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https://mathoverflow.net/questions/380402 | 1 | The question is quite "simple". Let $\lambda^\*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we **always** have
$$
\lambda^\*(E) +\lambda^\* ([0,1]\backslash E) >1?
$$
Are there examples of non-measurable sets such that equality $\lambda^\*(E) +\lambda^\... | https://mathoverflow.net/users/121404 | Summability issues of measure when we decompose a measurable set into two non-measurable parts | Suppose $E\subseteq[0,1]$, $\ F=[0,1]\setminus E$, $\ \lambda^\*(E)=a$, $\ \lambda^\*(F)=b$, $\ a+b=1$.
There are Lebesgue measurable sets ($G\_\delta$ sets) $A,B\subseteq[0,1]$ such that $E\subseteq A$, $\ \lambda(A)=\lambda^\*(E)=a$, $\ F\subseteq B$, $\ \lambda(B)=\lambda^\*(F)=b$.
Now $\lambda(A\cap B)=\lambda(... | 5 | https://mathoverflow.net/users/43266 | 380411 | 158,385 |
https://mathoverflow.net/questions/380236 | 3 | I am trying to understand an article by Gibbons, Rychenkova and Goto, called ["Hyperkähler quotient construction of BPS Monopole Moduli Spaces"](https://link.springer.com/article/10.1007/s002200050121). I will paraphrase the relevant notions and formulas in order to get to my question.
Let $M = \mathbb{H}$ and $q$ be... | https://mathoverflow.net/users/81645 | Stuck on a computation with quaternions and moment maps | I was able to finally prove that
$$d(\omega.d\mathbf{r}) = - \frac{1}{2r^3}(d\mathbf{r}\,\mathbf{r} \wedge d\mathbf{r}).$$
In the process, I have learned a lot. The main issue for me was that I was dealing with differential forms with values in $\mathbb{H}$, the latter being of course non-commutative. Now I am much m... | 1 | https://mathoverflow.net/users/81645 | 380422 | 158,388 |
https://mathoverflow.net/questions/380442 | 1 | Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma\_1,\dots,\sigma\_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq s<T$, the follwoing SDE with data has a strong solution $X\_t$:
$$
X\_t^{x,s} = x + \int\_0^t \mu(s,X\_s)ds + \sum\_{k=1}^... | https://mathoverflow.net/users/36886 | Conditions for Gaussianity of SDE | From the definition of the [Itô stochastic integral](https://en.wikipedia.org/wiki/It%C3%B4_calculus#Integration_with_respect_to_Brownian_motion), it is clear that the process $(X\_t)$ will be Gaussian if (i) $\mu(s,\cdot)$ is affine -- that is, $\mu(s,x)=a(s)+b(s)x$ for some regular enough functions $a$ and $b$ and al... | 2 | https://mathoverflow.net/users/36721 | 380446 | 158,393 |
https://mathoverflow.net/questions/380381 | 6 | I have the following PDE in two dimensions
$$
2\partial\_x\partial\_y\sqrt{1-u^2}+\left(\partial^2\_x-\partial^2\_y \right)u=0,
$$
with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively
$$
2\partial\_x\partial\_y\sin2\theta(x,y)+\left(\partial^2\_x-\partial^2\_y \right)\cos2\theta(x,y)=0,
$$
with re... | https://mathoverflow.net/users/171439 | Non-linear hyperbolic PDE | As I understand it, the equation you are imposing on the function $\theta(x,y)$, defined on a domain $D\subset\mathbb{R}^2$ in the $xy$-plane is that, for some positive constants $\lambda\_1\not=\lambda\_2$, the metric
$$
g = \lambda\_1\,(\cos\theta(x,y)\,\mathrm{d}x+\sin\theta(x,y)\,\mathrm{d}y)^2
+ \lambda\_2\,(\sin... | 7 | https://mathoverflow.net/users/13972 | 380448 | 158,395 |
https://mathoverflow.net/questions/380436 | 4 | In [this talk](https://www.youtube.com/watch?v=k68Mf8VAQz8&ab_channel=ENLASeminar) delivered by professor N. Trefethen it is stated that the condition number of a Vandermonde matrix of degree n verifies:
$$\kappa\sim(1+\sqrt{2})^{n}$$
It is based on [this paper](https://link.springer.com/article/10.1007/BF01437212) by ... | https://mathoverflow.net/users/170879 | monomial basis conditioning | Gautschi's paper can be downloaded from [here.](https://www.cs.purdue.edu/homes/wxg/selected_works/section_01/052.pdf) You may find [An elementary proof of the exponential conditioning of real Vandermonde matrices](https://www.researchgate.net/publication/237836551_An_elementary_proof_of_the_exponential_conditioning_of... | 4 | https://mathoverflow.net/users/11260 | 380452 | 158,396 |
https://mathoverflow.net/questions/380245 | 5 | I read the following claim in Z.Frolik's article "A generalization of realcompact spaces" on page 135.
Two subset $M$ and $N$ of a space $X$ are called completely seperated if there exists a real valued continuous function $f$ on $X$ with $f(M)\subset \{0\}$ and $f(N)\subset\{1\}$.
**Claim:** Let $X$ be a normal sp... | https://mathoverflow.net/users/86099 | Countable open covering of normal space | The claim is false it would imply that normal spaces are countably paracompact and hence that normality of $X$ would imply normality of $X\times[0,1]$. The latter is not the case, see [Mary Ellen Rudin, *A normal space $X$ for which $X\times I$ is not normal*, Fundamenta Mathematicae, **73** (1971/72), 179-186](https:/... | 3 | https://mathoverflow.net/users/5903 | 380456 | 158,397 |
https://mathoverflow.net/questions/380383 | 8 | Recently, I asked a somewhat related question [here](https://mathoverflow.net/questions/380342/estimating-the-size-of-omega-r-x-in-omega-textdistx-partial-omegar). In the comment section, I found the formula
$$
\lim\_{r\to 0}\frac{1}{r}\int\_{\Omega\_r} f(x)\,dx = \int\_{\partial \Omega}f(\sigma)\,d\mathcal{H}^{n-1}(\s... | https://mathoverflow.net/users/80191 | Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$ | In Evans & Gariepy's "Measure theory and fine properties of functions", Sec. 5.3., they construct the trace operator on a bounded Lipschitz domain $\Omega$ for BV-functions (and thus by inclusion for the subspaces $W^{1,p}$) in a similar fashion, using averages. In their case, they simply consider each part of the boun... | 4 | https://mathoverflow.net/users/51695 | 380462 | 158,399 |
https://mathoverflow.net/questions/380459 | 11 | Before the proliferation of computers in the 1950s, did human computers use floating-point formats for their computations?
Floating-point calculation was reportedly implemented already in the 1910s ([Wikipedia](https://en.wikipedia.org/wiki/Floating-point_arithmetic#History)), so one might assume the idea must have b... | https://mathoverflow.net/users/2082 | Did human computers use floating-point arithmetics? | In the field of hydrodynamics the first calculation by a human computer was carried out around 1920 for a [project](https://ilorentz.org/history/zuiderzee/zuiderzee.html) to transform an open sea into a closed lake, with the aim to protect Holland from flooding. The physicist Hendrik Lorentz headed a task force to calc... | 23 | https://mathoverflow.net/users/11260 | 380463 | 158,400 |
https://mathoverflow.net/questions/380460 | 4 | $\DeclareMathOperator\Sl{Sl}\DeclareMathOperator\PSl{PSl}\DeclareMathOperator\Isom{Isom}$Let $\widetilde{\Sl\_2}$ be the Thurson geometry that can either be described as the universal cover of $\PSl(2,\mathbb{R})$, or as the twisted line bundle over the hyperbolic plane $\mathbb{H}^2\mathbin{\tilde{\times}}\mathbb{E}$.... | https://mathoverflow.net/users/99898 | An explicit description of $\operatorname{Isom}(\widetilde{\operatorname{Sl}_2})$ | Surely the group $\tilde {SL}(2,\mathbb R)$ maps into this isometry group of the manifold $\tilde {SL}(2,\mathbb R)$, and in such a way that the composed map $\tilde {SL}(2,\mathbb R)\to {PSL}(2,\mathbb R)\cong Isom^+(\mathbb H^2)$ is the usual projection. So I imagine that your central extension by $\mathbb R$ comes f... | 5 | https://mathoverflow.net/users/6666 | 380469 | 158,401 |
https://mathoverflow.net/questions/380239 | 2 | $\DeclareMathOperator\End{End}$Following the deduction by John W. Morgan in his book [The Seiberg–Witten equations and applications to the topology of smooth four manifolds](https://press.princeton.edu/books/paperback/9780691025971/the-seiberg-witten-equations-and-applications-to-the-topology-of), an almost complex man... | https://mathoverflow.net/users/131004 | Clifford multiplication formula on an almost complex manifold | I think I figured this out and the calculations in the Kähler case in the book are misleading in a sense and the multiplication formula mentioned in page 109 is wrong.
As calculated in page 52, the action of a REAL one form $\alpha\in\Omega^1(X;\mathbb{R})$ on a spinor $\nu$ is indeed given by the formula:
\begin{equ... | 0 | https://mathoverflow.net/users/131004 | 380472 | 158,403 |
https://mathoverflow.net/questions/380473 | 3 | $\DeclareMathOperator\Var{Var}$Let $K\_{0}(\Var\_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of a variety, $X$, in $K\_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are varieties then we say that they are *piecewise isomorphic* if there are finite locally closed stratifications, $\{X\_... | https://mathoverflow.net/users/nan | Piecewise isomorphism versus equivalence in Grothendieck ring | There are no simple examples as yet; it's been an open question going back to at least [Larsen and Lunts - Motivic measures and stable birational geometry](https://arxiv.org/abs/math/0110255), which has been open for about 15 years, and some of us believed that it should be true.
The first counterexample for smooth n... | 4 | https://mathoverflow.net/users/111491 | 380493 | 158,410 |
https://mathoverflow.net/questions/380489 | 0 | I have a question like this:
Consider $N$ samples $X\_1, X\_2, ..., X\_N$ that uniformly random generated from standard basis $\{e\_i, i=1,2,...,d\}$, i.e. $(1,0,0,\cdots,0),(0,1,0,\cdots,0),(0,0,1,0,\cdots,0),\cdots,(0,0,0,\cdots,1)$. Dimension $d$. My question is how can I get an upper bound on $\|\sum\_{i=1}^N X\_i\... | https://mathoverflow.net/users/171775 | Bound the norm of sum of random vector that generated from standard basis | Let $n:=N$ and $V:=\|\sum\_{i=1}^n X\_i\|$.
Using [Talagrand's concentration inequality](https://en.wikipedia.org/wiki/Talagrand%27s_concentration_inequality) as applied to linear functions in [Section 2.2](https://ocw.mit.edu/courses/sloan-school-of-management/15-070j-advanced-stochastic-processes-fall-2013/lecture-... | 1 | https://mathoverflow.net/users/36721 | 380500 | 158,414 |
https://mathoverflow.net/questions/380488 | 3 | I have run into a real problem that is actually a sort of *assignment problem*. I am describing it here because I am interested in knowing whether this problem already has a name (and whether there is an algorithm to efficiently solve it). It can be stated as follows:
We have $n$ candidates $C\_1, \dots, C\_n$ (worke... | https://mathoverflow.net/users/30494 | Assignment problem with priorities and scores | This seems to be the assingment/matching problem for the [National Resident Matching Program](https://en.wikipedia.org/wiki/National_Resident_Matching_Program#Matching_algorithm) which is closely related to the [stable marriage problem](https://en.wikipedia.org/wiki/Stable_marriage_problem).
