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https://mathoverflow.net/questions/320832 | 1 | Is there a good reference for facts and theorems about BV real valued functions? I’m looking for something with much more than say Stein and Shakarchi 3, or Evans and Gariepy. Thanks!
| https://mathoverflow.net/users/132446 | Reference request: Functions of bounded variation in one real variable | The following book has a nice and long chapter of BV functions in one variable:
**G. Leoni**, *A First Course in Sobolev Spaces* Graduate Studies in Mathematics
Volume: 181.
Also an excellent book covering a great deal of the material is:
**I. P. Natanson,**
*Theory of functions of a real variable.*
Translated b... | 2 | https://mathoverflow.net/users/121665 | 320938 | 138,567 |
https://mathoverflow.net/questions/320931 | 3 | I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince myself that a "concrete" example exists (beside the "trivial" function, $x \mapsto \alpha \vert x \vert$ for some $\alp... | https://mathoverflow.net/users/119793 | Example of convex functions fulfilling a (strange) lower bound | Let $G$ be the set of functions $g\colon\mathbb R\to\mathbb R$ such that for some strictly positive real $a$ and $b$ and all real $x$ we have $g(x)=-ax$ if $x\le0$ and $g(x)=bx$ if $x\ge0$. Let $l\_1,\dots,l\_N$ be any linearly independent linear functionals on $\mathbb R^N$. Then any function $f$ on $\mathbb R^N$ of t... | 4 | https://mathoverflow.net/users/36721 | 320941 | 138,569 |
https://mathoverflow.net/questions/320900 | 4 | Is there any research on getting upper bound of the maximal possible number of consecutive positive integers which are less than $n!$ and NOT coprime with $n!$?
Easy to see that lower bound $\ge n$, (example that $n= \text{odd prime}$). Does upper bound $<2n$ strictly?
Note that the discussion is in range of $<n!$... | https://mathoverflow.net/users/44685 | upper bound of consecutive integers which are not coprime with $n!$ | More generally, for an integer $n \gt 1$, let $L(n)$ be the largest number $k$ so that there is an integer $m$ where each of the $k$ consecutive integers $m+1, \cdots, m+k$ has a prime factor in common with $n$. $L(n) \geq \omega(n)$, the number of distinct prime factors of $n$, and if the least prime factor $p$ of $n$... | 4 | https://mathoverflow.net/users/3402 | 320956 | 138,571 |
https://mathoverflow.net/questions/320965 | 30 | I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ and three with $-2$) and these are
$$
\begin{bmatrix}1 & 1 & 0\\ 0 & 1 & 1\\ 1 & 0 & 1\end{bmatrix}
$$
and the different p... | https://mathoverflow.net/users/9652 | Determinants of binary matrices | This question is too hard, because easier questions are already known to be hard.
The middle column is [A046747](http://oeis.org/A046747) in the OEIS, which is essentially equivalent to [A057982](http://oeis.org/A057982), the number of singular {±1}-valued matrices. The asymptotic behavior of this number was a longst... | 21 | https://mathoverflow.net/users/3106 | 320978 | 138,578 |
https://mathoverflow.net/questions/320943 | 3 | Sporadic groups have very few outer automorphisms (in fact, $|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable outer automorphism) one can get a twisted group or the same group over a subfield, but what about sporadic groups? I consu... | https://mathoverflow.net/users/5018 | Fixed points of the automorphisms of sporadic groups | I think the information you are looking for is in Table~5.3 of GLS3:
>
> Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald.
> *The classification of the finite simple groups*. Number 3. Part I.
> Chapter A. American Mathematical Society, Providence, RI, 1998.
> xvi+419 pp. ISBN: 0-8218-0391-3
>
>
>
For ... | 8 | https://mathoverflow.net/users/116714 | 320980 | 138,579 |
https://mathoverflow.net/questions/320948 | 0 | Suppose that $a\_1<1$, $a\_1+a\_2+a\_3>1.$ For $x,y,z>0,$
(1) define a fucntion
$$H(x,y,z)=\frac{x^{\frac{1}{2}}\int\_0^{\infty}\frac{1}{t^{a\_1}~ (1+t)^{a\_2+1}~
(1+t+z)^{a\_3}}\exp\big\{-\frac{x}{1+t}-\frac{ y}{1+t+z}\big\}dt }{\int\_0^{\infty}\frac{1}{t^{a\_1}~ (1+t)^{a\_2}~(1+t+z)^{a\_3}}\exp\big\{-\frac{x}{1+... | https://mathoverflow.net/users/134602 | Uniformly Bounded (updating) | We may assume that $x>100(1+|a\_2|+|a\_3|)^2$, for other $x$ simply $C=10(1+|a\_2|+|a\_3|)$ works perfectly.
Partition the integral in the numerator onto two parts: $I\_1$ over $(0,\sqrt{x}]$ and $I\_2$ over $[\sqrt{x},\infty)$. The second part does not exceed the denominator, since $\sqrt{x}(1+t)^{-1}<1$ for all $t\... | 2 | https://mathoverflow.net/users/4312 | 320985 | 138,580 |
https://mathoverflow.net/questions/320991 | -4 | If $G=(V,E)$ is a simple, undirected graph and $v\in V$, we set $N(v) = \{w\in V:\{v,w\}\in E\}$.
Is there an integer $k>1$ and a connected $k$-[regular](https://en.wikipedia.org/wiki/Regular_graph) graph $G=(V,E)$ such that there are $v\neq w \in V$ with $N(v) = N(w)$?
(Note that the definition of $N(\cdot)$ imp... | https://mathoverflow.net/users/8628 | Regular graph such that $2$ distinct vertices have same neighborhood set | An example with $k = 4$ is the octahedron.
| 0 | https://mathoverflow.net/users/41139 | 320992 | 138,582 |
https://mathoverflow.net/questions/320950 | 14 | I am new to H-spaces, delooping, etc. I know that not every $H$-space has a delooping (e.g. Stasheff's theorem, one needs a group-like $A\_\infty$ space). I also know that the same space can have inequivalent deloopings corresponding to different $H$-space structures. An example is $BU$, which has two $H$-space structu... | https://mathoverflow.net/users/134600 | Does an H-space have at most one delooping? | Another example is $S^3$. If I am not mistaken, there are exactly $12$ H-space structures on $S^3$. Indeed, we can consider the long exact sequence
$$[S^4\vee S^4, S^3] \to [S^6, S^3] \to [S^3\times S^3, S^3] \to [S^3\vee S^3, S^3]$$
of groups (using an arbitrary loop structure on $S^3$). We have to count the preimages... | 18 | https://mathoverflow.net/users/2039 | 320996 | 138,584 |
https://mathoverflow.net/questions/320878 | 7 | Consider an optimization problem over infinite variables:
$$
\begin{align}
\min\_{x}~& {\left\lVert{x}\right\rVert }\_p
\\
\text{s.t}~& \left\langle x, a\_n\right\rangle \ge 1~,~\forall n=1,\dots,N
\end{align}
$$
where $N\in\mathbb{N}$, $x$ and $\left\{a\_n\right\}\_{n=1}^{N}$ are all **infinite-length** vectors,
th... | https://mathoverflow.net/users/100796 | Proving an infinite norm minimization problem has finite support (non-convex p-norms) | If $p=1$, $N=1$ and $a\_1=(1/2,2/3,3/4,4/5,\ldots)$, the infimum equals 1 and is not achieved on a finitely supported vector (moreover, it is not achieved at all).
However if $0<p<1$ and the minimizer $x$ exists, it must have finite support (namely, of size at most $N$). To prove this, assume the contrary. Without l... | 4 | https://mathoverflow.net/users/4312 | 321004 | 138,589 |
https://mathoverflow.net/questions/321008 | 1 | For a real function $f$ on $\mathbb{R}$, define $e\_n(f)$ to be the infimum of the $L\_1$ distance between $f$ and piecewise constant functions on the subdivision of $\mathbb{R}$ into intervals of length $1/n$ ($0$ being one of the intervals bound). For $c>0$, does the space of $f$ such that $e\_n(f) = O(n^{-c})$ have ... | https://mathoverflow.net/users/112954 | Name of a function space | I think this is answered
>
> DeVore, Ronald A. "Nonlinear approximation." Acta numerica 7 (1998): 51-150
>
>
>
in Section 3.1: The space of functions for which $e\_n(f) = O(n^{-c})$ is the space $\mathrm{Lip}(c,L^1(0,1))$ and this space consists of the function $f$ for which
$$
\|f(\cdot+h)-f\|\_{L^1(0,1-h)} ... | 1 | https://mathoverflow.net/users/9652 | 321010 | 138,590 |
https://mathoverflow.net/questions/320999 | 3 | Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B\_p(r)$ be a geodesic open ball.
**Q** Can we find a constant $C=C(\kappa,r,m)$(independent on the point $p$), such that
$$
\left(\frac{1}{Vol(B\_p(r))}\int\_{B\_p(r)}\phi^{2^\*}\right)^{\frac1... | https://mathoverflow.net/users/95296 | Local Sobolev embedding on complete Riemannian manifold | The following result is Theorem 1.1 in:
**P. Maheux, L. Saloff-Coste,**
Analyse sur les boules d'un opérateur sous-elliptique.
*Math. Ann.* 303 (1995), 713–740.
>
> **Theorem.** Let $(M,g)$ be a complete Riemannian $n$-manifold, such that $Ric\geq kg$ for some $k\in\mathbb{R}$. Let $1\leq p<n$ and
> $p^\*=np/(... | 3 | https://mathoverflow.net/users/121665 | 321012 | 138,592 |
https://mathoverflow.net/questions/320987 | 0 | The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
then there is a unitary map $W:L^2(\mathbb R) \rightarrow H$ such that
$$
W^\*U(t)W = e^{itx}, \qquad W^\*V(s)W = e^{isp}... | https://mathoverflow.net/users/134620 | Stone–von Neumann theorem? | My earlier answer was incorrect. The question is ill-conceived because $T\_2$ is not a representation: it fails to satisfy $T\_2(s)T\_2(s') = T\_2(s + s')$. So of course it cannot be unitarily equivalent to a one-parameter group.
