# All Questions

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### Numerical Laplace Transform

I want to compute the Laplace transform of data vectors. I have tried the usual numerical software and I'm surprised to see that does not have this operation available. I wonder if there is a straight ...
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### Does this density limit exist?

Suppose that $T:\mathbb{R}^k\rightarrow\mathbb{R}^k$ is a transformation which is differentiable at a point $x\in\mathbb{R}^k$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^k$. Denote by ...
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### Regularization of a (divergent) cosine series

What would be a suitable regularized value for the following divergent series: $$S(y) = \sum_{k=1}^{\infty} \cos(k y) \quad y \in R\\$$ By way of added context, this series arises from a formal ...
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### Spherically wrapped gaussian distribution

I want to rotate a 3D vector by a random angle from a spherically wrapped gaussian distribution, but i dont know what exactly this distribution is?
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### How to derive Int f'(x)[f(x)]dx

Using clear explanations in the form of words and diagrams, explain how the formula for Int f'(x)[f(x)]dx is derived.
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### Intersection product of line bundles with $\mathcal{F}$.

I am having trouble with computing the intersection number using the following definition [Ref Vakil Chapter 20]: Suppose $\mathcal{F}$ is a coherent sheaf on $X$ with proper support of ...
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### I'm searching cutting points of a central angle of a circumference.

I'm working in a 2D drawing software and I need draw points of the cutting central angle to the circumference. I have a circumference, I know how its center point (2D coordinates), I know the radius, ...
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### If $G$ is abelian, it has a subgroup in every order of $|G|'s$ divisors?

Assume that $G$ is an abelian group, I read somewhere that it can be derived from Lagrange's theorem that it has a number of subgroups that is equal to the number of $G$'s divisors. Why does it hold? ...
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### Basis of (first de Rham) cohomology: $y^n=f(x)$

Let $K$ be a field, $f(x) \in K[x]$ be a monic polynomial with distinct roots, $\deg(f)=d$. Let $R=K[x,y]/(y^n-f(x))$ and $C=Spec(R)$. $\:\:\;\:\:\:\:\quad$ ($n>1$ integer) What is the basis ...
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### Finding subgroups of $\mathbb{Z}_{13}^*$

I need to find all nontrivial subgroups of $G:=\mathbb{Z}_{13}^*$ (with multiplication without zero) My attempt: $G$ is cyclic so the order of subgroup of $G$ must be $2,3,4,6$ Now to look for ...
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### Doubt about about claim about complexity of edge coloring powers of the line graph

Likely I am misunderstanding/missing something, but a claim in a paper appears wrong to me. According to Coloring Graph Powers: Graph Product Bounds and Hardness of Approximation p. 2 Unless ...
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### Nodes of the Dynkin diagram for even-dimensional orthogonal groups

I'm reading Wilson's book "The Finite Simple Groups", specifically the sections on the orthogonal groups. In section 3.7.4, he discusses subgroups of the orthogonal groups. The Dynkin diagram for the ...
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### Quasi-Banach algebras

We know that the space $Lp(0, 1)$, when $0<p<1$ is quasi-Banach spaces and has a trivial dual; $L_{p}(0,1)^{*}=\{0\}.$But its not algebra. Is there any quasi-Banach algebra with trivial dual? ...
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### What is the difference between the slope and the angular coefficient?

What is the difference between the slope and the angular coefficient?
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### Proof for the upper bound and lower bound for binomial coefficients.

I have seen the bounds $\left(\frac{n}{k}\right)^k \leq {n \choose k} \leq \left( \frac{en}{k}\right)^k$ for integers $n \geq k >0$ for the binomial coefficient. I can prove the upper bound in this ...
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### Inverse function simple question

Please explain question no. c in detail
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### Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
Consider a random $n$ by $n$ matrix $M$ chosen uniformly over all $n$ by $n$ $(0,1)$ matrices and a random vector $v \in \{-1,0,1\}^n$ chosen uniformly as well. Let $X = Mv$. What is $$P(X_i = 0 ... 3answers 49 views ### 0,1,0,1,0,1… has only 2 limit points Prove that the sequence 0,1,0,1,0,1... has only 2 limit points : 0 and 1. To be frank, I know the solution to the above particular problem. What I am interested is in knowing a general ... 1answer 25 views ### Power series representation of gamma function? I am looking for a power-series expression of the form \Gamma(z)=b+\sum_{k=0}^\infty a_kz^k where the a_k can be calculated as some function of k. 1answer 37 views ### Solving an exponential function I have the below exponential function which I wish to solve it for b. Other than resorting to the Lambert W function, is there alternative way of representing the solution?$$ \frac{(1+a)(1-b)}{ab ...
What is a valid parameterization for a general, real intersection of two surfaces: $$f(x,y,z) = 0, \, g(x,y,z) =0 ?$$ For particular cases we eliminate a coordinate if possible and use the form ...