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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136 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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65 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 ...
4
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128 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 ...
3
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55 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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91 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 ...
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130 views

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 ...
2
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13 views

Integers characterizing singularities of algebraic curves

The question in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...
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35 views

Understanding singularities through Jacobian

In kinematics of mechanisms we derive the constraint equations depending on the architecture and then analyse the singularities of the mechanisms by deriving their Jacobian matrices. For example: ...
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39 views

Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude ...
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44 views

How to determine the type of singularities in affine varieties

Let $X$ be the affine variety in $\mathbb C^3$ (coordinates $a, b, c$) defined as the zero set of $ac - b^2$. The variety has a "double point singularity" at the origin. This is somewhat intuitive, ...
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27 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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95 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 ...
2
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51 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 ...
2
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102 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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60 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 $\operatorname{mult}(f,P) = \max\{n\in\mathbb{N}\mid f\in\mathfrak{m}_P^{n}\}$, ...
2
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58 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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52 views

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 ...
2
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72 views

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 ...
2
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0answers
61 views

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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31 views

Gimbal lock easier to control with quaternions?

Using quaternions doesn't resolve the issue of gimbal lock, but make is more controllable... how come? They use less memory, and are commutable, and provide an smooth rotation along nonlinear ...
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35 views

On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
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42 views

How to solve a singularity problem for diferential equation?

Good day, Here is my equation $$ (x-1)^2 (x-h)^2q_1(x,w)y''(x)+(x-1)(x-h)q_2(x,w)y'(x)+q_3(x,w)y(x)=0 $$ (with $q_1(x,w), q_2(x,w), q_3(x,w)$ regular functions). The boundary conditions are $$ ...
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19 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 ...
1
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0answers
45 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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41 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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69 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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0answers
29 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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120 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 ...
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66 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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71 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 ...
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0answers
82 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
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0answers
43 views

What's a algebraically isolated singularirty?

What's its formal definition and how does it differ from a "simple" isolated singularity? I'm thinking about integrable $1$-forms here, though the definition may be more general.
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84 views

Topology of singularities

In the theory of surface singularities, it is well known that the topology of singularities is determined by its resolution dual graph. What`s the meaning of the topology of singularities? Here the ...
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115 views

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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32 views

Topology of Isolated Singularities

In Singular Points of Complex Hypersurfaces, Milnor explains that if $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated singularity at the origin, then $f^{-1}(\epsilon) \cap ...
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25 views

$A$ is nonsingular. eigenvalues of $A$ are unequal to zero

Assume that $A$ is nonsingular. Show that all eigenvalues of $A$ are unequal to zero. Express the eigenvalues and eigenvectors of $A􀀀^{-1}$ in terms of those of $A$.
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26 views

Difficulty with Milnor number

I am reading through the wikipedia page of Milnor number : https://en.wikipedia.org/wiki/Milnor_number#Examples I am reading example 2 where they calculate the Milnor number of $f(x,y)=x^3+xy^2$. So ...
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16 views

$p$ a nonsingular point in $V$ and $V(f)\cap T_p(V)$ then $f|_{T_p(V)}$ has a factor of multiplicity $\geq 2$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. at. Let $V \subset k^n$ be a hypersurface with $I(V ) = \langle f \rangle$. Show that if $V$ is not a hyperplane and $p \in V$ ...
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0answers
29 views

Local ring in singularity theory and algebraic geometry

In the book Singularity theory 1 by Arnold et al., page 13, there is a definition of a local algebra of singularity, which I tried and failed to rephrase in the language of algebraic geometry. It is ...
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13 views

How to numerically solve equation with regular singularity problem?

Good day, Here is my equation $$ (x-1)^2 (x-h)^2q_1(x,\Omega)y''(x)+(x-1)(x-h)q_2(x,\Omega)y'(x)+q_3(x,\Omega)y(x)=0 $$ ( with $q_1(x,\Omega)$[polynomial function 38 degree], $q_2(x,\Omega)$ ...
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0answers
58 views

How to get asymptotic form of the integrals with special functions?

I got difficulty when I try to plot I(x) for $m=1$ and $t=0.2$. The questions is how to get the asymptotic form of the following integral? $I(x,t)=\int_{0}^{\infty} \frac{f(y)}{2 \sqrt{\pi t}} ...
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15 views

Identification of pole

I would like to know whether the following statement is true or not If $f$ is an analytic function satisfied $|f(z)|\to\infty$ when $z\to z_0$, then $f$ is a pole at $z=z_0$ I would say the ...
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0answers
15 views

complexe analysis. Singularity points

What kind of singularity exists for sin[1/sin(1/x)]? I feel it is an essential singularity but I failed in the demontration. To develop in Laurent serie would be the solution but how??
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13 views

How to understand multifractal and multifractal spectrum?

A case I encountered with to illustrate multifractal is to assign different subintervals of Cantor set with different mass. In detail for each time, the interval is divided into 3 equal-length parts ...
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0answers
41 views

Is the complement of the singular locus of an algebraic variety a topological manifold?

Could someone explain to me clearly, please, why is the complement of the singular locus of an algebraic variety a topological manifold ? A lot of thanks for your help.
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39 views

About singularity

Let $X$ be a normal affine variety over $\mathbb{C}$. Q1. Let $x\in X$ be a singularity with a Cartier divisor $x\in D$. Then $\mathcal{O}_{X,x}$ is Cohen-Macaulay if and only if $\mathcal{O}_{D,x}$ ...
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39 views

Does a function with an exponential growth condition necessarily have infinitely many zeros?

This is part (2) of a question that I am working on. In part(1), I have constructed an entire function $f:=\cosh(\sqrt{z})$ that grows like $$\lim_{r \to \infty} \frac {\log M(r)}{\sqrt{r}}=1$$ ...
0
votes
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20 views

Derivative of a definite improper integral

-The derivative with respect to beta, for the following definite integral is required. g = $\int_\beta^{\sqrt(\beta^2 +1}$ $erfc(\gamma z)/\sqrt(z^2 - \beta^2)$dz -I am using the leibniz formula ...
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votes
0answers
30 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 ...
0
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0answers
30 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 ...