The singularity-theory tag has no wiki summary.
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Singularity models of the Ricci flow
I faced this sentence in my studies on Ricci flow:
The Bryant soliton is a singularity model for the degenerate neckpinch.
Q1: What is the definition and meaning of singularity model? Can one model ...
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Germ determination
Is it true that $R$-equivalent germs $f,g: \mathbb{R}^n,0\rightarrow \mathbb{R}$ have the same determinacy?
Note: two germs $f,g: \mathbb{R}^n,0\rightarrow \mathbb{R}$ are said to be $R$-equivalent ...
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1answer
49 views
Distribution with singularities.
I need some help to prove that $f$ defined by $\langle f,\psi\rangle:= \sum_{n=0} ^\infty
\psi^{(n)}(n)$ is a distribution which has singularities of infinite
order. Here $\psi$ is a test function ...
4
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2answers
74 views
Singularities of $f(z)=z/\cos(z)$
Regarding complex functions (in complex variables), I was wondering why the function $g(z)= \cos(z)$ has a singularity at $z = \infty$ but $f(z)= \dfrac{z}{\cos(z)}$ does not.
I am a bit confused ...
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How many types of surface singularities multiplicity two exist?
All varieties are over $\mathbb{C}$.
Let $S$ be a reduced algebraic surface in $\mathbb{P}^3$ with a singular point $p$ of multiplicity two. The question is local so we reduce to $S \subset ...
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Puiseux series and Resolution of Singularities
I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities.
So the Newton's method gives us a Puiseux series parametrizing the branches of ...
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1answer
151 views
Is there a classification of isolated essential singularities?
In the thread Why do we categorize all other (iso.) singularities as "essential"?, here is one of the questions that was asked:
Do we not care about essential singularities to classify ...
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I have an infinite solution to an ODE even though it has only a regular singular point
I have the ODE:
$\displaystyle y''(x)+\frac{y'(x)}{x+1}+y(x)=0$
I know that this has a regular singular point at $x=-1$, as $(1+x)^{-1}$ has only a first order pole, and $1$ has no pole at all, and ...
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1answer
34 views
Removable Singularity with bound on derivative
Here is a question from a practice exam:
Suppose $g(z)$ is a holomorphic function everywhere except the origin. Also suppose
$$
|g'(z)|\leq \frac{1}{|z|^{3/2}} \quad \text{ for } 0<|z|\leq 1
$$
...
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Algorithm for checking that a singularity is an ordinary double point
Let a complete intersection $d$-fold singularity be cut out by $k$ polynomials $\{f_1,\ldots,f_k\}$ in $\mathbb{C}^{k+d}$. Is there a relatively simple algorithm to check whether this is analytically ...
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55 views
Local algebra of a function at a point
The multiplicity of a $C^{\infty}$ germ at $0$, $f:(\mathbb{R}^m,0) \rightarrow (\mathbb{R}^n,0)$, is the dimension over $\mathbb{R}$ of its local algebra $Q_f=A_x/I_f$, where $A_x$ is the algebra of ...
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1answer
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Do singularities always appear on all Riemann sheets?
Consider a function $$f(z) = \frac{\ln z}{z^2+1}.$$ Besides the branching point $z=0$, the function also has singularities at $z = \pm i$. This singularities should appear on all Riemann sheets.
Is ...
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How do I check whether an orbifold admits deformations?
Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive $n^\text{th}$ root of unity, admit smooth deformations. This is because ...
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1answer
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questions about singularities and complex functions having poles of order k , proofs and examples
1)i need an example of a non isolated singularity
2) also i need an entire function which assumes every complex value but the number 1+2i
and i want to know the way in order to solve some other ...
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1answer
101 views
Contour integral $\int_{|z|=1}\exp(1/z)\sin(1/z)dz$
Evaluate the contour integral $$\int_{|z|=1}\exp(1/z)\sin(1/z)dz$$ along the circle $|z|=1$ counterclockwise once.
The singularities are $\dfrac1{\pi k},k\in\mathbb{Z}$ plus the limit point $0$. So I ...
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2answers
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Removable singularity and laurent series
How to show $z=\pm\pi$ is a removable singularity for $\frac1{\sin z}+\frac{2z}{z^2-\pi^2}$?