(Also, note that the Stone-von Neumann theorem only applies to irreducible representatio... | 2 | https://mathoverflow.net/users/23141 | 321015 | 138,594 |
https://mathoverflow.net/questions/321019 | -3 | Let $c\_k$ be the maximum chromatic number that a $k$-regular graph can have. What is $\lim\sup\_{k\to\infty}\frac{c\_k}{k}$?
| https://mathoverflow.net/users/8628 | Maximum chromatic number of a $k$-regular graph | The [complete graph](https://en.wikipedia.org/wiki/Complete_graph) with $k+1$ vertices (which is strongly $k$-regular) has degree $k$ and chromatic number $k+1$. On the other hand, by [Brook's theorem](https://en.wikipedia.org/wiki/Brooks%27_theorem) this is the maximum possible value for the chromatic number of a $k$-... | 2 | https://mathoverflow.net/users/7460 | 321022 | 138,596 |
https://mathoverflow.net/questions/136185 | 13 | The [Klee Trick](http://books.google.com/books?id=qYqd12vt2x8C&pg=PA66&lpg=PA66&dq=The%20Klee%20Trick&source=bl&ots=dh9LHDkfNr&sig=ZMinrf9nTN8C4FtpoKxeK6qC-Bw&hl=en&ei=dbv5TNqWPIGKlwfT3s3iBw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDMQ6AEwBA#v=onepage&q=The%20Klee%20Trick&f=false) allows one to find an $\mathbb{R}^... | https://mathoverflow.net/users/27453 | Is the dimension given by Klee trick ever sharp? | Are the embeddings required to be isometric embeddings? If not, then what about including three points into $\mathbb{R}$ in two ways, so that the middle point of the three changes? More explicitly, let $K=\{a,b,c\}$ and define $f(a)=0$, $f(b)=1$, $f(c)=2$, whereas $g(a)=1$, $g(b)=0$, $g(c)=2$. There isn't any self-home... | 6 | https://mathoverflow.net/users/124004 | 321024 | 138,597 |
https://mathoverflow.net/questions/321032 | 5 | Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f\_A$ be the associated *Minkowski functional*
$$
f\_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which turns to be a convex, homogeneous function defined on $\mathbb R^n$.
Given a Lipschitz regular domain $\Omega \subset ... | https://mathoverflow.net/users/119793 | Uniqueness of minimizers in a problem in the Calculus of Variations - Part II |
>
> *Existence of minimizers should not be a severe issue...*
>
>
>
The proof of the existence follows form the Arzela-Ascoli theorem, but the proof is not entirely obvious. This is Proposition 1.1 in:
**E. Giusti**, *Direct methods in the calculus of variations.* World Scientific Publishing Co., Inc., River E... | 5 | https://mathoverflow.net/users/121665 | 321039 | 138,601 |
https://mathoverflow.net/questions/320994 | 0 | What is the smallest cardinal $\beta$ such that it is provable in ${\sf (ZFC)}$ that $2^{\aleph\_\beta} > 2^{\aleph\_0}$?
| https://mathoverflow.net/users/8628 | Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$ | Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph\_0}>\aleph\_\delta$ holds. If you rather talk of models than Boolean-valued extensions, what this says is that we can force while preserving all ordinals, and in fact al... | 17 | https://mathoverflow.net/users/6085 | 321043 | 138,602 |
https://mathoverflow.net/questions/321033 | 11 | Let $\square\_2=\{(x,y): 0\leq x, y\leq1\}$ be the unit square in $\mathbb{R}^2$. Take $n>1$ points $P\_1, \dots, P\_n\in\square\_2$.
Denote the distances $d\_j=\min\{\Vert P\_k-P\_j\Vert: k\neq j\}$, for $j=1,\dots, n$.
>
> **QUESTION.** Is it true that $d\_1^2+\cdots+d\_n^2\leq 4$? The bound is tight when $n=2... | https://mathoverflow.net/users/66131 | Sum of squared nearest-neighbor distances between points in a square | We shall prove the more general result: for $n\ge2$ distinct points and any positive real $a,b$, we have $D(R):=d\_1^2+\cdots+d\_n^2\le 2a^2+2b^2$, where the points $P\_j$ are now in an $a\times b$ rectangle $R$ and the distances $d\_j$ are defined as in the OP. Let us use induction on $n$. For $n=2$, the result is obv... | 4 | https://mathoverflow.net/users/36721 | 321048 | 138,603 |
https://mathoverflow.net/questions/320951 | 6 | Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $I\subseteq m$ a proper homogeneous ideal in $S$. Is this true that we always have:
$$[Im:(x)][Im:(y,z)]\subseteq Im \ ?$$
In a paper we needed this statement for monomial ideals and it is easy to prove, but I can not see either way for general ideals. Has an... | https://mathoverflow.net/users/2083 | Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$? | Perhaps it is not so bad once we are willing to get our hand dirty a little. Let $f\in Im:(x)$. Then $xf = xf\_1+yf\_2+zf\_3$ with $f\_i \in I$. Rewriting, we have $x(f-f\_1) = yf\_2+zf\_3$. Since $x,y,z$ form a regular sequence we must have $f=f\_1+h$, with $h\in (y,z)$.
Now let $g\in Im:(y,z)$. Since $I$ is proper... | 7 | https://mathoverflow.net/users/2083 | 321060 | 138,606 |
https://mathoverflow.net/questions/227283 | 6 | Does **Con(ZFC+ there exists a strongly compact cardinal)** imply **Con(ZFC+ there exists a strongly compact cardinal $\kappa+ 2^\kappa > \kappa^+$)**?
| https://mathoverflow.net/users/11115 | Failure of GCH at a strongly compact cardinal | The answer to the question is yes, as it is proved by Usuba in the paper
[Strongly compact cardinals and the continuum function](https://arxiv.org/abs/1901.05313).
In fact Usuba proves the following more general result:
**Theorem**. Assume $\kappa$ is a strongly compact cardinal. Then there exists a generic extensi... | 4 | https://mathoverflow.net/users/11115 | 321078 | 138,611 |
https://mathoverflow.net/questions/320787 | 2 | I asked [this question](https://math.stackexchange.com/q/3062641/660) on Mathematics Stack Exchange but got no answer.
Here is the question:
Let $A$ be a domain (that is, a commutative ring with one in which the condition $ab=0$ implies $a=0$ or $b=0$).
Assume that $A$ has the following property:
If $\mathfrak ... | https://mathoverflow.net/users/461 | Are unique prime ideal factorization domains locally noetherian? | No, this property does not imply that $A$ is locally Noetherian.
For example, let $F\subset L$ be an extension of fields, and let $A=F+XL[[X]]$ (that is, $A$ is the set of power series over $L$ whose constant term belongs to $F$). Then, $A$ is a one-dimensional local domain with maximal ideal $\mathfrak{m}=XL[[X]]$. ... | 5 | https://mathoverflow.net/users/125073 | 321090 | 138,613 |
https://mathoverflow.net/questions/321071 | 6 | The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on *String Phenomenology*. It was brief and gave only basic outline of how to construct this homology.
Now de Rhams theorem asserts that there is an isomorphism between de Rham cohomology of smooth manifolds and that of singu... | https://mathoverflow.net/users/35706 | Is there a theorem showing that de Rham homology is isomorphic to singular homology? | I guess that by *de Rham homology* you mean the homology groups $H\_{k, \, \mathrm{dR}}(X)$ constructed on a closed manifold $X$ by using the complex of currents.
In that case, [**1**, Theorem 2 page 582] shows that there is an isomorphism between
$H^{n-k}\_{\mathrm{dR}}(X)$ and $H\_{k, \, \mathrm{dR}}(X)$, where th... | 17 | https://mathoverflow.net/users/7460 | 321094 | 138,615 |
https://mathoverflow.net/questions/321085 | 7 | Ingham has shown that there is a prime between $n^{3}$ and $(n+1)^{3}$ for large enough $n.$
Legendre's conjecture about the existence of primes between consecutive perfect squares is of course open.
What, if anything, is known about the existence of primes in the intervals
$$
[n^{2+\epsilon},(n+1)^{2+\epsilon}],
... | https://mathoverflow.net/users/17773 | Near-Legendre Conjecture | Baker, Harman and Pintz [showed in 2001](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.360.3671&rep=rep1&type=pdf) that for $\theta=0.525$, the interval $(x,x+x^\theta)$ contains at least one prime provided $x$ is sufficiently large. This is equivalent to the interval $[n^{2+\epsilon},(n+1)^{2+\epsilon}]$ to... | 13 | https://mathoverflow.net/users/9924 | 321097 | 138,617 |
https://mathoverflow.net/questions/321107 | 13 | For pedagogical reasons, I got interested in the equation $y^4-x^3=a$ over $\mathbf F\_p$.
To my surprise (maybe I'm naive), there is only one couple $(p,a)=(13,7)$ for which there is no solution, at least for $p\leq 2000$.
My question :
Is $(p,a)=(13,7)$ the only couple for which $y^4-x^3=a$ has no solution ov... | https://mathoverflow.net/users/39552 | Universality of $y^4-x^3$ mod $p$ | The curve $C:y^4-x^3z=az^4$ is nonsingular over $\mathbb F\_p$ for $p\ge5$ and $a\ne0$. It has genus $3$. So Weil's theorem says that
$$ \bigl| \#C(\mathbb F\_p) - p - 1 \bigr| \le 6\sqrt{p}. $$
There is only one point with $z=0$, namely $[1,0,0]$, so you'll always get solutions to your original equeation provided $p>6... | 34 | https://mathoverflow.net/users/11926 | 321109 | 138,622 |
https://mathoverflow.net/questions/321098 | 6 | On the basis of my computation, here I pose my following conjecture involving the cosine function.
**Conjecture**. For any positive integer $n$, we have the identity
$$\frac1{2n}\det\left[\cos\pi\frac{jk}n\right]\_{0\le j,k\le n}=\det\left[\cos\pi\frac{jk}n\right]\_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2... | https://mathoverflow.net/users/124654 | A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$ | First of all, we use the formula
$$
D:=\det [x\_j^k+x\_j^{-k}]\_{j,k=0,\dots,m-1}=\prod\_{l<j}(x\_j+x\_j^{-1}-x\_l-x\_l^{-1})=\prod\_{l<j} (x\_j-x\_l)(1-x\_j^{-1}x\_l^{-1}).
$$
This follows from the observation that $x^k+x^{-k}=p\_k(x+x^{-1})$ for a polynomial $p$ of degree $k$ with leading coefficient 1, so our matri... | 13 | https://mathoverflow.net/users/4312 | 321110 | 138,623 |
https://mathoverflow.net/questions/320890 | 2 | Let $E$ be a vector bundle over a compact connected Hausdorff space $X$. To an endomorphism $\alpha \in End(E)$, we associate a $C(X)-$ module $\Gamma(E,\alpha)$ consisting of all $\beta\in End(E)$ such that there exists a $\gamma \in End(E)$ with $\beta=\alpha \circ \gamma$.
$$\require{AMScd}
\def\diaguparrow#1{\sma... | https://mathoverflow.net/users/36688 | A module associated to an endomorphism of a vector bundle | It seems that your $C(X)$-module $\Gamma(E,\alpha)$ is just the image of $\alpha\_\*\colon \hom(E,E)\rightarrow \hom(E,E)$. The module $\Gamma(E,\alpha)$ is a direct summand of $\Gamma(E',\alpha')$ for $\alpha'\colon E'\rightarrow E'$ an extension of $\alpha$ to a trivialization $E\subset E'$ of $E$. Therefore, we can ... | 4 | https://mathoverflow.net/users/12166 | 321114 | 138,625 |
https://mathoverflow.net/questions/321112 | 6 | Sierpiński's theorem states that nonatomic probability measures take a continuum of values. What if I assume that $\mu$ is a countably additive probability measure on $(X,2^X)$ and further that $\mu(\{x\})=0$ for all $x\in X$ (a weaker assumption than non-atomicity). Does it follow that $\mu$ takes on every value in $[... | https://mathoverflow.net/users/12518 | Variant of Sierpiński's result on non-atomic measures | If $|X|$ is a [measurable cardinal](http://cantorsattic.info/Measurable), then there is a $\sigma$-complete ultrafilter $\mathcal U$ on $X$. You may define a countably additive probability measure $\mu$ on $(X,2^X)$ by setting $\mu(Y) = 1$ if $Y \in \mathcal U$ and $\mu(Y) = 0$ if $Y \notin \mathcal U$. This measure ta... | 12 | https://mathoverflow.net/users/70618 | 321117 | 138,626 |
https://mathoverflow.net/questions/321093 | 12 | There is a classical theorem that [no field can be expressed as finite union of proper subfields](https://mathoverflow.net/questions/39605/).
