Tagged Questions

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A Homeomorphism that is not unique even upto Isotopy

I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism ...
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why some solid can have no surface? [closed]

For solid construction, I can understand the closed surface has no edges. But i cannot understand why some solid can have no surface (except just lines?), any other solid which can have no surface?
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Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
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What does it mean to say “Resolving intersections”

Consider a surface (with boundary) $S$ with marked points on the boundary such that we may may triangulate the surface. Call a line joining two marked points in a triangulation an arc. Consider a ...
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Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-))$$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
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Cw complex $\Sigma_g$

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it ...
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How to distinguish walking on a sphere or on a torus?

Imagine that you're a flatlander walking in your world. How could you distinguish if the world is a sphere or a torus ? I can't see the difference from this point of view. If you are interested, this ...
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Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...
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Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
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Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
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Definition singular manifold

I'm looking for the definition of a singular manifold. I haven't found it yet. For instance, in $\mathbb{R}^4$, with $f(x,y,z,t)=xy-zt$, $f^{-1}(0)$ is a singular submanifold. I only found a few ...
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Covering of Riemann sphere

The question consists of several parts: What is the simply connected ramified covering of the Riemann sphere with ramification indexes {2. 3. 5} over three points of RS in every preimage of these ...
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Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
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Let $X:\mathbb R^2\rightarrow\mathbb R,$ be the map defined by $(x,y)\mapsto y-x.$ Let $h:\mathbb R\rightarrow\mathbb R$ be Borel measurable. Let $\mu$ be a Borel probability measure on $\mathbb ... 1answer 92 views Independence of$H(f)=\int_M \alpha \wedge f^* \beta$on choice of$d\alpha=f^*\beta$? I came across the following UCLA qual question while studying for my upcoming qual: Let$f: M^{4n-1} \to N^{2n}$be a smooth map between closed connected oriented manifolds of the indicated ... 1answer 73 views A norm on$\mathbb{R}^2$such that$\partial C$is the unit sphere? Suppose we are on$\mathbb{R}^2$. Assume that$C \subset \mathbb{R}^2$is a convex bounded neighborhood of the origin invariant by central symmetry. Let$\partial C$denote the boundary of$C$. My ... 1answer 141 views Fractals - when the number of seed shapes that can fit into the scaled-up copy is non-integer. I've heard people say (for eg. here) that the dimension of fractal patterns (particularly, in this question, Lindenmayer fractals) can be formulated as follows: $$D=\frac{\ln N}{\ln S}$$ Where$N$... 1answer 34 views Transforming a curve on an arc to a line I have a function, actually a point cloud, (similar to a sine wave) on an arc with a known radius of curvature. I need to remove the curvature to regenerate the original function (or point cloud). ... 1answer 19 views Relation between dense subsets in the product map and dense subsets in each component Let$\Omega$bea Polish space and$X_1,\dots,X_n:\Omega\rightarrow\mathbb R^d$be Borel measurable maps. Consider now the map$X:\Omega\rightarrow(\mathbb R^d)^n$defined by ... 1answer 44 views Constructing The Cayley Graph and quasi-isometry to$\mathbb{Z}$If we have a group$G$defined by:$G=\langle a,b\mid b^2=1\rangle$then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ... 1answer 40 views $T_3$is quasi isometric to$T_4$I have a question which asks me to show that$T_3$is quasi isometric to$T_4$, that is the three and 4 valence trees. I know that this means that I have to define a map$f:T_3\rightarrow T_4$such ... 0answers 17 views What is the nth barycentric simplicial subdivision? I read it in a paper, but, unfortunately, am not familiar with it. Here's my guess: For example, in$\mathbb{R}^2$, a simplex is a triangle, and if we divide it by three lines connecting each vertex ... 3answers 55 views What is a “control point”? I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ... 1answer 121 views How can a simple closed curve not look locally like the rotated graph of a continuous function? A simple closed curve is a continuous closed curve without self-intersections. The question of whether you can inscribe a square in every simple closed curve is currently an open problem, but this ... 1answer 52 views Is it true there exists$f:S^{2n}\longrightarrow S^{2n}$making the diagram commutative? Let$g:\mathbb R\mathbb P^{2n}\longrightarrow \mathbb R\mathbb P^{2n}$be a continuous map where$\mathbb R\mathbb P^{2n}=\mathbb S^{2n}/\{\pm x\}$. Is it true there exists$f:\mathbb ...
Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...