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https://mathoverflow.net/questions/291811 | 2 | I had asked [this](https://math.stackexchange.com/questions/2626310/confusion-in-definition-of-peripheral-spectrum) question on [Mathematics Stack Exchange,](https://math.stackexchange.com/) $2$ days ago but it got no response so I'm asking here.
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If $A$ is a closed operator, then the peripheral spectrum of $A... | https://mathoverflow.net/users/119514 | Confusion in definition of peripheral spectrum | These are indeed two different definitions, see the discussion in [One-parameter Semigroups of Positive Operators](https://books.google.nl/books?id=GTh7CwAAQBAJ&pg=PA169). That reference also gives two different names for the two definitions, peripheral spectrum versus boundary spectrum, and shows they are not equivale... | 2 | https://mathoverflow.net/users/11260 | 291816 | 128,551 |
https://mathoverflow.net/questions/291815 | 8 | We have a structure theorem for f.g. modules over $R$, whenever $R$ is a PID. In the case of $R=\mathbb{Z}/p\mathbb{Z}[x]$ ($p$ a prime), the structure theorem can be used to obtain the rational canonical form for matrices over the finite field $\mathbb{Z}/p\mathbb{Z}$.
>
> I am interested in some kind of canonical... | https://mathoverflow.net/users/31469 | Rational Canonical Form over $\mathbb{Z}/p^k\mathbb{Z}$ | The problem is open, and not because nobody tried. For instance, it is known that the number of similarity classes in $M\_n(\mathbf Z/p^2 \mathbf Z)$ is equal to the number of simultaneous conjugacy classes of pairs of *commuting* matrices in $M\_n(\mathbf Z/p\mathbf Z)$ (S. J AMBOR and W. PLESKEN , *Normal forms for m... | 10 | https://mathoverflow.net/users/9672 | 291823 | 128,552 |
https://mathoverflow.net/questions/291814 | 5 | Let $X$ be a del Pezzo surface (over $\mathbb{C}$), which is obtained by a blow up $\pi: X \rightarrow \mathbb{P}^{2}$ in a collection of points. Let $H$ be the hyperplane class of $\mathbb{P}^{2}$.
**Question**: Consider smooth, irreducible curves $\Sigma$ satisfying the inequality $-K\_{X}\cdot \Sigma > \frac{1}{2}... | https://mathoverflow.net/users/99732 | Curves in del Pezzo surfaces satisfying certain intersection inequality | Here is one proof that the genus is bounded (although presumably a more elegant solution exists).
Write $\Sigma=d\pi^\*H - \sum\_{i=1}^nm\_iE\_i$ in the usual basis for $\operatorname{NS}(X)$ (i.e. $\Sigma$ is the strict transform of a curve of degree $d$ in $\mathbb{P}^2$, with multiplicity $m\_i$ at each of the poi... | 6 | https://mathoverflow.net/users/104695 | 291833 | 128,554 |
https://mathoverflow.net/questions/291836 | 1 | Let $(\mathbb{Z}/n\mathbb{Z})^\*$ denote the multiplicative group of units of $\mathbb{Z}/n\mathbb{Z}$. Is there a finite commutative group $G$ such that for all $n\geq 2$, there is no injective group homomorphism $f:G\to (\mathbb{Z}/n\mathbb{Z})^\*$?
| https://mathoverflow.net/users/8628 | Subgroups of $(\mathbb{Z}/n\mathbb{Z})^*$ | No. By Dirichlet's theorem on primes in progressions, for each $n$ and $k$, there are at least $k$ primes $p\_1,\dots,p\_k$ congruent to $1$ modulo $n$, and so $$\left(\mathbb Z/ \left(\prod\_{i=1}^k p\_i\right) \mathbb Z\right)^\times$$ contains $(\mathbb Z/n)^k$ as a subgroup.