In the resident matching ... | 4 | https://mathoverflow.net/users/51668 | 380501 | 158,415 |
https://mathoverflow.net/questions/380385 | 1 | I'm trying to compute special values of Hecke L-function for the field $K=\mathbb{Q}(\sqrt[5]{1})$ using Magma (more exactly, I need $L(k, \chi^k)$, $k$ - integer, $\chi$ - Hecke character for the field $K$). However, I'm very confused, because the text [http://magma.maths.usyd.edu.au/~watkins/papers/hecke.pdf](http://... | https://mathoverflow.net/users/171707 | How to find an explicit value of a Hecke L-function using Magma? | The linked Magma documentation notes that HeckeCharacterGroup can have the real infinite places ("oo") omitted, corresponding to no real infinite places being ramified (as is trivially the case in your example).
```
HeckeCharacterGroup(I) : RngOrdIdl -> GrpHecke
HeckeCharacterGroup(I, oo) : RngOrdIdl, SeqEnum -> Grp... | 0 | https://mathoverflow.net/users/171793 | 380508 | 158,418 |
https://mathoverflow.net/questions/380506 | 5 | Suppose that a finite-dimesnional Hopf $C^\*$-algebra $H$ acts on a type $II\_1$ factor $N$ minimally (that is, $N^{\prime}\cap (N\rtimes H)=\mathbb{C}$). Is it true that there always exists a minimal action of the dual Hopf algebra $H^\*$ on $N$?
| https://mathoverflow.net/users/164194 | Action of a dual Hopf algebra on a factor | No, there might be no minimal action at all of $H^\*$ on $N$. By [Theorem A in this paper of Falguières and Raum](https://arxiv.org/abs/1112.4088) (see also [this paper](https://arxiv.org/abs/0811.1764) from which other examples may be deduced), for any rigid C$^\*$-tensor category $\mathcal{C}$ with finitely many irre... | 8 | https://mathoverflow.net/users/159170 | 380514 | 158,421 |
https://mathoverflow.net/questions/380511 | 15 | I'm now attending a reading seminar on the algebraic topology.
The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).
In those books, theorems on the Riemannian manifolds are frequently just mentioned and used.
To mention some examples
1. Riemann... | https://mathoverflow.net/users/123226 | Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology | The proof of the existence of good covers is contained in Bott & Tu on pages 42-43, though that proof does refer out to Spivak (see below).
In general, if your main goal is to study (algebraic) topology of manifolds, you probably don't need to know much about metrics and connections and that sort of thing that typica... | 15 | https://mathoverflow.net/users/6646 | 380531 | 158,427 |
https://mathoverflow.net/questions/380525 | -1 | My goal is to obtain the Big-Oh bound of the following recursive function with two variables:
$$T(n,m) = T(n, m-1) + T(n-1,m)+1$$
As initial conditions, $T(0,m)=1$ and $T(n, 0)=1$ for $m \geq 0$ and $n \geq 0$, respectively. Then, I think $T(n,m) \in O(2^{n+m})$ which can be proved as follows:
Proof:
* Let's pr... | https://mathoverflow.net/users/134666 | Big-Oh bound of a recursive function with two variables | A simple and exact value of $T(n,m)$ can be computed by the following simple substitution.
$$
C(n+m,m):=\frac{T(n,m)+1}2
$$
Then your recursive function becomes
$$
C(n+m,m)=C(n+m-1,m-1)+C(n+m-1,m)
$$
with the initial condition $C(m,m)=C(m,0)=1$. As you can see, this is the recursive formula for the [binomial coefficien... | 4 | https://mathoverflow.net/users/171820 | 380537 | 158,430 |
https://mathoverflow.net/questions/379585 | 9 | A *convex* polytope $P\subset\Bbb R^d$ is *centrally symmetric* if $-P=P$. It is *self-dual* (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\smash{\Bbb R^d})$ with $\smash{P^\circ}=XP$.
>
> **Question:** Are there centrally symmetric self-dual polyto... | https://mathoverflow.net/users/108884 | Are there centrally-symmetric self-dual polytopes in dimension $d> 4$? | There are centrally symmetric self-dual polytopes in every dimension. This follows from Proposition 3.9 in
*Reisner, S.*, [**Certain Banach spaces associated with graphs and CL-spaces with 1- unconditional bases**](http://dx.doi.org/10.1112/jlms/s2-43.1.137), J. Lond. Math. Soc., II. Ser. 43, No. 1, 137-148 (1991). [ZB... | 7 | https://mathoverflow.net/users/908 | 380543 | 158,433 |
https://mathoverflow.net/questions/380547 | 1 | Suppose that $f$ is the density of a high(-$d$)-dimensional Gaussian measure with mean $\mu$ and non-singular covariance matrix $\Sigma$. Let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous function for which the integral $\int f(x)g(x)dx<\infty$.
Are there known "efficient" quadrature rules specifically for c... | https://mathoverflow.net/users/36886 | Quadrature methods for high-dimensional Gaussian integration | You may want to use a stochastic algorithm. Entry points to the literature (which is large) could be
* [A stochastic
algorithm for high-dimensional integrals over unbounded regions with
Gaussian weight](https://core.ac.uk/download/pdf/81977086.pdf) (1999)
* [Higher-Dimensional
Integration with Gaussian Weight for App... | 2 | https://mathoverflow.net/users/11260 | 380551 | 158,435 |
https://mathoverflow.net/questions/380554 | 3 | Suppose that $G$ is a finite irreducible reflection group with irreducible orthogonal representation $\rho: G\rightarrow \mathrm{O}(d)$, and let $\rho^+: G^+\rightarrow \mathrm{SO}(d)$ be its restriction to the rotation subgroup $G^+$ of $G$.
**Question:** For what $G$ (respectively, $G^+$) is $\rho$ (respectively, $... | https://mathoverflow.net/users/53199 | Absolutely irreducible finite reflection/rotation groups | For $\rho$, take a non-trivial complex (absolutely) irreducible constituent $\chi$ of the character $\theta$ afforded by $\rho.$ Then for some reflection $t \in G$, we have $\chi(t) = \chi(1)-2,$ and ${\rm Res}^{G}\_{\langle t \rangle }(\chi)$ contains the non-trivial linear character $\lambda$ of $\langle t \rangle$ w... | 4 | https://mathoverflow.net/users/14450 | 380569 | 158,441 |
https://mathoverflow.net/questions/380589 | 6 | Von Neumann algebras have the following form of interpolation property: let $(x\_n)\_n$ and $(y\_n)$ be increasing and decreasing, respectively, sequences of self-adjoint elements in a von Neumann algebra $M$ such that $x\_n \leqslant y\_n$ for all $n$. Then there is $z$ such that $x\_n \leqslant z \leqslant y\_n$ for ... | https://mathoverflow.net/users/15129 | Certain interpolation property of von Neumann algebras | It does not hold for matrices. Let $P\_1 := \left[\begin{array}{cc} 1& 0 \\ 0& 0 \end{array}\right]$, $P\_2:= \left[\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right]$, and $X:= \left[\begin{array}{cc} 3 & 2,5 \\ 2,5 & 4\end{array}\right]$, and take the sets $A:=\{P\_1, P\_2\}$ and $B:=\{X, Id\}$. Both $X$ and $Id$ ar... | 6 | https://mathoverflow.net/users/24953 | 380595 | 158,450 |
https://mathoverflow.net/questions/380577 | 0 | This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well known result? I can provide a proof sketch if you are interested, and it has been checked numerically.
If $\zeta(s)=0$, w... | https://mathoverflow.net/users/140356 | On some property of the zeros of $\zeta(s)$ in the complex plane | That looks like just one of the two components of a "rotated" Dirichlet Eta function (sometimes called Alternate Zeta function):
$$e^{i\theta} \; \eta (s)= e^{i\theta} \; \sum \_{n=1}^{\infty}{\frac {(-1)^{n+1}}{n^{s}}}$$
It cannot hence help, as it is always possible to find a rotation angle $\theta$ that will bring t... | 3 | https://mathoverflow.net/users/15020 | 380596 | 158,451 |
https://mathoverflow.net/questions/380264 | 7 | What is the name of the function space formed by solutions to algebraic linear differential equations? Where can I find a discussion of its properties?
By an algebraic linear differential equation I mean a linear partial differential equation in $n$ variables whose coefficients are polynomials in those variables over... | https://mathoverflow.net/users/168668 | Spaces of solutions to algebraic linear differential equations | Further to Sam Gunningham's comment, Frédéric Chyzak's thesis, [Fonctions holonomes en calcul formel](https://tel.archives-ouvertes.fr/tel-00991717/document) appears to throw some light on this question.
| 3 | https://mathoverflow.net/users/106467 | 380600 | 158,452 |
https://mathoverflow.net/questions/380578 | 6 | Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $K:=\overline{\mathbb{F}\_p}$.
Let $b$ be a $p$-block of $G$ with abelian defect group $D$. Let $H:=N\_G(D)$. Let $c$ be the Brauer correspondent of $b$.
>
> M. Broué conjectured in the 90's that $b$ and $c$ are derived equivalent under these assum... | https://mathoverflow.net/users/12826 | Rickard's strengthening of Broué's abelian defect group conjecture and the lifting of some equivalences up to splendid derived equivalences | In almost all cases I know of where people have proved derived equivalences between blocks of finite groups, the proof hasn't really gone that way (i.e., finding a virtual bimodule and refining it to a splendid tilting complex). In fact, usually the virtual bimodule doesn't appear explicitly at all, although in most ca... | 6 | https://mathoverflow.net/users/22989 | 380603 | 158,453 |
https://mathoverflow.net/questions/380604 | 3 | Let $K$ be a slice knot in $S^3 = \partial B^4$. Then $K$ bounds a smoothly properly embedded disk $D$ in $B^4$. Let $\nu(D)$ denotes the tubular neighborhood of $D$.
Or we consider ribbon disks by excluding index two critical points.
Do we know $4$-dimensional handle decompositions of the slice disk exterior $B^4 ... | https://mathoverflow.net/users/nan | Handle decompositions of slice and ribbon disk exteriors | We can produce Kirby diagrams for the complement of a ribbon surface $S$.
Indeed, there is a procedure that is described in *Gompf-Stipsicz "4-manifolds and Kirby calculus"*.
I'll briefly explain that, for more details see pg 211-213 of that book.
First we perturb the projection $B^4 \simeq \mathbb{D}^3\times I \to I... | 1 | https://mathoverflow.net/users/158806 | 380607 | 158,454 |
https://mathoverflow.net/questions/380546 | 2 | Let the sequence $(a(n,k))\_{ n \in \mathbb{Z}}$ satisfy $$\sum\_{j=0}^k c(k,j)a(n-j,k)=0$$ with $c(k,j)=c(k,k-j)$ and $c(k,0)=1$ and with initial values $a(-n,k)=0$ for $1\leq n\leq{k-1}$ and $a(0,k)=1.$
For example for the binomial coefficients $c(k,j)=\binom{k}{j},$ we get $a(n,k)=\binom{-k}{n}.$
Let $A\_k(n)$ b... | https://mathoverflow.net/users/5585 | Some determinants which are closely related to recurrences | The question concerns the determinant of a [Hankel matrix](https://en.wikipedia.org/wiki/Hankel_matrix), or a fixed element of a Hankel matrix transform of a shifted sequence $a(n,k)$ for a fixed $k$, although I do not see how this fact alone can be useful. I give a standalone proof below.