I tried to compute the Laurent series, specifically the coefficients for $1/z,1/z^2,...$ since if we can ...
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1answer
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Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$?
Does $\frac {z^5}{\sin z^2-z^2}$ have a non-isolated singularity at $0$? If so, is it not meaningful to discuss its residue at $0$?
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1answer
56 views
Local rings and classifying singularities
My query is a little vague, but I'll try to be as concrete as possible.
Is there some sense in which the local ring of an algebraic variety (or more general complex space) at a point depends only on ...
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votes
2answers
105 views
Do the polynomial germs generate all the ring of germs?
I'm trying to understand some equality that comes up in stability theory involving sets of germs and I think I need a result like the next one, so if anyone knows anything about this and helps me it ...
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resolution of curve singularities
Let $\pi:X\longrightarrow C$ be the minimal good resolution of the curve singularity (C,o) with exceptional set $E$, where $C:=\{x_1x_2(x_1^{a_1}+x_2^{a_2})=0\}\subset \mathbb C^2$. Let $\bar C_i$ be ...
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What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C$?
What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C^*$? I need it to determine the type of the singularity at $z = 0$.
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1answer
143 views
Prove that $f(z)$ can not be a polynomial
Suppose $f(z)$ and $g(z)$ are entire functions and that $f(z)$ is not constant. If $|f(z)| < |g(z)|$ for all $z \in \mathbb C, $ prove that $f(z)$ can not be a polynomial.
I was thinking what I ...
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3answers
155 views
Singularity - Removable or Pole?
For the complex-valued function
$$f(z) = \left(\frac{\sin 3z}{z^2}-\frac{3}{z}\right)$$
classify the singularity at $z=0$ and calculate its residue.
Attempt at Solution
Rewriting $f(z) = ...
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'Ordinary $n$-tuple point' in higher dimensions
Everything above the line is just to provide context and motivation.
For an algebraic curve, we can define the multiplicity of a point $p$ (somewhat simplistically) as follows:
Choose coordinates ...
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1answer
56 views
Invertibility of matrix with each element equal to cofactor
I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the ...
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1answer
120 views
Question on the Proof of Casorati Weierstrass
The Casorati Weierstrass Theorem:
Let $f:X\subseteq \mathbb{C}\to \mathbb{C}$ have an essential singularity at $w\in \mathbb{C}$. Then,
\begin{equation}\forall \epsilon,\delta>0,\zeta\in ...
5
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2answers
245 views
Singularities of $e^{z - \frac{1}{z}}$
I believe $e^{z - \frac{1}{z}}$ has essential singularities at $z = 0$ and $z = \infty$ (in both cases because of a $\frac{1}{z}$ in the exponent) but I'm having a hard time proving this. How can one ...
0
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1answer
64 views
Supremum and Infinmum of function with singularity
My original approach was to let $f = \sin (1/x)$ and do regular calculus. However, I found that it wasn't so simple. I graphed the function on Mathematica and it was even worse!
It seems like the ...
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Question on definition of 3-fold cDV singularities
A 3-fold compound Du Val singularity of type $A_{k} \ (D_{k}, E_{k})$ is defined as a singular point on a 3-fold which is locally given by
$$
F(x,y,z,w)=f(x,y,z)+wg(x,y,z,w)=0,
$$
where $g(x,y,z,w)$ ...
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1answer
86 views
germ finitely determined
Does anyone know any result on finitely determined germs to help me prove that the germ $f(x,y)=x^3+ xy^3$ is $4$- determined? I tried using the definition of germs finitely determined, which is:$f: ...
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1answer
615 views
Singular points of ODE
My friend and I have conflicting answers and since his phone is off, I can't get his full solution and I don't understand his argument.
Consider this ODE
$$(x+1)y''+\frac{1}{x}y' + (x+3)y= 0$$
...
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2answers
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The notion of a germ in singularity theory
I quote from my lecture:
Let $X$ be a topological space (think of $X=\mathbb{C}^n$ with the classical topology), $p\in X$, $A,B\subseteq X$. Then $A\sim B$ if there exists an open subset ...
