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0answers
10 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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3answers
27 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
26 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
41 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 ...
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0answers
27 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?
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0answers
20 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
36 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) ...
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0answers
34 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 ...
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0answers
22 views

Classification singularity

I have to classify the singularity of the complex function $$f(z) = z \sin(1/z).$$ I already saw that zero is a essential singularity of $f$. But I can't determine the $$f(\{z \in \mathbb{C} : 0 ...
1
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1answer
32 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
14 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{}{} \\ ...
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0answers
25 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
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1answer
28 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
41 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
125 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
93 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 ...
1
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1answer
43 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
75 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
58 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
41 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 ...
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1answer
55 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
61 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
101 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
79 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
40 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??
1
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1answer
28 views

Essential singularity $z=a$

Suppose $f$ has an essential singularity at $z=a$, is it true that $f(z)$ is not equal to 3 for $z$ not equal to $a$, then $\frac{1}{f(z)-3}$ is bounded in some punctured disk $D'(a,s)$ for some ...
0
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1answer
53 views

Recommend resources on dynamical systems and singularities

I'm looking for resources on bifurcation theory and systems of non-linear differential equations, but am very particular about the way it is taught/explained. I would like the approach to be based on ...
1
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1answer
25 views

Generating function which has no singularity

We can know the growth rate of coefficients from singularities of generating functions, but if a generating function which has no singularity at all, for example, the exponential function. What ...
1
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2answers
52 views

How many $3 \times 3$ matrices are singluar?

How many $3 \times 3$ matrices are singluar? Describe the methodology used to achieve the result.
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0answers
25 views

How to ensure a matrix of a special rank

As described in the subject, how can I ensure a matrix of a special rank. for example, given matrix A of m*n and m>n; Then, how can I mathematically constrain the matrix A to be rank n? As we all ...
0
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1answer
27 views

Question about non essential singulariy

When reading Ahlfor's Complex Analysis book, I came across the notion of non essential singularity. I know that for a function $f(z)$ an element $a\in\mathbb C$ is a non essential singularity iff ...
0
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1answer
48 views

Branch locus of a projection of algebraic set

Let $X$ a algebraic cone in $\mathbb{C}^n$ with $\dim_0 X=p$. def.: if $f:A\to B$ is smooth map between smooth manifolds, then $br(f)$ is the points $x$ that $df_x$ is not surjective. def.: if $\pi: ...
3
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0answers
56 views

Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
1
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1answer
50 views

Singularity Classification

Suppose that I have the following function: $$f(z)=\sin\left(\frac{1}{\sin(\frac{1}{z})}\right)$$ If I'm trying to characterize singularities, I know that singularities will be found whenever ...
2
votes
1answer
113 views

Removable singularities of a holomorphic function

So, I'm a little confused about removable singularities. Consider the function below: $$f(z)=\frac{1}{(1+z^2)^{2/3}}$$ Obviously, we have isolated singularities at the points $z = \pm i$. ...
3
votes
1answer
91 views

Classify Singularities

So, I'm trying to classify the singular points of the following function: $$ f(z)=e^{\cot(\frac {1}{z})} $$ Obviously, when z is zero, the function tends to approach infinity, so that must be a ...
2
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0answers
61 views

necessary and sufficient conditions for having an irreducible hyperplane section

Let $S \subset \mathbb{P}^3$ be a surface different to a cone. If $S$ is a smooth variety, then its generic hyperplane section is irreducible. I can imagine a similar result holds for a surface with ...
2
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0answers
55 views

Relation of two definitions of 'Singular Point'

In my lecture we defined: For $(a,b)=P\in V(f)$, $P$ is a singular point if $\operatorname{mult}(f,P)>1$, where $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, ...
3
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1answer
50 views

Why is $P$ singular

This is from Shafarevich's 'Basic Algebraic Geometry 1': Let $P=(\alpha,\beta)\in X$, and suppose that the equation of $X$ is written in the form $f(x,y)=a(x-\alpha)+b(x-\beta)+g$, where $g$ is a ...
2
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1answer
71 views

Removable singularities for continuous functions

Let $f: D - K \rightarrow \mathbb{C}$ be holomorphic, where $D$ is a planar domain and $K$ is a compact subset of $D$. Suppose that $f$ extends continuously to all of $D$. On which conditions on $K$ ...
0
votes
2answers
392 views

Determining whether the system will have a nontrivial solution?

Say I have a 3x3 matrix (a1 = 3a2 - 2a3), Will they system Ax=b have a nontrivial solution? Is it non-singular? I realize nontrivial means an answer that is not a zero vector. It must be the ...
1
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2answers
59 views

Classifying singularity

Having trouble classifying a singularity... $f(z)=$$z^2-1\over z^6+2z^5+z^4$ with $z_0=0$ and $z_0=-1$ The $z_0=0$ is pretty simple, just need to put $z^4$ in evidence. But $z_0=-1$ I can't seem to ...
1
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1answer
42 views

Finding the types of singularities of $f(z)=\frac{1}{z\cdot (e^z -1 )}$

I am getting trouble to find the types of singularities of $$f(z)=\frac{1}{z\cdot (e^z -1 )}$$ What I tried to do is: $z=0$ $z=2\pi k i$ for $z=2\pi k i$ its in order 1, but for the first one I ...
2
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0answers
55 views

The multiplicity of $X$ at $x$ does not change when $X$ is cut by a generic hypersurface: what are those generic conditions?

Given an algebraic variety $X$ with a point $x \in X$, the multiplicity of $X$ at $x$ is defined as the multiplicity of the maximal ideal of $x$ in the local ring $\mathcal{O}_{X,x}$. In ...
2
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1answer
40 views

Conditions for the number of degenerated fibers of a morphism to be finite.

I'm having trouble to find a theorem about this: Let's $F_z$ be a family of curves defined by $$F_z : F(x,y) + z = 0$$ where $F$ is an irreducible polynomial and $z \in \mathbb{C}$. My question is: ...
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0answers
55 views

Do there exist double points on a surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all. I'm familiar with the du val singularities on surfaces, also apparently known as rational double points (http://en.wikipedia.org/wiki/Du_Val_singularity). In ...
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vote
0answers
61 views

A singular boundary value problem

Is there any numerical approach to solve a BVPs for ODEs of the form: $y'=\frac{ky^2-y^{3/2}-y}{\beta t}$ with initial point $(0,y0)$? I know a problem of the form $y' = \frac{S}{t}y+f(t,y)$ with ...
2
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1answer
93 views

Singularities in (Elementary) Real Algebraic Geometry

I've taken an introductory course in algebraic geometry, and am currently studying Sumio Watanabe's book, "Algebraic Geometry and Statistical Learning Theory". In this book Watanabe gives a, possibly ...
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0answers
105 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
1
vote
1answer
53 views

coefficients for getting a smooth equation

I'm studying the equation $$ x_0q_0 + x_1q_1 + x_2q_2 = 0$$ where $q_i$ is a homogeneous polynomial of degree two in the variables $x_0,\ldots,x_5$. I would like to have some simple choices for the ...