---
Let $k$ be fixed. F... | 2 | https://mathoverflow.net/users/7076 | 380615 | 158,457 |
https://mathoverflow.net/questions/380626 | 2 | Let $C(K)$ be the algebra of continuous functions on Cantor set. Is it possible to prove that $C(K)$ forms an AF-algebra without Bratteli diagram?
| https://mathoverflow.net/users/137242 | The algebra of continuous functions on Cantor set | Sure. Regard $K$ as $\{0,1\}^N$ and let $E\_n$ for $n\in N$ be the functions in $C(K)$ that depend only on the first $n$ components.
| 9 | https://mathoverflow.net/users/2554 | 380643 | 158,465 |
https://mathoverflow.net/questions/380513 | 5 | Let $E$ be a Banach space. Let $F$ be the collection of all $f\in E^\*$ such that $\left<f,e\_n\right>\to 0$, for every normalized basic sequence $\{e\_n\}$. It is easy to see that $F$ is a closed subspace of $E^\*$.
>
> Does $F$ separate points of $E$?
>
>
>
Note that if $E$ is reflexive, then $F=E^\*$, since... | https://mathoverflow.net/users/53155 | Is there a topology that makes every basic sequence null? | The answer is negative in every non reflexive space. If $X$ is non reflexive, there is a normalized basic sequence $(z\_n)$ in X s.t. $(z\_1 - z\_n)\_{n=2}^\infty$ and $(z\_1 + z\_n)\_{n=2}^\infty$ are both basic sequence (necessarily semi normalized). If $x^\*$ tends to zero along both of these basic sequences, then $... | 5 | https://mathoverflow.net/users/2554 | 380645 | 158,466 |
https://mathoverflow.net/questions/380639 | 6 | Let $G$ be a finitely presented group. It is clear that if the profinite completion $\widehat{G} $ of $G$ is finite, then any finite dimensional complex linear representation $\rho: G\to \text{GL}(m, \mathbb{C})$ is finite, i.e., $\rho(G)$ is a finite group. For the converse, is there a counterexample for such $G$ such... | https://mathoverflow.net/users/128887 | profinite completion and linear representations of finitely presented groups | Yes, there exists such a finitely presented group.
Let $\Gamma$ be a cocompact arithmetic lattice in a product of $\ge 2$ rank 1 groups simple groups (with trivial center) over locally compact fields of finite characteristic. So $\Gamma$ is finitely presented (it is even CAT($0$)). By Malcev, $\Gamma$ is residually f... | 10 | https://mathoverflow.net/users/14094 | 380648 | 158,468 |
https://mathoverflow.net/questions/380628 | 13 | Recall there are multiple ways to define the unit sphere bundle of a vector bundle. One is by constructing a fiberwise vector space metric and declaring the sphere bundle to have fibers the unit spheres in each of the vector space fibers. The other way is to use the equivalence of vector bundles and principal $O(n)$ bu... | https://mathoverflow.net/users/134512 | Do $\mathbb{R}^n$ bundles have unit sphere bundles? | It is not true in general that there is a subgroup $H$ of $Homeo(\mathbb{R}^n)$ such that the inclusion is a homotopy equivalence and $H$ preserves the unit sphere. If there was, then $H$ would also preserve the unit disk and thus every topological $\mathbb{R}^n$-bundle would contain a $D^n$-bundle. But it is not the c... | 15 | https://mathoverflow.net/users/798 | 380650 | 158,469 |
https://mathoverflow.net/questions/380649 | 12 | This is [cross-posted from MSE](https://math.stackexchange.com/questions/3962953/can-the-category-of-s-local-objects-be-reflective-but-not-a-localization-by-s) (and substantially re-written) after receiving no answers.
Suppose $\mathcal C$ is a category and $S \subseteq \operatorname{Mor}(\mathcal C)$ is some collect... | https://mathoverflow.net/users/149197 | Can the category of S-local objects be reflective but not a localization by S? | Not in general, no - there must be some additional conditions on $S$, such as a saturation condition.
Consider for instance the presentable case. Then if $S$ is small, $Loc(S) $ is always reflexive, and is always a localization of $C$ at the *saturated class generated by $S$*, but there is no reason to expect it to b... | 14 | https://mathoverflow.net/users/102343 | 380653 | 158,471 |
https://mathoverflow.net/questions/379899 | 1 | Recall that a Hausdorf topological space $X$ is called compactly generated if any set whose intersections with compacts are compact is closed. Locally compact and first countable spaces are compactly generated.
>
> Let $E$ be a Banach space with the norm $\|\cdot\|$ and the unit ball $B\_E$. Let $|||\cdot|||\le \|\... | https://mathoverflow.net/users/53155 | Is a topology sandwiched between two norms compactly generated? | Let $\tau$ be the weak topology on the Banach space $\ell\_1$. It is known that each weakly convergent sequence in $\ell\_1$ is norm convergent (i.e., $\ell\_1$ has the Shur property). This property implies that $\tau$ is not compactly generated (otherwise it would be equal to the norm topology). Now consider the norm
... | 3 | https://mathoverflow.net/users/61536 | 380657 | 158,473 |
https://mathoverflow.net/questions/380658 | 2 | Consider the linear constant coefficient differential operator
$P$ on the Hilbert space $L^2([0,1]^2;\mathbb C^2)$
$$P= \begin{pmatrix} D\_{z}+c & a \\ b & D\_{z}+c \end{pmatrix}$$
where $D\_z=-i \partial\_z =- i(\partial\_{x\_1} -i \partial\_{x\_2}).$
Here, $a,b,c$ are just some complex numbers.
I wonder whether o... | https://mathoverflow.net/users/119875 | Diagonalise self-adjoint operator explicitly? | By noting that $-i\partial\_{x\_1} $ is diagonalized by $e^{ik\_1 x\_1} $ and $-i\partial\_{x\_2} $ by $e^{ik\_2 x\_2} $, the problem reduces to a $2\times 2$ diagonalization for each $(k\_1,k\_2)$-block. The resulting eigenvalues are (denoting $k=k\_1-ik\_2 $)
$$
\frac{1}{2} (|a|^2 + |b|^2 ) +|k+c|^2 \pm\frac{1}{2} \s... | 6 | https://mathoverflow.net/users/134299 | 380665 | 158,475 |
https://mathoverflow.net/questions/16857 | 80 | Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes:
Given a finite (nonempty) set of real numbers, $S=\{a\_1,a\_2,\dots, a\_n\}$, with the property that for each $i$ there exist $j,k$ (not necessarily distinct) so that $a\_i=a\_j+a\_k$ (i... | https://mathoverflow.net/users/2384 | Existence of a zero-sum subset | The answer is in the affirmative; indeed,
>
> If $S$ is a finite non-empty subset of *any* abelian group such that every element of $S$ is a sum of two other (possibly, equal to each other) elements, then $S$ has a non-empty, zero-sum subset.
>
>
>
For a complete proof, see this [recent preprint](http://arxiv.... | 26 | https://mathoverflow.net/users/9924 | 380683 | 158,480 |
https://mathoverflow.net/questions/365563 | 5 | In [this paper](https://perso.univ-rennes1.fr/matthieu.romagny/articles/group_actions.pdf), M. Romagny defines for an action of a group scheme $G$ on a stack $X$ the fixed point stacks $X^G$ associated
to the group action on a stack and in Theorem 3.3 he proves that if
1. the group $G$ is proper and flat of finite re... | https://mathoverflow.net/users/109370 | Fixed point stack for a torus action | I hadn't seen this question until Arkadij contacts me directly. The answer is yes: if $G$ is a group scheme of multiplicative type then the fixed point stack is algebraic. This is now here : <https://arxiv.org/abs/2101.02450>.
| 4 | https://mathoverflow.net/users/17988 | 380684 | 158,481 |
https://mathoverflow.net/questions/358470 | 7 | I have found this lower bound for the size of minimal vertex cover (and proved it).
If a simple connected graph G on n vertices has largest and smallest eigenvalues $\lambda\_1,\lambda\_n$, respectively, and $\theta\_{n-1}$ is the second smallest Laplace eigenvalue, then
$$
\tau(G)\geq\frac{n\theta\_{n-1}^{2}}{\theta... | https://mathoverflow.net/users/156518 | Is this lower bound for the size of minimal vertex cover new/interesting? | I found a better bound for regular graphs- the Hoffmann bound:
Let G be a d-regular graph on $n$ vertices with minimal eigenvalue $\lambda\_{min}$. Then $$
\alpha\left(G\right)\leq\frac{-n\lambda\_{min}}{d-\lambda\_{min}}
$$
It is known that for a graph $G$ on $n$ vertices, $\alpha\left(G\right)+\tau\left(G\right)... | 0 | https://mathoverflow.net/users/156518 | 380693 | 158,485 |
https://mathoverflow.net/questions/380696 | 8 | Consider the set of continuous maps $C^0([0,1],[0,1])$ equipped with the compact-open topology. It is metrisable, and therefore sequential. It is also a k-space: see <http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf> Proposition 1.6. The proof relies on the facts that every k-closed subset is in partic... | https://mathoverflow.net/users/24563 | Compact-open topology and Delta-generated spaces | The mapping space $C([0,1],[0,1])$ in the compact-open topology is in fact $\Delta$-generated.
The reason for this is that every locally path-connected first-countable space is $\Delta$-generated. This was proved by Christensen, Sinnamon, and Wu in Proposition 3.11 of their paper [*The D-Topology for Diffeological Sp... | 11 | https://mathoverflow.net/users/54788 | 380699 | 158,486 |
https://mathoverflow.net/questions/372271 | 3 | I am considering large integer values of $N$ (100 or more digits in base-$10$).
