Questions tagged [winding-number]
For questions about winding numbers. The winding number of a continuous curve counts how many times it "loops" around a given point.
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Getting a point in the interior of a polygon without relying on winding order?
I am given an arbitrary set of points embedded in 3D.
The points are guaranteed to be ordered such that their order yields a simple closed polygon, but there is no information about whether they wind ...
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Soft question - Index theory in nonlinear dynamics vs Complex analysis
The video https://www.youtube.com/watch?v=wZvFKcQ_3Rc&t=8s mentioned something called the Index Theory. I can't find it on wikipedia. Where could I find more about the theory?
Here index is just ...
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Winding number of composition equaling zero implies existence of complex logarithm
I have an open and connected set $V$ that is also simply connected. Let $a_1, \ldots, a_n \in V$ and set $U = V \setminus \{a_1, \ldots, a_n\}$. Let $B_{R_i}(a_i)$ be disjoint balls around each $a_i$ ...
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Geometric characteristics or combinatorics analog for homotopy groups of spheres other than $\pi_n(S^n)$
The fact that $\pi_1(S^1) = \mathbb Z$ is geometrically intuitive.
It is linked to a wealth of concepts and results, notably the winding number, residue theorem on $\mathbb C$, etc.
There is a ...
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Winding number of points on a path
I know that for $|z| = 1$ circular path with counter-clockwise rotation, the winding number of any point within the circle is 1(or more if the path is circulated more than 1 time. and also negative if ...
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Homology in a picture? (Is this picture just metaphorical, or a rigorous example that can be formalized?)
A post-doc colleague showed me this picture and said: going from the diagram No.2 to No.3 and to No.4 is taking the homology.
I did not quite understand this comment. For me, if I take simplicial ...
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Winding Numbers and Simply Connected Sets [closed]
Suppose $D$ if a path connected open subset of $\mathbb C$ so that the winding number of any $x$ in $\mathbb C -D$ with respect to any loop lying in $D$ is 0. Why does this imply that every loop lying ...
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Application of generalized Cauchy integral theorem
The following is a question from chapter 8 (Cauchy's Theorem) of Ian Stewart and David Tall's textbook on complex analysis. This is not a homework problem, I am self-teaching.
Let $$\gamma_1(t)=-1+e^{...
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Winding number of external point
Let $γ$ be a closed (C1-)curve whose image is contained in ${z: |z| < R}$ for some $R > 0$. Show that for any $z$ with $|z| > R$ we have $\operatorname{Ind}(γ,z) = 0$. This case is clear to ...
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Index (winding number) of vector field depending on sign of gradient
Consider a vector field $F(x,y):\mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose $F(x_0)=0$ and det $nabla F(x_0) \neq 0$, prove that the index (winding number) of the vector field F at $x_0$ is 1 or -...
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Winding number of two curves defined on the same interval
This question is about an observation regarding winding numbers.
Let $\gamma_1$ and $\gamma_2$ be two closed curves defined on the same interval $[a,b]$ onto $\mathbb{R}^2\setminus \{(0,0)\}$, each ...
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calculating winding number around zero [duplicate]
let $m, n \in \mathbb{Z}$ be fixed and let $ 0 < r \neq 1$, determine the winding number of the closed curve $\gamma(t) : = e^{imt} + re^{int}$ $\gamma: [0, 2\pi] \to \mathbb{C} \backslash 0$ ...
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Compute an oriented angle in an abstract plane.
I am coding something a bit esoteric.
I have a stream of points which are guaranteed to be planar, however the points are embedded in some high dimensional space (e.g. 3D).
I need to define a notion ...
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Why is the winding number of a holomorphic function the number of solutions to $f(z)=f(z_0)$?
In class the following situation was presented: suppose $f:\Omega\to \mathbb{C}$ is holomporphic. Let $z_0\in\Omega$ and suppose $f'(z_0)=0$ but $f''(z_0)\not=0$.Now consider the circle $C$ of radius $...
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Winding number of a loop in $S^1$ is same as the end point of its lift to $\mathbb{R}$
Let $\alpha\in \Omega(S^1,1)$ be a closed, piecewise $C^1$ curve at $1$, then winding number of $\alpha$ around $0$ $=n(\alpha,0):=\frac{1}{2\pi i}\int\limits_\alpha \frac{dz}{z}\in\mathbb{Z}$
Let $p:\...
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Mapping with infinite values - Winding number and complex analysis
Is there a continuous closed curve $\gamma$ in $\mathbb{C}$ with the property that the mapping
$$
\nu_{\gamma}: \mathbb{C} \backslash \operatorname{im}(\gamma) \rightarrow \mathbb{Z}
$$
takes ...
