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https://mathoverflow.net/questions/361962 | -3 | Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, $B\subseteq X$ be a non-empty Borel subset of a polish space $X$, $A$ be an uncountable indexing set, and $\{X\_{\alpha,n}\}\_{a \in A, n \in \mathbb{N}}$ a set of $X$-valued random elements with the following properties:
* For every pair of distinct $... | https://mathoverflow.net/users/36886 | Getting almost certainty from uncountably many low-probability events | Here is a "very regular" counterexample:
Let $X=\mathbb R$, $A:=(0,1)=:B$, and $X\_{a,n}:=Z+a$ for all $a\in A$ and $n\in \mathbb R$, where $Z\sim N(0,1)$. Then all your conditions hold. However,
$$P\Big(\bigcup\_{n\in\mathbb N,a\in A}\{X\_{a,n}\in B\}\Big)=P(|Z|<1)\ne1.$$
| 1 | https://mathoverflow.net/users/36721 | 362007 | 152,278 |
https://mathoverflow.net/questions/359904 | 3 | Let $H$ be a Hopf algebra with invertible antipode. Let $A$ be a braided Hopf algebra in the Yetter-Drinfeld category ${}\_H^H\mathcal{YD}$ over $H$.
A nonzero left integral in $A$ is a nonzero element $x\in A$ such that $yx=\epsilon(y)x$ for all $y\in A$. A nonzero right integral in $A$ is a nonzero element $x\in A$... | https://mathoverflow.net/users/66288 | Integrals and finite dimensionality in braided Hopf algebras | I think the answer to your question is affirmative.
Consider the Radford biproduct $A\rtimes H$ defined in [1], which is an ordinary Hopf algebra over $\Bbbk$ defined on the vector space $A\otimes\_{\Bbbk} H$. This construction does not require $A$ to be finite-dimensional over $\Bbbk$. The following was shown in [2,... | 2 | https://mathoverflow.net/users/33854 | 362010 | 152,279 |
https://mathoverflow.net/questions/362008 | 2 | Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X) $
and $H^{p,q}\_{(2)}(X)$ respectively.
As is well known, on a compact complex manifold $X$, $H^{p,q}(X) \cong H^{p,q}\_{(2)}(X)$ by the Hodge isomorphism.
My question is : **On a noncompact complex manifold, can we compare... | https://mathoverflow.net/users/124749 | Relations between Dolbeault cohomology and the corresponding $L^2$-cohomology | I don't know if that was a typo (did you want isomorphism, or did you really mean inclusion/injection?). In any case, the question in either interpretation has a negative answer. Let $X$ be a compact Riemann surface $\overline{X}$ minus a postive but finite number of points $p\_1, p\_2,\ldots$.
An element of $H^{1,0}\... | 4 | https://mathoverflow.net/users/4144 | 362016 | 152,283 |
https://mathoverflow.net/questions/362018 | 5 | Despite the catastrophe for the world and the many victims, at least the lockdown is favorable to two activities: Mathematics and Gardening. The difficulty of handling hedge trimmer wires and garden hoses calls to some mathematical questions relative to the (un)knots in $R^3$,
which are very natural and which I could n... | https://mathoverflow.net/users/105095 | The metric difficulty of unknotting unknots | The answer is yes for all I think, provided there is a finite width isotopy, as you can just scale the curve to make it arbitrary close to the origin, do the isotopy to a tiny circle there, and expand it back. So the infimum of $A$ or $B$ should be $2$.
Now for 1) if you work in an annulus instead of a disk the answe... | 3 | https://mathoverflow.net/users/112954 | 362021 | 152,286 |
https://mathoverflow.net/questions/330968 | 4 | I am reading the paper by Kuznetsov and Lunts, *Categorical resolutions of irrational singularities*, and I’m struggling with a few things. The definition of gluing of DG-categories $\mathcal{D}\_1$ and $\mathcal{D}\_2$ along a bimodule $\phi \in \mathcal{D}\_2^{op} \otimes \mathcal{D}\_1$ is the following: they say th... | https://mathoverflow.net/users/91572 | Definition of gluing of dg categories | This gluing can be viewed as a dg-category of "generalized morphisms". If you take $\mathcal D\_1 = \mathcal D\_2 = \mathcal D$ and $N$ to be the diagonal bimodule, then what you get is the dg-category of morphisms described for example by Drinfeld in *Dg-quotients of dg-categories*. The point is that closed degree $0$... | 1 | https://mathoverflow.net/users/20883 | 362028 | 152,288 |
https://mathoverflow.net/questions/362031 | 2 | I'm sorry if this is a trivial question, but it seems I can't find a clear answer.
I have a finitely generated Poisson algebra $A$, the Poisson scheme $X=Spec(A)$ and an automorphism $g$.
What is the definition of the fixed point subscheme $X^g$?
| https://mathoverflow.net/users/155568 | Fixed point scheme definition | This might be an unhelpful answer, but have you already considered the paper [John Fogarty - Fixed points schemes](https://www.jstor.org/stable/2373642)?
| 5 | https://mathoverflow.net/users/126773 | 362032 | 152,289 |
https://mathoverflow.net/questions/362033 | 3 | Start with some notations: $(a,q)\_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$, shortened by $(a)\_n$, and $(a)\_{\infty}=\prod\_{k=0}^{\infty}(1-aq^k)$.
It's easy to verify (using algebraic means) that, for each $m\in\mathbb{Z\_{\geq0}}$,
$$\sum\_{n\geq0}\frac{q^{n^2}q^{mn}}{(q)\_n(q)\_{n+m}}=\frac1{(q)\_{\infty}}. \tag1$$
... | https://mathoverflow.net/users/66131 | Seeking a combinatorial proof for the invariance of a $q$-series | The RHS is the (size) generating function for all integer partitions.
The LHS is a modification of the idea of keeping track of the [Durfee square](https://en.wikipedia.org/wiki/Durfee_square) of a partition.
Namely, for $m\in\mathbb{N}$, let us define for a partition $\lambda$ the $m$-Durfee square of $\lambda$ to... | 10 | https://mathoverflow.net/users/25028 | 362035 | 152,290 |
https://mathoverflow.net/questions/361955 | 0 | For any cardinal $\kappa$, let $[\kappa]^{<\kappa}$ denote the collection of subsets of $\kappa$ with cardinality $<\kappa$. Is there an infinite cardinal $\kappa$ and ${\cal C}\subseteq [\kappa]^{<\kappa}$ with the following properties?
1. $c \neq d \in {\cal C} \implies |c\cap d|= 1$,
2. For all $\alpha\in \kappa$ ... | https://mathoverflow.net/users/8628 | Subset of $[\kappa]^{<\kappa}$ with linear intersection | There is no such $\kappa$. Consider an infinite cardinal $\kappa$ and assume for a contradiction that $\mathcal C$ has the stated properties. For $\alpha\in\kappa$ let $\mathcal C(\alpha)=\{c\in\mathcal C: \alpha\in c\}$.
**Claim 1.** $|\mathcal C|\ge\kappa$.
**Proof.** The map $\{c,d\}\mapsto c\cap d$ is a surject... | 3 | https://mathoverflow.net/users/43266 | 362039 | 152,291 |
https://mathoverflow.net/questions/361920 | 5 | I apologise for the confusion of the following sentences. I'm lazy to give more information about *Rough path theory* as Is a fairly broad subject.
On page 14 of *"A Course on Rough Paths
With an Introduction to Regularity Structures*" by **Peter K. Friz & Martin Hairer** has written:
For $\alpha \in (1/ 3; 1 /2]$... | https://mathoverflow.net/users/116415 | Intuition behind Gubinelli derivative | In a way it is very much like a usual derivative. Recall first that for a regular function $Y$, its derivative $Y'\_s$ at a point $s$ is the (unique) number such that
$$
Y\_{t,s}=Y'\_s(t-s)+ R\_{s,t},
$$
where $R\_{s,t}\to0$ faster than linearly. If $Y$ is twice differentiable, then $R\_{s,t}\lesssim |t-s|^2$. That is,... | 5 | https://mathoverflow.net/users/100941 | 362042 | 152,292 |
https://mathoverflow.net/questions/362043 | 2 | I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and speculative, if this post, that is crossposted on Mathematics Stack Exchange (I've asked it as [**MSE 3636345**](https://math.stac... | https://mathoverflow.net/users/142929 | Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes | This should be provable by standard although laborious methods. What follows is a proof sketch (I have not checked all the computational details but this method should work).
We recall a few basic facts whose proofs will be omited:
Let $h(n) = \sigma(n)/n$, and let $H(n) = = \prod\_{p|n} \frac{p}{p-1}$.
1. For al... | 2 | https://mathoverflow.net/users/127690 | 362048 | 152,294 |
https://mathoverflow.net/questions/362050 | 0 | If $V,W$ are infinite-dimensional vector spaces with basis {${v\_i}$} and {${w\_j}$} respectively it holds that $V\otimes W$ has as basis {${v\_i⊗w\_j}$}.
What about the reciprocal? That is: if {${v\_i}$} and {${w\_j}$} are families of vectors in $V$ and $W$ respectively such that the family {${v\_i⊗w\_j}$} is a basi... | https://mathoverflow.net/users/60493 | Reverse the construction of a basis for a tensor product of vector spaces | I am afraid this question might be downvoted for not being research-level, but let me quickly expand the comment by Padraig Ó Catháin. For a fixed $k$, consider $W\_k=\operatorname{Span}(w\_k)$ and the projection $W\to W\_k$; by composing with the map $(v,\alpha w\_k)\mapsto \alpha v$, you get a bilinear mapping $V\tim... | 2 | https://mathoverflow.net/users/126773 | 362057 | 152,299 |
https://mathoverflow.net/questions/361936 | 3 | I am learning about the so-called "Mackey Machine" for unitary irreps of semidirect products of locally compact groups. Let $G = N \rtimes H$ where $N$ is a closed normal abelian subgroup and $H$ is a closed subgroup which acts on $N$ by $\phi: H \to Aut(N)$. Given an unitary irrep $\sigma$ of $H$ and a irreducible rep... | https://mathoverflow.net/users/16988 | Mackey theory for semidirect products: equivalence between constructions for modules | By [request](https://mathoverflow.net/questions/361936/mackey-theory-for-semidirect-products-equivalence-between-constructions-for-mod#comment912779_361936), I post my comments ([1](https://mathoverflow.net/questions/361936/mackey-theory-for-semidirect-products-equivalence-between-constructions-for-mod#comment912511_36... | 3 | https://mathoverflow.net/users/2383 | 362058 | 152,300 |
https://mathoverflow.net/questions/337303 | 15 | One of the most celebrated results in algebraic combinatorics is the Hook Length Formula of Frame-Robinson-Thrall which counts the number of standard Young tableaux of given partition shape. Such SYTs can be viewed as linear extensions of a poset (namely, the poset of the shape). There are also product formulas enumera... | https://mathoverflow.net/users/25028 | Unified framework for posets with order polynomial product formulas | I wrote a [survey of posets with order polynomial product formulas](https://arxiv.org/abs/2006.01568). It does not provide a "unified framework" for these posets, but does put forward a heuristic that they are the posets with good dynamical behavior.
| 2 | https://mathoverflow.net/users/25028 | 362059 | 152,301 |
https://mathoverflow.net/questions/361928 | 1 | Let $k(t,x)$ be the transition density of Brownian motion $$ k(t,x) := \frac{1}{\sqrt{2 \pi t}} \exp \left\{ \frac{-x^2}{2t} \right\} , \quad t \geq 0, x \in {\mathbb R.}$$
**Question**
Let $0 < x < a$. Show that
$$ \lim\_{x \nearrow a}\int\_0^t \frac{a-x}{s} k(s,x-a)k(t-s,a)ds = k(t,a).$$
Can someone offer some in... | https://mathoverflow.net/users/130369 | Limit of an integral / Boundary behaviour of a Gaussian convolution / single layer potential | First note that
$$ \partial\_x k(s, x) = -\frac{x}{s}k(s,x),
\quad s > 0, x \in {\mathbb R},$$
so your integral becomes
$$
q\_a(t,x) := \int^t\_0 \partial\_x k(s,x-a)k(t-s,a)ds.
$$
Now suddenly your integral $q\_a(t,x)$ becomes a representation of the unique classical solution to the boundary value problem for the... | 1 | https://mathoverflow.net/users/127334 | 362063 | 152,302 |
https://mathoverflow.net/questions/361977 | 2 | Let $A$ be a $C^\*$-algebra.