In my algorithm, I need to be able to compute the reciprocal of $N$ with enough precision that the repetend will have been produced exactly. (I estimate this to be to $\lfloor \log N \rfloor$ digits or $\lfloor \log\_{2} N \rfloor$ bits)
... | https://mathoverflow.net/users/143543 | What is the big-O time complexity of computing $1/N$ to $\log_{2}(N)$ bits of precision? | Denote by $M\_b$ the complexity of multiplying two $b$-digit integers $z = xy$. One easily sees that this is essentially obtained by convolving the $b$-dimensional vectors of digits $x\*y$. The school algorithm is a "slow convolution" algorithm that takes $O(b^2)$, but fast convolution algorithms give rise to $M\_b = O... | 6 | https://mathoverflow.net/users/73890 | 380704 | 158,487 |
https://mathoverflow.net/questions/380713 | 7 | There is a theorem attributed to Hahn that every ordered field $F$ containing $\mathbb R$ is a subfield of a formal power series field $\mathbb R[[X^\Gamma]]$, where $\Gamma$ is an ordered abelian group. Can you give a nice reference in English for a proof of this theorem? Or if it is not too hard, please sketch a proo... | https://mathoverflow.net/users/11145 | Hahn’s theorem on ordered fields | This theorem, which extends Hahn's embedding theorem for ordered abelian groups to ordered fields, has a complicated history that makes it difficult to attribute it to any single author. However, by the early 1950s, as a result of the work of Kaplansky (1942), it appears to have assumed the status of a ``folk theorem''... | 11 | https://mathoverflow.net/users/18939 | 380715 | 158,491 |
https://mathoverflow.net/questions/380080 | 0 | If we use CG elements (continuous Galerkin), the boundary integration in FEM can be easily converted to sum over quadrature points using node basis functions of the edges. However, in DG elements (discontinuous Galerkin), there is no shared node basis and each elements have its own node basis. So each edge has multiple... | https://mathoverflow.net/users/171509 | Boundary integration of weak form in FEM using DG elements | Discontinuous Galerkin is the name, not for a single method, but for an extremely broad family of methods. Consider the BVP $$\nabla \cdot a\nabla u = f \text{ in } \Omega \text{ and } u=0 \text{ on } \partial \Omega.$$
Assume $\bar{\Omega} = \cup\_k \bar{K}\_k$ is a triangulation of $\bar{\Omega}$. Multiply the BVP by... | 1 | https://mathoverflow.net/users/73890 | 380725 | 158,495 |
https://mathoverflow.net/questions/380728 | 1 | Let $w\colon [0,T]\times\mathbb{T}^d \to \mathbb{R}^n$ be such that
$$ \|w\|\_{L^\infty(BMO)} := \sup\_{t\in[0,T]}\|w(t,\cdot)\|\_{BMO} \leq C $$
and $\int\_{\mathbb{T}^d} w(t,x)\mathrm{d}x = 0 $ for all $t$.
The corollary from the John-Nirenberg inequality states that
$$ \int\_{\mathbb{T}^d} e^{p|w(t,x)|} \mathrm{d}... | https://mathoverflow.net/users/171944 | Integrability of $\exp\left(p\int_0^t |w(s,x(s,y))| \mathrm{d}s\right)$ for $w\in L^\infty(0,T;BMO(\mathbb{T}^d))$ | I don't know much about BMO things, but I do know the following version of Jensen's inequality:
$$\phi\left(\int f(s) d\mu(s)\right) \leq \int \phi(f(s)) d\mu(s),$$
provided $\mu$ is a probability measure. From that point of view, I'm not sure part 2 of your argument is correct. You should instead put $d\mu(s) = ds/t$ ... | 4 | https://mathoverflow.net/users/73890 | 380732 | 158,498 |
https://mathoverflow.net/questions/380673 | 2 | Are there algebraic projective curves over finite fields other than $\mathbb{P^1}$ that if a vector bundle on it, is stable under Frobenius i.e. $F^\*E\cong E$ implies that $E$ is a trivial bundle? If so, does every algebraic curve admit an etale cover of this form?
| https://mathoverflow.net/users/127776 | Vector bundles that are fixed under pull-back by the absolute Frobenius | For a finite flat cover $\pi:Y\to X$ the pushforward $E:=\pi\_\*\mathcal{O}\_Y$ comes with a morphism $F^\*E\to E$ induced by the Frobenius on $Y$. If $\pi$ is etale this morphism is an isomorphism: over affine charts $X=Spec\, A$, $Y=Spec\, B$ we want to show that the map $B\otimes\_{A,F\_A}A\xrightarrow{b\otimes a\ma... | 4 | https://mathoverflow.net/users/39304 | 380737 | 158,499 |
https://mathoverflow.net/questions/380734 | 9 | This question is motivated by recent work of R P Stanley, [Theorems and conjectures on some rational generating functions](https://arxiv.org/abs/2101.02131). Consider the polynomials
$$P\_n(x)=\prod\_{i=1}^{n-1}(1+x^{3^{i-1}}+x^{3^i}).$$
Define the sequence $a\_n$ to count the number of monomials of $P\_n(x)$. For exam... | https://mathoverflow.net/users/66131 | Counting monomials in product polynomials: Part I | Yes, it is true. In other words, you ask whether $|X\_n|=F\_{2n}$ where $$X\_n:=\sum\_{i=1}^{n-1}\{0,3^{i-1},3^i\}.$$
We have $$X\_n=X\_{n-1}\cup Y\_{n-1}\cup Z\_{n-1},\quad (1)$$ where $Y\_{n-1}=X\_{n-1}+3^{n-1}$, $Z\_{n-1}=X\_{n-1}+3^n$. We have $(X\_{n-1}\cup Y\_{n-1})\cap Z\_{n-1}=\emptyset$, since $\min Z\_{n-1}=3... | 15 | https://mathoverflow.net/users/4312 | 380738 | 158,500 |
https://mathoverflow.net/questions/380752 | 4 | Let $K/F$ be a finite extension of local fields (of characteristic 0). Is it true that the quotient group $K^\times/ F^\times$ is always compact?
I understand that if the extension is cyclic, it is compact by Hilbert 90. But does it hold in general?
| https://mathoverflow.net/users/32746 | Is $K^\times/ F^\times$ compact for local fields? | $O\_K^\times$ is compact thus so is $$K^\times/ \pi\_F^\Bbb{Z}=\pi\_K^{\Bbb{Z/eZ}} \times O\_K^\times, \qquad e=\frac{v(\pi\_F)}{v(\pi\_K)}$$ Being a quotient of a compact group by a closed subgroup $$K^\times/F^\times=(K^\times/ \pi\_F^\Bbb{Z})/(F^\times/ \pi\_F^\Bbb{Z})$$ is compact.
Otherwise you can use the isomo... | 6 | https://mathoverflow.net/users/84768 | 380754 | 158,507 |
https://mathoverflow.net/questions/380655 | 2 | I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S\_4 \subset \mathbb{P}^{15}$ I'm finding some hard stuff that I'm not able to figure it out.
Let me briefly recall the notations. Set $T=\mathbb{P}(W)=\mathbb{P}^4$ and $\ma... | https://mathoverflow.net/users/146431 | Help about "Varieties with small Dual Varieties" by L.Ein | It seems to me that these various (somehow independent questions) could be asked to your master thesis advisor. It's basically what such an advisor is made for, I guess. A few hints:
1- This is an obvious consequence of the Euler exact sequence which identifies $T\_{\mathbb{P}(V)}(-1)|\_{\ell}$ with $V/\ell$. See pag... | 1 | https://mathoverflow.net/users/37214 | 380755 | 158,508 |
https://mathoverflow.net/questions/380776 | 4 | Let $M$ be a complex manifold and $X \subset M$ a complex submanifold. We may assume that $X$ is compact, if that's helpful.
Can we always find a neighbourhood $U$ of $X$ in $M$ together with a holomorphic map $r : U \to X$ which restricts to the identity map on $X$?
In the $C^\infty$-case, any tubular neighborhood... | https://mathoverflow.net/users/123207 | Existence of holomorphic retraction | No, this is actually very rare. Indeed the existence of such a retraction implies that the exact sequence $$0\rightarrow T\_X\rightarrow T\_{M|X}\rightarrow N\_{X/M}\rightarrow 0$$ splits. In particular, the coboundary map $H^0(N\_{X/M})\rightarrow H^1(X,T\_X)$ is zero, which means that first order deformations of $X$ ... | 10 | https://mathoverflow.net/users/40297 | 380777 | 158,513 |
https://mathoverflow.net/questions/380766 | 0 | Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$.
I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists \alpha > 0:\quad \lVert y\rVert \le \alpha\lVert y+z\rVert \forall y∈Y,\forall z∈Z$.
However, reading this question [A criterion for the sum of two closed sets to be closed... | https://mathoverflow.net/users/171981 | Specific criterion for the sum of two closed sets to be closed | Let $\alpha>1$ such that
$$
\frac1{\alpha-1}=d(S(Y),S(Z)).
$$
Then for all $0\ne y\in Y$, $0\ne z\in Z$ we have
\begin{multline\*}
\frac1{\alpha-1}\le\left\lVert \frac{y}{\lVert y\rVert}-\frac{z}{\lVert z\rVert}
\right\rVert
\le \left\lVert \frac{y}{\lVert y\rVert}-\frac{z}{\lVert y\rVert}
\right\rVert
+
\left\lVert\fr... | 3 | https://mathoverflow.net/users/nan | 380781 | 158,515 |
https://mathoverflow.net/questions/380783 | 4 | Let $L$ be a number field, $\mathcal{O}\_L$ its ring of integers, and $\mathcal{Cl(O}\_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}\_L)$. By $r(n)=r([c], n)$, I mean the number of ideals of norm $n$, that belong to the class $[c]$,
$$r(n)=r([c], n)= \sharp\bigg\{ \mathfrak{I} \subse... | https://mathoverflow.net/users/68462 | What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"? | It sounds like you're looking for something like the function
$$\zeta\_C(s) = \sum\_{\mathfrak{a} \in C} N(\mathfrak{a})^{-s} = \sum\_{n \ge 1} r([c], n) n^{-s}.$$
These functions are sometimes called "ideal class zeta functions" and they come up from time to time in the literature. See e.g. this paper in J London Math... | 6 | https://mathoverflow.net/users/2481 | 380788 | 158,517 |
https://mathoverflow.net/questions/380785 | 5 | I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
Now clearly this is very difficult, indeed it is equivalent to find the integer solutions of an elliptic curve $E$ defined over $\mathbb{Z}$, in particular $E$ i... | https://mathoverflow.net/users/146431 | Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$ | Technically speaking, an elliptic curve is a genus 1 curve with a choice of rational point. The automorphism group of an elliptic curve is the subgroup of automorphisms of the genus 1 curve that fix that rational point. So there is no contradiction with the facts you looked up, it just means that no point is fixed by a... | 12 | https://mathoverflow.net/users/949 | 380793 | 158,519 |
https://mathoverflow.net/questions/380099 | 5 | For $a,b \in \mathbb{Z}$ we define the binary quartic form
$$\displaystyle F\_{a,b}(u,v) = a(u^2 - v^2)^2 + 4bu^2 v^2.$$
We shall assume throughout that the discriminant
$$\Delta(F\_{a,b}) = 4096a^2 b^2 (a-b)^2$$
of $F\_{a,b}$ is non-zero; that is, the form $F\_{a,b}$ is non-singular. Consider the twist family of g... | https://mathoverflow.net/users/10898 | Rank of jacobians of twists of hyperelliptic curves of genus one | The answer is yes, and it's fairly elementary. By the usual 2-descent, the curve $C$ gives a class $c$ in $H^1(\mathbb{Q},E[2])$, where $E$ is the Jacobian you wrote down. As you vary $d$, the groups $H^1(\mathbb{Q},E\_d[2])$ are canonically isomorphic, and $c$ is also the class of $C\_d$. To answer your question, you ... | 2 | https://mathoverflow.net/users/949 | 380794 | 158,520 |
https://mathoverflow.net/questions/380805 | 0 | I've encountered a definition in several papers, but literally none of them define the term. They all instead reference a book by Menger that has never been printed in English. The term is "rim-type" of a topological space; the context I'm running into it in is the theory of curves/one-dimensional spaces.