In contrast, there is an [example](http://matwbn.icm.edu.pl/ksiazki/cm/cm7/cm717.pdf) of an integral domain that can be expressed as finite union of proper subrings.
Therefo... | https://mathoverflow.net/users/129541 | PID expressed as finite union of subrings | Consider the following subrings of $\mathbb{F}\_2^3$:
$$
V\_1=\{(0,0,0),(1,1,1),(1,0,0),(0,1,1)\}\\
V\_2=\{(0,0,0),(1,1,1),(0,1,0),(1,0,1)\}\\
V\_3=\{(0,0,0),(1,1,1),(0,0,1),(1,1,0)\}
$$
Then $\mathbb{F}\_2^3=V\_1\cup V\_1\cup V\_3$.
Let $R$ be a PID admitting $3$ distinct surjective ring maps $f\_1,f\_2,f\_3$ onto $... | 10 | https://mathoverflow.net/users/86006 | 321120 | 138,627 |
https://mathoverflow.net/questions/321129 | 3 | Let $X$ be a regular affine $\mathbb{C}$-scheme, $A$ a (finitely-generated) $\mathbb{C}$-algebra. Let $Y \subset X \times \mathrm{Spec}(A)$ be a closed subscheme of codimension $1$ such that for each $t \in \mbox{Spec}(A)$, the fiber $Y\_t$ is an effective Cartier divisor of $X \times \{t\}$. Is $Y$ flat over $\mbox{Sp... | https://mathoverflow.net/users/45397 | Effective Cartier divisor is an open property | I assume that "scheme" means $\mathbb{C}$-scheme, and that the product is over $\mathbb{C}$. I also assume that $Y$ is a closed subscheme.
First (trivial) counterexample: take for $Y$ an effective Cartier divisor in $X\times t$ ($t\in \mathrm{Spec}(A)(\mathbb{C})$), viewed as a subscheme of $X\times \mathrm{Spec}(A)$... | 2 | https://mathoverflow.net/users/7666 | 321131 | 138,631 |
https://mathoverflow.net/questions/165037 | 6 | Let U be a simply connected domain with smooth boundary in the complex plane, and let $\mathbb D$ be the unit disc. Is there a nice sufficient condition for the existence of a biholomorphic map $f:\mathbb{D} \xrightarrow{\simeq} U$ with $$\sup\_{z \in \mathbb{D}}\,\,\,\, \left| f^\prime(z)\right| \le 1$$
For compari... | https://mathoverflow.net/users/2085 | Bounds on the derivative of a Riemann map | I wrote a [short paper](https://arxiv.org/abs/1601.02711) to answer this question. Here's a summary. Let's shift $U$ so it contains $0$ and normalize the biholomorphic map $f : \mathbb D\to U$ by $f(0)=0$.
By the maximum principle, it suffices to have $|f'|\le 1$ near the boundary of $\mathbb D$, which can be expres... | 1 | https://mathoverflow.net/users/nan | 321140 | 138,635 |
https://mathoverflow.net/questions/320817 | 1 | For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that
$$
\frac{P(X>c)}{P(X>c-1)}=1-o(1)
$$
uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note that $c$ will vary with $n$). The $o(1)$ rate is meant as $n\to\infty$.
Now I have the following results (Note that $q=1-p$... | https://mathoverflow.net/users/65953 | Showing $o(1)$ convergence for ratio of successive binomial tail probabilities | I think, if $c=np+o(n)$, you may simply use $P(X=c+1)=P(X=c)(1+o(1))$, and so
$P(X\in \{c,c+1,\dots,c+M-1\})=(M+o(1))P(X=c)$ for any fixed $M$.
| 1 | https://mathoverflow.net/users/4312 | 321144 | 138,637 |
https://mathoverflow.net/questions/321141 | 3 | In the presence of a calculus of (right) fractions, one may prove that every equivalence class of the general localization---the quotient of $F(UC +\_{obj W} W^{op})$, the free category on the amalgamation of the underlying quiver of $C$ and $W^{op}$ with the objects of $W$, tautologously inverting $W$---is represented... | https://mathoverflow.net/users/31420 | Why isn't the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions? | If $C$ is small, $C[W^{-1}]$ is again small, even without a calculus of fractions. The problem in general is that when $C$ is a locally small large category, the zig-zag sequences can range over finite sequences of objects of $C$, which form a large set.
Essentially, the problem is that the hom-sets in the localizati... | 13 | https://mathoverflow.net/users/1353 | 321145 | 138,638 |
https://mathoverflow.net/questions/321116 | 3 | I am sorry, but I am quite new to Ext groups of sheaves. However, I have a closed embedding of projective $\mathbb{C}$-schemes $\iota : X \hookrightarrow Y$ and was wondering if
$$\iota\_\*:\mathrm{Ext}^\*\_X(\mathcal{F},\mathcal{G}) \to \mathrm{Ext}^\*\_Y (\iota\_\*\mathcal{F},\iota\_\*\mathcal{G})$$ was an iso, respe... | https://mathoverflow.net/users/130550 | is the induced map of an embedding an Iso on Ext-groups? | First, there is an adjunction isomorphism
$$
Ext^\bullet(i\_\*F,i\_\*G) \cong Ext^\bullet(Li^\*(i\_\*F),G),
$$
where $Li^\*$ is the derived pullback functor. Furthermore, if $X$ in $Y$ is a locally complete intersection, then
$$
L\_pi^\*(i\_\*F) \cong F \otimes \Lambda^pN^\vee\_{X/Y}.
$$
These two observations combine ... | 2 | https://mathoverflow.net/users/4428 | 321149 | 138,639 |
https://mathoverflow.net/questions/321118 | 1 | Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p\in X$.
If $\dim X>0$, then does there necessarily exist a smooth curve $C$ on $X$ such that $f(p)$ is a smooth point of the closure of $f(C)$ in $Y$ ?
| https://mathoverflow.net/users/127118 | Image of smooth curve containing the image of a point as smooth point | That is not true. Let $k$ be an algebraically closed field of characteristic prime to $34$. Let the domain of the morphism be a Zariski open neighborhood of the origin in $\mathbb{A}^2\_k$ with coordinates $(s,t)$. Let the target of the morphism be a Zariski open neighborhood of the origin in $\mathbb{A}^2$ with coordi... | 1 | https://mathoverflow.net/users/13265 | 321158 | 138,641 |
https://mathoverflow.net/questions/321151 | 4 | Can there exist non-trivial elementary embeddings $j,k:V\_{\lambda+1}\rightarrow V\_{\lambda+1}$ along with a strictly increasing function
$r:\omega\rightarrow\omega$ such that
$j^{r(2n)}(\mathrm{crit}(j))>k^{r(2n)}(\mathrm{crit}(k))$ and
$j^{r(2n+1)}(\mathrm{crit}(j))<k^{r(2n+1)}(\mathrm{crit}(k))$ for all $n$?
| https://mathoverflow.net/users/22277 | Are there any I1 embeddings with interweaving critical sequences? | One can easily arrange this for embeddings $V\_\lambda\to V\_\lambda$, and I think the same method works for $V\_{\lambda+1}$.
To do it, start with any $j:V\_\lambda\to V\_\lambda$. The various compositions $j\circ j\circ\cdots\circ j$ with itself push up the target $j(\kappa)$ as high as desired. And the applicatio... | 4 | https://mathoverflow.net/users/1946 | 321170 | 138,644 |
https://mathoverflow.net/questions/321168 | 2 | In the paper "Quantum state transformations and the Schubert calculus" by Sumit Daftuar and Patrick Hayden (Annals of Physics 315 (2005) 80-122) on page 91, we have following notations:
$A\_r$ denotes the $r$-dimensional vector space spanned by eigenvectors corresponding to the $r$ largest eigenvalues $\lambda\_1 < \... | https://mathoverflow.net/users/119246 | Typo in a paper definition of Schubert cells? |
>
> I am baffled by the definition of $S\_\pi$ since to me it seems as if there is always only one element contained in $S\_\pi$, namely the subspace spanned by the eigenvectors corresponding to the eigenvalues $\lambda\_{\ell\_j}$, $j = 1,\ldots,r$, of $A$.
>
>
>
This logic is wrong. Try the simplest nontrivial... | 2 | https://mathoverflow.net/users/2530 | 321174 | 138,648 |
https://mathoverflow.net/questions/321176 | 0 | Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'\psi)'\;\;\;\text{for }\psi'\in C^2(\mathbb R).$$ Note that $L^\ast e^{-h}=0$.
>
> Are we able to show that there i... | https://mathoverflow.net/users/91890 | Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz | Suppose that $h'(x)$ satisfies a *dissipativity condition*: there exists $K>0$ and $A \ge 0$ such that for all $x \in \mathbb{R}$ we have $x h'(x) \ge K x^2 - A$. Consider as a candidate function $v(x) = x^2$. Then, $$
L v(x) = - x h'(x) + 1 \le - K x^2 + A + 1 \;.
$$ So, (1) is satisfied with $\lambda=K$ and $c=A+1$. ... | 1 | https://mathoverflow.net/users/64449 | 321182 | 138,649 |
https://mathoverflow.net/questions/321069 | 4 | Let $X\_1,\dots,X\_k$ denote a collection of independent samples of a Poisson random variable whose mean also happens to be equal to $k$. Does the quantity $$k\boldsymbol{E}\min\left\{ \frac{1}{1+X\_{1}},\dots,\frac{1}{1+X\_{k}}\right\} $$ have a strictly positive limit as $k$ becomes large?
| https://mathoverflow.net/users/125803 | The minimum of the reciprocals of some Poisson random variables | Let
\begin{equation}
Y\_i:=\frac1{1+X\_i}, \quad Y:=\min(Y\_1,\dots,Y\_k).
\end{equation}
Then
\begin{equation}
EY=\int\_0^\infty P(Y>y)\,dy=\int\_0^\infty P(Y\_1>y)^k\,dy.
\end{equation}
Next, for $y\in(0,\frac1{1+2k})$ and $x:=\frac1y-1>2k$, we have
\begin{equation}
P(Y\_1>y)=1-P(X\_1>x),\quad P(X\_1>x)\le P(... | 3 | https://mathoverflow.net/users/36721 | 321186 | 138,652 |
https://mathoverflow.net/questions/320713 | 9 | A topological space has *countable spread* if every discrete subspace is at most countable.