If $I$ is an essential two-sided ideal in $A$, then it is fact that for every $a \in A$ we have $\|a\| = \sup\_{x \in I, \|x\|=1} \|xa\|$. The argument is that we have an injective (since the ideal is essential) $C^\*$-map of $A$ into the multiplier algebra of $I$, which due to injectivit... | https://mathoverflow.net/users/13356 | Norm of a multiplier of a right-ideal in C*-algebras | Yes. The $\sigma(A^{\*\*},A^\*)$-closure of $I$ in the second dual von Neumann algebra $A^{\*\*}$ is an ultraweakly closed right ideal, which is of the form $pA^{\*\*}$ for some projection $p$. (In fact $p$ is the ultrastrong limit of any left approximate unit of $I$.) Thus,
$$\sup\_{x\in I,\ \|x\|=1}\| xa \| = \sup... | 3 | https://mathoverflow.net/users/7591 | 362065 | 152,303 |
https://mathoverflow.net/questions/362041 | 1 | I would like to be able to express the coefficients of $(2+x+x^2)^n$ in terms of the trinomial coefficients studied by Euler, ${n \choose \ell}\_2 = [x^\ell](1+x+x^2)^n$ where $[x^\ell]$ denotes the coefficient of $x^\ell$. The triangle of these numbers is given in OEIS A027907 and begins
\begin{matrix}
1 \\
1 & 1 & 1 ... | https://mathoverflow.net/users/14807 | Coefficients of $(2+x+x^2)^n$ from trinomial coefficients | Using Abdelmalek's tip in the comments, here's a solution to a more general version of the "larger program" mentioned at the end. For an arbitrary constant $c$,
\begin{align}
[x^\ell](c+x+\cdots+x^k)^n & = [x^\ell] \left((c-1) + (1+x+\cdots+x^k)\right)^n \\
& = \sum\_{m=0}^n {n \choose m}(c-1)^{n-m}[x^\ell](1+x+\cdots+... | 2 | https://mathoverflow.net/users/14807 | 362066 | 152,304 |
https://mathoverflow.net/questions/362061 | 4 | In Deligne-Mumford's "[The irreducibility of the space of curves of given genus](http://www.numdam.org/item/?id=PMIHES_1969__36__75_0)", the authors use the "Schlessinger's theory",
and refer his "thesis".
Where can I read it?
It seems to be different from his paper "[Functors of Artins rings](https://doi.org/10.1090/S... | https://mathoverflow.net/users/128235 | Schlessinger's thesis | I think by 'Schlessinger's theory' D-M mean both the foundations of deformation theory as described in the functor of Artin rings paper (which can nowadays be found in any book on deformation theory, e.g., Hartshorne's "Deformation Theory" or Sernesi's "Deformation of Algebraic Schemes"), and more specifically the defo... | 5 | https://mathoverflow.net/users/104669 | 362067 | 152,305 |
https://mathoverflow.net/questions/362054 | 1 | Let $X$ be a separable Banach space such that $X$ and its dual $X^\*$ have Radon-Nikodym property. Let $C$ be a convex, closed and bounded subset of $X$.
Can we say that $C$ is **weakly compact** or **weakly locally compact**?
An idea please
| https://mathoverflow.net/users/152650 | $C$ is **weakly compact** or **weakly locally compact**? | The closed unit ball of a Banach space $X$ is weakly compact if and only if it is reflexive. So if $X$ is a non-reflexive space such that $X$ and its dual have RNP, take $C$ to be the unit ball to see that the answer to your first question is negative.
For instance, takes $X$ to be the classical James space; its dual... | 3 | https://mathoverflow.net/users/763 | 362072 | 152,307 |
https://mathoverflow.net/questions/362037 | 6 | I wonder if anyone could find the following unpublished paper of Bloch-Kato:
Spencer Bloch and Kazuya Kato, $p$-divisible groups and Dieudonné crystals, unpublished.
A similar question is here while both links in the question are failed right now.
[An unpublished note by Spencer Bloch and Kazuya Kato](https://ma... | https://mathoverflow.net/users/159023 | An unpublished note by Bloch-Kato on p-divisible groups and Dieudonné crystals | The unpublished note was no longer on the Web (also [archive.org](https://archive.org) did not have it cached). I asked professor Moonen for a copy to share with you, here it is: [p-divisible groups and Dieudonné crystals](https://www.ilorentz.org/beenakker/MO/pDivDieudCryst.pdf) by Bloch and Kato.
 with the same topology?
A quasi-Banach space is, of course, just like a Banach space except the triangle inequality requireme... | https://mathoverflow.net/users/73784 | quick question about renorming quasi-Banach spaces into p-Banach spaces | Ben,
$$\|x\|^\prime = \inf\Big\{ \big(\sum\_{i=1}^n \|x\_i\|^p\big)^{1/p}\colon \sum\_{i=1}^n x\_i = x, x\_i\in X, n\in \mathbb N \Big\}\qquad (x\in X)$$
is the standard $p$-convex renorming. The hardish part is to find a suitable $p$. You will find more details in Kalton & Peck's *An F-space sampler*.
| 6 | https://mathoverflow.net/users/15129 | 362090 | 152,313 |
https://mathoverflow.net/questions/362075 | 0 | Let $f$ be a trancendental meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Pi$ be the stereoprojection map from the north pole on the unit sphere. My question is the following:
For any two points $P,Q \in \mathbb{C}$, can we find a curve $\gamma$ connecting $P$ and $Q$, such that $\Pi^{... | https://mathoverflow.net/users/51546 | Image of transcendental meromorphic functions | No. A simple example is
$f(z)=e^z$, $P=0$, $Q=10\pi i$. For any curve from $P$ to $Q$,
the image is a closed curve which winds $5$ times around zero. So it
cannot correspond to an arc of the great circle traversed once.
| 2 | https://mathoverflow.net/users/25510 | 362098 | 152,316 |
https://mathoverflow.net/questions/362070 | 2 | Let $v\_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v\_{j}\}\_{j=1}^{m}$ and $n>0$.
>
> Q: Can we say anything about the m roots $w\_{1},...,w\_{m}$ of
>
>
>
$$p(w)=n+\sum\_{j=1}^{m}\frac{v\_{j}}{w-v\_{j}}=0?$$
Can we say anything about their approximate location in $\mathbb{C}\setminus... | https://mathoverflow.net/users/99863 | Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$ | As Alexandre said, there is not enough guidance to as to what kind of info about these equations the OP is after. In any case, something of interest for equations given in the above form is that there is a nice formula for the resultant of two such equations. It involves the *matrix-tree theorem*. The formula was [disc... | 1 | https://mathoverflow.net/users/7410 | 362100 | 152,317 |
https://mathoverflow.net/questions/263909 | 5 | On pp. 152-3 of Hydon's [*Symmetry Methods for Differential Equations*](https://www.kent.ac.uk/smsas/personal/ph282/sym.htm) (2000 ed.), he lists some computer packages for symmetry-finding. [This related Mathematica StackExchange question](https://mathematica.stackexchange.com/q/20438/47125) mentions the [SYM](https:/... | https://mathoverflow.net/users/69318 | Symmetry-finding with SAGE? | Although not specifically for SAGE (which is a conglomeration of several math packages), Dr. Willy Hereman's "[24 [symmetry-finding] software packages](https://inside.mines.edu/~whereman/software.html)" is a continually updated list of software (last updated: March 18, 2020).
| 0 | https://mathoverflow.net/users/69318 | 362113 | 152,318 |
https://mathoverflow.net/questions/362040 | 6 | I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to reformulate my question and exclude the previous one, since now I can pose it in a more concrete way. The problem is ba... | https://mathoverflow.net/users/150264 | Reformulation - Construction of thermodynamic limit for GFF | For $x\in\mathbb{Z}^d$ I will denote by $\bar{x}$ the corresponding equivalence class in the discrete finite torus $\Lambda\_{L}=\mathbb{Z}^d/L\mathbb{Z}^d$.
I will view a field $\phi\in\mathbb{R}^{\Lambda\_L}$ as a column vector with components $\phi(\bar{x})$ indexed by $\bar{x}\in\Lambda\_L$.
The discrete Laplacian ... | 3 | https://mathoverflow.net/users/7410 | 362118 | 152,320 |
https://mathoverflow.net/questions/362101 | 2 | Consider a smooth projective morphism of schemes $X \rightarrow S$ with relative dimension $n$ (the application I have in mind is with $S$ = an open subset of $\text{Spec } \mathbb{Z}$) and assume that $V$ and $W$ are two closed subscheme of $X$, flat over $S$, such that $\text{codim}(V) + \text{codim}(W) = n$.
Deno... | https://mathoverflow.net/users/158892 | Upper semi-continuity of intersection numbers | In fact the intersection numbers are *constant*: by [Fulton, Cor. 20.3], the specialisation maps $\sigma \colon A^\*(X\_\eta) \to A^\*(X\_s)$ are ring homomorphisms if $S$ is a Dedekind scheme with generic point $\eta$ and closed point $s$. By [Fulton, §20.2] the intersection product for flat cycles is just the fibre p... | 2 | https://mathoverflow.net/users/82179 | 362120 | 152,321 |
https://mathoverflow.net/questions/362121 | 5 | In analytic number theory we like to weigh our counting functions with a smooth function $f$, so that we may apply Poisson's summation formula and take advantage of Fourier transforms. Typically the weight function $f$ will be a Schwartz type function with the following properties:
1) $f(x) \geq 0 $ for all $x \in \... | https://mathoverflow.net/users/10898 | A question about Schwartz-type functions used in analytic number theory | The answer to your question is yes, and it is a pretty well-understood topic.
First of all $X$ is more or less irrelevant for the bounds in 4) so let us take $X = Y$, say, for convenience.
Second, we can always scale everything down by $Y$. So without loss of generality $Y = 1$.