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index of a curve equal to the winding number
Why is the index of a holomorphic function $f:\mathbb{C}\to\mathbb{C}$ along a $\mathcal{C}^1$-curve $\Gamma$ equal to the winding number of $F(\Gamma)$ around $0$? So why is
$\displaystyle\frac{1}{2\...
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Does the path $t \mapsto \zeta(\frac 12 + it)$ eventually stay clockwise?
When we plot the path $\gamma(t):= \zeta (\frac 12+it)$ starting at $t=0$ and letting $t$ increase, we get the following beautiful image (courtesy of 3b1b's Riemann zeta function video's thumbnail), ...
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Proof of the Whitney-Graustein theorem
I'm researching about the Whitney-Graustein themorem, which states that two closed curves are regularly homotopic iff their winding numbers are equal. I've found a few references, like the original ...
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Find a closed path so that the number of zeroes is equal to the winding number
Let $D= \{ z\in \mathbb{C}: |z|<1\}$ and $T= \{ z\in \mathbb{C} : |z|=1\}$.
Let $f: \Omega \rightarrow \mathbb{C} $ be analytic on an open set $\Omega \supset D \cup T$, and suppose that $f(z)\ne0$,...
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What type of Mathematics, if any, is this? (On curiosities associated with a logo.)
I'm not sure whether I have articulated my curiosity well enough here. Please, therefore, bear with me if I need to edit the question, and please forgive me if this is otherwise a nonsense question ...
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Is the winding number of a null-homologous cycle outside an open ball zero?
Let $U\subset{\mathbb{C}}$ be open and $c=n_1•c_1+…+n_k•c_n$ (formal sum with $c_i:[0,1] \to U$ continuous, $n_i \in \mathbb{Z}$) be a null-homologous cycle (closed 1-chain with $Ind_c(z)=0 \text{ for ...
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Winding number of an ellipse $\gamma$ at $0$ using $\omega(\gamma,0)=\frac{1}{2i\pi}\oint_{\gamma}\frac{1}{z}dz$?
In this exercise, we are supposed to firstly find a path that parametrizes the following ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ for $a,b \in \mathbb{R}$
$\textit{I have found the following path: ...
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How to show that the winding number must be $0$ or $\pm$ $1$ in this case?
Prerequisite:
$K: [0,1] → \mathbb{C}$ piecewise, continuously differentiable, closed path that meets the non-positive real axis at only at one point
$K^{-1} (\{x|x \le 0\})= \{ t_0 \}$ with $0 < ...
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Regarding the winding number
My main question is about part B, but I would also be grateful if you can tell me what you think about part A.
Define a smooth vector field
$X$ on $S^1$ as follows:
$X(x,y)=(-y,x)$. For a smooth map $...
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A question about the formula for winding numbers in SMT 2021 Power Round
I was looking at the Power Round of Stanford Math Tournament and I came across this formula for a winding number of a curve:
$$w(\gamma)=\frac{1}{2πi}\int_0^{2π}\frac{\gamma’(s)}{\gamma(s)} \mathrm {...
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Maps that preserve winding numbers
Update: now crossposted to MathOverflow: https://mathoverflow.net/questions/419705/maps-that-preserve-winding-numbers
I am looking for a characterisation of the continuous maps on some subset of $A\...
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Closed rectifiable curve with arbitary winding number.
This is an exercise from Conway's Functions of One Complex Variable(page 83, exercise 2).
Give an example of a closed rectifiable curve $\gamma$ in $\mathbb C$ s.t. for any integer $k$ there is a ...
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Representations of $\mathfrak{u}(1)$ on $\mathbb{C}$ and $\bar{\mathbb{C}}$ with winding number $k\neq0$ are not isomorphic.
In the text I'm reading, $U(1)$ has a representation on $\mathbb{C}$ with winding number $k$ given by:
$$z\mapsto z^k$$
I'm supposed to show that the representations of $\mathfrak{u}(1)$ with winding ...
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"Winding number" of the map $g: T^{2n+1} \to U(N)$
For $n\in \mathbb Z_+$, let $T^{2n+1}$ be the torus and $U(N)$ be the unitary group, where $N$ is sufficiently large.
A physics paper on topological insulators claims the following:
For a map $g: T^{...
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How do I compute the local degree of the following map?
I have the following porblem.
Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a linear mapping, represented by a $2\times 2$ matrix. Moreover let the determinant be nonzero. Show that the local ...
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How do I show that the winding number of this closed continuous path is zero?
I have the following problem:
Show that if $\gamma:[a,b]\rightarrow U$ is a closed continuous path, $p\in \mathbb{R}^2$ and $U\subset \mathbb{R}^2\setminus \{p\}$ is an open subset on which there is ...