A curve $X$... | https://mathoverflow.net/users/110965 | (Seeking Definition) What Does it Mean for a Space to have Rim-Type $\alpha$? Or the 'derivative' of a Countable Set? | I believe (based on e.g. [this source's use of "derived set"](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.560.4518&rep=rep1&type=pdf)) this refers to the [Cantor-Bendixson derivative](https://en.wikipedia.org/wiki/Derived_set_(mathematics)). This does indeed consist of throwing out the isolated points. We... | 3 | https://mathoverflow.net/users/8133 | 380807 | 158,524 |
https://mathoverflow.net/questions/380786 | 11 | Encouraged by the responses to [my earlier MO question](https://mathoverflow.net/questions/380734/counting-monomials-in-product-polynomials), here is a follow up and upgraded quest.
Let $e\geq2$ be an integer. Define the polynomials
$$P\_{n,e}(x)=\prod\_{i=1}^{n-1}\left(1+x^{e^{i-1}}+x^{e^i}+\cdots+x^{e^{i+e-3}}\righ... | https://mathoverflow.net/users/66131 | Counting monomials in product polynomials: Part II | The answer to Question 1 is positive. In Question 2 it is true that
**Claim 1.** $a\_{n,e}$ equals to the number of walks of length $e+2(n-1)$ in the path graph $P\_{e+1}$ from one end to the other one.
I start with general reformulations, then prove Claim 1, then deduce the generating function for $a\_{n,e}$ (Ques... | 5 | https://mathoverflow.net/users/4312 | 380813 | 158,526 |
https://mathoverflow.net/questions/380816 | 6 | To begin, let us set
$$A\_Q(n):=\sum\_{d|n \\ d<Q}\mu(d)$$
If we fix $Q$ and let $n$ vary, we get a very surprising amount of cancellation. For instance, the trivial bound
\begin{align\*}
\mathbb{E}\_{n\in\mathbb{N}}\left[|A\_Q(n)|\right]&\leq\mathbb{E}\_{n\in\mathbb{N}}\left[\sum\_{\substack{d|n \\ d<Q}}1\right]... | https://mathoverflow.net/users/159298 | Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$ | This is not true. In fact
$$
x(\log x)^{-1+1/\pi} \gg \sup\_n \Big| \sum\_{\substack{ d|n \\ d\le x}} \mu(d) \Big| \gg x (\log x)^{-1+1/\pi}.
$$
The upper bound is due to [Montgomery and Vaughan](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43188/10998_2004_Article_400315.pdf?sequence=1&isAllowed=y) (see T... | 12 | https://mathoverflow.net/users/38624 | 380826 | 158,532 |
https://mathoverflow.net/questions/380828 | 23 | In my recent researches, I encountered functions $f$ satisfying the following functional inequality:
$$
(\*)\; f(x)\geq f(y)(1+x-y) \; ; \; x,y\in \mathbb{R}.
$$
Since $f$ is convex (because $\displaystyle f(x)=\sup\_y [f(y)+f(y)(x-y)]$), it is left and right differentiable. Also, it is obvious that all functions o... | https://mathoverflow.net/users/40520 | Are such functions differentiable? | Replace $x$ with $x+y$ to get
$f(x+y)\ge f(y)(1+x)$ or $f(x+y)-f(y)\ge xf(y)$.
Replace $y$ with $x+y$ and then interchange $x$ and $y$ to get $f(x+y)-f(y)\le xf(x+y)$.
Together,
$$
xf(y)\le f(x+y)-f(y)\le xf(x+y).
$$
Dividing by $x$ and taking the limit as $x\to0$ implies that $f$ is differentiable with $f'=f$.
| 43 | https://mathoverflow.net/users/nan | 380832 | 158,535 |
https://mathoverflow.net/questions/380835 | 12 | $\DeclareMathOperator{\Diff}{Diff}$
From the work of Galatius - Randall-Williams and Berglund - Madsen we have homological stability (with respect to g) of $B\Diff\_\partial (W\_{g,1})$ and rational homological stability of $B\widetilde\Diff\_\partial(W\_{g,1})$, where $W\_{g,1}$ is $(\#\_g S^d \times S^d ) \setminus D... | https://mathoverflow.net/users/134512 | Homological stability and Waldhausen A-theory | I don't think that you can deduce homological stability of the coinvariants from the Serre spectral sequence as you suggest. But this precise situation was studied in my paper "An upper bound for the pseudoisotopy stable range", which may be useful.
To answer your last question, A-theory only describe these groups in... | 10 | https://mathoverflow.net/users/318 | 380846 | 158,539 |
https://mathoverflow.net/questions/380847 | 11 | Let $K$ be a compact subset of $\mathbb R^n$ with $n\ge 2$ (say if you like $n=2$, which is possibly sufficiently representative).
**Q:** *Does there exist a closed simple curve $u:\mathbb S^1\to\mathbb R^n $ such that $K\cup u(\mathbb S^1 )$ is connected?*
The set $K$ may have uncountably many connected components... | https://mathoverflow.net/users/6101 | Connecting a compact subset by a simple curve | Not always.
Let $K$ be a subset of an ambient space $V$ ($V=\mathbf{R}^2$ is fine, but doesn't matter) that is the closure of a discrete subset $D$, such that $K-D$ is homeomorphic to a segment. This exists in $\mathbf{R}^n$ for $n\ge 2$.
Then every closed subset of $V$ that meets every component of $K$ has to cont... | 11 | https://mathoverflow.net/users/14094 | 380851 | 158,540 |
https://mathoverflow.net/questions/380848 | 3 | What is the paper where the [Liouville theorem](https://en.wikipedia.org/wiki/Harmonic_function#Liouville%27s_theorem) for harmonic function was first stated? Did it come before or after (or in the same paper) as the [Liouville theorem in complex analysis](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_an... | https://mathoverflow.net/users/nan | Origin of the Liouville theorem for harmonic functions | References to Liouville go back to his 1847 result that a doubly periodic function without poles is identically constant, which does not yet contain the generalization to either harmonic functions or holomorphic functions.
I quote from [Barry Simon, Harmonic Analysis: A Comprehensive Course in Analysis, Part 3](https... | 5 | https://mathoverflow.net/users/11260 | 380852 | 158,541 |
https://mathoverflow.net/questions/340189 | 4 |
>
> **Problem.** Assume that a metrizable separable space $X$ is the countable union $X=\bigcup\_{n\in\omega}X\_n$ of pairwise disjoint $G\_\delta$-sets $X\_n$ in $X$ such that each $X\_n$ is an absolute $F\_{\sigma\delta}$-set. Is $X$ an absolute $F\_{\sigma\delta}$?
>
>
>
| https://mathoverflow.net/users/61536 | The Borel class of a countable union of $G_\delta$-sets, which are absolute $F_{\sigma\delta}$ | The answer to this question is negative and follows from
**Theorem.** Each $G\_{\delta\sigma}$-subset $A$ of a Polish space $X$ can be written as the union $\bigcup\_{n\in\omega}A\_n$ of a sequence $(A\_n)\_{n\in\omega}$ of pairwise disjoint $G\_\delta$-sets in $X$.
*Proof.* Write the set $A$ as the union $A=\bigcu... | 2 | https://mathoverflow.net/users/61536 | 380853 | 158,542 |
https://mathoverflow.net/questions/380858 | 0 | In [Brubeck, Nakatsukasa, and Trefethen - Vandermonde with Arnoldi](https://arxiv.org/abs/1911.09988) (example 3) they solve the following linear system:
$$\operatorname{Re}\left(\begin{array}{ccc}1 & \cdots & z\_{1}^{n} \\ 1 & \cdots & z\_{2}^{n} \\ \vdots & \ddots & \vdots \\ 1 & \cdots & z\_{m}^{n}\end{array}\right)... | https://mathoverflow.net/users/170879 | Solution of complex linear system | No, the first line is not equivalent to what you claim in the text, I believe: that's *one* linear system containing a subtraction and not two separate linear systems.
In detail, the matrix of this linear system is obtained by concatenating horizontally `M = real(A)` and `N = imag(A(:,2:n+1))`. If you split the unkno... | 1 | https://mathoverflow.net/users/1898 | 380859 | 158,545 |
https://mathoverflow.net/questions/376447 | 6 | Suppose $X$ and $Y$ are schemes of finite type over $\mathbb{Z}$.
How is the arithmetic zeta function of their product, $\zeta\_{X \times Y} (s)$, related to their individual zeta functions, $\zeta\_X(s)$ and $\zeta\_Y(s)$? More generally if $Z$ is a fiber bundle over $X$ with fiber $Y$, does a similar relation hold?... | https://mathoverflow.net/users/168668 | Arithmetic zeta functions of products and fibrations | We have the relation $\zeta\_{X\times Y}(s) = \zeta\_X(s)\* \zeta\_Y(s)$ where $\*$ is the Witt product in the Witt ring of $\mathbb Z[[t]]$. For any commutative ring A, the (big) Witt ring $W(A)$ is defined by:
* Under addition, $W(A), +$ is isomorphic to the group $(1 + tA[[t]],\times)$
* The multiplication $\*$ is... | 4 | https://mathoverflow.net/users/58001 | 380870 | 158,550 |
https://mathoverflow.net/questions/380498 | 1 | Over a smooth algebraic curve, do all vector bundles admit a finite resolution by semi-stable bundles? Or is there a characterization of the vector bundles that do?
Edit: As an example on $\mathbb{P}^1$, it is possible. For a vector bundle tensor it with the right $\mathcal{O}(n)$ so that the smallest summand in the ... | https://mathoverflow.net/users/127776 | Finite resolution by semi-stable bundles | Devlin Mallory's approach is essentially correct, and in fact the situation is even a little better than he suggested.
Let $V$ be an arbitrary vector bundle of rank $n$. Let $W$ be a stable bundle of rank $n+1$ (if one exists) or a semistable bundle. Fix an ample bundle $\mathcal O(1)$.
I claim that for $m$ suffici... | 3 | https://mathoverflow.net/users/18060 | 380872 | 158,551 |
https://mathoverflow.net/questions/380874 | 1 | If $X, Y$ are topological, and $R\subseteq X\times Y$ we say that $R$ is *continuous (from $X$ to $Y$)* if for every $V\subseteq Y$ with $V$ open, we have $$R^{-1}(V) = \{u\in U: \exists v\in V:(u,v)\in R\}$$ is open in $X$.
Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters](https://en.wikipedia.org/w... | https://mathoverflow.net/users/8628 | Is the Rudin-Keisler ordering a continuous relation? | I assume that, in your definition of continuity of relations, the unspecified $V$ is intended to be an open subset of $Y$. With this assumption, the answer to your question is affirmative. If $V$ is any nonempty open subset of NPU$(\omega)$ then $(\leq\_{RK})^{-1}(V)$ is all of NPU$(\omega)$.
To prove it, first notic... | 4 | https://mathoverflow.net/users/6794 | 380879 | 158,553 |
https://mathoverflow.net/questions/380723 | 8 | $\DeclareMathOperator\Card{Card}$The book *Forcing Eine Einführung in die Mathematik der Unabhängigkeitsbeweise* by Hoffmann provides an intuition behind boolean valued models of set theory which I will explain below. But when I try to make use of the intuition, as I understand it, I run into problems.
**On pages 272... | https://mathoverflow.net/users/171884 | Intuition behind Boolean-valued models of set theory | Due to the insightful comments made by @AlexKruckman and @AndreasBlass, we can give an answer to the question.
Let $\mathcal{M}$ be a countable, transitive standard model of ZFC.
Further, let $B \in \mathcal{M}$ be a boolean algebra that is complete in $\mathcal{M}$.
