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0answers
33 views

Tangent Cone of a Complete Intersection

Can you give me an example of an affine variety $X \subseteq \mathbb{A}^n_{\mathbb{C}}$ over the complex numbers which is a complete intersection such that the reduced tangent cone at some point $p ...
0
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0answers
17 views

Laurent series about which point?

I have been given the question: "Find the laurent series of $1/(1-z^3)$". No other information. The question was posed in the process of finding the series' residual and in the answer I can see that ...
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2answers
24 views

Why does the function $f(z) = 1/\sin(\pi/z)$ have isolated singular points?

In the complex analysis text book "Complex Variables and Applications 8th edition", it states the function $1/sin (\pi/z)$ has singular points $z = 0$ and $z = 1/m; (m = \pm 1,2,3,4,\dots.) $ I sort ...
1
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1answer
45 views

Residue theorem application [demonstration]

I really don't know how to solve this problem! Consider $F$, an analytic fuction, so that, $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial So, I tried to ...
-1
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1answer
60 views

Residue theorem application

How could we use the Residue theorem to calculate the following integral: $$\int_0^{2\pi} \frac{1}{1-2p\cos{x} + p^2} dx$$ where $p$ is a real constant, such that $p\in ]0,1[$ Thank you!
0
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0answers
9 views

Quadratic singularities and local curves.

Everything is to be understood over the complex field. Assume you have two finite dimensional $\mathbb{C}$-vectorial space $V$ and $W$. You are given a bilinear form : $$\phi:V\times V\rightarrow W ...
0
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0answers
25 views

Singularities and Residues [Demonstration]

How could I solve the following problem: "Consider $F$, an analytic function, so that $$f(z)=F(\frac{1}{z-1})$$ has a pole. Demonstrate that F(z) is a polynomial." I know that an analytic function ...
0
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0answers
24 views

Du Val singularities, Reid‘s D$_4$

Maybe this is a stupid question, but I feel puzzled. In Reid's note: Chapters on algebraic surfaces, in 4.2 Du Val singularities, example D$_4$, he writes the equation of D$_4$: x$^2$+y$^3$+z$^3$=0, ...
7
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1answer
75 views

Showing a complex analytic function is unbounded

This was one of the problems on a previous year's Complex Analysis final exam. Assume $f\in \mathcal O (\mathbb H )$, non-constant, and $f(\frac {i}{\sqrt n})=0, \forall n\in \mathbb N$. Prove that ...
0
votes
1answer
40 views

Power Series with singularities in {z: |z|=1}

Prove that all the points in $D=\left\{ z \in \mathbb{C} : \mid z\mid=1 \right\}$ are singularities of the function $$ f(z)=\sum_{n=0}^{\infty} \frac{z^{n!}}{n!} $$ This was easy for the ...
0
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0answers
16 views

$a$ is a non isolated singularity of $f (z)$

If a point $a$ is a non-isolated singularity of a function $f(z)$ and $|f(z)|$ is bounded in some neighbourhood of $a$ then what kind of singularity of $f(z)$ occurs at $a$? Removable singularity ...
4
votes
1answer
47 views

Resolution of singularities of the determinant hypersurface

Let $$\det\nolimits_n=\sum_{\pi\in\mathfrak{S}_n} \operatorname{sgn}(\pi)\cdot x_{1,\pi(1)}\cdots x_{n,\pi(n)} \in \mathbb{C}[x_{ij}\mid 1\le i,j\le n]$$ be the determinant polynomial. It defines a ...
3
votes
1answer
55 views

Help with understanding the notation $\mathbb{C}\{f\}$

I am reading an article "Relative Cohomology and volume forms" of J. P. Francoise. Here the author considers the germ of a function $f\colon(\mathbb{C}^n,0) \to (\mathbb{C},0)$, and he speaks of the ...
1
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0answers
26 views

Residue Theorem: can I say this?

