work I'm doing with Steimer Symmetrization: If we consider an n-dimensional ellipsoid with semi-axes of radius $a_1< ... < a_n$, ...
0
votes
0answers
21 views

Polar set and adjoint operator

I'm trying to understand the following statement: A bounded operator between Banach spaces $u:X\rightarrow Y$ satisfies: $$\textrm{inf}\{\;||u(B_X \cap S)||:S\subset X,\,\textrm{codim}S=k \; ...
-1
votes
1answer
118 views

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 ...
3
votes
1answer
51 views

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 ...
2
votes
1answer
52 views

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 $E^d$ ...
0
votes
0answers
36 views

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 ...
0
votes
1answer
28 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
46 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
346 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
21 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
35 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
23 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
63 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
35 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
19 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
23 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
78 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
14 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
69 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
2answers
92 views

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 ...
3
votes
0answers
53 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
38 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
44 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
69 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
40 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
53 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
33 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
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 ...
1
vote
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 ...
-1
votes
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 ...
0
votes
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. ...
1
vote
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
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
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 ...
9
votes
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 ...
2
votes
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!
1
vote
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 ...
0
votes
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?
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
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 $k\rightarrow\infty$ ?
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
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 ...
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
20 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 ...