Then we can understand the intuition given i... | 1 | https://mathoverflow.net/users/171884 | 380892 | 158,556 |
https://mathoverflow.net/questions/380719 | 4 | Say $X\_1, \cdots, X\_n$ are i.i.d random variables with mean zero, let $S\_n = \sum\_{i=1}^n X\_i$, we know by SLLN $$\frac{S\_n}{n}\rightarrow 0\text{ a.s}$$
We could further know that the sequence of random variables $\{\frac{S\_n}{n}\}$ are uniformly integrable. Hence u.i + converge in prob(weaker than a.s) impli... | https://mathoverflow.net/users/160836 | Sample average L1 convergence speed | $\newcommand{\ep}{\epsilon}$Somehow, I have only now recalled about Latala's inequalities for moments of the sums of positive independent random variables (r.v.'s), which, in particular, allow one to easily obtain the order of magnitude of $E|S\_n|$.
Indeed, by the [Marcinkiewicz--Zygmund inequalities](https://en.wik... | 1 | https://mathoverflow.net/users/36721 | 380893 | 158,557 |
https://mathoverflow.net/questions/379802 | 0 | Let $X\_1,...,X\_n$ be $n$ gaussian random variables $N(0,1)$ not necessarily independent or jointly correlated, $S=\sum\_{i=1}^n w\_i X\_i$ be the weighted sum of these gaussian variables (because $(X\_i)\_{i=1,..,n}$ are not jointly correlated, $S$ can be non normally distributed)
1/ What are the upper bound and/or... | https://mathoverflow.net/users/62193 | Bounds for the sum of dependent gaussian random variables | I found the answer in the theorem 4.9, Bounds for Distribution Functions of Sums of n Random Variables, [chapter 4 of Cherubini, Copula Methods in Finance](https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118673331.ch4) (there is no proof).
Given $S = \sum\_{i=1}^n w\_i X\_i $, the term $P(S \leq s) $ has the lowe... | 0 | https://mathoverflow.net/users/62193 | 380909 | 158,561 |
https://mathoverflow.net/questions/380908 | 0 | Suppose that $X\geq0$, and that the moment generating function of $X$ exists in an interval around 0. Given some $\delta>0$ and integer $k=1,2,...$, show that
$$\inf\_{k=0,1,...}\frac{E(|X|^k)}{\delta^k} \leq \inf\_{\lambda>0} \frac{E(e^{\lambda X})}{e^{\lambda \delta}}. $$
Consequently, an optimized bound based on p... | https://mathoverflow.net/users/163923 | Polynomial Markov versus Chernoff Bound for random variables | Let $b$ denote the LHS. Expanding $e^{\lambda X}$ in a power series you can deduce that
$$E(e^{\lambda X}) \ge \sum\_{k \ge 0} \frac {b \lambda^k \delta^k}{k!}=b e^{\lambda \delta} \,.$$
| 4 | https://mathoverflow.net/users/7691 | 380910 | 158,562 |
https://mathoverflow.net/questions/380911 | 3 | I've been teaching myself quantum mechanics, and I realized that I'm missing something fundamental. Namely, there are two pictures that I don't know how to reconcile:
1. Quantum Mechanics generalizes Hamiltonian dynamics in the following sense. In classical mechanics the set of compactly supported real-valued functio... | https://mathoverflow.net/users/98901 | What is the precise relationship between real Poisson algebras and commutative $C^*$ algebras? | Quantum mechanics is not just noncommutative probability; a commutative $C^{\ast}$-algebra alone corresponds via Gelfand duality to some (locally) compact Hausdorff space $X$, which is not equipped with a notion of dynamics. The role of the Poisson bracket on smooth functions on phase space is to provide dynamics, sinc... | 2 | https://mathoverflow.net/users/290 | 380912 | 158,563 |
https://mathoverflow.net/questions/380882 | 7 | $\newcommand{\bf}[1]{\mathbb #1}\newcommand{\sc}[1]{\mathscr #1}$
A *duality* between two vector spaces $E$ and $F$ over $\bf K$ ($= {\bf R}$ of ${\bf C}$)
is, by definition, a bilinear form
$$
\langle \cdot, \cdot\rangle :E\times F\to \bf K,
$$
such that, if $\langle x, y\rangle =0$ for every $x$ in $E$, then $y=0$.... | https://mathoverflow.net/users/97532 | Is every sequentially $\sigma(E',E)$-continuous linear functional on a dual Banach space $E'$ necessarily a point evaluation? | Mikael de la Salle points out this is true when $E$ is separable, as shown in Corollary V.12.8 of Conway, *A Course in Functional Analysis, 2e*.
For a non-separable counterexample, consider the uncountable ordinal space $[0, \omega\_1]$, which is compact Hausdorff, and $E = C([0, \omega\_1])$. By the Riesz representa... | 7 | https://mathoverflow.net/users/4832 | 380921 | 158,567 |
https://mathoverflow.net/questions/380849 | 1 | This question again might be silly, like the last post([deleted](https://mathoverflow.net/questions/380672/a-self-homotopy-equivalence-of-a-closed-surface-preserves-geometric-intersection)). Let me know I will delete it.
>
> **Problem:** Let $\Sigma$ be a surface without boundary and $f:\Sigma\to \Sigma$ be
> a *pr... | https://mathoverflow.net/users/129539 | Homotopy equivalence preserving all geometric intersection numbers | The answer is "no". $\newcommand{\CC}{\mathbb{C}}$
Consider the map $f \colon \CC \to \CC$ given by $f(z) = z^2$. All geometric intersection numbers in $\CC$ are zero, so the extra assumption holds automatically. Note that $f$ is proper, and is a homotopy equivalence.
However, there is no proper homotopy of $f$ to ... | 3 | https://mathoverflow.net/users/1650 | 380930 | 158,568 |
https://mathoverflow.net/questions/380899 | 6 | I have a reference request on following comment I found in
[nLab article](https://ncatlab.org/nlab/show/Karoubian+category) on Karoubian categories & envelopes. It states:
>
> The Karoubian envelope is also used in the construction of the
> category of pure motives, and in K-theory.
>
>
>
Almost every introduc... | https://mathoverflow.net/users/108274 | Idempotent completions in K-theory | In Schlichting's paper [Negative K-theory of derived categories](https://link.springer.com/article/10.1007/s00209-005-0889-3). Math. Z. 253, 97–134 (2006), a definition of negative $K$-theory of triangulated categories is given. These are abelian group valued functors $\mathbf{K}\_{i}$ with $i\leq 0$.
These groups ag... | 3 | https://mathoverflow.net/users/142783 | 380932 | 158,569 |
https://mathoverflow.net/questions/380923 | 7 | I posted this [question](https://math.stackexchange.com/questions/3967715/prove-0-sum-k-1n1-frac1a-k-prod-j-1-j-neq-kn1-fraca-k) on Math StackExchange but did not get any answer. I am trying my luck here.
>
> Let $a\_{1},a\_{2},\dotsc,a\_{n+1}$ be a sequence of distinct non-zero real numbers with
> $$\sum\_{j=1}^{n... | https://mathoverflow.net/users/38620 | How prove this Webb inequality? | Here is the proof that the sum is non-negative.
Denote $f(x)=x^{n-2}|x|=x^n/|x|$. Then
$$
A:=\sum\_{k=1}^{n+1}\dfrac{1}{|a\_{k}|}\prod\_{j=1,j\neq k}^{n+1}\dfrac{a\_{k}}{a\_{k}-a\_{j}}=
[x^n]\sum\_{k=1}^{n+1}f(a\_k)\prod\_{j\ne k}\frac{x-a\_j}{a\_k-a\_j}=:[x^n]h(x),
$$
where the polynomial $h(x)=Ax^n+\ldots$, $\deg h... | 4 | https://mathoverflow.net/users/4312 | 380953 | 158,577 |
https://mathoverflow.net/questions/380961 | 3 | Let us define this sum as a function of $z \in \mathbb{C}$ with some positive parameter $a$
$$
f(z; a) = \sum\limits\_{n = 0}^{\infty}\frac{|z|^{2n}}{n!}e^{-ian^2}.
$$
Probably, it can be expressed (or somehow related) in terms of theta-function.
| https://mathoverflow.net/users/152731 | Can the following sum be counted or expressed in terms of special functions? | Probably the answer is negative. Your series is a restriction of the analytic function in two complex variables:
$$F(\zeta,q)=\sum\_{n=0}^\infty\frac{q^{n^2}}{n!}\zeta^n,\quad |q|\leq1,$$
obtained by setting $q=\exp(-ia)$ and $\zeta=|z|^2>0$. Function $F$ is continuous, entire with respect to $\zeta$ and analytic for $... | 6 | https://mathoverflow.net/users/25510 | 380986 | 158,591 |
https://mathoverflow.net/questions/378266 | 1 | Suppose $H$ is a closed subgroup of a Lie group $G$. Then in Lee's book [Introduction to Smooth Manifolds](https://doi.org/10.1007/978-0-387-21752-9) (Ch. 9) he showed that the action $H\times G\to G$ $(h,g)\mapsto gh$ is a smooth, free, proper action. I have a small problem regarding showing the action is proper. Note... | https://mathoverflow.net/users/136860 | A question regarding the action of a Lie subgroup | As @Ramiro Lafuente pointed out in a [comment](https://mathoverflow.net/questions/378266/a-question-regarding-the-action-of-a-lie-subgroup#comment960592_378266), there's a gap in this proof.
You're apparently using the first edition of my *Smooth Manifolds* book. The problem is fixed in the second edition, because th... | 5 | https://mathoverflow.net/users/6751 | 380990 | 158,593 |
https://mathoverflow.net/questions/380971 | 26 | First, some context. Ever since I was a high schooler, I have been fascinated with large numbers. As I have grown in mathematical maturity, I have become both disappointed and fascinated to see that the process of naming larger and larger numbers requires a sort of philosophical tradeoff.
For example, an ultrafinitis... | https://mathoverflow.net/users/115247 | What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers? | There's a certain confusion underlying your question, which Andreas Blass's answer is trying to point out. Let me see if I can explain it in different words.
You say, “the negation of Con(ZFC) proves it halts in finite time” and you are trying to use this fact to argue about which axioms beyond ZFC to accept. The bes... | 22 | https://mathoverflow.net/users/3106 | 380992 | 158,595 |
https://mathoverflow.net/questions/380957 | 3 | $\DeclareMathOperator\Nil{\mathsf Nil}\DeclareMathOperator\ker{ker}$I was reading through The $K$- book by Charles A. Weibel. There I found a very interesting category $\Nil(R)$, which consists of pairs like $(P , \nu)$, where $P$ is a finitely generated projective module and $ \nu : P \rightarrow P$ is a nilpotent end... | https://mathoverflow.net/users/nan | $K_0(\mathsf{Nil}(R))$ when $R$ is a field | The answer is **yes**, and this follows essentially from the Jordan decomposition of nilpotent endomorphisms.