By the classification of finite commut... | 7 | https://mathoverflow.net/users/18060 | 291838 | 128,555 |
https://mathoverflow.net/questions/291834 | 4 | Let $I$ be an arbitary index set, $((A\_i)\_i,\|.\|\_i)\_{i\in I}$ be a family of Banach algebras, with approximate diagonal $(m^{i}\_α)\_α\subseteq A\_i\hat\otimes A\_i$, and $B=\{(x\_i)\_i\in {\displaystyle \prod \_{i\in I}}A\_i|\, \sum\_i\|x\_i\|\_i<\infty\}$.
Is there any approximate diagonal for $B$?
**My idea:*... | https://mathoverflow.net/users/76115 | approximate diagonal | Unfortunately this will not work, but the counterexamples I know rely on results that are either difficult or unpublished.
1) Take $A\_n = M\_n{\mathbb C})$ with usual multiplication and $C^\*$-norm. Each $A\_n$ is amenable with constant $1$, so there exists an actual diagonal element in $A\_n\hat\otimes A\_n$ which ... | 5 | https://mathoverflow.net/users/763 | 291843 | 128,557 |
https://mathoverflow.net/questions/291846 | 1 | Let $\mathbb{Z}\_p$ the ring of $p$-adic numbers. It's known that the multiplicative unit group $\mathbb{Z}\_p ^\times$ can be set theoretically described as $\bigcup \_{1 \le a \le p-1} a+ p\mathbb{Z}\_p$.
I want to know how to see that we have also the group isomorphism $\mathbb{Z}\_p ^\times \cong C\_{p-1} \times... | https://mathoverflow.net/users/108274 | The Unit Group of $\mathbb{Z}_p$ | This is probably more suitable to MathStackExchange, although there may well be researchers in allied areas that aren't aware of this fact. So in that spirit, here's the standard proof. There is an exact sequence
$$ 1 \to 1+p\mathbb Z\_p \to \mathbb Z\_p^\times \to \mathbb F\_p^\times \to 1 .$$
So one needs split this ... | 15 | https://mathoverflow.net/users/11926 | 291849 | 128,558 |
https://mathoverflow.net/questions/291864 | 8 | The density of squarefree numbers is well known.
I am wondering about the squarefree numbers in $[2,n]$ namely the second and higher moments of the gaps. Is anything known about these?
| https://mathoverflow.net/users/17773 | Gaps in Squarefree numbers | A number of authors e.g. Hooley, Filaseta, Trifonov have considered this problem of moments of gaps between square-free numbers. For example, [Filaseta and Trifonov](http://people.math.sc.edu/filaseta/papers/distribpaper.pdf) (paper in Proc. London Math. Soc. (1996); see Theorem 4) showed that for all $0\le \gamma < 43... | 17 | https://mathoverflow.net/users/38624 | 291869 | 128,566 |
https://mathoverflow.net/questions/291872 | 3 | The modular group $\Gamma\_{g}$ of isotopy classes of diffeomorphisms of a genus $g$ surface $S$ acts on $H^1(S,\mathbb{Q})$ (or $H^1(S,\mathbb{Z})$) respecting the intersection pairing. This gives a homomorphism $\Gamma\_{g}\to Sp(2g)$. We also know that this homomorphism is surjective. Therefore, to understand $H^1(S... | https://mathoverflow.net/users/120250 | Symplectic representation of modular group | For your last question: the map from the hyperelliptic modular group to $\operatorname{Sp}(2g,\mathbb{Z}) $ is *not* surjective as soon as $g\geq 3$. This was proved by V. Arnold,
*A remark on the branching of hyperelliptic integrals as functions of the parameters*,
Functional Anal. Appl. 2 (1968), 187–189.
| 3 | https://mathoverflow.net/users/40297 | 291889 | 128,569 |
https://mathoverflow.net/questions/291878 | 2 | Since $\mathbb R$ is a topological ring, the representable contravariant functor $\mathrm{Hom}\_{Top}(-,\mathbb R)$ sends topological spaces to (unital, commutative and associative) $\mathbb R$-algebras. Consequently, it determines a covariant functor from $\mathrm{Top}$ to affine $\mathbb R$-schemes.
It is a straigh... | https://mathoverflow.net/users/75650 | Which completely regular Hausdorff spaces admit a proper map to $\mathbb R$? | The paper [Characters on algebras of smooth functions](http://www.mat.univie.ac.at/~michor/characters.pdf) might help here.