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If $ |\phi_1| <| \phi_2| $ and $\phi = \phi_1 + \phi_2 $ then $\mathrm {Ind} _{\phi} (0) = \mathrm {Ind} _{\phi_2} (0)$
Let $ \phi1, \phi2: [0,1] \to \mathbb {C} $ be Paths, such that $ |\phi_1 (t) | <| \phi_2 (t) | $ for all $ t \in [0,1] $. If $ 0 \leq t \leq 1 $, we set $\phi (t) = \phi_1 (t) + \phi_2 (t)$. Prove ...
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Small perturbations of a loop in $\mathbb C$ does not change its winding number
From Visual Complex Fuctions by Elias Wegert, Lemma 2.7.19, it says that
Let $\gamma_0 : [0,1] \to \mathbb C - \{0\}$ be a loop, i.e. a continous function with same endpoints. Denote by $d$ the ...
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How can I prove that this defintion of a winding number is valid?
I have read a definition of a winding number on wikipedia and it involves finding the continuous polar parametrization of the curve, but then the question arises, why does such a parametriation always ...
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Does injective continuous / holomorphic functions preserve winding number?
I am trying to find if the winding number is invariant to some transformations. I already know that it is invariant respect to translation, rotation and scaling. Also, it is not invariant respect to ...
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Can I compute contour orientation without using polygon area sign?
Most of the times, I determine contour orientation generating 2D points and computing the closed polygon area. Depending on the area value sign I can understand if the contour is oriented clockwise or ...
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How do I prove this statement about winding numbers and continuous maps?
I have the following problem:
Let $D$ be a disk with boundary circle $C$ and let $f:D\rightarrow \mathbb{R}^2$ be a continuous map. Suppose $P\in \mathbb{R}^2\setminus f(C)$ and the winding number of ...
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Winding Number of Linear Map
Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a linear map, represented by a matrix $A \in \mathbb{R}^{2 \times 2}$, with non-zero determinant.
Further, define the path $\gamma:[0;1]\to \mathbb{R}^2$,
$$\...
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$W(\gamma +v , p+ v) = W(\gamma, p)$ why is this true?
I don't understand why this statement in lecture is true :
let $\gamma : [a,b] \rightarrow \mathbb{R}^2 \setminus \left \{ p\right \} $ a continuous path, and v any vector in the plane. Moreover, ...
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Embedding of graphs in topological spaces
I am studying a problem concerned with the embedding of a graph $G=(V,E)$ in spaces that are not simply connected, e.g. $(\mathbb{C}\setminus\{0\})\times\mathbb{R}$. My understanding is that an ...
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usage of Rouche's theorem for winding number in proof of Residue theorem
A doubt from proof of residue theorem from Tao's notes.
notes 4 Theorem 21 (Residue theorem) Let $U\subset \mathbb{C}$ be a simply connected open set, and let $f: U\setminus S\rightarrow \mathbb{C}$ ...
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A question about such integrations relating to winding numbers
I am wondering, since
$$
\frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z
$$
can offer information about winding numbers by
$$
\frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} ...
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Why is $n(F,D)$ odd, when $F$ is odd?
This is a question from Do Carmo's Differential Forms and Applications (question 8, chapter 2). Actually, this question was made and answered here. The problem is: The answer redirects the OP to here (...
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Winding number is locally-constant for general curves (not $C^1$) using variation of argument definition
I've been searching for this proof, here and on the web too, but it seems like the answer is harder to find than expected. There are many similar questions about this but they all (implicitly or ...
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Calculate the integral $ \int_{0}^{1} \frac{1}{z_0+t(z_1-z_0)-z} \,dt $.
Let $z, z_0, z_1$ be three different complex numbers.
I am trying to calculate the integral $ \int_{0}^{1} \frac{1}{z_0+t(z_1-z_0)-z} \,dt $. Is there a closed form? Any hints? Thank you.
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Proving that the winding number is $k$
I am trying to prove that the winding number $n(p\circ\gamma_r,0)$ is $k$, when $p(z)=a\prod_{j=1}^k(z-z_j)$ is a complex polynomial of degree $k$ and $\gamma_r:[0,2\pi] \rightarrow \mathbb{C}, \...
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How winding number in a boundary point of a curve is defined?
We know that, the winding number counts the number of rounds of a curve
around a point. It is a positive integer for positive oriented curves and a negative integer
for curves with negative ...
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Is this set a star domain?
Definition:
A set S in $\mathbb{C}$ is called a star domain if there exists an $a \in S$ such that for all $x \in S$, the line segment from a to x is in S. And the point $a \in S$ is called a star ...
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if $f\in C(\mathbb{\overline{D}})$ and $\forall z\in\partial\mathbb{D}.f(z)=z$ then $\exists z\in\mathbb{D}.f(z)=0$
if $f\in C(\mathbb{\overline{D}})$ and $\forall z\in\partial\mathbb{D}.f(z)=z$ then $\exists z\in\mathbb{D}.f(z)=0$
f is not analytical only Continuous .
solve with complex analysis tools
I thought ...
user876781