In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. (Def: ...

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14 views

PDE Singularity and time of shock

Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$ It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ ...
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34 views

A Question About Poles.

I have some questions in my mind bothering me to understand poles. Let $z_0$ be a pole of order $m$ for $f(z)$. Does that mean: 1- $(m+1)$ is the smallest positive integer such that: ...
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1answer
36 views

Finding singularities of A Function

I want to find singularities of $$f(z)=\frac{{z}^{2}}{e^z + {e}^{-z} - 2}$$ I solved this problem but I am not sure about it. Is it correct? $${e^z + {e}^{-z} - 2}= 0$$ Then I divide by $$e^{-z}$$ to ...
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22 views

asymptotic matched expansion with transiently blowing inner solution

I have been trying to solve following set of equations with method of matched asymptotic expansion, $\frac{dy(t)}{dt}=k z(t) - 3 \alpha y(t) - y(t)^2 + \mu (M-z(t))^2$ $\epsilon ...
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41 views

General Fiber in Positive Characteristic

It is well-known that a complex polynomial, considered as a function $f:\mathbb{C}^n\to\mathbb{C}$, is a fiber bundle over a cofinite set of "atypical values" which include the singular values of $f$. ...
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1answer
32 views

Properties of resolution of singularities

Let $X$ be a complex algebraic varieties and $\pi:X' \to X$ be a resolution of singularities of $X$. Let $Y$ be a smooth (irreducible) subvariety of $X$. Is $\pi^{-1}(Y)$ smooth and irreducible? What ...
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3answers
49 views

Obtaining a Non-Singular Matrix from a Singular one by Perturbation

In a paper "http://www.math.cornell.edu/~nussbaum/papers/08-1.pdf" (page 264 Lemma 2) I encountered the following way of obtaining an invertible (non-singular) matrix from a non-invertible (singular) ...
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107 views

Singularities on a weighted projective curve

Let $C$ be the curve of degree $3$ defined over $\mathbb{C}$ given by $$x(y+z)=y^3-z^3$$ which lives in the weighed projective space $\mathbb{P}(x,y,z)=\mathbb{P}(2,1,1)$. Is the curve singular ?
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1answer
43 views

Is an analytic one-to-one function on the whole plane necessarily a polynomial? (Can it be disproved?)

I had to show what a one-to-one analytic function from the plane to itself could possibly be. So, I studied the behavior of such a function at infinity: Case 1: Such a function cannot have no ...
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19 views

Second Intrinsic Derivative

I'm trying to understand the second intrinsic derivative but I just don't get it. For example for an $A_r$ singularity $y_{1}=t_{1}, y_{2}=t_{2}, y_{r-1}=t_{r-1}, \cdots , y_{r}=t_1x + t_2x^2+\cdots ...
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39 views

Desingularization of curves

Let $C$ be a reduced, reducible curves over an algebraically closed field with at worst nodal singularities. Does there exist an irreducible non-singular curve $\tilde{C}$ and a finite, surjective ...
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14 views

Eliminate singularity from a PDE system

I need some help with the theory of PDE systems, which I am not very familiar with. Assume $f$ and $g$ are two functions of $x,y$ in the real plane. I have the equation ...
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2answers
114 views

How to get the asymptotic form of this oscilatting integral?

So the integral is like this: $$\int_1^\infty \frac{\cos xt}{(x^2-1)\left[\left(\ln\left|\frac{1-x}{1+x}\right|\right)^2+\pi^2\right]}\mathrm{d}x$$ The question is how to get the asymptotic form of ...
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34 views

magma: singularity type (IsNode?)

Does magma know the singularity type of a point in a scheme? I've seen some functions but they all seem to be about curves. This is my context: I have a surface in $P^2\times P^1$ with zero ...
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62 views

resolution of a surface singularity

Let $f: Y \to (0\in X)$ be a resolution of a surface singularity and $R^1f_*\mathcal{O}_Y=0$. Then $H^1(Y,\mathcal{O}_Y)=0$. Why? Is it clear? How can I show it?
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1answer
30 views

Singularity of Product of two complex function $f$ and $g$

Suppose $f$ has an essential Singularity at $z = a$ and $g$ has a pole at $z = a$. Then the product $fg$ has an essential Singularity at $z = a $. Is this hold if $g$ has removable ...
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27 views

Singular values and trace?

Given that $X$ and $Y$ are positive definite matrices, how can I bound the singular values $\sigma(X+Y)$ in terms of the trace of $X$ and $Y$?
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2answers
84 views

Preimage of singular points of smooth map between manifolds

Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ where $V$ is a (finite dim, real) vector space (of potentially very large dimension) and $SU(n)$ is the special unitary Lie group, what ...
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10 views

question on singularity theory and normal forms of local function germs

Hello I realize this might be a difficult topic but I was just trying to practice for my advanced class on singularities I wanted to compute normal forms for corank 2 singularities of the function ...
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1answer
35 views

Question about finding Laurent Series over closed region and classifying singularity

Represent $\sin(\pi x/(x+1))$ Laurent Series about the region $0<|x+1|<2$: Its true that $$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ So the $$\sin(\pi x/(1+x))=\sum (-1)^{n-1} \frac{(\pi ...
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50 views

Is exceptional divisor always a Projective bundle over the centre?

Let $f:\tilde X \rightarrow X$ be a blow up at center Z. Is $f^{-1}(z)=\mathbb{P}^{k}$ ?for some $k$, $\forall$ $z\in Z$ In the case of blow-up of $\mathbb{A}^{n}$ at origin it is very clear that the ...
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2answers
33 views

Properties of the ring of smooth function germs, question on proof.

Let us denote by $C_n$ the ring of $C^{\infty}$ smooth function germs $f : (\mathbb R^n, 0) \to \mathbb R$ or the ring of analytic functions germs $f : (\mathbb C^n, 0) \to \mathbb C$. Denote by ...
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23 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
62 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 ...
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1answer
48 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 ...
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1answer
77 views

Residue theorem application [duplicate]

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!
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15 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 ...
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30 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 ...
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47 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, ...
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1answer
94 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 ...
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1answer
51 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 ...
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17 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 ...
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1answer
62 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 ...
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1answer
57 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 ...
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28 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 ...
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1answer
39 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: ...
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1answer
33 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 $$ R(x) = (T- x)^{-1}.$$ ...
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1answer
100 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 ...
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1answer
36 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 ...
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21 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 ...
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87 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 ...
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1answer
28 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
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97 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 ...
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1answer
49 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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64 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
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1answer
89 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, ...
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21 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.
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31 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 ...
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1answer
37 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 ...
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48 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 ...