Put $g = f'$. Let us also for simp... | 6 | https://mathoverflow.net/users/104330 | 362125 | 152,325 |
https://mathoverflow.net/questions/362115 | 3 | If $H\_i = (V\_i, E\_i)$ are [hypergraphs](https://en.wikipedia.org/wiki/Hypergraph) for $i=1,2$ then we say they are *isomorphic* if there is a bijection $f: V\_1 \to V\_2$ such that for $A \subseteq V\_1$ we have $$A\in E\_1 \text{ if and only if } f(A) \in E\_2.$$
We say that $H=(\omega, E)$ is a *complete regular l... | https://mathoverflow.net/users/8628 | Are complete regular linear hypergraphs on $\omega$ isomorphic? | If $K$ is a field of cardinality $\aleph\_0$, then the points and lines of the projective plane over $F$ constitute a complete regular linear hypergraph. The field $K$ can be recovered (up to isomorphism) from the hypergraph, so this produces lots of non-isomorphic such hypergraphs.
| 4 | https://mathoverflow.net/users/6794 | 362129 | 152,327 |
https://mathoverflow.net/questions/362137 | 2 | Given $a\_1,a\_2,...,a\_m$ positive integer. Denominator $d$ is smallest positive integer for $b\_l$ integer coefficient.
$$\sum\_{k=1}^m\binom{n}k a\_k= \frac{1}d\sum\_{l=1}^mb\_ln^l$$
Now consider $n=dt+r$ where $d>r\ge 0$.
can it be shown that, above equation transform as
$$\frac{1}d\sum\_{l=1}^mb\_l(dt+r)^l... | https://mathoverflow.net/users/149083 | Problem related to transforming polynomial | Denote $\sum b\_l(x+r)^l=\sum c\_l x^l$. The numbers $c\_l$ are still integers, $c\_0=\sum b\_l r^l$ is divisible by $d$, and we have, denoting $dt+r=z$,
$$
\frac1d\sum b\_l(dt+r)^l=\frac1d\sum c\_l (dt)^l=\frac{c\_0}d+\sum\_{l>0} c\_l t\cdot (dt)^{l-1}=\\
\frac{c\_0}d+\sum\_{l>0} c\_l t\cdot (z-r)^{l-1}=
\frac{c\_0}d+... | 2 | https://mathoverflow.net/users/4312 | 362141 | 152,333 |
https://mathoverflow.net/questions/362132 | 3 | Suppose $Z=X+Y$ where $X$ is independent of $Y$ and $Y\sim N(0,1)$. I would like to compare $\text{var}(E(X|Z))$ to $\text{var}(E(Z|X))$. Obviously, $\text{var}(E(Z|X))=\text{var}(X)$.
In particular, my guess is that $\text{var}(E(Z|X)) > \text{var}(E(X|Z))$. If $Z$ is normal, then this is easy to prove directly from... | https://mathoverflow.net/users/159075 | Reversing the order of conditioning in a sum to compare conditional variances | **EDIT**: It seems that one can remove the integrability condition.
Suppose that $\text{Var}[X]<\infty$ and $Z=X+Y$, where $Y\sim\mathcal{N}(0,1)$ and $X$ and $Y$ are independent.
Because of the law of total variance,
$$\text{Var}\{E[Z|X]\}=\text{Var}\{X\}\\=E[\text{Var}\{X|Z\}]+ \text{Var}\{E[X|Z]\}\geq \text{Var}\{... | 3 | https://mathoverflow.net/users/69603 | 362142 | 152,334 |
https://mathoverflow.net/questions/360599 | 4 | I'm reading Heinonen's book on metric measure spaces. He writes that for general domains $\Omega \subset \mathbb{R}^n$, $M^{1,p}(\Omega) \subset W^{1,p}(\Omega)$ where the former are Hajlasz-Sobolev spaces (defined in 5.4) and the latter are Sobolev spaces.
Later in 5.17, he remarks on a way to see that $M^{1,p}(\Ome... | https://mathoverflow.net/users/69441 | Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same | Your argument is not correct. If a property $P$ fails for $Y$ and $X\subset Y$, it does not follow that it fails for $X$. For example $X=\{0\}\subset\mathbb{R}=Y$ but there are many properties true for $X$ and not true for $Y$.
You always have $M^{1,p}(\Omega)\subset W^{1,p}(\Omega)$ for all $1\leq p\leq\infty$. Howe... | 5 | https://mathoverflow.net/users/121665 | 362146 | 152,336 |
https://mathoverflow.net/questions/362153 | 4 | Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between *even power* (2,4,...) sums (traces) of the *imaginary part* of the complex zeros above the real axis of the Riemann zeta function $\zeta(s)$ and the derivatives evaluated at $t =0$ of... | https://mathoverflow.net/users/12178 | Derivatives of Riemann $\xi$ and traces of zeros | It is not a good idea to compute
$$\sum\_{n=1}^\infty \gamma\_n^{-k}$$ by computing a partial sum of several thousands of terms. The series converges but too slowly for this.
(see "Computation of the secondary zeta function" is on arXiv now: <https://arxiv.org/abs/2006.04869>).
For simplicity assume that the Rieman... | 4 | https://mathoverflow.net/users/7402 | 362160 | 152,338 |
https://mathoverflow.net/questions/361725 | 4 | Let $\pi: E \to B$ be a fiber bundle of (topological or differentiable) manifolds. Denote by $[B, E]\_{\pi}$ the set of all homotopy classes of sections of the bundle, i.e
\begin{align}
[B, E]\_\pi &= \{\sigma: B \to E \ | \ \pi\sigma = \text{id}\_B \}/\sim \\
\sigma \sim \sigma' &\iff \exists H: I \times B \to E \ |... | https://mathoverflow.net/users/137622 | Set of all sections of a fiber bundle up to homotopy equivalence | The relevant (simple) obstruction theory mentioned in the comments is contained in G. W. Whitehead's *Elements of Homotopy Theory*, Section VI.6. There an answer to the more general lifting problem is obtained under certain conditions, one of which is that the fibre $F$ is $q$-simple for certain values of $q$ (meaning ... | 5 | https://mathoverflow.net/users/8103 | 362167 | 152,341 |
https://mathoverflow.net/questions/362168 | 3 | Question 1. Let $\Gamma=(V,E)$ be a connected
graph with $n$ vertices, all of degree $d\geq 4$. Assume every vertex has $d$ distinct neighbors. (We can think of $d$ as being much smaller than $n$, but not necessarily bounded.)
As is customary, for a set of vertices $W\subset V$, we define the *boundary* $\partial W$... | https://mathoverflow.net/users/398 | Existence of connected component with large boundary? | If all degrees are at least 3, there exists a spanning tree with at least $n/4+2$ leaves (D. J. Kleitman and D. B. West, [Spanning trees with many leaves](https://doi.org/10.1137/0404010), SIAM J. Disc. Math. 4(1991), 99-106), the сomplement of these leaves gives you a connected set with boundary of size at least $n/4+... | 6 | https://mathoverflow.net/users/4312 | 362176 | 152,344 |
https://mathoverflow.net/questions/362181 | 0 | Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric polynomials), with the band width (degree of the trigonometric polynomial) along any direction, being non decreasing, in such a way that the sequence converges pointwise ... | https://mathoverflow.net/users/14414 | Can we construct a sequence of trigonometric polynomials that converges pointwise to a given continuous function on the torus? | The multidimensional Fejer series, i.e the Cesaro averages of the Fourier series of f, will converge uniformly to f. See <https://arxiv.org/pdf/1206.1789.pdf> or <https://www.sciencedirect.com/science/article/pii/S0022247X12000546> for a lot more detailed information.
| 5 | https://mathoverflow.net/users/7691 | 362184 | 152,346 |
https://mathoverflow.net/questions/362178 | 7 | Suppose I have a distribution $E$ such that $\phi \ast E$ is square-integrable for all $\phi \in C\_c^\infty \left( \mathbb{R}^d \right)$. Is it possible to prove that $E$ is tempered? It seems plausible to me, but I only get so far:
For brevity, define
\begin{equation}
G\_\phi = \phi \ast E
\end{equation}
for an... | https://mathoverflow.net/users/18936 | Prove that a given distribution is tempered | In Laurent Schwartz's *Théorie des Distributions* (page 245, chap. VII, §5) you can find something similar: A distribution $T\in \mathscr D'(\mathbb R^d)$ is tempered if and only if all regularizations $T \ast \varphi\in \mathscr O\_M$ for
$\varphi\in\mathscr D(\mathbb R^d)$, where $\mathscr O\_M$ is the space of slowl... | 10 | https://mathoverflow.net/users/21051 | 362192 | 152,347 |
https://mathoverflow.net/questions/362133 | 4 | This is a question that I suspect is simply a matter of technical issues written down or clarified somewhere in the literature, but which I can't find.
Suppose we're working over an arbitrary base scheme $S$, maybe with some unspecified basic niceness assumptions. Following, e.g., the terminology used on [p. 493 of H... | https://mathoverflow.net/users/120548 | When is a formal group smooth? | I think an adaptation of Schlessinger's argument from *Functors of Artin rings* should work. Let $\Lambda$ be a Noetherian ring, and suppose we have a connected formal group $\mathcal{G}$ formally smooth over $\Lambda$ with relative tangent space $\mathcal{G}(\Lambda[t]/t^2)\cong \Lambda^n$. We also assume that at any ... | 1 | https://mathoverflow.net/users/120548 | 362197 | 152,349 |
https://mathoverflow.net/questions/362210 | 8 | Let $C\_n = C\_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^3$. Is there a Riemannian metric $g$ on $C\_n$ such that given any two configurations in $C\_n$, there is a unique geodesic joining them?
In addition, it would be nice if $g$ was also geodesically complete, and if $g$... | https://mathoverflow.net/users/81645 | Is there a Riemannian metric on the configuration space of $n$ distinct points with "nice" geodesics? | The answer is no. This relies on two things:
1. A uniquely geodesic proper metric space is contractible; see [here](https://math.stackexchange.com/questions/479022/is-a-uniquely-geodesic-space-contractible-i) for a proof.
2. $C\_n$ is not contractible. Indeed, it has many nontrivial homology groups (there is a huge l... | 17 | https://mathoverflow.net/users/317 | 362211 | 152,355 |
https://mathoverflow.net/questions/361892 | 8 | Given a $\mathbb{C}$-linear braided fusion category $\mathcal{C}$ containing a fusion rule of the form e.g.
\begin{equation}X\otimes Y\cong A\oplus B \oplus C\end{equation}
(where $A,B, C, X$ and $Y$ are all simple objects with $A, B, C$ non-isomorphic), we can write the $R$-matrix $R^{XY}=\text{diag}(R^{XY}\_{A}, R^{X... | https://mathoverflow.net/users/135817 | R-matrices and symmetric fusion categories | Basically any representations of any groups will give you counterexamples in the symmetric case. The simplest case, where X and Y are the defining representation of SU(2), already works. For non-symmetric categories you'd expect that "generically" they'll look different, e.g. if you look at the category of representati... | 3 | https://mathoverflow.net/users/22 | 362212 | 152,356 |
https://mathoverflow.net/questions/362221 | 0 | For integers $a,b$ define $$\mathcal R(a,b)=\{q\in\mathbb Z\cap[1,\min(a,b)]: a\equiv b\bmod q\}$$ and $\mathsf{LCM}(\mathcal R(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal R(a,b)$.
>
> How big can $\mathsf{LCM}(\mathcal R(a,b))$ be if $a,b\in\big[\frac r2,r\big]$ hold and are coprime?