I know that $f(z)$ is analitic everywhere except of $z=0$, where it has a pole of order $k$. Can I say that $f'(z)$ will have a pole of order $k+1$? For example, if $f(z)=\frac{1}{z^k} \rightarrow ...
0
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1answer
34 views

Series simplification

I had three problems to work on and I was able to solve the third summation problem. The first two, I am having difficulty understanding as to how to proceed. Here are the questions: ...
2
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1answer
29 views

Decomposition of resolvent in projections

I am reading the book Perturbation theory for linear operators from Kato. He defines (§5 Section 3) for an operator $T : X\to X$ on a finite Banach Space the resolvent as $$ R(x) = (T- x)^{-1}.$$ ...
6
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1answer
68 views

Books or texts on singularity theory [closed]

So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
1
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1answer
31 views

How do I compute this Milnor number

I need to compute $\mu (x^5+y^5)=5$ on the point $p=(0,0)\in\mathbb{C}^2$. By definition, for $f\in\mathbb{C}[x,y]$, I have $$ \mu(f)=\dim\dfrac{\mathcal{O}_{(0,0)}}{<\dfrac{\partial f}{\partial ...
0
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0answers
16 views

show that a function on annulus has a removable singularity

How can I show, that for $f \in \mathcal{O} \cap L_2(U \setminus \{z_0\})$, f has a removable singularity at $z_0$? I should use this fact: if $$f = \sum_{n= - \infty} ^{\infty} a_n (z-z_0)^n \in ...
2
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0answers
77 views

Manifolds, isolated singularities, open map

I can't figure out how to solve the following question: 1) Let $M$ and $N$ be $C^k$-manifolds, such that $\dim M = \dim N = n>1$, and let $f:M\rightarrow N$ be a $C^k$-function. Show that if the ...
0
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1answer
23 views

Singular values of rectangular matrix

can any one explain me the need for singular values of a matrix. Explanation with a practical example will be appreciated
3
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1answer
74 views

What is the value of this integral (using the Argument Principle),

F(z) is given by $$F(z) = e^zz^{-2}(z-1)(z^2-4)(z+8)^7$$ What is the value of the integral $$\int_0^{2\pi} \frac{F'(3e^{i\theta})}{F(3e^{i\theta})}d\theta \space \space ?$$ I think the relevant ...
1
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1answer
44 views

singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
4
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0answers
53 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
3
votes
1answer
71 views

Meromorphic function with a simple pole and a simple zero, and satisfies an inequality. What can it be?

Describe all meromorphic functions f(z) in the complex plane with a simple pole at z=1, a simple zero at z=-1, and for which $$|f(z)|\le M|z|,$$ for $|z|\ge 2$ for some $M>0$. I know that, ...
0
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0answers
22 views

Find the isolated singularities of a given function

I am trying to find and classify the isolated singularities of: $g(z)=z/cos(z)$ I'm unsure of how to approach this, I was thinking Laurent expression of g may be useful, however I don't know how to ...
0
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0answers
17 views

How to determine singularities of a series?

Given a double Fourier series, how do we determine its singularities ? PS: I wonder how we find singularities(mathematically) if a function cannot be expressed in a closed form.
0
votes
3answers
30 views

Questions about poles

Find order of pole for $$f(z)=\frac{1}{e^{z}-1}$$ at $z=0$. Now I turned the function into this: $$\frac{1}{\sum_{1}^{\infty}x^k/k!}$$ I think the pole has order $1$ but $\lim(z(f(z)))$ seems to be ...
0
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1answer
33 views

Singularities of $\frac{1}{e^{\frac{1}{z}}+2}$ and classification?

I'm considering the function $$\frac{1}{e^{\frac{1}{z}}+2}$$ Clearly, it has a unique singularity at $z=0$. I feel like its an essential but I can't find the Laurent expansion or any other way of ...
2
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0answers
44 views

Requesting information on constructed discontinuous functions (from any perspective)

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Define the function $F:\mathbb{R}\rightarrow \mathbb{R}$ as $$F(x)=f(x)\prod_{n=1}^\infty\frac{x-\frac{1}{n}}{x-\frac{1}{n}}$$ I ...
0
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0answers
36 views

Term for Multiple Functions that Share Critical Points?

Is there a term for when multiple functions share each other's critical points? Or, in general, when one function has a subset of the critical points of another?
1
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0answers
52 views

If Gaussian random vector has singular covariance matrix, isn't there probability density function?

I got a complicated problem. Suppose that Gaussian random vector having Covariance matrix; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 ...
0
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1answer
52 views

Negative index coefficients of Laurent series for 1/sin(z)

Given $f(z) = \dfrac{1}{\sin(z)}$ a) Give singularities b) Determine coefficients $a_{-1}$ and $a_{-3}$ of the Laurent series So I thought: a) $n \pi$, where $n$ is an integer b) ...
0
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0answers
77 views

Cusp singularity

What is the most general way of defining cusps singularities for a continuous curve $\mathbf{r}:(a,b)\to\mathbb{R}^2$ and how can we define them? What are the minimum requirements for the curve ...
1
vote
1answer
61 views

How to find tangent cone in singular point?