By Theorem 8.10 in Todorcevic's book "[Partition Problems in Topology](https://books.google.com.ua/books/about/Partition_Problems_in_Topology.html?id=k6UZnZ_qkgIC&redir_esc=y)", PFA implies that each regular space $X$ with cou... | https://mathoverflow.net/users/61536 | What is known about topological groups of countable spread in ZFC? | The answer is no.
In the paper *"[A separable normal topological group need not be Lindelöf](https://www.sciencedirect.com/science/article/pii/0016660X76900337)"* (General topology and its applications, 1976), Hajnal and Juhász use the continuum hypothesis to construct a hereditarily separable topological group $G$ w... | 8 | https://mathoverflow.net/users/17836 | 321208 | 138,659 |
https://mathoverflow.net/questions/321201 | 22 | A length space is a metric space $X$, where the distance between two points is the infimum of the lengths of curves joining them. The length of a curve $c: [0,1] \rightarrow X$ is the sup of
$$ d(c(0), c(t\_1)) + d(c(t\_1), d(t\_2)) + \cdots + d(c(t\_{N-1}), c(1)) $$
over all $0 < t\_1 < t\_2\cdots < t\_{N-1} < 1$ and ... | https://mathoverflow.net/users/613 | Is the metric completion of a Riemannian manifold always a geodesic space? | I have been thinking about this since Deane and I discussed it this morning, and I came up with the following idea. Let $\Sigma:=\{1,\tfrac{1}{2},\tfrac{1}{3},\ldots\}\cup \{-1,-\tfrac{1}{2},-\tfrac{1}{3},\ldots\}$. The set $\Sigma\cup \{0\}$ is closed in $\mathbb{R}$.
Let $(M,g)$ be the complement of $[0,1]\times (\... | 17 | https://mathoverflow.net/users/2819 | 321214 | 138,661 |
https://mathoverflow.net/questions/321132 | 14 | Mochizuki's notion of a *Frobenioid* introduced in [The geometry of Frobenioids I](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Geometry%20of%20Frobenioids%20I.pdf) is rather elaborate. However, he also introduces a myriad of further properties that a Frobenioid may satisfy, and his main results in that paper are al... | https://mathoverflow.net/users/2362 | Are fully general Frobenioids necessary? | From [the horse's mouth](http://www.kurims.kyoto-u.ac.jp/~motizuki/Responses%20to%20questions%20on%20Frobenioids.pdf):
>
> Indeed, the only Frobenioids that are used in the IUTeich papers are the
> following:
>
>
> (F1) *tempered Frobenioids*, i.e., a generalization developed in [EtTh], §3, §4,
> §5, of the geo... | 8 | https://mathoverflow.net/users/4177 | 321216 | 138,662 |
https://mathoverflow.net/questions/321130 | 6 | I apologize in advance if this is too broad and off-topic here. I have seen some papers in the field of vertex operator algebras (VOA) theory about simple current extensions. As far as I understand simple currents are invertible objects in some vertex tensor category (for example representation categories of certain VO... | https://mathoverflow.net/users/nan | Simple current extensions in VOA theory and CFTs | I'm not a physicist, so I can't say anything authoritative on your first question, but I can say something about questions 2. and 3.
You are correct that in VOA theory, simple currents are invertible irreducible objects in a vertex tensor category of representations of a VOA (although if one does not know that the ve... | 5 | https://mathoverflow.net/users/118337 | 321222 | 138,664 |
https://mathoverflow.net/questions/321217 | 2 | If $(X,d\_X)$ is a compact metric space, and $(Y,d)$ is another metric space. Moreover, suppose that the metric capacity of $(Y,d)$ is at-least that of $(X,d\_X)$, that is
$$
\kappa\_X(\epsilon)\leq \kappa\_Y(\epsilon) ; (\forall \epsilon \in (0,1]
.
$$
Here the metric capacity of a metric space $(X,d\_X)$ is defined... | https://mathoverflow.net/users/36886 | Continuous inclusion of metric spaces of smaller capacity | The answer is **no**. You can have $X$ path connected and $Y$ totally disconnected satisfying your condition. Then any continuous map from $X$ to $Y$ is necessarily constant so there are no injective maps from $X$ to $Y$.
Here is another example: $X=S^1$ is a unit circle of with the geodesic metric and $Y=\mathbb{R}... | 4 | https://mathoverflow.net/users/121665 | 321223 | 138,665 |
https://mathoverflow.net/questions/321108 | 10 | A simple, undirected graph $G = (V,E)$ is said to be *strongly rigid* if the identity is the only [graph endomorphism](https://en.wikipedia.org/wiki/Graph_homomorphism).
For which positive integers $k>2$ is there a strongly rigid $k$-[regular](https://en.wikipedia.org/wiki/Graph_homomorphism) graph?
| https://mathoverflow.net/users/8628 | Strongly rigid regular graphs | Let $X(\mathcal{S)}$ be the block graph of a Steiner triple system $\mathcal{S}$ on $v$ points. The triple system consists of $b=v(v-1)/6$ triples from a set $V$ of size $v$ such that each pair of points from $V$ lies in exactly one triple. Necessarily $v\equiv1,3$ mod 6 and, if this condition holds, triple systems on ... | 8 | https://mathoverflow.net/users/1266 | 321225 | 138,666 |
https://mathoverflow.net/questions/321215 | 6 | Given a matrix $W\_{n,m}$ whose each entry $w\_{ij}$ is 1 or -1 or 0 with probability $p$, $p$ and $1-2p$ respectively, $0<p<0.45$
Let $R\_i$ be the sum of $i^{th}$ row and $C\_{j}$ is the sum of $j^{th}$ column.
What is the probability that row sums are greater than 0 and column sums are greater than 0, i.e
$$P(... | https://mathoverflow.net/users/134727 | Probability that sum of each row and sum of each column is greater than 0 for a random matrix | To the best of my knowledge, there is no known result that applies to this case immediately. I'll mention some articles which demonstrate techniques that could be used.
[Riordan and Selby (2000)](https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/maximum-degree-of-a-random-graph/2... | 6 | https://mathoverflow.net/users/9025 | 321226 | 138,667 |
https://mathoverflow.net/questions/321224 | 1 | Suppose $0<\alpha<\beta<1$, and $\Omega$ is a bounded subset of $\mathbb{R}^n$. Then the Holder space $C^{\beta}(\Omega)$ is **compactly embedded** into $C^{\alpha}(\Omega)$. But if $\Omega=\mathbb{R}^n$, then the compact embedding is not true.
However, if we consider the weaker weighted Holder space $C^{\alpha, -\de... | https://mathoverflow.net/users/35702 | Reference for compact embedding between (weighted) Holder space on $\mathbb{R}^n$ | Since the norm is quite specific, I am not sure if you can find it in any book. However, you can prove compactness of the embedding directly. Given a sequence $f\_k\in C^{\beta}(\mathbb{R}^n)$, the compactness for bounded domains and a standard diagonal argument shows that you can find a subsequence $f\_{k\_\ell}$ that... | 3 | https://mathoverflow.net/users/121665 | 321229 | 138,669 |
https://mathoverflow.net/questions/321221 | 9 | In quantum field theory Feynman has invented a diagrammatic method to encode various terms in the Taylor decomposition of integrals of the following form below which I will write in a baby version as finite dimensional integral rather than path integral (and using "imaginary time"):
$$Z(j\_1,\dots,j\_n):=\frac{\int\_{\... | https://mathoverflow.net/users/16183 | One particle irreducible Feynman diagrams | Section 5 of Borcherds, Barnard, [Lectures on Quantum Field Theory](https://arxiv.org/abs/math-ph/0204014) is a discussion of the 0-dimensional spacetime case, which gives finite dimensional integrals.
| 5 | https://mathoverflow.net/users/121 | 321233 | 138,672 |
https://mathoverflow.net/questions/321234 | 4 | Given a morphism of Lie groups $ \theta:G\rightarrow H$ and a principal $G$ bundle $ \pi:P\rightarrow M$ there are (at least) two ways to assign a principal $ H$ bundle.
1. See that the morphism of Lie groups $ \theta:G\rightarrow H$ gives an action of $ G$ on $ H$ by $ g.h=\theta(g).h$. Given an action of $ G$ on m... | https://mathoverflow.net/users/118688 | Morphism of Lie groups $\theta:G\rightarrow H$ giving an equivalence of categories $BG\rightarrow BH$? | It is a diffeomorphism, since there is an equivalence of bicategories $DifferentiableStacks \simeq LieGroupoids[W^{-1}]$ where the RHS is the [bicategorical localisation](https://ncatlab.org/nlab/show/bicategory+of+fractions) of the usual 2-category of Lie groupoids a la Pronk. Because of the special nature of the doma... | 5 | https://mathoverflow.net/users/4177 | 321235 | 138,673 |
https://mathoverflow.net/questions/320523 | 2 | I have a question of a conditioned diffusion processes.
This question is somewhat related to an argument which appears [in this article](https://www.ams.org/journals/tran/2000-352-06/S0002-9947-00-02594-0/S0002-9947-00-02594-0.pdf):
Let $D=\{z=(x,y) \in \mathbb{R}^2 \mid |y|<1\}$ and $K=\{(x,y) \in D \mid x<1\}$. We ... | https://mathoverflow.net/users/68463 | On hitting times of conditioned diffusions | Let $h(x,y)$ denote the probability that a Brownian motion started at $(x,y)$ hits $K = \{x < 1, |y| < 1\}$ before it leaves $D = \{|y| < 1\}$. That is, $h = 1$ on $K$, $h = 0$ on $\partial D$ and $h$ is harmonic in $D \setminus K$ (plus the usual continuity condition on regular boundary points of $D \setminus K$). The... | 1 | https://mathoverflow.net/users/108637 | 321236 | 138,674 |
https://mathoverflow.net/questions/321241 | -1 | Let $a$ and $b$ be two real numbers and $p\_n(x,y)$ the polynomial: $$p\_n(x,y)=\sum\_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer.
In a [previous post](https://mathoverflow.net/questions/315532/p-nx-y-sum-i-0n-1xn-1-iyi-is-always-an-integer) I asked if $p\_n(a,b)$ was a rational number (or an integer... | https://mathoverflow.net/users/70464 | If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is | It's false in general: consider $a=\sqrt[4]{2},b=-\sqrt[4]{2}$. Then $p\_4(a,b)=p\_6(a,b)=0,p\_5(a,b)=2$ yet $p\_3(a,b)=\sqrt{2}$.
| 7 | https://mathoverflow.net/users/30186 | 321244 | 138,675 |
https://mathoverflow.net/questions/321246 | 12 | Let $\sum\_{j=1}^{\infty}a\_{j}$ be a convergent series of positive numbers and $\{z\_{j}\}\_{j=1}^\infty$ a closed discrete subset of the open unit disc $\mathbb{D}$. Then $h(z):=\sum\_{j=1}^{\infty}\frac{a\_{j}}{z-z\_{j}}$ is a meromorphic function on $\mathbb{D}$.