Let $(F^n,\nu)$ be an $n$-dimensional vector space and a nilpotent endomorphism. Then $\nu^n=0$ and we can write a filtration
$$ F^n=\ker\nu^n\supseteq \ker \nu^{n-1} \supseteq \ker \nu^{n-2} \supseteq \cdots... | 1 | https://mathoverflow.net/users/43054 | 380997 | 158,598 |
https://mathoverflow.net/questions/43445 | 11 | Consider the Baker–Campbell–Hausdorff formula ([Wikipedia page](https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula)):
$$Z(X,Y) := X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \dotsb$$
Many sources, including the Wikipedia page, have an explicit expression for ... | https://mathoverflow.net/users/3040 | Baker–Campbell–Hausdorff formula: prime divisors of denominators | If the homogeneous component $Z\_n(X,Y)$ of $Z(X,Y)$ of degree $n$ is represented in the Lyndon basis $\mathcal{L}\_n$, or in any basis $\mathcal{B}\_n$ whose transformation
matrix $T\_{\mathcal{L\_n}\to\mathcal{B\_n}}$ has determinant $\pm 1$, then an explicit formula for the exponent $f(p,n)$ of the highest power of ... | 9 | https://mathoverflow.net/users/172140 | 381001 | 158,600 |
https://mathoverflow.net/questions/381002 | 3 | If I have a smooth positive scalar function $f$ defined on a 2-dimensional manifold $M$, then $f:M\rightarrow (0, \infty)$, where the metric of $M$ is $g=\frac{dx^2+dy^2}{y^2}$, i.e., $M$ is Poincare' half-plane.
$f$ must satisfy the following PDEs:
$\begin{cases}
\Delta f=f/2 \\ |\nabla f|^2=\frac{(f^2+3f)}{2}+1
... | https://mathoverflow.net/users/111304 | Solution existence in a pde system | Yes. Here is a general approach to this problem: Suppose that one has two functions a>0 and b on some interval $I\subset\mathbb{R}$ and one wants to know whether there is a solution $f$ to the system
$$
|\nabla f|^2 = a(f)^2,\qquad \Delta f = a(f)b(f)
$$
on some (nonempty) open set in the Poincaré upper half plane (i.e... | 10 | https://mathoverflow.net/users/13972 | 381012 | 158,603 |
https://mathoverflow.net/questions/380897 | 1 | In Hirsch's Differential Topology there's the following :
>
> Suppose a compact $n$-manifold can be expressed as $A\cup B$ where $A,B$ are compact $n$-dimensional submanifolds and $A\cap B$ is an $(n-1)$-dimensional submanifold. Then $\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B)$.
>
>
>
Trying to solve this a q... | https://mathoverflow.net/users/155363 | Vector field tangent to a submanifold and transverse to the zero section | Choose an open cover of $M$ where, for each open set $U$ in the cover that intersects $N$, there is a chart $U \subseteq \mathbb R^n$ that sends $N$ to $\mathbb R^{n-1}$.
Choose a partition of unity for this open cover.
For each open set, define a vector field on $\mathbb R^n$ as $\sum\_i f\_i \frac{d}{dx\_i}$ wher... | 2 | https://mathoverflow.net/users/18060 | 381023 | 158,608 |
https://mathoverflow.net/questions/321484 | 3 | In the book *3264 and All That* by Eisenbud & Harris, the authors claim that for smooth projective varieties admitting an affine stratification, the algebraic equivalence relation and the rational equivalence relation define the same intersection theory (p. 553). Anyway, they do not give an explicit reference where one... | https://mathoverflow.net/users/80084 | Equivalence relations among algebraic cycles | In fact more is true: the cycle class map $CH^\*(X)\rightarrow H^\*(X)$
is an isomorphism. See e.g. Fulton's Intersection theory, Examples 1.9.1 and 19.1.11 (b).
| 4 | https://mathoverflow.net/users/40297 | 381029 | 158,610 |
https://mathoverflow.net/questions/370239 | 13 | The following definition has arisen naturally in two papers of mine. The papers are on rather unrelated topics; of course they are within my narrow interests, so there's some symbolic dynamics connection, but really in both cases I just needed to find good "generic elements / finite subsets" of the group, and I can't h... | https://mathoverflow.net/users/123634 | Splendid groups | **edit January 18th, 2021**
Final nail in the f.g. splendid coffin. Apparently, the dihedral group and its variants are indeed the only non-splendid groups. So maybe I should've called non-splendid groups splendid and vice versa, because this would roll off the tongue better then. I of course originally thought non-s... | 5 | https://mathoverflow.net/users/123634 | 381031 | 158,611 |
https://mathoverflow.net/questions/381022 | 8 | In the paper "Classification of $(n - 1)$-Connected $2n$-Manifolds" by C.T.C.Wall (Annals of Mathematics , Jan., 1962, Second Series, Vol. 75, No. 1 (Jan., 1962), pp. 163-189), Wall studies $(n - 1)$-Connected $2n$-Manifolds with a small ball removed and proves a classification result for such manifolds in terms of alg... | https://mathoverflow.net/users/99732 | On the state of the art on closed $(n-1)$-connected $2n$ manifolds | The classification problem of smooth oriented closed $(n-1)$-connected $2n$-manifolds for $n\ge3$ splits into three parts.
1. Classify smooth almost closed compact oriented $(n-1)$-connected $2n$-manifolds, where almost closed means that the boundary is a homotopy sphere.
2. Understand those homotopy spheres that ari... | 11 | https://mathoverflow.net/users/32022 | 381036 | 158,613 |
https://mathoverflow.net/questions/380904 | 8 | There's a large countable ordinal which has cropped up (as a *lower* bound!) in a computable structure theory problem I'm playing with. At present I don't really understand how big it is, and I'm curious where it fits in amongst better-understood ordinals. (I do have a kind of upper bound, but it's weird and not very h... | https://mathoverflow.net/users/8133 | How big is the least non-$\Sigma^1_1$-pointwise-definable ordinal? | Claim: Let $\kappa$ be least such that $L\_\kappa$ is admissible and
$L\_\kappa\models$``$\omega\_1$ exists''
and let $\alpha=\omega\_1^{L\_\kappa}$. Then $\alpha$ is the least
non-$\Sigma^1\_1$-pd ordinal. Moreover, the 1st projectum of $L\_\kappa$ is
$\rho\_1^{L\_\kappa}=\omega\_1^{L\_\kappa}=\alpha$, and the 1st sta... | 9 | https://mathoverflow.net/users/160347 | 381041 | 158,616 |
https://mathoverflow.net/questions/380940 | 4 | Let $\Sigma^k$ be a $k$-dimensional Stein manifold with embedding as a real manifold (let's assume that that embedding is analytic if it makes things easier) $\Sigma^k \hookrightarrow \Bbb R^{2k}$.
*Main question.* Is it true that $\Sigma^2$ holomorphically embeds in $\Bbb C^2$?
*Addendum.* Is there an example of $... | https://mathoverflow.net/users/81055 | Stein manifolds with "wrong" minimal dimension of embedding | A result of [Gompf](https://arxiv.org/abs/1110.1865) shows that, in the case $k=2$, if the complex structure on the domain induced by a smooth embedding in $C^2$ is homotopic to a Stein structure, then the embedding is isotopic to a complex embedding. "Homotopic" means homotopic through almost-complex structures.
| 2 | https://mathoverflow.net/users/172172 | 381043 | 158,618 |
https://mathoverflow.net/questions/381058 | 2 | I have just started reading through the paper of Cattani--Kaplan--Schmid -- Degeneration of Hodge structures (Annals of Mathematics, 123 (1986), 457--535). For the purposes here, take $f : X \to S$ to be a surjective holomorphic map from a compact Kähler manifold onto a complex manifold $S$ of strictly lower dimension.... | https://mathoverflow.net/users/172177 | Is the Hodge bundle a holomorphic vector bundle? | Perhaps my original answer was a bit technical. So let add a few comments at the beginning. The first question, is how does one define $H^{p,q}(V\_s)$? Initially, it's defined as the space of $(p,q)$ forms which are harmonic with respect to a K"ahler metric. There is no reason why this would vary holomorphically. Howev... | 4 | https://mathoverflow.net/users/4144 | 381061 | 158,626 |
https://mathoverflow.net/questions/381070 | 4 | Higman proved the existence of a finitely generated simple group here:
Higman, Graham
A finitely generated infinite simple group.
J. London Math. Soc. 26 (1951), 61–64.
It is a quotient of what is called Higman's group, which is an amalgamated free product.
Question: is either group isomorphic to a (non-trivial) ... | https://mathoverflow.net/users/172183 | Is Higman's group a free product? | As pointed out by @BenjaminSteinberg in the comments, a simple group cannot be a non-trivial free product.
It's also true that the [Higman group](https://en.wikipedia.org/wiki/Higman_group) cannot be a non-trivial free product. It has a presentation
$$G=\langle a,b,c,d | a^{-1}ba=b^2, b^{-1}cb=c^2, c^{-1}dc=d^2, d^... | 13 | https://mathoverflow.net/users/1345 | 381072 | 158,629 |
https://mathoverflow.net/questions/380994 | 2 | Let $\log \_b^ac$ denote an [iterated](https://en.wikipedia.org/wiki/Iterated_function) base-$b$ logarithm function. For example, $$\log \_2^3({2^{65536}}) = {\log \_2}({\log \_2}({\log \_2}({2^{65536}}))) = 4.$$
Pick an *arbitrary* model M of Turing machines, assuming that a machine operates with the two-symbol alph... | https://mathoverflow.net/users/122796 | Uncomputability of a function based on the Busy Beaver function | The nested logarithm doesn't really do much here. It is a computable function with a computable inverse, and thus the functions $n \mapsto x\_n$ and $f$ are not only Turing equivalent, but related via computable rescaling.
As such, the answer to [this question](https://mathoverflow.net/questions/137421/busy-beaver-mo... | 3 | https://mathoverflow.net/users/15002 | 381077 | 158,630 |
https://mathoverflow.net/questions/381057 | 7 | Is there a triple of nonzero even integers $(a,b,c)$ that satisfies the following infinite system of congruences?
$$
a+b+c\equiv 0 \pmod{4} \\
a+3b+3c\equiv 0 \pmod{8} \\
3a+5b+9c\equiv 0 \pmod{16} \\
9a+15b+19c\equiv 0 \pmod{32} \\
\vdots \\
s\_na + t\_nb + s\_{n+1}c \equiv 0 \pmod{2^{n+1}} \\
\vdots
$$
where $(s\_n)$... | https://mathoverflow.net/users/48162 | Is there a nonzero solution to this infinite system of congruences? | Let $u\_n = a s\_n + b t\_n + c s\_{n+1}$. The stronger claim is true: for large enough values of $n$,
the number $u\_n$ will be exactly divisible
by a fixed power of $2$ that doesn't depend on $n$.
Let $u\_n = a s\_n + b t\_n + c s\_{n+1}$ then (by induction)
$$u\_{n} = u\_{n-1} + 2 u\_{n-2} + 4 u\_{n-3}.$$
The... | 9 | https://mathoverflow.net/users/172190 | 381079 | 158,631 |
https://mathoverflow.net/questions/381082 | 1 | Let $V$ be a vector space over a field $\mathbb F$ and $k$ some natural number.
It isn't hard to show that the space of multiaffine maps $V^{[k]}\to\mathbb F$ decomposes as a direct sum of vector spaces $\bigoplus\_{I\subset [k]} M\_I $ where $M\_I$ is the space of multilinear maps $V^I\to\mathbb F$ (thought of as maps... | https://mathoverflow.net/users/170979 | Algebraic structure of the space of multiaffine maps | Let $\newcommand{\FB}{\mathrm{FB}}\newcommand{\FI}{\mathrm{FI}}\newcommand{\Vect}{\mathrm{Vect}}\FB$ be the category of finite sets and bijections, and $\FI$ the category of finite sets and injections, considered as symmetric monoidal categories under disjoint union.