Despite its title it is applicable to completely regular spaces, see 1.3.1.
The main theorem gives sufficient conditions for the evaluation property not involving proper mappings.
See also [thi... | 3 | https://mathoverflow.net/users/26935 | 291898 | 128,573 |
https://mathoverflow.net/questions/291751 | 20 | Define,
$$\lambda\_n =\frac{(\tfrac12)\_n}{(1)\_n} =\frac{(\tfrac12)\_n}{n!} =\frac{\tbinom{2n}{n}}{2^{2n}} =\binom{n-\tfrac12}{n}$$
with *Pochhammer symbol* $(x)\_n$ and *binomial* $\tbinom{n}{k}$. I noticed that the following 14 formulas have a nice "affinity".
>
> **Level 3:**
>
>
>
$$\sum\_{n=0}^\infty \... | https://mathoverflow.net/users/12905 | On a pattern for upside-down Ramanujan pi formulas | All these pairs of formulas are examples of the transformation $n \to n + \frac12$ removing a factor which does not depend on n. The upside-down transformation is essentially $n \to -n$ (therefore it changes $z$ to $z^{-1}$) reinterpreting $(a)\_{-n}$ as $\frac{(-1)^n}{(1-a)\_n}$ if $a \neq 1$, and $(1)\_{-n}$ as $\fra... | 10 | https://mathoverflow.net/users/120265 | 291899 | 128,574 |
https://mathoverflow.net/questions/291818 | 4 | Let $k$ be an arbitrary field with $\operatorname{char}(k) \neq 2$. Let $G$ be a linear algebraic group over $k$. Let $X$ be the conjugacy class of a semisimple element $s \in G(k)$ of order 2 (or a union of such sets).
Can we say something about the subgroup of $G(k)$ generated by $X$? Is it closed under the Zarisk... | https://mathoverflow.net/users/120232 | Is the subgroup generated by a conjugacy class of semisimple elements Zariski closed? | We can assume that the $G(k)$-conjugacy class of $s$ generates a Zariski-dense subgroup $Q$ of $G$.
Let me assume that $k$ has characteristic zero and provide consequences. So $G=U\rtimes S$ for the unipotent radical $U$, which is defined over $k$, and some $k$-defined reductive subgroup $S$, and $G(k)=U(k)\rtimes S... | 4 | https://mathoverflow.net/users/14094 | 291902 | 128,575 |
https://mathoverflow.net/questions/291911 | 11 |
>
> Is $[0,1]$ a disjoint union of $\aleph\_1$ compact subsets with empty interior?
>
>
>
The answer is obviously yes assuming the continuum hypothesis. Also, by [Baire's lemma](https://en.wikipedia.org/wiki/Baire_category_theorem), the answer is negative if one replaces $\aleph\_1$ with $\aleph\_0$.
Does anyb... | https://mathoverflow.net/users/36952 | Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior? | It is independent of ZFC. As you mention, CH implies the answer is yes. A different axiom, MA+$\neg$CH, implies the answer is no.
A little more precisely, there is a cardinal number denoted $\mathrm{cov}(\mathcal M)$, one of the so-called ``small cardinals'', that is defined to be the smallest number of meager sets n... | 19 | https://mathoverflow.net/users/70618 | 291915 | 128,579 |
https://mathoverflow.net/questions/291827 | 1 | Let us denote by $Mor\_3(\mathbb{P}^1,\mathbb{P}^3)$ the spaces of degree three morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^3$,
$$f(x\_0,x\_1)=[f\_0(x\_0,x\_1):f\_1(x\_0,x\_1):f\_2(x\_0,x\_1):f\_3(x\_0,x\_1)]$$
where $f\_0,f\_1,f\_2,f\_3$ do not have common factors. Then $Mor\_3(\mathbb{P}^1,\mathbb{P}^3)$ sits ins... | https://mathoverflow.net/users/nan | Compactifications of spaces of morphisms | I started writing an answer about the generators and relations for the $\mathbb{Q}$-Picard group of the stack of stable maps of genus $0$ curves to an arbitrary projective homogeneous variety, but it quickly got too long. So here is an explanation of the extra divisor class relation that I was missing in my comments. T... | 1 | https://mathoverflow.net/users/13265 | 291921 | 128,583 |
https://mathoverflow.net/questions/291914 | 1 | What is the CW-complex of Eilenberg-MacLane space $K(\mathbb{Z}\_2,2)$?
