# Tagged Questions

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: http://en.m....

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### Prove that $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ is an integral domain [closed]

Let $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ be an analytic algebra. I am trying to prove that $R$ is an integral domain. Basically I know that if $\langle x^2 - yz\rangle$ is a prime ...
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### A complement of Milnor's book on singularities with exercises/examples

I'm currently reading "Singular Point of Complex Hypersurfaces" by Milnor. This is really a great book, but I did realize I didn't saw any concrete examples, computations etc ... I was wondering if ...
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### 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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### Classify singularities - Hint [duplicate]

I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point?
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### How to determine the residues of $\frac{z}{\sinh(kz)}$?

What methods can I use to determine the residues of $\frac{z}{\sinh(kz)}$? Singularities occur at $z=\frac{i n \pi}{k}$ for $k \neq 0$ and $n$ an element of the integers. I've attempted a series ...
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### Canonical scheme structure on the singular locus of a variety

Let $X$ be subvariety of affine space $\mathbb{A}_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations $$X:(F_1=\cdots = F_m=0)\subset \mathbb{A}^n.$$ Then $X$ is singular at $p\in X$...
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### Simple singularities of a vector field

Let $M^n \subset \mathbb{R}^{n+1}$ a compact and orientable hypersurface of even dimension, with Gauss map $\gamma : M \to S^n$, that is, $\gamma(p)$ is normal to $M$ at $p \in M$. Let $a, b \in S^n$ ...
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### 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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### Quick question on the roots and poles of a meromorphic function,

Does the degree of the polynomial in the numerator always equal the degree of the polynomial in the denominator? In other words, the number of zeros, counting multiplicity, equals the number of poles,...
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### 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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### How can a positive integrand integrate to 0? [duplicate]

I integrated $\dfrac{\log x}{1+x^2}$ from $0$ to infinity with residue calculus and got... $0$. This also agrees with Wolfram Alpha. How can this be? Is it due to the behavior of $\log(x)$ near ...
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### Why must a meromorphic function, bounded near infinity, have the same number of poles and zeros?

Writing down some easy rational functions to check this, I don't see why this must be the case. Although if the function had 3 simple zeros and 2 simple poles its rational form would be in the form ...
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### Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
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### Do the singular matrices form a topological manifold

So the definition of manifold I'm using is that of a topological manifold (a topological space with an atlas of homeomorphisms to $\mathbb{R}^n$). I have two related questions: Is the set of ...
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### 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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### Finding the order of pole of $f(z)=\frac{\sin z}{z-\pi}$

The problem is Kreyszig 10ed international edition : 16.2 #9. What is the order of the pole at $z=\pi$ of the function $f(z)$ below? $$f(z)=\frac{\sin z}{z-\pi}$$ I thought that it will be a simple ...
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? $I(x,t)=\int_{0}^{\infty} \frac{f(y)}{2 \sqrt{\pi t}} \... 1answer 70 views ### Singularities of$\sin(z)/(1-\cos(\sqrt{z}\,))\displaystyle f(z) = \frac{\sin(z)}{1-\cos(\sqrt{z}\,)}$. The assignment is to find all the singularities of$f$, determine the type of them and the residue. It is clear that the singularities are$...
I am trying to evaluate$$\int_{-\infty}^{\infty} \frac {x^2 -x^4}{1-x^6}\,dx,$$ which is an old exam problem. There is a special note on this problem that reads: Note: Your answer need not be a ...