# Tag Info

### What is the origin/history of the following very short definition of the Lebesgue integral?

This approach was used in the German Analysis (Calculus) textbook MR0222221 Hans Grauert and Ingo Lieb, Differential- und Integralrechnung. Band I: Funktionen einer reellen Veränderlichen, ...
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### What is the origin/history of the following very short definition of the Lebesgue integral?

This definition is due to Jan Mikusiński, see Mikusiński, Jan, The Bochner integral. Basel, Stuttgart: Birkhauser, 1978. Mikusiński has co-authored another book on integration with Hartman in 1961, ...
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### Uniqueness of solutions of Young differential equations

For such small $\beta$, we need to use Rough path theory to make sense of the integral and so below I go over that. (Indeed for $\beta\in (\frac{1}{2},1]$, there is an ODE theory for Young integrals ...
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### Elementary convexity example

Here is a much simpler proof, actually of the more general fact that $$f(x):=x^p(1+\ln^+ x)^s$$ is convex in $x\ge0$ for any real $p\ge1$ and $s\ge0$, where $\ln^+ x:=\ln\max(1,x)$. For the left and ...
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### Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$?

Comment Let $S_n = \prod_{k=1}^n f(k/n)$ and $g(x) = \log f(x)$. Then $$\frac{1}{n} \log S_n = \frac{1}{n}\sum_{k=1}^n g\left(\frac{k}{n}\right)$$ which should [improper integral, so not certain] ...
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### Quantitative analytic continuation estimate for functions small except on a small set

This conjecture is correct. Take $K=e$, and let $\gamma\leq 1/4$; we will fix $\gamma$ later. First we give a crude estimate of $c_0$. Let $g(z)=\sum_{1}^\infty c_nz^n.$ Since $|c_n|\leq e^n$, we ...
• 81.9k
1 vote
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### Continuity of a reaching time of a submanifold

If $V$ is closed in $\mathcal{O}$, then $\tau^V$ is continuous at $x_0$. If $V$ is not closed, then it is not hard to find counterexamples (e.g. imagine $\mathcal{O}=\mathbb{R}^3$, $V$ is an open disk ...
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### Second order differentiability of convex functions

The following argument is used in my recent paper with D. Azagra and A. Cappello (work in progress). The answer is yes. I asked about whether the proof of Theorem 1 can be modified so to cover Theorem ...
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### Entire function with almost periodic boundary condition?

The answer seems to be negative. Suppose that an entire function $f$ satisfies $f(z+v_i)=e^{A_iz}f(z)$, where $v_1$ and $v_2$ generate a lattice. Let $\Pi$ be the fundamental parallelogram of this ...
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### Does gravity constant affect boundedness of solution?

$\newcommand\la\lambda$No. E.g., if $n=1$, $x_0\ne0$, and $f(u)=-u^2/2$ for all real $u$, then the solution x(t)=\frac{x_0}{2} \, \Big(\frac{e^{\la_+ t}-e^{\la_- t}}{\sqrt{4 g+1}}+e^{\la_- t}+e^{\...
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### Quantitative analytic continuation estimate for a function small on a set of positive measure

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### Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

I can achieve $L(f - g) \leq (\frac{1}{2} + \frac{\pi}{4})\epsilon = (1.285\ldots)\epsilon$. Two reductions: (1) we can assume $|f(t)| < c$ for all $t \in (a,b)$ and (2) we can take $\epsilon = 1$. ...
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### General version of Weyl's lemma

unfortunately Weyl lemma cannot be generalized to any divergence form elliptic operator. The problem is given by the smoothness of the coefficients $a_{ij}$. There is a huge literature on the subject. ...

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