>
>
>
| https://mathoverflow.net/users/136553 | On $\mathsf{LCM}$ of a set of integers | In other words,
$$\mathcal R(a,b)=\{q\in[1,\min(a,b)]: q\mid(b-a)\}.$$
It is easy to see that for any $x,y\in \mathcal R(a,b)$, we have $\mathrm{LCM}(x,y)\mid(b-a)$, and therefore $\mathrm{LCM}(\mathcal R(a,b))\leq |b-a|$.
| 0 | https://mathoverflow.net/users/7076 | 362222 | 152,359 |
https://mathoverflow.net/questions/362106 | 2 | Let $\mathscr{T}$ be a triangulated category, and $\mathscr{A}$ be a right admissible subcategory, which means that $i\_{\mathscr{A}} : \mathscr{A} \rightarrow \mathscr{T}$ has a right adjoint $i\_{\mathscr{A}}^R$. Let $\mathscr{B}$ another subcategory of $\mathscr{T}$ (not necessarily right/left admissible). Consider ... | https://mathoverflow.net/users/91572 | Admissibility of intersection of subcategories | It is possible. Given $\mathscr{T}$, $\mathscr{A}$ and $i\_\mathscr{A}^R$, it is often possible to find a subcategory $\mathscr{B}$ of $\mathscr{T}$ so that $\mathscr{A}\cap\mathscr{B}=0$ (and so certainly $\mathscr{A}\cap\mathscr{B}$ is a right admissible subcategory of $\mathscr{B}$), but $i\_\mathscr{A}^R(\mathscr{B... | 3 | https://mathoverflow.net/users/22989 | 362227 | 152,361 |
https://mathoverflow.net/questions/362226 | 2 | Is there $n\in\mathbb{N}$ and a collection ${\cal C}$ of subsets of $\{1,\ldots,n\}$ with the following properties?
1. $|{\cal C}| = n$,
2. $|c| > 1$ for all $c\in {\cal C}$,
3. $c\neq d \in {\cal C} \implies |c\cap d|=1$, and
4. $\big|\{|c|: c\in {\cal C}\}\big| > 2$.
| https://mathoverflow.net/users/8628 | Intersecting subsets of $\{1,\ldots,n\}$ | No, there isn't. This is essentially the dual version of the [De Bruijn-Erdos theorem](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(incidence_geometry)) if the elements of $\mathcal C$ are the points, and the elements from $\{1,\ldots,n\}$ are the lines. The original proof is [here](https://www.r... | 7 | https://mathoverflow.net/users/955 | 362229 | 152,362 |
https://mathoverflow.net/questions/362234 | 7 | Given a (smooth, orientable) n-dimensional manifold $M$ with two (pseudo-)Riemannian metrics $g\_{1}$ and $g\_{2}$ of the same signature that induce the same Levi-Civita connection and satisfy $\sqrt{|\det{g\_1}|}\mathrm{d}x^{1}\wedge...\wedge\mathrm{d}x^{n} = \sqrt{|\det{g\_2}|}\mathrm{d}x^{1}\wedge...\wedge\mathrm{d}... | https://mathoverflow.net/users/159134 | Are two metrics with the same Levi-Civita connection and the same volume form identical? | The standard Minkowski metric, transformed by scaling one variable by 1/2, and another by 2, has the usual volume form, and the same Levi--Civita connection (which depends only on the affine structure).
| 15 | https://mathoverflow.net/users/13268 | 362235 | 152,363 |
https://mathoverflow.net/questions/362237 | -2 | Hoping someone may be able to point me in the right direction so I can research this topic further.
Scenario: You have a vector field (either 2D or 3D) and you wish to find the shortest path between two points located within the vector field.
For example imagine an object floating on the ocean surface. The object h... | https://mathoverflow.net/users/159135 | Shortest Path finding in vector fields (2D and 3D) | The problem you are struggling to express here is known as Zermelo's navigation problem, and you are not correctly writing it down. You can read about it in Vel Jurdjevic's excellent book **Geometric Control Theory**.
| 2 | https://mathoverflow.net/users/13268 | 362249 | 152,365 |
https://mathoverflow.net/questions/362250 | 1 | Let $\nu$ be a *finite* Borel measure on $\mathbb{R}^n$ and define the shift operator $T\_a$ on $L^p\_{\nu}(\mathbb{R}^n)$ by $f\to f(x+a)$ for some fixed $a\in \mathbb{R}^n-\{0\}$. Suppose moreover that
$\nu$ is absolutely continuous wrt the Lebesgue measure $m$ and let
$
\frac{d \nu}{dm}(x)= h(x).
$
In this case, ... | https://mathoverflow.net/users/36886 | Operator norm of shift operator for finite measure spaces | Well, for large $a$ the norm goes to infinity. Find a ball $B$ such that $\nu(B) > \nu(\mathbb{R}^n) - \epsilon$ and consider the characteristic function of $B$ shifted by $-a$, for any $a$ greater than the radius of $B$. Its $L^2$ norm is at most $\sqrt{\epsilon}$, but after shifting by $a$ its norm is $> \sqrt{\nu(B)... | 6 | https://mathoverflow.net/users/23141 | 362260 | 152,368 |
https://mathoverflow.net/questions/362258 | 1 | Let $\lambda\_k,\mu\_k\in\mathbb R\_{\ge0}$ $(k\ge1)$ be nonnegative real numbers, let $S=\mathbb Z\_{\ge0}$ be the nonnegative integers, let $T=\mathbb R\_{\ge0}$ be the nonnegative real numbers and consider the continuous-time Markov chain $X=(X\_t)\_{t\in T}$ on $S$ with rates
$$Q(n,n+k)=(n+1)\lambda\_k\quad(k\ge1),... | https://mathoverflow.net/users/110883 | Existence of Markov chain on nonnegative integers with specified rates | Define first the modified rates
$$ \tilde Q(n,m) = \frac{Q(n,m)}{n + 1} \, . $$
Clearly, $\tilde Q(n, n+k) = \lambda\_k$, and $\tilde Q(n, n-k) \leqslant \mu\_k$. Assuming that $\lambda\_k$ is summable (otherwise the problem is clearly ill-posed), $\tilde Q$ corresponds to a unique conservative continuous-time Markov c... | 1 | https://mathoverflow.net/users/108637 | 362263 | 152,370 |
https://mathoverflow.net/questions/362254 | 5 | Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive degrees. Using K-flat resolutions we can define the [derived tensor product](https://stacks.math.columbia.edu/tag/06XY):
$$
... | https://mathoverflow.net/users/20883 | Cohomology of derived tensor product of complexes and Künneth spectral sequence | This isn't a complete answer, but here are some thoughts. I'm sceptical that it is often true.
If $R$ is semisimple then the result holds, because every module over a semisimple ring is projective.
We may as well take $V$ and $W$ to be complexes of projectives, and then ask when is $H(V\otimes W)\cong H(V)\otimes ... | 5 | https://mathoverflow.net/users/159143 | 362267 | 152,373 |
https://mathoverflow.net/questions/362271 | 1 | Fix $\epsilon>0$ and let $(\Omega,F,F\_t\mathbb{P})$ be a stochastic base. Is there a (Markov) diffusion process $X\_t$ satisfying an SDE of the form:
$$
d X\_t = \mu(t,X\_t)dt + \Sigma(t,X\_t)dW\_t, X\_0^x
$$
such that the (random) function $f\_X:x\to X\_1^x$ satisfies
$$
\mathbb{P}\left(
\int\_{x \in \mathbb{R}^n} |... | https://mathoverflow.net/users/36886 | Probability that a stochastic flow is near $0$ | $\newcommand\ep\epsilon$ $\newcommand\R{\mathbb R}$ $\newcommand\Si{\Sigma}$
Let
$$X^x\_t:=xe^{-ct|x|}$$
for some real $c>0$ and allreal $t\ge0$ and $x\in\R^n$. Then $X^x\_0=x$ for all $x$ and
your SDE holds with $\mu(t,x)=-c|x|xe^{-ct|x|}$ and $\Si(t,x)=0$. Moreover,
$$\int\_{\R^n}|X^x\_1|\,dx=\int\_{\R^n}|x|e^{-c|... | 1 | https://mathoverflow.net/users/36721 | 362272 | 152,376 |
https://mathoverflow.net/questions/362281 | 0 |
>
> Let two collections of random variables $\{X\_i\}$ and $\{Y\_i\}$ be independent and let $\{Y\_i\}$ be i.i.d. Then
> $$\max\_{1\leq i\leq n}(X\_i+Y\_1)\preceq \max\_{1\leq i\leq n}(X\_i+Y\_i).$$
> where $\preceq$ is stochastic domination.
>
>
>
I think it needs to find a coupling such that $X\_i'=\max\_{1\... | https://mathoverflow.net/users/168083 | How to show $\max_{1\leq i\leq n}(X_i+Y_1)\preceq \max_{1\leq i\leq n}(X_i+Y_i)$? | Conditioning on the $X\_i$'s and using the independence of the $Y\_i$'s from the $X\_i$'s, we reduce the consideration to the case when $X\_i=x\_i$ for some real $x\_i$'s and all $i$. So, the statement of interest reduces to this:
$$P(x\_i+Y\_1\le y\quad \forall i)\ge P(x\_i+Y\_i\le y\quad \forall i)$$
for all real $y... | 3 | https://mathoverflow.net/users/36721 | 362284 | 152,379 |
https://mathoverflow.net/questions/362288 | 12 | I have been trying to read a physics textbook on Quantum Field theory. There seems to me to be a bit of a disconnect in most texts I have looked at between quantum mechanics and quantum field theory, in the passage from multiparticle wave functions to fields. I'm curious if there is a physical reason for this. I'm aski... | https://mathoverflow.net/users/7108 | Is there a physical reason that fields in QFT are globally defined? | You are describing what is commonly known as *second quantization* (as you probably already realize). In a nutshell, the main mathematical statement behind second quantization is the following: the algebra $\mathcal{A}\_{particles}$ generated by creation/annihilation operators acting on the Fock space $\mathfrak{F}\_{p... | 9 | https://mathoverflow.net/users/2622 | 362290 | 152,383 |
https://mathoverflow.net/questions/361800 | 1 | How one can solve the following stochastic optimization problem?
\begin{align}
\max\quad& \mathbb{E}[\mathbf{1}^{\mathrm{T}}X]\\
\text{s.t.} \quad& \mathbb{E}[\mathbf{A}X]\leq\mathbf{1}\_{m\times 1}\qquad(\ast),
\end{align}
where $\mathbf{A}$ is a random matrix and consequently the solution vector $X$ is also a random ... | https://mathoverflow.net/users/68835 | How to solve this stochastic optimization problem? | This is a slight addition to RaphaelB4's answer, to explain one approach to extending the discrete case to the general case. This approach begins by thinking about the discrete case in slightly more detail.
Note that RaphaelB4's optimization problem only has one input parameter: the matrix $\hat{A}$ formed by concate... | 3 | https://mathoverflow.net/users/2363 | 362292 | 152,384 |
https://mathoverflow.net/questions/362275 | 3 | Let $\mathcal{V}$ be a monoidal closed (complete, cocomplete, reasonable...) category.
Let $\mathsf{T}$ be an enriched monad over $\mathcal{V}$. The forgetful functor $\mathsf{U}: \mathsf{Alg}(\mathsf{T}) \to \mathcal{V}$ is tautologically monadic in $\mathcal{V}$-Cat. If we pass to the [underlying categories](https:... | https://mathoverflow.net/users/104432 | Is monadicity preserved by the underlying functor? | Following up on my comment, I think the direct proof is easiest - it’s just an exercise in translating classical definitions into the enriched setting. Let’s call the underlying monad $T\_0$.
For objects, observe that an algebra of $T$ is given by $a: I \to V(TA, A)$ (which is a map $TA \to A$ in the underlying categ... | 5 | https://mathoverflow.net/users/75783 | 362296 | 152,385 |
https://mathoverflow.net/questions/362294 | 3 | I apologize if this notion is well-known, but I couldn't find anything useful and I am not sure what key words to look for.
