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# MathOverflow : Recent Questions and Answers (Page 5)

## Does the following sum converge?

Does the sum$$lim_{ntoinfty}sum_{k=0}^{lflooralpha n rfloor}C_n^k(-1)^kleft(1-frac{k}{alpha n}right)$$converge, where $C_n^k$ is the binomial coefficient and $0 <alpha <1$?The above question has been...

Asked on 12/21/2021 by Ryan Chen

## Is a homotopy sphere with maximum Morse perfection actually diffeomorphic to a standard sphere?

The Morse perfection of a closed differentiable manifold $Sigma^n$ is defined to be the largest integer $k$, such that there exists a smooth mapping$$p:S^ktimesSigma^nrightarrowmathbb{R}$$where ...

## Differential birational equivalence

Suppose the base field algebraically closed and of zero characteristic. There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion is...

## Calculating $n$-dimensional hypervolumes ($n sim 50$), for example

I have a question regarding efficient and possibly simple algorithms for computing volumes of $n$-dimensional polytopes. The polytope of concern isn't arbitrary: it is obtained by applying a linear...

Asked on 12/20/2021 by Luka Klinčić

## Definition of a system of recurrent events

[I asked a version of this question on MSE a few weeks ago and didn't get any useful feedback. Apologies if I am just being stupid.] I am reading...

Asked on 12/20/2021 by Rob Arthan

## Regularity of a conformal map

Let $D$ be a domain in $mathbb{C}$ with $n$ boundary components. From the work of Koebe, we know that $D$ can be conformally mapped to a...

## Is there a standard definition of weak form of a nonlinear PDE?

Comments on the question Are those distributional solutions that are functions, the same as weak solutions? suggest there might not be a standard definition of the weak form of...

## Is there an algebraic version of Darboux's theorem?

Let $M$ denote a smooth manifold, and $omega in Omega^2(M, mathbb{R})$ a symplectic form. The classical version of Darboux's theorem states that for any $x in M$,...

## The strength of "There are no $Pi^1_1$-pseudofinite sets"

For $Gamma$ a set of second-order sentences in the empty language, say that a set $X$ is $Gamma$-pseudofinite if $X$ is infinite but for every sentence...

## Are groups with the Haagerup property hyperlinear?

In his 2008 paper Hyperlinear and Sofic Groups: A Brief Guide, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question...