The assignment $I \mapsto M\_I$ is a functor $\FB\... | 4 | https://mathoverflow.net/users/1310 | 381089 | 158,634 |
https://mathoverflow.net/questions/381101 | 2 | Let $\Omega\subset\mathbb{R}^2$ be open and of class $C^1$. The Sobolev embedding theorem implies that if $u\in W^{k,2}(\Omega)$ and if $k\in\mathbb{N}: k\geq 2$, then $u$ is continuous.
>
> **Question.** Does there exist a similar result for fractional Sobolev Spaces? For example, if $u\in W^{1+\theta,2}(\Omega)$ ... | https://mathoverflow.net/users/44981 | Sobolev embedding for fractional Sobolev spaces | If $\Omega$ is a "nice" domain in $\mathbb R^n$ and $u \in W^{1+\theta,p}(\Omega)$ with $\theta \in (0, 1)$, then both $u$ and the weak gradient $\nabla u$ are in $W^{\theta,p}(\Omega)$, and hence, by the Hardy–Littlewood-Sobolev inequality (Theorem 6.7 in the *Hitchhiker's guide to the fractional Sobolev spaces*), $u$... | 3 | https://mathoverflow.net/users/108637 | 381106 | 158,640 |
https://mathoverflow.net/questions/381093 | 2 | I would like to prove Chebyshev's sum inequality, which states that:
If $a\_1\geq a\_2\geq \cdots \geq a\_n$ and $b\_1\geq b\_2\geq \cdots \geq b\_n$, then
$$
\frac{1}{n}\sum\_{k=1}^n a\_kb\_k\geq \left(\frac{1}{n}\sum\_{k=1}^n a\_k\right)\left(\frac{1}{n}\sum\_{k=1}^n b\_k\right)
$$
I am familiar with the non-... | https://mathoverflow.net/users/172210 | How can I prove Chebyshev's sum inequality with probabilistic methods? | Let $A$ be the random variable attaining the values $a\_1,\dotsc,a\_n$ with equal probabilities, and define $B$ similarly, subject to $\mathbb P(B=b\_i|A=a\_i)=1$. Then $\mathbb E(A)=\frac1n\,\sum\_{1\le i\le n} a\_i$, $\mathbb E(B)=\frac1n\,\sum\_{1\le i\le n} b\_i$, and $\mathbb E(AB)=\frac1n\,\sum\_{1\le i\le n} a\_... | 7 | https://mathoverflow.net/users/9924 | 381116 | 158,642 |
https://mathoverflow.net/questions/381111 | 6 | In *Curvature and symmetry of Milnor spheres*, Grove and Ziller construct metrics of non-negative sectional curvature on $S^3$-bundles over $S^4$ (by using a cohomogeneity one action). In the same paper, they ask whether this can be done in other dimensions (see Problem 5.1). Does anybody know whether there has been pr... | https://mathoverflow.net/users/147200 | Metrics of non-negative sectional curvature on $S^7$-bundles over $S^8$ | My understanding is that this is generally unknown. Of course, a few of the total spaces (e.g., $S^7\times S^8$, $S^{15}$, and the unit tangent bundle of $S^8$) are homogeneous spaces, so admit a non-negatively curved metric.
For the most interesting class of $S^7$ bundles over $S^8$ (the exotic $15$-dimensional sphe... | 8 | https://mathoverflow.net/users/1708 | 381119 | 158,644 |
https://mathoverflow.net/questions/381103 | 6 | Let $X$ be a scheme. Is the category $QCoh(X)$ of quasi-coherent sheaves on $X$ locally presentable? If so, can we say anything about the $\kappa$ for which $QCoh(X)$ is locally $\kappa$-presentable? (e.g. is it always finitely presentable? Or related to the $\kappa$ of [Gabber's result](https://stacks.math.columbia.ed... | https://mathoverflow.net/users/172222 | Is Qcoh(X) locally presentable? | Zariski descent tells us that
$$\operatorname{QCoh}(X)=\lim\_{U\subseteq X} \operatorname{QCoh}(U)$$
where $U$ ranges through all open affines and the limit is taken in the $(2,1)$-categorical sense. Since small limits of presentable categories are presentable and $\operatorname{QCoh}(\operatorname{Spec}R)=\operato... | 11 | https://mathoverflow.net/users/43054 | 381121 | 158,645 |
https://mathoverflow.net/questions/381114 | 0 | From the book **Billingsley - Convergence of probability measures, 1999,** we have the following definitions of tightness and relative compactness and the Prohorov's theorem:
**Tightness:** Let $\Pi$ be a family of probability measures on $(S,\mathcal{F})$. The family $\Pi$ is tight if for every $\epsilon$ there is a... | https://mathoverflow.net/users/117762 | Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$? | $\newcommand\F{\mathcal F}\newcommand\G{\mathcal G}\newcommand\ep{\epsilon}\newcommand\si{\sigma}$The notion of "tightness of family $\Pi$ in some set $A$" is in general undefined. Instead, you can define it as follows:
>
> Let $\Pi$ be a set of probability measures over $(S,\F)$, where $S$ is a topological space a... | 2 | https://mathoverflow.net/users/36721 | 381127 | 158,647 |
https://mathoverflow.net/questions/381126 | 2 | In page 27 in HTT of J.Lurie, the expression
$$\text{Map}\_S(X,Y):=Y^X\times\_{S^X}\{\phi\}\in \text{Set}\_\Delta$$
appears for simplicial set $X,Y,S$ in Warning 1.2.2.2. However, I couldn't understand two notation in this expression, first one is exponential of simplicial set and the second one is product which have l... | https://mathoverflow.net/users/164702 | question about notation in HTT of J.Lurie | Just think in terms of ordinary sets for the moment. We have sets and maps $X\xrightarrow{\phi}S\xleftarrow{\psi}Y$ and we want to think about the set
$$ \text{Map}\_S(X,Y) = \{f\colon X\to Y: \psi f=\phi\}. $$
We can think of $f$ as an element of $Y^X$, and composition with $\psi$ gives a map $\psi\_\*\colon Y^X\to S^... | 9 | https://mathoverflow.net/users/10366 | 381134 | 158,650 |
https://mathoverflow.net/questions/381132 | 7 | Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to
$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable mappings }T: X \rightarrow Y \text{ and } T\_\#\mu = \nu$$
has a relatively straightforward interpretation: We try to fi... | https://mathoverflow.net/users/157982 | How to interpret couplings in optimal transport? | Of the mass $\mu(A)$ in $A$ a fraction $\pi(A \times B)$ is transported to $B$, so you can think of this as a randomized transport map. A basic example to think of is $\mu=\delta\_0$ and $\nu=(\delta\_1+\delta\_{-1})/2$. Half the mass at 0 is sent to 1 and half is sent to -1. You can get a better intuition from reading... | 5 | https://mathoverflow.net/users/7691 | 381145 | 158,656 |
https://mathoverflow.net/questions/381088 | 2 | General polytopes are [not determined by their edge-graph](https://mathoverflow.net/questions/309826/can-two-non-equivalent-polytopes-of-same-dimension-have-the-same-graph/380096#380096) (up to combinatorial equivalence). But I came accross the statement that [zonotopes](https://en.wikipedia.org/wiki/Zonohedron#Zonotop... | https://mathoverflow.net/users/108884 | Are zonotopes determined by their edge-graph? | Yes, the face lattice of a zonotope is determined by its graph. This is Theorem 6.14 of Bjorner, A., Edelman, P. H., and Ziegler, G. M. (1990). Hyperplane arrangements with a lattice of regions. Discrete Comput. Geom., 5(3):263–288.
The result uses the relation between hyperplane arrangements and zonotopes.
The oth... | 5 | https://mathoverflow.net/users/11134 | 381166 | 158,664 |
https://mathoverflow.net/questions/376509 | 11 | *I realize this is long, but hopefully I think it may be worth the reading for people interested in combinatorics and it might prove important to Covid-19 testing.* **Slightly reduced in edit**.
The starting point of this question is this [important article by Mutesa et al.](https://nature.com/articles/s41586-020-288... | https://mathoverflow.net/users/4961 | What are efficient pooling designs for RT-PCR tests? | I add this answer so as to be able to mark this question as answered.
As I should have guessed, these problems have been studied for more than 70 years, and the questions I asked are probably either solved or known to be open, up to minor changes.
One reference relevant to the questions I asked here (pertaining to "com... | 1 | https://mathoverflow.net/users/4961 | 381187 | 158,671 |
https://mathoverflow.net/questions/381200 | 2 | I'm trying to get an exact solution to this second order inhomogeneous PDE:
$$
\frac{\partial^2}{\partial{x}^2} y(x, z) - \frac{\partial^2}{\partial z^2} y(x, z)=k^2y(x, z)-\frac{1}{3}e^{4(x-2z)}y(x, z)
$$
where $k^2$ is a constant. No boundary conditions.
Any ideas? I tried with $t=(x-cz)$ and with variable separa... | https://mathoverflow.net/users/172296 | Second order inhomogeneous PDE | I am writing my comment as answer. If we change the variables $x,z \rightarrow u,v$ such that $u=x-2y$ (Just to make the exponential term a single variable function), and $v=ax+by$, then
Then, to eliminate $\frac{\partial^2 y}{\partial u\partial v}$ term we need to choose $\frac{a}{b}=-2$. Choose, $a=-2, b=1$.
Then... | 4 | https://mathoverflow.net/users/156029 | 381206 | 158,679 |
https://mathoverflow.net/questions/381129 | 3 | We can naively consider an operad as a collection $\{P(n)\}\_{n\geq 0}$ of vector spaces $P(n)$ consisting of "functions" with $n$ inputs and one output, equipped with a number of compositions
$$P(m)\times P(n)\to P(m+n-1)$$
given by attaching the output of an element of $P(n)$ to one of the inputs of an element of $P(... | https://mathoverflow.net/users/161009 | Generalised operad structures | There is a general framework for working with these generalized operad structure in the book [*A Foundation for PROPs, Algebras, and Modules*](https://bookstore.ams.org/surv-203/) (called *Foundation* below). A pre-publication version is [here](https://u.osu.edu/yau.22/main/). In Foundation Chapter 10, the concept of a... | 4 | https://mathoverflow.net/users/53034 | 381207 | 158,680 |
https://mathoverflow.net/questions/381202 | 10 | I consider a finite irreducible matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^d)$. I am interested in the maximal size of $\Gamma$ depending on $d$. But this question makes only sense if there is an upper limit.
In even dimension there is no such limit. This is easiest seen in dimension $d=2$, where we have the cyc... | https://mathoverflow.net/users/108884 | Arbitrarily large finite irreducible matrix groups in odd dimension? | Indeed, in odd dimension it's bounded.
Indeed, let $\Gamma$ be such a matrix group. By Jordan's theorem, it has a normal abelian subgroup $\Lambda$ of index $\le c\_d$. (An explicit bound for $d\ge 71$ is $c\_d=(d+1)!$, by work of Collins and Weisfeiler, see [Breuillard - An exposition of Jordan's original proof of h... | 13 | https://mathoverflow.net/users/14094 | 381209 | 158,681 |
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