Suppose we have a lattice $\Lambda \subset \mathbb{Z}^n$, given by in the form
$$\displaystyle \Lambda = \left\{M \mathbf{u} : \mathbf{u} \in \mathbb{Z}^n \right \}$$
for some matrix $M$ wi... | https://mathoverflow.net/users/10898 | Given a lattice in $\mathbb{Z}^n$, what can be said about its 'transpose' lattice? | Let $V = \mathbb{Z}^n/(M \mathbb{Z}^n), V' = \mathbb{Z}^n/(M^T \mathbb{Z}^n)$. I claim that $V \simeq V'$ as abelian groups.
By the [classification of finite abelian groups](https://en.wikipedia.org/wiki/Finitely_generated_abelian_group#Classification), $V \simeq \oplus \mathbb{Z}/d\_i \mathbb{Z}$ for some $d\_1 | d\... | 3 | https://mathoverflow.net/users/44191 | 362297 | 152,386 |
https://mathoverflow.net/questions/362299 | -1 | Let $X$ be separable Banach space and $\{x\_n\}$ be a bounded sequence, relatively weakly compact sequence in $X$. we set $y\_n=\frac{1}{n}\sum\_{i=1}^{n}{x\_i}$, then (together with the Krein and Eberlein-Smulian theorems), we can assume that there exists a **subsequence** of $\{y\_n\}$ converges weakly to some elemen... | https://mathoverflow.net/users/152650 | Existence of weak limit for bouded sequence $\{y_n\}$ such that for every weak limit point $\{y_n\}$ must equal $y$ | If $y\_n$ does not converge weakly to $y$ then there is a weakly open neighborhood $U$ of $y$ and a subsequence $y\_{n\_k} \notin U$. By weak compactness this subsequence has a weak limit point $z \notin U$. But $z$ is then also a weak limit point of the original sequence $y\_n$, contradicting uniqueness.
| 1 | https://mathoverflow.net/users/4832 | 362300 | 152,387 |
https://mathoverflow.net/questions/360771 | 7 | Let $M$ be a von Neumann algebra, and let $\Delta$ be a unital normal $\*$-homomorphism $M \rightarrow M \mathbin{\bar\otimes} M$ that satisfies the coassociativity condition $(\Delta \mathbin{\bar\otimes} \mathrm{id}) \circ \Delta = (\mathrm{id} \mathbin{\bar\otimes} \Delta ) \circ \Delta$. Assume that $M$ is an $\ell... | https://mathoverflow.net/users/66833 | Characterizing discrete quantum groups | Also the implication (2) $\Rightarrow$ (1) holds and can be proven as follows.
Denote by $\mathcal{C}$ the category of all finite dimensional, nondegenerate $\*$-representations of $M$. The morphisms are the intertwining linear maps. Turn $\mathcal{C}$ in a $C^\*$-tensor category by defining $\pi\_1 \otimes \pi\_2$ t... | 9 | https://mathoverflow.net/users/159170 | 362309 | 152,391 |
https://mathoverflow.net/questions/361970 | 11 | For a set $X$ we endow the set $\omega^X$ of all functions from $X$ to $\omega$ with the natural partial order $\le$ defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$.
A function $f:\omega^\omega\to\omega^X$ is called *monotone* if for any $\alpha\le\beta$ in $\omega^\omega$ we have $f(\alpha)\le f(\beta)$ in $... | https://mathoverflow.net/users/61536 | A monotone countably cofinal function from $\omega^\omega$ to $\omega^{\omega_1}$ | The answer is no, if you require that the monotone function $F \, \colon \, \omega^\omega \rightarrow \omega^{\omega\_1}$ is total. I will use fairly standard notation, i.e. $f,g \in \omega^\omega$, $\alpha, \beta \in \omega\_1$ and $k,m,n \in \omega$.
The proof works as follows: Towards a contradiction assume that s... | 5 | https://mathoverflow.net/users/134910 | 362314 | 152,393 |
https://mathoverflow.net/questions/346219 | 5 | Let $A$ be an [abelian von Neumann algebra](https://en.wikipedia.org/wiki/Abelian_von_Neumann_algebra) and $G$ a countable group acting on $A$. In the literature we meet usually two kinds of crossed product $A \rtimes G$ being a factor:
* if the action is (essentially) free then $A \rtimes G$ is a factor iff the act... | https://mathoverflow.net/users/34538 | Unusual crossed product constructions being factors | The following provides a necessary and sufficient condition for an arbitrary crossed product von Neumann algebra $L^\infty(X) \rtimes G$ to be a factor. As a corollary, I include a simpler criterion for actions that preserve a probability measure.
First note that a criterion for arbitrary actions necessarily has to r... | 8 | https://mathoverflow.net/users/159170 | 362329 | 152,395 |
https://mathoverflow.net/questions/362336 | 3 | I've came across an identity once (I don't remember where) concerning convolutions of Gaussian measures. If I'm not mistaken, this identity was
\begin{eqnarray}
(\mu\_{C}\*f)(y) = \exp\bigg{[}\frac{1}{2}\bigg{(}\frac{\partial}{\partial x}, C\frac{\partial}{\partial x}\bigg{)}\bigg{]}f(x)\bigg{|}\_{x=y} \tag{1}\label{1... | https://mathoverflow.net/users/150264 | Identity on convolution with Gaussian measure | Upon Fourier transformation the convolution becomes a product of the Fourier transform ${\cal F}[f]$ of the function $f$ and the Fourier transformed Gaussian measure, which is again a Gaussian with covariance matrix $C^{-1}$,
$${\cal F}[\mu\_{C}\*f](k) = \exp\left(-\tfrac{1}{2}\sum\_{n,m}k\_n C\_{nm} k\_m\right){\cal F... | 1 | https://mathoverflow.net/users/11260 | 362341 | 152,400 |
https://mathoverflow.net/questions/362322 | 2 | It is well known that given a Fibered category $P\_F: E \rightarrow C$ with a *cleavage* $K$ we can construct a pseudofunctor $F\_K: C^{op} \rightarrow Cat$. Now if one chooses a **different cleavage** $L$ but consider the **same fibered category** $P\_F$ then how do $F\_K$ and $F\_L$ are related? (Note here $F\_L$ is ... | https://mathoverflow.net/users/86313 | Is there any relation between two pseudofunctors associated to two different cleavages of the same fibered category? | Two different cleavages produce isomorphic pseudofunctors.
This follows immediately from Theorem 8.3.1
in Borceux's Handbook of Categorical Algebra 2.
Specifically, part (1) of this theorem states
that for any pseudofunctors P and Q we have an isomorphism
of categories PsFun(P,Q)→Cart(φ(P),φ(Q)).
Now if P and Q a... | 1 | https://mathoverflow.net/users/402 | 362342 | 152,401 |
https://mathoverflow.net/questions/361798 | 18 | Let $\mathbb{O}$ be the octonion algebra (say over $\mathbb{R}$) and let $J\_{3}(\mathbb{O})$ be the set of $3 \times 3$ hermitian matrices with octonion coefficients, that is:
$$ J\_3(\mathbb{O}) = \left\{ \begin{pmatrix} \lambda\_1 & a & b \\ \overline{a} & \lambda\_2 & c \\ \overline{b} & \overline{c} & \lambda\_3... | https://mathoverflow.net/users/37214 | Subgroup $\mathrm{E}_6$ generated by $\mathrm{Spin_7}$ and $\mathrm{SL}_3$ | **N.B.:** I am revising my response for clarity. (The actual answer to the question asked by the OP is still the same, but I think that this re-organization, particularly at the end, makes the structure of the argument for the answer more clear. I was inspired to do this because some people had some difficulty followin... | 6 | https://mathoverflow.net/users/13972 | 362343 | 152,402 |
https://mathoverflow.net/questions/357043 | 2 | My questions concerns Definition 1.2 of an orbital integral in the paper [Orbital integrals on General Linear Groups](http://rcluckers.perso.math.cnrs.fr/prints/CluDen.pdf) by Cluckers and Denef. I will recall the definition below, but my question is: how does Definition 1.2 relate to the usual definition of an orbital... | https://mathoverflow.net/users/48554 | Orbital integral in Cluckers and Denef | My sense is that this paper is actually not (at least, not in any straightforward way) related to the orbital integrals one thinks of in connection with the Fundamental Lemma, for example. You see, the key assumption in the Cluckers-Denef paper is that $X$ is a homogeneous $G$-space, that is, $G({\mathbb C})$ acts on $... | 3 | https://mathoverflow.net/users/15073 | 362352 | 152,406 |
https://mathoverflow.net/questions/362338 | 4 | I have a very basic question regarding algebraic model theory. I am trying to read [*Espaces de Berkovich, polytopes, squelettes et théorie des modèles*](https://doi.org/10.1142/S1793744212500077) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=3020334)) by Antoine Ducros. The relevant section is Section 0.31. ... | https://mathoverflow.net/users/112369 | What is the definable functor associated to an algebraic scheme (model theory of valued fields) | This follows by elimination of imaginaries in algebraically closed fields. Given a finite affine cover of your scheme, one obtains a functor to Set by gluing the affine charts. This is indeed not definable, but a quotient of a definable set by a definable equivalence relation. Elimination of imaginaries tells you preci... | 3 | https://mathoverflow.net/users/8145 | 362357 | 152,408 |
https://mathoverflow.net/questions/362317 | 5 | Given a $n \times n$ real Cauchy like matrix $C$, i.e., for real vectors $r$, $s$, $x$, $y$
$$
C\_{ij} = \frac{r\_i s\_j}{ x\_i - y\_j}
$$
>
> Can a Cauchy-like $C$ be orthogonal, i.e., $C C^T = I$ for $n > 2$?
>
>
>
There exists such an orthogonal $C$ for $n = 2$ , $x = [1,0.4]$, $y = [6.25,0.625]$, $r = [... | https://mathoverflow.net/users/51478 | Orthogonal Cauchy-like matrix | Please give a look at [this short note](https://www.researchgate.net/publication/341979812_Do_orthogonal_Cauchy-like_matrices_exist).
| 4 | https://mathoverflow.net/users/159193 | 362361 | 152,409 |
https://mathoverflow.net/questions/362367 | 1 | Do Fourier transform properties still hold in the case of fractional derivatives ?
i.e I have seen many times that some lectures define fractional derivative as :
$$\frac{d^{\alpha}}{dx^{\alpha}}f=\mathscr{F}^{-1}\big[\mathscr{F}[f(x)](w)\cdot w^{\alpha}\big](x)$$
Indeed fractional derivatives of exponentials d... | https://mathoverflow.net/users/158886 | Fourier analysis and fractional calculus | Too long for a comment. Your spelling of the name of the mathematician Joseph Fourier is incorrect. Also your formula is almost impossible to read: your fractional derivative on the lhs is the Fourier multiplier $(i\xi)^\alpha$ and thus formally you find
$$
\left(\left(\frac{d}{dx}\right)^\alpha f\right)(x)=\int e^{i x... | 3 | https://mathoverflow.net/users/21907 | 362372 | 152,413 |
https://mathoverflow.net/questions/362369 | 5 | A binar is simply a set $S$ equipped with a single binary operation $\*$. A power-associative binar is a binar where the subalgebra generated by a single element is associative. Equivalently, they can be axiomatized with the infinite set of equations, $\{(x\*x)\*x=x\*(x\*x), ... \}$. Is there some finite set of axioms ... | https://mathoverflow.net/users/43439 | Is the class of power-associative binars finitely axiomatizable? | No.