If we only consider the case of infinite sum, does... | https://mathoverflow.net/users/134494 | Zeros of an infinite series | This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z\_k$ tend to a limit on the u... | 21 | https://mathoverflow.net/users/25510 | 321247 | 138,676 |
https://mathoverflow.net/questions/321164 | 1 | Let $a,b : V\to W$ be two morphisms of **smooth** complex analytic spaces.
Assume $a$ and $b$ are local analytic isomorphisms.
>
> * Does the equalizer $U$ of $a,b$ exist as a smooth complex analytic space?
>
>
>
It feels the answer should be “no”, even though $a$ and $b$ are local analytic isomorphisms.
A... | https://mathoverflow.net/users/134710 | Equalizer of local analytic isomorphisms | Take, say, $W=\mathbb{C}^n$, $U=$ some neighborhood of 0 in $\mathbb{C}^n$, $a=$ the inclusion. Now fix some analytic $h:U\to W$ such that $h(0)=0$ and $Z:=h^{-1}(0)$ is not smooth at $0$. For small engouh $\varepsilon>0$, $b:=a+\varepsilon h$ is a local isomorphism (possibly after shrinking $U$), but the equalizer of ... | 1 | https://mathoverflow.net/users/7666 | 321254 | 138,678 |
https://mathoverflow.net/questions/300645 | 6 | The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee\_p C\mathbb H$ of two coni over the Hawaiian earring by the bad point $p\in\mathbb H$.
However, as stated at the bottom of [this post](https://wildtopology.word... | https://mathoverflow.net/users/61824 | Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$ | No such homomorphism is possible. The prototypical nature of the Griffiths twin cone guarantees this.
Let $X,Y$ be contractible with basepoints $x,y$ respectively. We only need to assume $\{x\}$ and $\{y\}$ are closed. Let $X\vee Y$ be the wedge with basepoint $\ast$.
Suppose $h:\pi\_1(X\vee Y)\to \mathbb{Z}$ is n... | 4 | https://mathoverflow.net/users/5801 | 321265 | 138,681 |
https://mathoverflow.net/questions/321263 | 6 | Motivating example:
Given a ring, we can construct its lattice of ideals functorially
(with morphisms mapping to the preimage maps on ideals).
Next, we may flatten this category of lattices, obtaining a category
of pairs $(R,I)$ with morphisms $(R,I)\to(S,J)$ being morphisms $f:R\to S$
such that $I=f^\*J$.
Finally, thi... | https://mathoverflow.net/users/134759 | Flattening categories | The construction you’re describing can be seen as the **[Grothendieck construction](https://ncatlab.org/nlab/show/Grothendieck+construction)**, turning the functor $\newcommand{\op}{\mathrm{op}}I : \mathrm{Rng}^\op \to \mathrm{Cat}$ into the total category $\int I$ (what you’ve called the *flattening*) with its project... | 3 | https://mathoverflow.net/users/2273 | 321266 | 138,682 |
https://mathoverflow.net/questions/321257 | 4 | These topics are outside of my area of research, so I am not quite sure where in the literature to find the answers.
In what follows, if $X$ is partially ordered and $n$ is a natural number, let $[[X]]^n$ denote the set of $S\subset X$ such that $|S|=n$ and $S$ is linearly ordered. A *tree* will be a partially ordere... | https://mathoverflow.net/users/134757 | Partition Calculus and Ramsey theory question | Question 1 is true for countable $\omega^\xi$ but it is at least consistent that this is false at $\omega\_1.$ There might be a ZFC counterexample but the first thing that popped to mind is a [Suslin tree](https://en.wikipedia.org/wiki/Suslin_tree). It is consistent with ZFC that there is a Suslin tree but it is also c... | 2 | https://mathoverflow.net/users/2000 | 321271 | 138,684 |
https://mathoverflow.net/questions/321258 | 6 | I am reading a paper that proves the representability for certain functors whose domain is the category of superschemes. The paper claims that to prove representability of functors (or possibly just the functor of interest) in this category we must first place an etale topology on the category.
More broadly, why is ... | https://mathoverflow.net/users/117411 | Motivation for using etale topology in representability of functors problems | I don't know much about superschemes, but would like to share some viewpoints for representability of functors on ordinary schemes. A theorem of Grothedieck states that a representable functor is a sheaf in fpqc topology, hence a priori it is also a sheaf in coarser topologies such as fppf topology and étale topology (... | 7 | https://mathoverflow.net/users/98049 | 321275 | 138,687 |
https://mathoverflow.net/questions/321260 | 14 | Let $X\_0$ be a smooth projective variety over $\mathbf{F}\_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}\_q$.
Crystalline cohomology $H^\*\_{\rm cris}(X) := H^\*((X/W(k))\_{\rm cris},\mathcal{O}\_{(X/W)\_{\rm cris}})[1/p]$ is known to be a Weil cohomology.
It is also known to be computed ... | https://mathoverflow.net/users/nan | Some basic questions on crystalline cohomology | 1)Yes, such decomposition follows from the fact that Frobenius on the de Rham-Witt differential forms acts in a way that slopes on $H^i(X, W\Omega^j)[1/p]$ are in the interval $[j,j+1)$. This forces the spectral sequence coming from the stupid filtration on the de Rham-Witt complex to degenerate at the first page and, ... | 8 | https://mathoverflow.net/users/39304 | 321280 | 138,688 |
https://mathoverflow.net/questions/320210 | 2 | I am adding some context here. I am reading [Introduction to Differentiable Stacks](https://webusers.imj-prg.fr/~gregory.ginot/papers/DiffStacksIGG2013.pdf) by Gregory Ginot.
In page no $7$, just before the remark $2.2$ he says the following.
>
> One shall be careful that fibre product of differentiable stacks is... | https://mathoverflow.net/users/118688 | Fibered product of stacks comes from a Lie groupoid | Think of $BG$ and $BH$ as *topological stacks*, whereby one can calculate a topological groupoid presenting the stack $BG\times\_{BH} BG$, namely the following: the object space is the space underlying $H$ and the morphism space is $G\times H \times G$. The source map $s\colon G\times H \times G \to H$ is the projectio... | 3 | https://mathoverflow.net/users/4177 | 321296 | 138,693 |
https://mathoverflow.net/questions/258108 | 7 | I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz").
The answer seems pretty clear, and proceeds like the definition of manifolds:
1) If $X$ is a topological space, a Lipschitz chart is a homeomorphism from an open subset of $X$ to a metric space.
2) Two L... | https://mathoverflow.net/users/57323 | Objects whose morphisms are Lipschitz maps | I randomly found an answer to my question two years later! The spaces I described in the question are called "locally metric spaces" in
J. Luukkainen, J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci. Fennicae 3 (1977), 85--122.
They are introduced, together with "local metrics", in their Section 3.4. As ... | 2 | https://mathoverflow.net/users/57323 | 321307 | 138,696 |
https://mathoverflow.net/questions/321288 | 3 | Let $3\leq k\leq 8$ be an integer. Suppose $M$ is a complex surface which has a Kahler-Einstein metric and has the same Betti numbers as $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$, i.e.
$$b\_0(M)=b\_4(M)=1,~~ b\_1(M)=b\_3(M)=0,~~b\_2^+(M)=1,~~ b\_2^-(M)=k.$$
Then can we conclude that $M$ is biholomo... | https://mathoverflow.net/users/107048 | Surface with Kahler-Einstein metric | There answer is no. The topological manifolds $\mathbb{CP}^2 \sharp k \overline{\mathbb{CP}^2}$ admit smooth structures that support KE with negative scalar curvature for k=5, 6, 7, 8
For k=8, see <https://arxiv.org/pdf/dg-ga/9705007.pdf>
For k=6, 7, see <https://arxiv.org/pdf/0806.1424.pdf>
For k=5, see <https:... | 2 | https://mathoverflow.net/users/127247 | 321309 | 138,698 |
https://mathoverflow.net/questions/321304 | 3 | Let $(X,\mu)$ be a measure space and $\phi=(\phi\_1,\cdots,\phi\_d)\in L^{\infty}(X)$.
Let
$$r=\max\left\{\sum\_{i=1}^d|z\_i|^2; (z\_1,\cdots,z\_d)\in \mathcal{C}(\phi)\right\},$$
where $\mathcal{C}(\phi)$ is consisting of all $z = (z\_1,\cdots,z\_d)\in \mathbb{C}^d$ such that for every $\varepsilon>0$
$$\mu \left(\... | https://mathoverflow.net/users/113054 | Prove that $\mu \left(\left\{t\in X\,;\;\sum_{i=1}^d|\phi_i(t)|^2>r \right\}\right)=0$ | It suffices to prove that, given $\rho>t$, $g(t) \leqslant \rho$ for almost all $t$, where $g$ is your sum of squares. Assume the contrary. Then the $\phi$-preimage of the set $\Omega=\{(z\_1,\dots,z\_d):\sum z\_i^2 >\rho\} $ has positive measure. The set $\Omega$ is a countable union of compact sets lying in $\Omega$.... | 1 | https://mathoverflow.net/users/4312 | 321310 | 138,699 |
https://mathoverflow.net/questions/321251 | 4 |
>
> **Question:**
>
>
> can the set of edges that resemble the convex hull ($CH$ for short) of $n$ points in the euclidean plane be determined in $O(n)$ time?
>
>
>
I know that the time complexity of determining the $CH$ of $n$ points is $O(n\log h)$ where $h$ is the number of points constituting to the $CH$ a... | https://mathoverflow.net/users/31310 | Complexity of Determining the Edges of Planar Convex Hulls | It's been known for some time that the *unordered convex hull problem*—"just identifying the vertices on the planar convex hull, takes $\Omega(n \log n)$ time":
>
> Ramaswami, Suneeta. "Convex hulls: Complexity and applications (a survey)." Technical Reports (CIS) (1993): 264. [PDF download](https://repository.upen... | 2 | https://mathoverflow.net/users/6094 | 321311 | 138,700 |
https://mathoverflow.net/questions/321062 | 2 | It is well known that the 2d incompressible Navier-Stokes equations under periodic boundary conditions always have global smooth solutions, given smooth initial conditions.
