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confusing limits

calculate this limit when x tend to infinity : f(x) = (x+5)^(1+1/x) - x^(1+1/x+5) I used Laurent series at infinity but in the middle I confuesd and can not solve it but I am sure the correct answer ...
  • 1
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0 answers
12 views

What is the regularized value of the numerocity of primes?

Let numerocity of a subset $S$ of integers to be a sum $\sum_{k=-\infty}^\infty p(k)$ where $p(k)$ is the indicator function of the set $S$. We also can write it as $p(0)+\sum_{k=1}^\infty (p(k)+p(-k))...
  • 8,620
1 vote
0 answers
13 views

A Lipschitzian's condition for the measure of nonconvexity

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by: $$\alpha(A)=\sup_{b\in\overline{\operatorname{...
  • 311
0 votes
0 answers
67 views

Why is $\mathrm{SO}(4)$ not a simple Lie group?

$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ ...
  • 61
0 votes
0 answers
44 views

Elements of $ K(x,y) $ in $ K[[x^{\pm {1}},y^{\pm {1}}]] $

Let $ K(x,y) $ be fraction field of $ K[x,y] $. Now take an arbitrary element $ f/g \in K(x,y) $ where $ f, g \in K[x,y] $. Suppose $ f = \sum_{m \in {\mathbb{Z}}^{2}} y^{m_{1}} z^{m_{2}} $ and $ g = ...
  • 689
1 vote
1 answer
25 views

About Palm distribution

Can someone explain the Palm distribution? Or provide some information about Palm distribution. The article called 《A tutorial on Palm distributions for spatial point processes》 is hard to understand.
  • 11
0 votes
0 answers
9 views

First order condition HJB only one solution in R for my domain

I am working on a macroeconomic model and I need to derive a first order condition from a HJB equation. Simplified, I can write the FOC as $ac+bc^{1/2}-1 = 0$ and want to solve for $c$. I have $a>0$...
  • 1
-1 votes
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26 views

On size of product of integer and inverse in a residue class?

$p$ is a large prime. Let $|a|=\min(a,p-a)$. Given a small $\epsilon>0$ and a real $t>1$ how many integers $a$ are there in $\{1,2,\dots,p-1\}$ such that $|a|>p^{1/t}$ and $|a||b|<p^{1+\...
  • 13.1k
4 votes
0 answers
40 views

Is there a connected Hausdorff anticompact space that is countably infinite?

Cross-posted from MSE. Following Bankston - The total negation of a topological property, a topological space is called anticompact if all its compact subsets are finite. The linked MSE post above ...
  • 141
4 votes
0 answers
79 views

Intuition for Luna's Étale Slice Theorem

I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $\mathbb{C}$. Here is the setup. Let $X$ be an affine algebraic variety and $G$ a reductive group ...
1 vote
0 answers
41 views

Resolving complexes of coherent analytic sheaves

Background Throughout, let $X$ be a smooth complex manifold. It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). ...
  • 821
2 votes
0 answers
34 views

Approximation by gaussian mollification in Sobolev spaces

I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify) $$\label{0}\tag{0} \|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
  • 202
-1 votes
0 answers
20 views

Is there an equation for the space bounded by multiple curves made of lines

https://www.desmos.com/calculator/jax0qssltb Basically, you can make an approximation for a curve by drawing many lines segments, in my problem, my curve approximation is given by the equation $y = (-\...
2 votes
0 answers
50 views

Manifestation of Hecke operator on the category of abelian varieties (or motives)

If we are given some postivie integer $N$ and a prime $p$, then we have the Hecke operator $T_p$ on modular forms, which is a cohomological manifestation of the Hecke correspondence $$X_0(N)\leftarrow ...
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0 answers
49 views

Approximate range of Radon-Nikodym derivative in a dynamical system

Suppose $(X, G, \Omega, \mu)$ is a dynamical system where $(X, \Omega, \mu)$ is a Borel measure space and $G$ is acting on $X$ such that each group action $x\mapsto g\cdot x$ defines a measurable ...

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