Indeed, let $\mathcal{V}\_n$ be the variety of magmas generated by the relating identities with variable $y$ saying that for every $k\le n$, all products of $k$ copies of $y$ are equal. Since the variety of power-associative magmas is $\bigcap\_n \mathcal{V}\_n$, a negative answer to the question is equivalent to... | 13 | https://mathoverflow.net/users/14094 | 362375 | 152,415 |
https://mathoverflow.net/questions/362370 | 2 | My question is whether there exist algorithms to determine a minimal (or close to minimal) cover of a Riemannian manifold (or some subset thereof) with balls of a fixed radius r>0.
| https://mathoverflow.net/users/nan | Algorithms for finding minimal ball covers | As it is a version of set-cover, the problem is NP-hard.
The following is a special case:
>
> **(1)** Chepoi, Victor, and Bertrand Estellon. "Packing and covering $\delta$-hyperbolic spaces by balls." In *Approximation, Randomization, and Combinatorial Optimization.* Algorithms and Techniques, pp. 59-73. Springer, ... | 1 | https://mathoverflow.net/users/6094 | 362376 | 152,416 |
https://mathoverflow.net/questions/362320 | 1 | Let $X\neq\varnothing$ be a set and let ${\cal E}\subseteq {\cal P}(X)\setminus\{\varnothing\}$ be a collection of non-empty subset. We say that a map $f: {\cal E}\to X$ is a *chromatic self-map* if
1. $f(e) \in e$ for all $e\in {\cal E}$, and
2. if $e\_1\neq e\_2 \in {\cal E}$ and $e\_1\cap e\_2 \neq \varnothing$, t... | https://mathoverflow.net/users/8628 | Chromatic self-maps on finite complete linear hypergraphs | By domotorp's answer <https://mathoverflow.net/q/362229> to a previous question, De Bruijn-Erdos theorem characterizes complete linear collections as near-pencils or finite projective planes.
The image of a chromatic self-map is a system of distinct representatives. This is exactly the setting of Hall's marriage theo... | 1 | https://mathoverflow.net/users/24076 | 362379 | 152,417 |
https://mathoverflow.net/questions/362383 | 2 | Let $A, A^{-1} \in \mathbb{R}^{n \times n}$ be known matrices. Suppose we have an invertible matrix $B \in \mathbb{R}^{(n+1) \times (n+1)}$ of the following form:
$$B = \begin{bmatrix}
A & b\\
b^T & 1
\end{bmatrix}$$
where $b$ is a column vector and $c$ is a row vector. **How can I calculate matrix $B^{-1}$ from... | https://mathoverflow.net/users/158426 | Inverse of a larger matrix where the inverse of the submatrix is known | You know $\begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix}^{-1} = \begin{pmatrix} A^{-1} & 0 \\ 0 & 1 \end{pmatrix}$, and from there you can make two successive rank-$1$ modifications, first adding $b$ along the last column, then $c$ along the last row. So, using Sherman–Morrison twice should work.
| 2 | https://mathoverflow.net/users/88133 | 362386 | 152,420 |
https://mathoverflow.net/questions/362381 | 5 | In JP May's paper *A general algebraic approach to Steenrod operations*, Steenrod operations are constructed in wide generality. In this context, it is not necessarily true that negative Steenrod operations are zero. However, I could not find any examples (not in May's paper, not even elsewhere), where one actually pro... | https://mathoverflow.net/users/151804 | Examples of non-zero negative Steenrod operations | Notice the switch in grading from homology to cohomology on May's page 182: $P^s(x) = P\_{-s}(x)$. Operations that raise degree when graded homologically lower degree when graded cohomologically. A large part of the motivation for the paper was to give a common framework for the Dyer-Lashof operations in the homology o... | 8 | https://mathoverflow.net/users/14447 | 362388 | 152,421 |
https://mathoverflow.net/questions/362395 | 3 | Let $R$ be a regular local ring and let $P$ be a prime ideal of height $h$ in $R$.
Is it always the case that $P$ contains a regular sequence of lenght $h$?
This is clear if $h$ is $0,1$ or $\dim R$.
Since $R$ is regular, the localization $R\_P$ is regular local of dimension $h$, and thus contains an $R\_P$-regular... | https://mathoverflow.net/users/86006 | Do height $h$ prime ideals in regular local rings contain regular sequences of length $h$? | The answer is yes. In fact, more generally, if $R$ is a Cohen-Macaulay ring, then the height of any prime ideal in it is equal to its depth, which is just the length of a maximal regular sequence contained in P. See for example Theorem 2.1.2 of the book
Cohen-Macaulay rings by Bruns and Herzog.
| 3 | https://mathoverflow.net/users/159211 | 362401 | 152,425 |
https://mathoverflow.net/questions/362408 | 1 | Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say that ${\cal A}\subseteq [\omega]^\omega$ is an *almost disjoint family* if $A \neq B \in {\cal A}$ implies $|A\cap B|< \aleph\_0$.
Let $X\neq\varnothing$ be a set and let ${\cal E}\subseteq {\cal P}(X)\setminus\{\varnothing\}$ be a co... | https://mathoverflow.net/users/8628 | Chromatic self-maps for almost disjoint families | If such an $f$ exists then $\mathcal A\_n=\{e\in\mathcal A:f(e)=n\}$ is a collection of pairwise disjoint subsets of $\omega$, so $\mathcal A\_n$ is countable, so $\mathcal A=\bigcup\_n\mathcal A\_n$ is countable. So the answer is "no" if $\mathcal A$ is uncountable. Of course it is "yes" if $\mathcal A$ is countable.
... | 2 | https://mathoverflow.net/users/43266 | 362410 | 152,427 |
https://mathoverflow.net/questions/362333 | 1 | The following is a question I posted about a week ago on Maths stackexchange [there](https://math.stackexchange.com/questions/3694924/about-the-type-of-the-polarization-of-an-abelian-variety), but it didn't bring any discussion nor comment. For this reason I am posting it here also.
Let $X$ be an abelian variety of d... | https://mathoverflow.net/users/125617 | About the type of a polarization of an abelian variety | Let $\lambda: A\rightarrow A^{\vee}$ be any polarization of degree prime to the characteristic, not necessarily self-dual.
There exists an $\lambda^{\vee} : A^{\vee}\rightarrow A$ such that $\lambda^{\vee}\circ \lambda = [n]$ for some integer $n$ where $n$ is invertible in $k$.
So $\ker(\lambda) \subset A[n] \simeq \l... | 3 | https://mathoverflow.net/users/110362 | 362432 | 152,433 |
https://mathoverflow.net/questions/339447 | 11 | I was surprised to learn [here](https://strangenewuniverse.wordpress.com/2011/10/29/infinite-trees-and-konigs-lemma/) that the man responsible for "König's Lemma" was Hungarian, and spelled his last name Kőnig (with a different accent on the o), presumably with the same accent that occurs in Paul Erdős' last name. I st... | https://mathoverflow.net/users/3199 | Spelling König's Lemma | In [A tale of three eras: The discovery and rediscovery of the Hungarian Method](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwiE36aWl_DpAhWNTsAKHbOZA6Y4KBAWMAV6BAgFEAE&url=http%3A%2F%2Fwww.math.toronto.edu%2Fmccann%2F1855%2FKuhnEJOR12.pdf&usg=AOvVaw3VvgdtGwwEQHNMUSQft7Qy)
Harold W. Kuhn ad... | 4 | https://mathoverflow.net/users/31310 | 362440 | 152,436 |
https://mathoverflow.net/questions/362415 | 1 | Let $k,m$ and $r$ be positive integers.
Define
$$\Omega(k,m,r) = \binom k {m-2r}\binom {k-m+2r} r$$
and
$$\Omega(k,m) = \sum\_{r=\max\{0,m-k\}}^{[\frac{m}{2}]}\Omega(k,m,r).$$
>
> **Question.**
> 1. Is $\Omega(k,m)$ has a simple formula?
> 2. Is $\frac{\sum\_{r=\max\{0,m-k\}}^{[\frac{m}{2}]}r\Omega(k,m,r)}{(... | https://mathoverflow.net/users/157823 | simple formula for a finite sum of multinomial numbers | First notice that
$$\Omega(k,m,r) = \binom{k}{m-2r,r,k-m+r}.$$
It follows that $\Omega(k,m)$ equals the coefficient of $x^{m-k}$ in $(1+x+x^{-1})^k$, which is the same as the coefficient of $x^{m}$ in $(1+x+x^2)^k$, also known as the [trinomial coefficient](https://en.wikipedia.org/wiki/Trinomial_triangle) $\binom{k}{m... | 2 | https://mathoverflow.net/users/7076 | 362445 | 152,437 |
https://mathoverflow.net/questions/362431 | 5 | (a) Do there exist integers $x$ and $y$ such that $x^3+x^2y^2+y^3=7$ ?
(b) Is this equation belongs to some family $F$ of equations for which there is a known algorithms for testing if they have an integer solution? For example, such algorithms are known for quadratic equations (in $n$ variables) and for cubic equati... | https://mathoverflow.net/users/89064 | $x^3+x^2y^2+y^3=7$, and solvable families of Diophantine equations | (a) No. There are no integer solutions. The curve $C$ you give has genus $3$ and it has an obvious automorphism $\phi(x,y) = (y,x)$. The quotient curve is an elliptic curve. In particular, if you let $X = -(x+y)$ and $Y = xy$, then the equation becomes $E : Y^{2} + 3XY = X^{3} + 7$. So any integer point on your curve g... | 16 | https://mathoverflow.net/users/48142 | 362446 | 152,438 |
https://mathoverflow.net/questions/362453 | 2 | Actually the question has more details than what it says in the title. Sorry about that I may described the question wrongly.
Let $X\_1^n, X\_2^n,\dots$ be i.i.d. Bernoulli random variables with parameter $\lambda/n$, i.e. $X\_1^n \overset{d}{=}\operatorname{Be}(\lambda/n)$ with fixed $\lambda > 0$. Consider $$
T\_i^... | https://mathoverflow.net/users/157350 | Does sum of i.i.d. Bernoulli random variables with parameter $\lambda/n$ asymptotically converge to Gamma distribution? | $\newcommand\la\lambda$ $\newcommand\nt{\lfloor nt\rfloor}$
For any natural $i,n,k$, let $S^n\_k:=X^n\_1+\dots+X^n\_k$, with $S^n\_0:=0$. Then for any real $t>0$
\begin{align}
P(T^n\_i/n>t)&=P(T^n\_i>\nt) \\
&=P(S^n\_{\nt}<i)\to P(S\_{\la t} <i) \\
&=\frac{\la^i}{\Gamma(i)}\int\_t^\infty u^{i-1} e^{-\la u}\,du
\end{a... | 3 | https://mathoverflow.net/users/36721 | 362463 | 152,444 |
https://mathoverflow.net/questions/362469 | 0 | Let the function $\Lambda : [0,T] \times \mathbb{R^n} \times \mathbb{R^n} \to \mathbb R$ be continuously differentiable. Then the integral functional $I(x) = \int\_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ is finite value for all $x \in C^{2}[0 ,2]$ .