I tried searching for a proof that the 2d incompressible Euler equations under periodic boundary conditions always have global smooth solutions,... | https://mathoverflow.net/users/7089 | 2d incompressible Euler equations under periodic boundary conditions | A famous result of Beale, Kato and Majda shows that the maximum norm of the vorticity has to become infinite if smooth solutions do not exist globally. This is true even in three dimensions. In two dimensions, the $L^\infty$ norm of the vorticity is conserved, so global existence follows immediately.
| 3 | https://mathoverflow.net/users/12120 | 321320 | 138,703 |
https://mathoverflow.net/questions/321303 | 11 | Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?
| https://mathoverflow.net/users/23542 | ZF(C) and category theory | Yes, this is a well-known fact that goes back to Cole, Mitchell, and Osius in the 70's. The relevant kind of category is, as Harry says in the comments, a [well-pointed topos](https://ncatlab.org/nlab/show/well-pointed+topos) with extra properties; Lawvere's [Elementary Theory of the Category of Sets](https://ncatlab.o... | 17 | https://mathoverflow.net/users/49 | 321322 | 138,704 |
https://mathoverflow.net/questions/321339 | -2 | Simplify the following:
\begin{equation}
\sum\limits\_{\ell =1}^n P(n,\ell) (e^x -1)^\ell
\end{equation}
to something like $n!n^n$. I got curious about this expression after going through this answer: math.stackexchange.com/q/3076350.
| https://mathoverflow.net/users/134779 | Sum of: k permutations of n $\times e^x$ | $$\sum\limits\_{l=1}^n {n! \over (n-k)!} (e^x -1)^l =$$
$$e^{\frac{1}{e^x-1}} \left(e^x-1\right)^n \Gamma \left(n+1,\frac{1}{-1+e^x}\right)-1$$
| 0 | https://mathoverflow.net/users/89654 | 321340 | 138,710 |
https://mathoverflow.net/questions/321291 | 1 | Let $\{a\_i\}$ be a sequence of reals such that $|a\_i|\geq|a\_{i+1}|$ for all $i$, and consider the following norm: $$\|\{a\_i\}\| = \sup\_k \frac{1}{\sqrt{k}}\sum\_{i=1}^k |a\_i|~.$$ One can see that -- among all decreasing sequences -- this norm is bounded above by the $\ell\_2$ norm as explained [here](https://math... | https://mathoverflow.net/users/70190 | Characterizing a norm on sequences | The spaces you are asking about are called `Marcinkiewicz sequence spaces', see, for example <http://mate.dm.uba.ar/~slassall/marcinkiewicz.pdf>
| 1 | https://mathoverflow.net/users/85406 | 321342 | 138,711 |
https://mathoverflow.net/questions/321080 | 6 | Fix integers $n\ge 3,d\ge 2$, and partitions $\lambda\_1,\ldots,\lambda\_n$ of $d$. Let $\mathcal{H}$ be the moduli space of degree $d$ covers $f:C\to\mathbb{P}^1$ that have ramification profiles $\lambda\_i$ over pairwise distinct marked points $x\_i\in\mathbb{P}^1$, where $C$ is smooth and projective and the $x\_i$ f... | https://mathoverflow.net/users/119354 | smoothness of Hurwitz spaces with arbitrary ramification profiles | **Edit.** Fixed some mistakes about Zariski tangent spaces. For all of the torsion, cyclic cotangent sheaves $\Omega\_u$ with support $R$, it is necesssary to pass to the $u$-relative dual sheaf $\Omega^\vee\_u := \textit{Hom}\_{\mathcal{O}\_C}(\Omega\_u, \omega\_u\otimes \mathcal{O}\_R)$, which is isomorphic to $\Omeg... | 13 | https://mathoverflow.net/users/13265 | 321343 | 138,712 |
https://mathoverflow.net/questions/321350 | 2 | I need to emulate this sequence for a program: <http://oeis.org/A025302>
Stuff that I've taken into account:
* After finding the prime divisors of a number. I take any divisor as **p** and apply the following rule:
**p is an odd prime divisor of n, then either p appears to an even power in n, or p≡1(mod4)**
* A... | https://mathoverflow.net/users/134789 | Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way | Copying from <http://mathworld.wolfram.com/SumofSquaresFunction.html>:
>
> To find in how many ways a positive integer $n>1$ can be expressed as a sum of $k=2$ squares ignoring order and signs, factor it as
> $$n=2^{a\_0}p\_1^{2a\_1}...p\_r^{2a\_r}q\_1^{b\_1}...q\_s^{b\_s}$$
>
> where the $p\_i$ are primes of... | 4 | https://mathoverflow.net/users/85967 | 321354 | 138,715 |
https://mathoverflow.net/questions/321345 | 2 | Suppose that $a, b$ and $c$ are constant.
Is there the **necessary and sufficient** conditions of $a ,b, c$ for the following integration is integrable? i.e.
$$\int\_0^\infty \int\_0^\infty \int\_0^\infty \frac{1}{(1+x)^a (1+y)^b (1+x+t)^c (1+y+t)^c} t^{-\frac{1}{2}}e^{-\frac{1}{t}} \, dx \, dy \, dt < \infty.$$
| https://mathoverflow.net/users/134602 | Necessary and Sufficient conditions for integrable function | Let us assume that $a,b,c$ are real numbers.
The integral in question equals
\begin{equation}
I=I\_1+I\_2,
\end{equation}
where
\begin{equation}
I\_1:=\int\_0^2 dt\; t^{-1/2}e^{-1/t} J\_a(t)J\_b(t),\quad I\_2:=\int\_2^\infty dt\; t^{-1/2}e^{-1/t} J\_a(t)J\_b(t),
\end{equation}
\begin{equation}
J\_a(t):=\int\_0^\in... | 2 | https://mathoverflow.net/users/36721 | 321356 | 138,716 |
https://mathoverflow.net/questions/300460 | 11 | Let $V$ be a vertex operator algebra with all the good finiteness properties that people usually assume (positively graded, $C\_2$-cofinite, $V\cong V'$, etc.)
Let $W$ be a module for $V$, not necessarily irreducible.
How does one prove, or where can I find a proof of the following fact:
>
> **Claim:**
> The m... | https://mathoverflow.net/users/5690 | Linear independence of genus-one correlation functions | A proof of this property can be found in Proposition 2.2 of Huang's paper *[Vertex operator algebras and the Verlinde conjecture](https://arxiv.org/abs/math/0406291)*. This proof actually uses Zhu's algebras to simplify the discussions (similar to Zhu's proof of the linear independence of characters in *[Modular invari... | 6 | https://mathoverflow.net/users/86652 | 321362 | 138,719 |
https://mathoverflow.net/questions/321368 | 18 | Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into.
Is $d(A)$ uniform over all abelian varieties of a fixed $g$? Or are there special ones that embed into even smaller projective spaces?
Can $d(A)$ be computed explicitl... | https://mathoverflow.net/users/126543 | Embedding abelian varieties into projective spaces of small dimension | Recall that any smooth projective variety of dimension $g$ embeds into $\mathbf{P}^{2g+1}$. Consider now an abelian variety $A$ of dimension $g$ which embeds into $\mathbf{P}^{2g}$. [Van de Ven](https://link.springer.com/article/10.1007/BF02414149) proves (essentially by applying the self-intersection formula to the no... | 30 | https://mathoverflow.net/users/104669 | 321372 | 138,722 |
https://mathoverflow.net/questions/321379 | 9 | In a physical problem I need to investigate the following nonlinear differential equation
$$\ddot x+\omega^2\left (1+\frac{m^2\dot x^2}{p^2}\right)x=0,$$
where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found the following:
* ... | https://mathoverflow.net/users/32389 | Nonlinear oscillator with velocity-dependent frequency | This type of ODE,
$$\ddot{x}+f(x)\dot{x}^2+g(x)=0$$
is known as a Liénard equation of the second kind. It has been studied for example in [Monotonicity of the period function of the Liénard equation of second kind](https://arxiv.org/abs/1608.02910) (2016).
A particular case with nice properties is
$$\ddot{x}-\frac{f'... | 14 | https://mathoverflow.net/users/11260 | 321382 | 138,723 |
https://mathoverflow.net/questions/321298 | 3 | Let $f:\mathbb{R}\_+\to\mathbb{R}\_+$ be an [Orlicz function](https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Orlicz_space), or sometimes referred to as an [Young function](https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Orlicz_space), i.e. it is a convex, non-decreasing function such that $f(0)=0$. I am trying to study ... | https://mathoverflow.net/users/64194 | Which Orlicz functions $f$ make the function $f^{-1}\left(\frac{\sum_{j=1}^s f(x_j)}{s}\right)$ convex? | It follows from Hardy--Littlewood--Polya (see section 3.16, page 86), that if for $x>0$ we have $f, f', f''>0$, and $f'/f''$ is concave, and $d\mu$ is a probability measure on the probability space $\Omega$, then the functional
$$
h \mapsto f^{-1}\left(\int\_{\Omega} f(h(\omega)) d\mu(\omega) \right) \qquad (\*)
$$
... | 2 | https://mathoverflow.net/users/50901 | 321397 | 138,728 |
https://mathoverflow.net/questions/321401 | 1 | I am working with a product of $n\times n$ matrices $A\_1,\ldots,A\_k$. Under which conditions can I assume that
$$\|A\_1\cdots A\_k\|\_\infty \leq \|A\_1\cdots \hat{A\_i}\cdots A\_k\|\_\infty \|A\_i\|\_\infty,$$
where $\|\cdot\|\_\infty$ denotes the operator norm, and $\hat{A\_i}$ denotes the omission of $A\_i$.
... | https://mathoverflow.net/users/111720 | Inequality for the operator norm of a product of matrices | For a simple counterexample, suppose $A\_1$, $A\_2$, $A\_3$ are the orthogonal projections
$$ A\_1 = \pmatrix{1 & 0\cr 0 & 0\cr},\ A\_2 = \pmatrix{1/2 & 1/2\cr 1/2 & 1/2\cr},\ A\_3 = \pmatrix{0 & 0\cr 0 & 1\cr}$$
Then $\|A\_1 A\_3\| = 0$ but $\|A\_1 A\_2 A\_3\| > 0$.
That covers all your cases except "unitary". Of co... | 5 | https://mathoverflow.net/users/13650 | 321403 | 138,729 |
https://mathoverflow.net/questions/321286 | 5 | In algebraic number theory, one constructs for each number field $K$ a subring $\mathcal{O}\_K$, the integral closure of $\mathbb{Z}$ inside $K$, which carries many of the properties which make $\mathbb{Z}$ a nice ring- it is a Dedekind Domain.
Hence, for a number field $K$, we have assigned an integral $\mathcal{O}\... | https://mathoverflow.net/users/30211 | What is the formality behind passing from Number Fields to Number Rings | Let me expand on my comment, with thanks to Daniel Loughran for corrections.
Of course,
>
> If $K$ is a number field, then $\mathcal{O}\_K$ is the unique ring with the following universal property:
>
>
> 1. $\mathcal O\_K$ is an integral domain.
> 2. The fraction field of $\mathcal O\_K$ is $K$
> 3. $\mathcal O... | 5 | https://mathoverflow.net/users/2362 | 321405 | 138,730 |
https://mathoverflow.net/questions/321284 | 5 | Let $G$ be a compact abelian group. The unitary characters of $G$ form an orthonormal basis of $L^2(G)$, so every square integrable function $f: G \rightarrow \mathbb C$ admits a Fourier expansion
$$f(x) = \sum\limits\_{\chi \in \hat{G}} c\_{\chi} \chi(x) \tag{1}$$
where the $c\_{\chi}$ are uniquely determined comp... | https://mathoverflow.net/users/38145 | Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$ | Observe that, because we can cover by open sets, on each of which the function is constant under translation by some open in the finite adeles, by compactness we can take a finite subcover, do the function is translation-invariant by the intersection, another open subgroup. Modding our the whole adele group by this, an... | 1 | https://mathoverflow.net/users/18060 | 321413 | 138,732 |
https://mathoverflow.net/questions/321407 | 0 | Suppose that $a, b, c\_1$ and $c\_2$ are real constant.