What is the CW-complex of Eilenberg-MacLane space $K(\mathbb{Z}\_n,d)$?
What is the CW-complex of Eilenberg-MacLane space $K(\mathbb{Z}\_n\times \mathbb{Z}\_m,d)$?
For example, I like to know the number of cells in each dimensions, and the ... | https://mathoverflow.net/users/17787 | CW-complex of Eilenberg-MacLane spaces | This is mostly an addendum to Denis' answer.
The "standard model" of the Eilenberg-Mac Lane space $K(A,n)$ that Denis mentioned has the following number of $k$-dimensional cells:
$$
\sum\_{m=0}^k \binom{k}{m} (-1)^{k-m} |A|^{\binom{m}{n}}
$$
When $n=1$, this is
$$
\sum\_{m=0}^k \binom{k}{m} (-1)^{k-m} |A|^m = (|A|-... | 11 | https://mathoverflow.net/users/360 | 291923 | 128,584 |
https://mathoverflow.net/questions/291819 | 6 | Consider a rational Quaternion algebra $M$ over $\mathbb{Q}$ that does not split at $\infty$. For example take the rational Hamilton quaternions $M=\mathbb{Q}(-1,-1)$.
For the adele ring $\mathbb{A}$ we define $M(\mathbb{A}):= M\otimes\_{\mathbb{Z}} \mathbb{A}$ and $G=G(\mathbb{A}):= M(\mathbb{A})^\times / \mathbb{A... | https://mathoverflow.net/users/113771 | Computing Tamagawa number of torus in Quaternion algebra | Here are some more details. As John Voight said, the quaternion algebra is kind of irrelevant here. If $\gamma$ is a regular semisimple element, then its centralizer is a torus ${\mathbf T}$ over ${\mathbb Q}$ satisfying
$${\mathbf T}({\mathbb Q}) = K^\times / {\mathbb Q}^\times,$$
where $K = {\mathbb Q}(\gamma)$ is th... | 6 | https://mathoverflow.net/users/3545 | 291925 | 128,585 |
https://mathoverflow.net/questions/291894 | 6 | In the framework of classical motives, there is no such thing as a motive $\mathbb Q(-\tfrac 12)$, i.e. a tensor root of $\mathbb Q(-1)$. There is one, however, in a more general setting of "exponential motives", over any number field containing $\sqrt{-1}$. There is even a motive $M$ over $\mathbb Q$ whose tensor squa... | https://mathoverflow.net/users/5952 | The L-function of Q(-1/2) and the "number of prime $p\equiv 3$ divisors" function | For your first question, note that one can write your $L$-series as
$$
L(s) = \zeta\big(s-\tfrac12\big)^{(1+i)/2} L\big(s-\tfrac12,\chi\_{-4}\big)^{(1-i)/2} M(s),
$$
where $\chi\_{-4}$ is the nontrivial character modulo $4$, and $M(s)$ is an Euler product that converges absolutely for $\Re s>1$. Therefore $L(s)$ does h... | 5 | https://mathoverflow.net/users/5091 | 291935 | 128,588 |
https://mathoverflow.net/questions/291887 | 4 | A classic formulation of the Bernstein inequality (from Wikipedia) is as follow:
Let $X\_1, \ldots, X\_n$ be independent zero-mean random variables. Suppose that $|X\_i|\leq M$ almost surely, for all $i$. Then, for all positive $t$,
$\mathbb{P} \left (\sum\_{i=1}^n X\_i > t \right ) \leq \exp \left ( -\frac{\tfrac{... | https://mathoverflow.net/users/101250 | Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences | It appears you want to have the following:
>
> Let $X\_1,\dots,X\_n$ be independent zero-mean random variables (r.v.'s ) (or, more generally, martingale-differences) with $S\_n:=X\_1+\dots+X\_n$, $B^2:=EX\_1^2+\dots+EX\_n^2$, and $M:=\frac1n\sum\_1^n M\_i$, where $M\_i:=\text{ess sup}|X\_i|$. Then
> \begin{equat... | 5 | https://mathoverflow.net/users/36721 | 291942 | 128,592 |
https://mathoverflow.net/questions/291883 | 9 | Paracontrolled calculus was developed by Gubinelli, Imkeller and Perkowski as a way of treating singular stochastic PDEs such as KPZ, $\Phi\_3^4$ or PAM, around the same time regularity structures were introduced by Hairer.