My question: Is the integral functional $I(x) = \int\_{0}^{... | https://mathoverflow.net/users/108824 | Is the integral functional $I(x) = \int_{0}^{T} \Lambda (t , x(t), \dot{x} (t)) \; dt $ locally lipschitz on the space $C^2 [0 ,T] $? | The answer is no. Indeed, let $n=1$, $T=1$, and $\Lambda(t,u,v)\equiv v^2$, so that
\begin{equation}
I(x)=\int\_0^1 x'(t)^2\,dt.
\end{equation}
Let $x\_0:=0$ and, for each real $b\ge1$ and all $t\in[0,1]$,
\begin{equation}
y\_b(t):=e^{-bt}.
\end{equation}
Then
\begin{equation}
\|y\_b-x\_0\|=\|y\_b\|=\int\_0^1 |y... | 2 | https://mathoverflow.net/users/36721 | 362471 | 152,446 |
https://mathoverflow.net/questions/362473 | 2 | As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form
$$\forall z \in \mathbb{C}, \quad f(z) = a z + b$$
with $a \in \mathbb{C}\backslash \{0\}$ and $b \in \mathbb{C}$. This gives an esp... | https://mathoverflow.net/users/39261 | 3D similarities and quaternions? | Call a quaternion whose scalar part is zero a vector quaternion. We shall denote the vector quaternions as $\mathbb R^3$. Given $q = w + xi + yj + zk$, we shall define $q^\*$ (called the "conjugate" of $q$) to be $w - xi - yj - zk$.
If $q$ is a unit quaternion, then $v \in \mathbb R^3\mapsto qvq^\*$ is a rotation. Al... | 3 | https://mathoverflow.net/users/75761 | 362495 | 152,450 |
https://mathoverflow.net/questions/361942 | 2 | Let $G=\operatorname{PGL}\_2(p)$, where $p\ge 5$ is a prime. Is there a generating triple of involutions $(x,y,z)$ of $G$ such that $|xy|=p$, $|xz|=p+1$ and $|yz|=p-1$? That means, $\langle x,y,z\rangle=G$ with
1. $x^2=y^2=z^2=1$;
2. $\langle x,y\rangle \cong D\_{2p}$;
3. $\langle x,z\rangle\cong D\_{2(p+1)}$;
4. $\l... | https://mathoverflow.net/users/131819 | Existence (or the number) of generating triple of involutions of $\operatorname{PGL}_2(p)$ with some conditions | Such triples exist, I think.
First, embed $PGL\_2(p)$ in $S\_{p+1}$ through its action on $1$-spaces from ${\mathbf F}\_p^2$. This maps elements of order $p+1$ in $PGL\_2(p)$ to $(p+1)$-cycles, and elements of order $p-1$ to $(p-1)$-cycles. In particular, every element of order $p+1$ or $p-1$ gets mapped to an odd p... | 4 | https://mathoverflow.net/users/36466 | 362504 | 152,451 |
https://mathoverflow.net/questions/362500 | 8 | Let $X,Y$ be path-connected finite CW complexes with base points $x\_0,y\_0$, let $f\colon X\to Y$ be a surjective continuous map, such that for every $y\in Y$, the fiber $f^{-1}(y)$ is path connected. In this case, is the induced map $$f\_\*\colon\pi\_1(X,x\_0)\to\pi\_1(Y,y\_0)$$ on topological fundamental groups nece... | https://mathoverflow.net/users/nan | Morphism with connected fibers induce surjection on fundamental groups? | This answer is a complement to Andy's one. If $X$ and $Y$ are complex algebraic varieties then you have the following fact (see more generally Kollár "Shafarevich maps and automorphic forms" Proposition 2.10.2):
If $X$, $Y$ are irreducible algebraic varieties with $Y$ normal and $f:X\to Y$ is a dominant morphism such... | 3 | https://mathoverflow.net/users/116075 | 362514 | 152,453 |
https://mathoverflow.net/questions/362511 | 1 | Fix $\mathbf t \in \mathbb{R}\_+^d$. I am looking for a r-atomic measure $\nu$ on $\mathbb{R}\_{+}^{d}$ that solves the following 'moment' conditions.
$$\forall\, \mathbf i \in \mathbb{N}^{d} \, s.t \, \lvert\mathbf i \rvert \le n \in \mathbb N, \; \mu\_{\mathbf i}^{\mathbf{t}} = \int\limits\_{\mathbb R\_+^d} \frac{\... | https://mathoverflow.net/users/143783 | Can this be translated to a truncated multivariate moment problem? | Changing the variables from $\mathbf r=(r\_j)$ to $\mathbf s=(s\_j)$, where
$$s\_j:=g(\mathbf r)\_j:=\frac{r\_j}{1+\sum\_i t\_ir\_i},$$
we have
$$r\_j:=\frac{s\_j}{1-\sum\_i t\_is\_i}.$$
So, the transformation $g$ is a homeomorphism of $\mathbb R\_+^d$ onto
$$\Sigma:=\Big\{\mathbf s\in\mathbb R\_+^d\colon \sum\_i t\... | 1 | https://mathoverflow.net/users/36721 | 362518 | 152,456 |
https://mathoverflow.net/questions/362502 | 8 | Fix a Young subgroup $H\_\lambda \subseteq \mathcal S\_n$, where $\lambda \vdash n$ is a partition of $n$ with $k$ blocks. Inside the group algebra $\mathbb C[\mathcal S\_n]$, consider the idempotent
$$\varepsilon = \frac{1}{|H\_\lambda|}\sum\_{h \in H\_\lambda} h.$$
The double cosets $H\_\lambda \backslash \mathcal... | https://mathoverflow.net/users/30138 | Structure constants for the double coset algebra of a Young subgroup | There is a combinatorial rule for the structure constants. It appears in Section 2 of <https://arxiv.org/abs/1104.1959>, although it is not immediately clear that it is indeed what you are looking for.
Suppose that $V$ is a vector space of dimension at least $l(\lambda)$. Then in the setting of Schur-Weyl duality, th... | 6 | https://mathoverflow.net/users/159272 | 362532 | 152,460 |
https://mathoverflow.net/questions/362509 | 2 | I'm not sure this is an appropriate question for this site but I've tried math stack exchange and got no answers. Also, this problem arose in one of my research problems, so I'm stating it here.
The strong operator topology is defined on [Simon and Reed's book](https://rads.stackoverflow.com/amzn/click/com/012585050... | https://mathoverflow.net/users/150264 | Strong topology on a topological vector space | This is a clash of two cultures which use the adjective "strong" for a topology with completely different meanings. I agree with Jochen's that this choice of terminology is quite unfortunate. I believe the question the OP is after is what is the correct topology on spaces of distributions like $\mathscr{D}'$, $\mathscr... | 2 | https://mathoverflow.net/users/7410 | 362536 | 152,462 |
https://mathoverflow.net/questions/362527 | 5 | Let $D\subseteq\mathbb{N}^+$, and consider the graph $G\_D$ with vertices set $\mathbb{N}$ and edges set $\{(x,y)\in\mathbb{N}\times\mathbb{N}\;s.t.\;|x-y|\in D\}$. I expect that if $D$ is dense enough in $\mathbb{N}^+$, then the chromatic number of $G\_D$ is large. As Wojowu pointed out in the comments, positive densi... | https://mathoverflow.net/users/54552 | Chromatic number of distance graphs over the integers | I'll show that if $G\_D$ has chromatic number $k$ then $D$ has upper Banach density at most $(k-1)/k$.
So suppose $G\_D$ has chromatic number $k$. Let $\mathbb{N}$ be partitioned into $P\_1,\ldots,P\_k$, where each $P\_i$ is independent with respect to $G\_D$. Without loss of generality, $P:=P\_1$ has upper Banach d... | 4 | https://mathoverflow.net/users/38253 | 362539 | 152,465 |
https://mathoverflow.net/questions/362528 | 1 | I want to show that there is a non-homogeneous Poisson process with a certain intensity function, but I have some problems while showing that this Poisson process satisfies the axioms(?). I am using the axioms as follows:
* $N(0) = 0$
* if $s\leq t$, then $N(s)\leq N(t)$
* etc
as usual. Now the problem is given by... | https://mathoverflow.net/users/157350 | Non-homogeneous Poisson process with intensity function $\lambda\cdot f_{X_1}$ | $\newcommand{\la}{\lambda}$
Take any $t\_0,\dots,t\_n$ such that $0=t\_0<\dots<t\_n=\infty$. For each $j\in[n]:=\{1,\dots,n\}$, let $I\_j:=[t\_{j-1},t\_j)$ and
\begin{equation}
\nu\_j:=\#\{i\in[Z]\colon X\_i\in I\_j\}.
\end{equation}
Then, for each nonnegative integer $z$, the joint conditional distribution of $(\nu... | 2 | https://mathoverflow.net/users/36721 | 362541 | 152,466 |
https://mathoverflow.net/questions/362451 | 2 | Let $V$ be a $\mathbb{F}\_p$-vector space of dimension $d$. Set $W=\bigoplus\_{1\leq i\leq n} V$ and let
$$S=\{w\_i=(v\_{i1},\dots,v\_{in}): 1\leq i\leq nd\},$$
be a basis for $W$. I am wondering if the following statement holds: $S$ can be partitioned into $n$ sets, $B\_1,\dots, B\_n$, each of size $d$, such that for... | https://mathoverflow.net/users/91357 | Partitioning bases of vector spaces | Use induction on $n$. For the step of induction, apply the following lemme to $V$ and $\bigoplus\_{2\leq i\leq n} V$.
**Lemma.** If $w\_i=(u\_i,v\_i)$ constitute a basis of $U\oplus V$, then the $w\_i$ can be split into two groups such that the $U$-components of the first group and the $V$-components of the second gr... | 1 | https://mathoverflow.net/users/17581 | 362542 | 152,467 |
https://mathoverflow.net/questions/362482 | 2 | I am self studying the book "Commutative Ring Theory" by H. Matsumura. The main theorem of section 22 is the [theorem 22.3](https://books.google.com/books?id=yJwNrABugDEC&pg=PA174), which characterizes flatness of a module $M$ over any ring $A$. The (part of the) theorem states :
Let $A$ be a ring, $I$ an ideal of $A... | https://mathoverflow.net/users/154028 | The local flatness criterion | edit: I apologize, the original answer was nonsense (I mixed different Tor functors in a silly way).
The following works, but uses the equivalent condition (1) of the question, and does not show $I\otimes M = IM \implies I^n \otimes M = I^n M$ unconditionally.
Consider the exact sequence:
$$ I^2\otimes M \righta... | 2 | https://mathoverflow.net/users/75344 | 362547 | 152,469 |
https://mathoverflow.net/questions/362470 | 7 | The following question is inspired by [Function with vector space](https://mathoverflow.net/questions/362400/function-with-vector-space) , which has been closed for unknown reason and which may have a wellknown answer. Is the following true?
Let $X$ be an uncountable set. Then there is a function $f \colon X \times X... | https://mathoverflow.net/users/100904 | On a property possibly separating countable and not countable cardinals | Yes, such a function $f:X\times X\to\mathbb N$ exists if $X$ is uncountable. It will suffice to prove it for $X=\omega\_1$. The following proof is based on the same idea as a comment by [Ashutosh](https://mathoverflow.net/users/2689/ashutosh) but uses elementary set theory instead of the result of Todorcevic.