How to find tangent cone in singular point of surface? For example, considering surface in $\mathbb{R}^3$ given by equation $x^2z=y^2$, what is it's tangent cone in the origin? UPD:By tangent cone ...
0
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0answers
36 views

Characteristics of Singularity Points

Determine the character of the singularity at $z=0$ for each of the following functions: \begin{align} &\frac{1/z^{7}}{e^{z}-1} \tag{1} \\ &\frac{}{} \\ ...
0
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0answers
30 views

The normalisation map is a bi-Lipschitz map?

Let $X$ a reduced analytic space, $n: W \rightarrow X$ the normalisation map, $W$ the normalisation of $X$ and $S$ the singular set of $X$. When we restrict $n$ to $W\setminus n^{-1}(S)$, we know that ...
0
votes
1answer
33 views

Relations between different but similar Milnor fibers

Let $f:\mathbb A^N\to\mathbb C$ be a holomorphic function and let $P$ be a critical point, i.e. $$P\in Z(\textrm df)\subset \mathbb A^N.$$ The Milnor fiber of $f$ at $P$ is the intersection of a ...
1
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2answers
57 views

Determine the nature of singularities and calculate the residue of $f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3}$

$$f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3},\;\;\;\;\;\;\; \mathrm{Res}[f(z),0]$$ I am having trouble determining the nature of singularities. This is what I managed to do: ...
6
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3answers
290 views

How to know if a tangent bundle is trivial from its defining equations

In this question, I am considering only regular manifolds. A Trivial Bundle The circle $S^1$ is known to have a trivial tangent bundle. As a subset of $\mathbb{R}^4$, the tangent bundle of $S^1$ ...
3
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2answers
170 views

Types of singularities of a function

How can I determine the type of a singularity of a function $$f(z)={e^{1/z} \over z-1}+{\pi z \over 2\sin(\pi z)}$$ at $z_0=0$? I don't see an easy way to represent it using Laurent series, neither ...
2
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1answer
55 views

Question regarding singularity of a complex function

Consider the function $$f(z) = {1 \over (z-i)(z+i)}$$ with a Laurent series expansion at $z_0=i$ on a domain $\;\Omega=\left\{z\in \mathbb{C}:2\lt\left|z-i\right|\right\}$ $$\begin{eqnarray}f(z)={1 ...
1
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1answer
89 views

Blowup of a (very) simple singularity

Take the action of $\mathbb{Z}_2$ on $\mathbb{C}^2$ given by $(-1) \cdot (z,w) = (-z,-w)$ and of course, $(1)\cdot (z,w) = (z,w)$. If you look at the resulting quotient space $\mathbb{C}^2 / ...
0
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0answers
62 views

Alternative Proof why algebraic groups are smooth

Definitions: Let $k$ be a field, by an affine algebraic group we will mean an affine group scheme $G$ over $Spec(k)$ with the property that the coordinate ring of the underlying affine scheme ...
0
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1answer
49 views

Finding solution to a unidirectional nonlinear wave equation

I can do the parts a), b) and c) and find that in part c) that the condition in which the solution will break down is when $1+tf'(x-tu)=0$ However I am unable to part d) I tried ...
1
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1answer
77 views

Slodowy slices for non ADE type Lie algebras

In the first answer to: http://mathoverflow.net/questions/16026/the-finite-subgroups-of-sl2-c the correspondence between transverse slices of subregular elements of the nilpotent cone for a simple Lie ...
1
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1answer
81 views

non-removable singularities of a function are essential singularities of the composition function

Let $f$ be a non-constant enrire function on $\mathbb{C}$ such that $f(z+i)=f(z)$ for all $z$. Let $U$ be an open subset of $C$ and $z_0\in U$. Let $g:U\setminus\{z_0\}\longrightarrow\mathbb{C}$ be ...
2
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1answer
106 views

Lipschitz manifold and semi-algebraic is Lipschitz graph?

It is known that there are Lipschitz manifolds that are not Lipschitz graphs and $C^1$ manifold is locally $C^1$ graph. If $M\subset \mathbb{R}^m$ is a Lipschitz manifold (with the outer distance) ...
2
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0answers
83 views

Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the ...
1
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1answer
44 views

What are some properties or characteristics of singular eigenvector matrices?

I'm trying to predict when a matrix $A$ will have an eigenvector matrix $X$ that is singular. What properties or characteristics would $A$ or $X$ have to make $X$ singular??