Is there the **necessary and sufficient** conditions of $a ,b, c\_1,c\_2 $ for the following integration is integrable? i.e.
$$\int\_1^{\infty}\int\_1^{\infty}\int\_1^{\infty}\frac{1}{~x^{a}~y^{b}~(x+y)^{c\_1}~(x+y+t)^{c\_2}}~t^{-\frac{1}{2}}e^{-\frac{1}{t}} dx... | https://mathoverflow.net/users/134602 | Integrable function | The factor $e^{-1/t}$ is $\asymp1$ for $t>1$, and so, it may be dropped. So, the integral in question is finite iff $I\_j<\infty$ for all $j=1,\dots,6$, where
\begin{equation}
I\_j:=\iiint\limits\_{R\_j}\frac{dx\,dy\,dt}{x^a\,y^b\,(x+y)^{c\_1}\,(x+y+t)^{c\_2}\,t^{1/2}},
\end{equation}
\begin{align}
R\_1&:=\{(x,y,t)\... | 3 | https://mathoverflow.net/users/36721 | 321420 | 138,735 |
https://mathoverflow.net/questions/321406 | 7 | Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper?
I am interested in particular in the case when $X$ and $Y$ are both complete metric spaces.
Any help will be very much appreciated!
EDIT: Just to be ... | https://mathoverflow.net/users/86341 | Proper homotopy | The following counterexample is stolen from page 1 of [this paper](https://arxiv.org/abs/1808.08073) by Thomas Rot. The map $[0,1]\times\mathbb R\to\mathbb R$ given by $(t,x)\to (1-t)x^2+x$ is not proper, e.g., the preimage of $\{0\}$ contains the sequence $(1-\frac{1}{n}, -n)\_{n\in\mathbb N}$. Of course, each parabol... | 11 | https://mathoverflow.net/users/1573 | 321424 | 138,736 |
https://mathoverflow.net/questions/320949 | 3 | Let $G$ be a connected solvable Lie group and let $H$ denote ist commutator subgroup. By definition, every element $g \in H$ can be written as a product of commutators and the minimal number of commutators needed to write $g$ is called the commutator length and denoted ${\rm cl}(g)$.
Since $G$ is amenable, it follows... | https://mathoverflow.net/users/134603 | Commutator length in connected solvable Lie groups |
>
> Proposition: for every connected Lie group $G$, the commutator length is bounded on the subgroup $[G,G]$.
>
>
>
(1) To start with, for arbitrary groups, the property P that the derived subgroup has bounded commutator length, passes to group quotients. Indeed, say that $G$ satisfies P$\_n$, $n\ge 1$, if every... | 2 | https://mathoverflow.net/users/14094 | 321425 | 138,737 |
https://mathoverflow.net/questions/321428 | 8 | Some graphs ($K\_n$ or $K\_n$ minus any one edge, for instance) only permit one minimal colouring up to different labels of the colours. Is there anything known about these kind of graphs? I can think of a number of examples of such graphs (mostly just $K\_{n,m}$ minus an edge or two), but I can't think of any general ... | https://mathoverflow.net/users/122533 | What is known about graphs that permit only one colouring? | It's a bit hard to give a comprehensive answer without knowing exactly what sort of properties you are after, but here is a start. Such graphs are called uniquely colorable graphs (see [here](http://mathworld.wolfram.com/Uniquelyk-ColorableGraph.html) and [here](https://en.wikipedia.org/wiki/Uniquely_colorable_graph)).... | 11 | https://mathoverflow.net/users/2384 | 321430 | 138,738 |
https://mathoverflow.net/questions/321432 | 6 | I'm looking for some $f(x)$ that has the following property:
$\sum\_{x=1}^\infty f(kx) = r^k$
for some real $0 < r < 1$, and at least for strictly positive integer $k$.
Does such an $f(x)$ exist?
This could also be thought of in terms of some sequence of real numbers $f[n]$.
I posted this at MSE but got no an... | https://mathoverflow.net/users/24611 | Looking for infinite series resembling an exponential | You can solve this system of equations explicitly in terms of the function
$$F(z)=\sum\_{m=1}^{\infty} \mu(m)z^m=z-z^2-z^3-z^5+z^6+\cdots$$
where $\mu(m)$ is the [Möbius function](https://en.wikipedia.org/wiki/M%C3%B6bius_function).
The function $F(z)$ can be shown to be transcendental (see the paper ["Transcendence of... | 14 | https://mathoverflow.net/users/2384 | 321433 | 138,740 |
https://mathoverflow.net/questions/312498 | 3 | Suppose that $\mathcal{C}$ is a *finitary* category, so for any two objects $A$ and $B$ we have that $|\mathrm{Ext}^i(A,B)| < \infty$ for $i\geq 0$, suppose $\mathcal{C}$ has *finite global dimension*, so $\mathrm{Ext}^i(A,B) = \{0\}$ for all sufficiently large $i$**\***. In such a category $\mathcal{C}$ we can define ... | https://mathoverflow.net/users/64073 | Intuition for the Euler form in a finitary category | This answer perhaps says things that are all obvious to the OP.
$\textrm{Ext}^i(A,B)$ is a vector space over the ground field, so its cardinality is $q^d$ where $d$ is the dimension of the vector space and $q$ is the cardinality of the field. The quantity in question is therefore $(q^{1/2})^{\langle A,B\rangle}$, whe... | 3 | https://mathoverflow.net/users/468 | 321440 | 138,742 |
https://mathoverflow.net/questions/321299 | 4 | I have discovered (and found an elementary proof of) the following
$$\zeta(2k)=(-1)^{k-1}\dfrac{\pi^{2k}}{2^{4k}-2^{2k}}\left[k\dfrac{2^{2k}}{(2k)!}+{\displaystyle \sum\_{l=1}^{k-1}(-1)^{l}\dfrac{2^{2k-2l}}{(2k-2l)!}\dfrac{2^{4l}-2^{2l}}{\pi^{2l}}\zeta(2l)}\right]$$
($k=1,2,3,\dots$)
My questions are:
1. Is this fo... | https://mathoverflow.net/users/nan | Is the following recursion formula for $\zeta(2n)$ known? | It's clearest to write the identities in terms of Bernoulli numbers $B\_n$ using
\begin{equation}
\zeta (2 n) = \frac{(-1)^{n - 1} (2 \pi)^{2 n}}{2 (2 n)!} B\_{2 n} .
\end{equation}
Remembering that $B\_1 = \frac{1}{6}$ and that $B\_n$ vanishes for all other odd $n$, the desired identity is
\begin{equation}
\sum\_{k = ... | 15 | https://mathoverflow.net/users/39284 | 321455 | 138,747 |
https://mathoverflow.net/questions/321450 | 5 | Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from [Random Diophantine polynomials: Percent solvable?](https://mathoverflow.net/questions/203981/random-diophantine-polynomials-percen... | https://mathoverflow.net/users/10035 | How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average? | $$\mathbb{E}[\text{#|roots of P|}]=\mathbb{E}(\sum\_{k\in \mathbb{Z}}1\_{k \text{ is a root of P}})=\sum\_{k\in\mathbb{Z}}\mathbb{P}(P(k)=0)$$
For $k=0$, $\mathbb{P}(P(0)=0)=\frac{1}{(2B+1)}$.
For $k\neq 0$, because the coefficients are independent (if $B$ and $d$ large) $P(k)$ should behave like a Gaussian if $d$ is... | 4 | https://mathoverflow.net/users/99045 | 321457 | 138,749 |
https://mathoverflow.net/questions/314888 | 9 | Simplicial sets and cubical sets (with or without connections) are defined as presheaves over some indexing categories. There is a full subcategory of simplicial sets that we can identify with the category of (oriented and abstract) simplicial complexes. An object in this category consists of a set together with a coll... | https://mathoverflow.net/users/43574 | Simplicial set are to cubical sets what simplicial complexes are to ...? | (This is rather an answer to the question "what is an abstract cubical complex" asked in the comments:)
In [this paper](http://www.users.miamioh.edu/farleyds/Far1.pdf), Farley defines an abstract cubical complex $C$ as a collection of subsets of a given vertex $V$ with:
* $C$ covers $V$,
* If $\sigma, \tau \in C$, ... | 2 | https://mathoverflow.net/users/130860 | 321458 | 138,750 |
https://mathoverflow.net/questions/313042 | 2 | I have spent an insane amount of time searching for a preprint I have printed a few months ago but misplaced. I cannot find it anymore and this drives me crazy.
It might not have been meant for publication, or might be more of a survey, I am not sure. I do not remember the authors (obviously) but remember the content... | https://mathoverflow.net/users/4961 | $C^1$ partially hyperbolic diffeomorphism have Hölder stable holonomies (reference request) | The preprint I looked for was ["FLAVORS OF PARTIAL HYPERBOLICITY"](http://w3.impa.br/~viana/out/flavors.pdf) by F. Abdenur and M. Viana.
| 1 | https://mathoverflow.net/users/4961 | 321468 | 138,753 |
https://mathoverflow.net/questions/321462 | 8 | By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G\_\delta$-subsets of the real line) it is possible to prove the following fact.
>
> **Theorem (CH).** Under CH the real line contains an uncountable subset $X$ admitting no strictly decreas... | https://mathoverflow.net/users/61536 | A strictly decreasing function between uncountable subsets of the reals | In Todorcevic's book ["Partition Problems in Topology"](https://books.google.com.ua/books/about/Partition_Problems_in_Topology.html?id=k6UZnZ_qkgIC&redir_esc=y) I have found Proposition 8.4(c) saying that under OCA for any uncountable sets $X,Y$ of reals there exists a strictly increasing function $f:Z\to Y$ defined on... | 3 | https://mathoverflow.net/users/61536 | 321469 | 138,754 |
https://mathoverflow.net/questions/321465 | 6 | Note: Here all functions are $\mathbb R \to \mathbb R$. $Id$ denotes the identity function.
Let $g\_i$ be a family of functions indexed by some (potentially uncountable) index set $I$. Given a function $f$, we say $g\_i$ are **uniformly $o(f)$** if for every $e > 0$ there exists some $d > 0$ such that $|g\_i(x)| \le ... | https://mathoverflow.net/users/132446 | Uniformly differentiable functions | At first, we should have $L\_x(h) =f'(x) h$ (this is the definition of the derivative, if we forget the uniformness.)
I claim that $f'$ must be uniformly continuous (seen from summing up the relations for $(x, h) $ and $(x+h, - h) $), and this is enough (seen from Lagrange intermediate value theorem $f(x+h) =f(x) +h... | 3 | https://mathoverflow.net/users/4312 | 321471 | 138,755 |
https://mathoverflow.net/questions/321416 | 5 | Let $U$ be a simple VOA which is self-dual and of CFT type (i.e., $U\simeq U'$, and $U$ has grading $U=\bigoplus\_{n\in\mathbb N}U(n)$ with $U(0)$ spanned by the vacuum vector $\Omega$). Let $V$ be a (conformal) extension of $U$ which is also simple, self-dual, and of CFT type.