For eac... | 8 | https://mathoverflow.net/users/43266 | 362558 | 152,473 |
https://mathoverflow.net/questions/360942 | 0 | I am trying to estimate the following expectation value in the multinomial probability distribution:
\begin{equation}
\mathbb{E}\_{P}\left[ \left( \frac{n!}{x\_1!..x\_k!}\right)^{\alpha - 1} \right]
\end{equation}
where $P$ the usual multinomial distribution, $\theta\_i \in [0,1] \forall i$, $\sum\_{i=1}^{k} \theta... | https://mathoverflow.net/users/158394 | Generalization of multinomial theorem for powers of multinomial coefficients | I doubt there exists an closed-form expression for
\begin{equation}
S\_\alpha := \sum\_{x\_1 + .. + x\_k = n} \left( \frac{n!}{x\_1!..x\_k!}\right)^{\alpha} \theta\_1^{x\_1}..\theta\_k^{x\_k}.
\end{equation}
Using the [power mean inequality](https://en.wikipedia.org/wiki/Generalized_mean#Generalized_mean_inequality), ... | 0 | https://mathoverflow.net/users/7076 | 362559 | 152,474 |
https://mathoverflow.net/questions/362554 | 3 | Let $(R, \mathfrak m)$ be a Noetherian local ring of depth at least $2$. Let $X=Spec(R)$ denote the affine- scheme with structure sheaf $\mathcal O\_X$ and $U=Spec(R)\setminus \{\mathfrak m\}$ be the punctured spectrum and write $\mathcal O\_U=\mathcal O\_X|\_U$. Then it is known that $\Gamma\_U(\mathcal O\_U)\cong R$.... | https://mathoverflow.net/users/135253 | Algebraic vector bundles on the punctured spectrum: an exact reference for a result | One modern account can be found in [this thesis of Majidi-Zolbani](https://academicworks.cuny.edu/cgi/viewcontent.cgi?article=2797&context=gc_etds), especially Chapter 1 and Appendix A. The point is that if $E$ is a vector bundle on $U$, then the global sections $\Gamma\_U(E)$ is a finite $R$ module that is locally fre... | 3 | https://mathoverflow.net/users/2083 | 362564 | 152,475 |
https://mathoverflow.net/questions/362562 | 6 | An inverse semigroup is an algebra with two operations: binary $\cdot$ and unary $^{-1}$ such that $\cdot$ is associative and $xx^{-1}x=x, xx^{-1}yy^{-1}=yy^{-1}xx^{-1}$. The Brandt semigroup with 1, $B\_2^1$, is the inverse semigroup of $2\times 2$-matrices consisting of 0, I, and the four matrix units $e\_{i,j}$, $i,... | https://mathoverflow.net/users/nan | Identities of finite inverse semigroups | I believe this is a well known open question. It has this property as a semigroup but it is not clear as an inverse semigroup. Mark Sapir showed it is contained in a finitely based locally finite variety of inverse semigroups. Your question is problem 3.10.13 in his book Combinatorial algebra: syntax and semantics.
| 4 | https://mathoverflow.net/users/15934 | 362568 | 152,477 |
https://mathoverflow.net/questions/63561 | 22 | There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets):
a) compositions with parts from {1,2}
(e.g., 2+2 = 2+1+1 = 1+2+1 = 1+1+2 = 1+1+1+1)
b) compositions that do not have 1 as a part
(e.g., 6 = 4+2 = 3+3 = 2+4 = 2+2+2)
c) compos... | https://mathoverflow.net/users/14807 | Fibonacci, compositions, history | Found it! (Sorry, Doug, ha ha.)
Augustus de Morgan added several appendices to his *Elements of Arithmetic* in the fifth edition, 1846 (available on Google Books). Appendix 10, pages 201-210, is "on combinations." The relevant paragraph is on 202-203.
>
> Required the number of ways in which a number can be comp... | 7 | https://mathoverflow.net/users/14807 | 362569 | 152,478 |
https://mathoverflow.net/questions/362501 | 1 | I want to find all cayley graphs on $Z\_{11}$. I know how many connected cayley graphs exist but i want to find all of them, connected or not, to find their eigenvalues. I found some of them and a theorem about isomorphism of caykey graphs on $Z\_p$, p is a prime number. Also I trid to work with GAP to construct these ... | https://mathoverflow.net/users/152342 | Cayley graphs on $Z_{11}$ and $Z_p$ | If you are interested in graphs (not digraphs), then the elements of the connection set must come in pairs, so you are only looking at subsets
$$
C \subseteq \{\pm1, \pm2, \pm3, \pm4, \pm5\}.
$$
Moreover, we know that the graph with connection set $C$ is isomorphic to the graph with connection set $kC$ ($k \ne 0$ an... | 5 | https://mathoverflow.net/users/1492 | 362572 | 152,479 |
https://mathoverflow.net/questions/362516 | 3 | Let's speak of the theory $\sf ZC + rank$ as the first order set theory with axioms of Extensionality, Separation, infinity, and choice (written as usual), plus iterative powers and foundation, those are:
**Iterative Power:** $\forall \text { ordinal } \alpha \exists x : x=P^\alpha(\emptyset)$
where $P^\alpha(\empt... | https://mathoverflow.net/users/95347 | Is every extension of ZFC interpretable in a finite extension of ZC + rank? | $\let\res\restriction\def\N{\mathbb N}$By the discussion in comments, it is missing from the question that all the theories are assumed **consistent** (otherwise the answer is trivially yes, as everything is interpretable in the inconsistent extension) and **recursively axiomatizable** (otherwise the answer is triviall... | 6 | https://mathoverflow.net/users/12705 | 362597 | 152,484 |
https://mathoverflow.net/questions/362603 | 3 | Does there exist a complex smooth proper variety whose fundamental group is finite non-abelian simple?
| https://mathoverflow.net/users/nan | A complex variety with a finite non-abelian simple fundamental group | Yes. In fact, Serre proved that any finite group is the fundamental group of a smooth projective complex variety. See Proposition 15 of:
J.-P. Serre, Sur la topologie des variétés algébriques en charactéristique $p$, Symposium Internacional de Topologia Algebraica, Universidad Nacional Autonoma de Mexico, 1958, pp. 2... | 14 | https://mathoverflow.net/users/317 | 362605 | 152,486 |
https://mathoverflow.net/questions/362563 | 9 | I was working with some Dirichlet series and I realized that I have never seen any general conditions under which
\begin{equation}
\sum\_{n=1}^{\infty}\frac{a\_n}{n}=\lim\_{s\to1^+}\sum\_{n=1}^{\infty}\frac{a\_n}{n^s}\label{1}\tag{1}
\end{equation}
holds. This is obviously not true in a general case since if so the... | https://mathoverflow.net/users/159298 | Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$ | Here are analogous Tauberian theorems for power series and Dirichlet series that involve a condition of *analytic continuation* to a boundary point plus one extra condition that is necessary for the series to converge at some point on that boundary.
Power series: If $c\_n \to 0$ then $\sum\_{n \geq 0} c\_nz^n$ conver... | 7 | https://mathoverflow.net/users/3272 | 362610 | 152,487 |
https://mathoverflow.net/questions/362608 | 6 | Let $X$, $Y$ be complex affine algebraic manifolds (closed submanifolds of $\mathbb{C}^n$), let $f\colon Y\to X$ be a finite covering. Let $\mathcal{L}$ be a holomorphic line bundle on $X$. Suppose $f^\*\mathcal{L}$ is an algebraic line bundle. Is $\mathcal{L}$ necessarily algebraic?
(This is true when $f$ admits a ... | https://mathoverflow.net/users/nan | "Holomorphic line bundle" + "algebraic after finite cover" implies "algebraic"? | Yes. This follows from P. Deligne "Equations differentielles..." LNM 163, Proposition II 2.22, since the notion of moderate growth at infinity will coincide for $X$ and $Y$. This works more generally for any coherent sheaf.
| 4 | https://mathoverflow.net/users/3847 | 362611 | 152,488 |
https://mathoverflow.net/questions/362335 | 0 | **Edit:** According to comment by Leo Monsaingeon I revise my question:
Is there a Riemannian metric on $M\_n(\mathbb{R})$ for which the function $trace$ is a bounded function on **every complete(whole)** geodesic? that is for every **complete** geodesic $\gamma$, the restriction $trace$ to $\gamma$ is a bounded func... | https://mathoverflow.net/users/36688 | Riemannian metrics on matrix space for which the restriction of trace function to each complete geodesic is a bounded function | It is possible to find such metrics. However, they are a bit unnatural, in that they aren't really constructed from any inherent properties of matrices. For concreteness, I'm going to restrict my attention to $M\_2$ and $Gl\_2$, but this construction can be modified to work for any larger $n$ as well.
We consider $2 ... | 3 | https://mathoverflow.net/users/125275 | 362614 | 152,490 |
https://mathoverflow.net/questions/362618 | -1 | Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-empty open subset of $X$. But how to show this?
| https://mathoverflow.net/users/36886 | Dense linear span implies closed convex hull has non-empty interior | The answer is no. E.g., let $X=\ell^2$ and
$$Y=\{x\in\ell^2\colon\sum\_n nx\_n^2\le1\}.$$
Then $Y$ is a closed convex set, spanning $X$, but the interior of $Y$ is empty.
---
**Details:** $Y$ is convex because the function $\mathbb R\ni u\mapsto u^2$ is convex. To show that $Y$ is closed one may use the Fatou l... | 3 | https://mathoverflow.net/users/36721 | 362621 | 152,493 |
https://mathoverflow.net/questions/362567 | 0 | Assume I have a sequence $\{a\_m\}$ that is vanishing and strictly positive:
$$
0<a\_{m+1}\leq a\_m\leq\ldots\leq a\_1<\infty, \quad \lim\_{m\to \infty}a\_m = 0
$$
Is it true or false that this has a subsequence $\{a\_{m\_n}\}$ such that
$$
a\_{m\_n}\asymp n^{-1}
$$
which is to say, there exists a constant $C>0$ such t... | https://mathoverflow.net/users/51335 | Vanishing sequence and subsequence with particular decay | The answer is no. E.g., let $a\_m:=1/(m!)^2$ and $n\_k:=k!(k+1)!$ for natural $m,k$. Your desired condition means that $|\ln na\_{m\_n}|$ is bounded. In our case, we have
$$a\_{k+1}<\frac1{n\_k}<a\_k$$
for all $k$.
So, for all $m\le k$ we have $1<n\_k a\_k\le n\_k a\_m$ and hence $|\ln(n\_k a\_m)|\ge|\ln(n\_k a\_k)|=... | 0 | https://mathoverflow.net/users/36721 | 362626 | 152,496 |
https://mathoverflow.net/questions/362631 | 2 | Let $A$ be a representation-finite algebra and $M$ an indecomposable module with finite projective dimension $g >0$.
>
> Question 1: Do we have $dim(Ext\_A^g(M, \tau\_g(M)))=1$? Here $\tau\_g(M)=\tau ( \Omega^{g-1}(M))$.
>
>
> Question 2: Could this even be true when $A$ is not represenation-finite? Probably not,... | https://mathoverflow.net/users/61949 | Ext between a module and its higher Auslander-Reiten translate | Question 1: I think that the answer is no. We have
$$\operatorname{Ext}^g\_A(M,\tau\_g(M)) \simeq \operatorname{Ext}^1\_A(\Omega^{g-1}\_A(M), \tau(\Omega^{g-1}\_A(M)).$$
Look at the following example in QPA:
```
gap> Q := DynkinQuiver( "A", 5, [ "l", "l", "r", "r" ] );
<quiver with 5 vertices and 4 arrows>
gap> K... | 2 | https://mathoverflow.net/users/130741 | 362638 | 152,501 |
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