Now suppose that $U$ is regular, which ... | https://mathoverflow.net/users/86652 | Are extensions of regular vertex operator algebras also regular? | **Update:** I think it will be useful to have a more coherently written answer, since I have learned more since my original response. Under the assumptions of the question, the answer is **yes**, the vertex operator algebra $V$ will be regular, **provided** that the categorical dimension $\mathrm{dim}\_{\mathcal{C}}\,V... | 4 | https://mathoverflow.net/users/118337 | 321473 | 138,756 |
https://mathoverflow.net/questions/321460 | 1 | Consider an even order, balanced(both partitions have same vertices) bipartite regular graph of order greater than or equal to $12$ and degree atleast six and divisible by $6$. Then is the graph of Type 1(totally colorable by $\Delta+1$ colors where $\Delta$ is the maximum degree)?
By petersen theorem, the graph has ... | https://mathoverflow.net/users/100231 | Total Coloring of even regular bipartite graphs | No it is not always possible to total color a bipartite graph with $\Delta+1$ colors, even with the given restrictions on $\Delta$ and the number of vertices. This is a counterexample. Let $G$ be a complete bipartite graph with $n$ vertices on each side, where $n$ can be any integer you want as long as it is sufficient... | 3 | https://mathoverflow.net/users/122188 | 321485 | 138,760 |
https://mathoverflow.net/questions/321051 | 13 | The *Sorgenfrey line* $\mathbb S$ is the real line endowed with the topology generated by the base consisting of all half-intervals $[a,b)$ for real numbers $a<b$.
The Sorgenfrey line is first-countable and non-metrizable and hence is not homeomorphic to a topological group.
On the other hand, the Sorgenfrey line... | https://mathoverflow.net/users/61536 | Topological groups containing the Sorgenfrey line | Now I have a (relatively simple) ZFC-answer to the initial problem, see [this preprint](https://arxiv.org/abs/1901.10727) for more details.
>
> **Theorem 1.** If a topological group $G$ contains a topological copy of the Sorgenfrey line, then it contains a discrete subspace of cardinality continuum.
>
>
>
*Pro... | 6 | https://mathoverflow.net/users/61536 | 321495 | 138,764 |
https://mathoverflow.net/questions/321501 | 0 | I have a question regarding an alternate form of the Bessel equation and how that alternate form translates to the modified Bessel equation and its solution. The modified form is from:
<http://mathworld.wolfram.com/BesselDifferentialEquation.html>
and looks like this:
$$ \frac{d^2 y}{dx^2}+\frac{1-2\alpha}{x}\frac{dy... | https://mathoverflow.net/users/134821 | Alternate forms of the Bessel equation | $x^\alpha I\_\alpha(\beta x)$, $x^\alpha K\_\alpha(\beta x)$ and
$x^\alpha I\_{-\alpha}(\beta x)$ are indeed solutions to your "modified" differential equation. Of course, there are only two linearly independent solutions on, say, $(0,\infty)$: if $\alpha$ is a non-integer, we have
$$ K\_\alpha(x) = \frac{\pi}{2 \sin(... | 1 | https://mathoverflow.net/users/13650 | 321509 | 138,769 |
https://mathoverflow.net/questions/321497 | 5 | This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.
Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be their von Neumann algebra tensor product.
>
> **Question:** Can every projection in $M\bar{\otimes}N$ be expressed as... | https://mathoverflow.net/users/25499 | Projections in the tensor product of von Neumann algebras |
>
> The answer is **no**.
>
>
>
*Proof.* Let $\mathcal{H}$ and $\mathcal{K}$ be any infinite-dimensional Hilbert spaces, and let $\{\xi\_n\}\_{n=1}^\infty$ and $\{\eta\_n\}\_{n=1}^\infty$ be sequences of any orthogonal vectors of norm $1$ in $\mathcal{H}$ and $\mathcal{K}$, respectively. Then $\zeta:=\sum\_{n=1}... | 7 | https://mathoverflow.net/users/25499 | 321521 | 138,772 |
https://mathoverflow.net/questions/321524 | 3 | Let $A$ a subset of $\mathbb R ^n$, $B=B(x,r) \subset \mathbb {R} ^n$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $\mathbb R ^n$ by $\mathcal H^{n-1}$. Also assume that $\mathcal H^{n-1} (\partial A) < + \infty$ (One can assume that $A$ is a set of finite perimeter in necessary).
In this ca... | https://mathoverflow.net/users/62739 | Hausdorff measure of intersection of a ball and a set in $\mathbb {R} ^ n$ | Let me state and prove the following:
>
> **Proposition.** Let $E \subset \mathbb R^n$ be a set of finite perimeter. For $\mathcal L^1$-a.e. $\rho>0$ the following equality holds:
> $$
> P(E \cap B\_{\rho}) = P(E, B\_{\rho}) + \mathcal H^{n-1}(E \cap \partial B\_{\rho}).
> $$
>
>
>
**Proof.**
Let $u \in BV(... | 5 | https://mathoverflow.net/users/100976 | 321525 | 138,773 |
https://mathoverflow.net/questions/312494 | 3 | If $X$ is a curve with a nodal singularity at $x$, it's referred to [here](https://mathoverflow.net/questions/317/dualizing-sheaf-on-singular-curves) and [here](https://mathoverflow.net/questions/58559/dualizing-sheaf-of-a-nodal-curve) that its dualising sheaf is
$$\omega\_X \ = \ \pi\_\*(\Omega\_{X}(p\_1+\cdots+p\_n)'... | https://mathoverflow.net/users/119012 | Why is this the dualising sheaf of a singular curve? | There are two ways to compute the dualising sheaf of a curve:
1. Mimick Serre's proof of Serre duality (p.8-13 of [this link](https://anagrams-seminar.github.io/grothendieck-duality/overview.pdf)) for nonsingular curves, which directly gives the result in the question and explains where the residue map comes from (it... | 6 | https://mathoverflow.net/users/119012 | 321529 | 138,774 |
https://mathoverflow.net/questions/321483 | 5 | A regular topological space $X$ is called
$\bullet$ *cosmic* if $X$ is a continuous image of a separable metrizable space;
$\bullet$ *[cometrizable](http://www.ams.org/journals/tran/1989-313-01/S0002-9947-1989-0992600-5/S0002-9947-1989-0992600-5.pdf)* if $X$ admits a weaker metrizable topology such that each point ... | https://mathoverflow.net/users/61536 | Is each cosmic space cometrizable? | Discussing this problem with [Alex Ravsky](https://mathoverflow.net/users/43954/alex-ravsky) we constructed the following
>
> **Example.** The Euclidean topology $\tau\_0$ on the set $\mathbb Q$ of rational numbers can be enlarged to a regular topology $\tau$ of weight $\omega\_1$ such that the countable (and hence... | 4 | https://mathoverflow.net/users/61536 | 321558 | 138,779 |
https://mathoverflow.net/questions/321380 | 4 | I asked this in [cstheory.SE](https://cstheory.stackexchange.com/questions/42204/reversible-polynomial-circuit-iff-polynomial-reversible-circuit) a week ago. Since there are no answers or comments, and since this is perhaps more about permutations than computation, I hope it is ok to cross-post here as well.
My quest... | https://mathoverflow.net/users/123634 | Reversible polynomial circuit = polynomial reversible circuit? | There is an easy trick that one can use to encode any function into a bijective function.
Suppose $f:2^{m}\rightarrow 2^{n}$ is an arbitrary function.
Define $L\_{f}:2^{m}\times 2^{n}\rightarrow 2^{m}\times 2^{n}$ by letting
$L\_{f}(x,y)=(x,y\oplus f(x))$.
For a more optimized argument, let us define the reversib... | 2 | https://mathoverflow.net/users/22277 | 321572 | 138,782 |
https://mathoverflow.net/questions/321547 | 24 | There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic).
Then there are several classes of groups like asynchronously automatic groups for which it is known that there is an exponential time algorithm to solve the word problem ... | https://mathoverflow.net/users/122275 | Are there any computational problems in groups that are harder than P? | An earlier reference for groups with this property is
J. Avenhaus and K. Madlener. Subrekursive Komplexität der Gruppen. I. Gruppen mit vorgeschriebenen Komplexität. Acta Infomat., 9 (1): 87-104, 1977/78.
There is a hierarchy of the recursive functions known as the (difficult to pronounce) *Grzegorczyk Hierarchy* $... | 25 | https://mathoverflow.net/users/35840 | 321575 | 138,783 |
https://mathoverflow.net/questions/321580 | 2 | Let $f:\mathbb{C}^n\to\mathbb{C}$ be an entire holomorphic function of $n$ complex variables. Then its Nevannlinna characteristic equals
$$
m\_f(r)=\int\_{\partial B(r)}\log^+|f(z)|d\eta(z),\quad\forall r>0,
$$
where $\partial B(r)$ is the sphere of radius $r$ and $\eta$ is the normalized Euclidean measure on it. In th... | https://mathoverflow.net/users/89313 | Modulus bounded by Nevanlinna characteristic in several variables | I belive the following should work: basically the only thing we know is 2-dimensional estimate that you wrote (I'll always talk about real dimension to avoid ambiguity). Then to get estimate of $\log |f(z\_0)|$ in terms of $\int\_{\partial B(r)} \log^{+}|f(z)|d\eta(z)$ let's take average over all 2-dimensional planes p... | 2 | https://mathoverflow.net/users/104330 | 321585 | 138,785 |
https://mathoverflow.net/questions/312418 | 10 | Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X\_1,X\_2$ such that $X\_1$ and $X\_2$ are homeomorphic but not diffeomorphic, and $\partial X\_1$ and $\partial X\_2$ are homeomorphic to $M$.
I imagine that all 3-manifolds have exotic pairs of ... | https://mathoverflow.net/users/99414 | Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds? | Here is a list of 3-manifolds $Y$ that are boundaries of exotic 4-manifolds <https://arxiv.org/pdf/1901.07964.pdf>
1. If either $Y$ or $-Y$ (i.e with reverse orientation) has a contact structure with non-trivial contact invariant.
2. If $Y$ or $-Y$ has weak symplectic filling.
3. If $Y$ bounds both positive and negat... | 11 | https://mathoverflow.net/users/33064 | 321590 | 138,788 |
https://mathoverflow.net/questions/321563 | 1 | On a degree $n$ Hirzebruch surface $F\_n$, suppose we have a very ample linear system. It is known that its generic smooth irreducible members give a Lefschetz pencil on $F\_n$. Let us take a member, $G$, in this pencil. And suppose we know that $G$ intersects the fiber, $F$, of $F\_n$ $m$ times. Generically they inter... | https://mathoverflow.net/users/134934 | Very ample linear systems - intersections with multiplicity >1 | A partial answer (too long for a comment): suppose $G$ is sufficiently ample so that $H^1(\mathbb{F}\_{n},\mathcal{O}(G-S))=0$. Then the restriction map $H^0(\mathbb{F}\_{n},\mathcal{O}(G))\rightarrow H^0(S,\mathcal{O}(G)\_{|S})$ is surjective. This shows that:
1) There exists $G'\in \lvert G \rvert$ such that $G\cdo... | 2 | https://mathoverflow.net/users/40297 | 321591 | 138,789 |
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