Regularity structures are now the standard treatment of singular stochastic PDEs although I h... | https://mathoverflow.net/users/nan | Is there any reason to use paracontrolled calculus over regularity structures? | I don't think that the reason given in the paper by Bailleul and Bernicot is a good one. Basically, they treat an example which is simple enough so that it is still manageable to describe the various bits and pieces needed to control their solutions "by hand" instead of combining them into a single object in a more coh... | 18 | https://mathoverflow.net/users/38566 | 291950 | 128,594 |
https://mathoverflow.net/questions/291969 | 0 | I guess that the number of integers $x$ which satisfy the condition $x\*p(x) \leq n$ is $O(n^{2/3})$ or $O(n^{3/4} / \ln n$), but I cannot prove it. I just write a program to count the number. The results are listed.
Note: $p(x)$ means the largest prime factor of $x$.
n the numbers
$10^5$ 1894
$10^6$ 9108 ... | https://mathoverflow.net/users/120302 | How many integers $x$ satisfy that $x*p(x) \leq n$, where $p(x)$ means the largest prime factor of $x$? | Quick observation. Choose $\alpha$ close to 1 and consider the number for which $x\leqslant n^{\alpha}=:N$, $p(x)\leqslant n^{1-\alpha}=N^{(1-\alpha)/\alpha}$. The number of such $x$ grows as $\rho(\frac\alpha{1-\alpha})N$, where $\rho$ is [Dickman's function](https://en.wikipedia.org/wiki/Dickman_function). It already... | 5 | https://mathoverflow.net/users/4312 | 291972 | 128,599 |
https://mathoverflow.net/questions/291970 | 2 | Consider a smooth curve $\gamma$ of finite length in the unit square $[0,1]\times[0,1]$. Is the following statement correct?
There exists a Lebesgue null set $N$ such that for all $x\in [0,1]\setminus N$ the set $\{y\in[0,1]:(x,y)\in\gamma\}$ has only finitely many points.
Generalization:
Consider a smooth manifo... | https://mathoverflow.net/users/79956 | Integral Geometry | By the theorem of Sard (Morse 1939, Sard 1942) the set of all singular values of a $C^k$-mapping $M\to N$ is of Lebesgue measure 0 in $N$ if $k> \max\{0, \dim(M)-\dim(N)$. Here $M$ and $N$ are smooth manifolds. A point $y\in N$ is a regular value of $f$ if $T\_xf$ is surjective for all $x\in f^{-1}(y)$. If not, then $y... | 3 | https://mathoverflow.net/users/26935 | 291974 | 128,601 |
https://mathoverflow.net/questions/291984 | 3 | It can be shown, by using the fact that $E\_4(z)^2 = E\_8(z)$ (where the $E\_k$ are Eisenstein series), that $$\sigma\_7(n) = \sigma\_3(n) + 120\sum\_{0 < m < n}\sigma\_3(m -n)\sigma\_3(n)$$ where $\sigma\_k(n) := \sum\_{d|n} d^k$. Is there a more elementary proof of this, and is there a simple reason one should expect... | https://mathoverflow.net/users/40983 | Elementary proof of certain divisor sum idenrity | I have asked a question about a *different* proof of such identities here: [Derivation of certain sums "the hard way"](https://mathoverflow.net/questions/282989/derivation-of-certain-sums-the-hard-way?rq=1), but I would say that the non-modular proofs are much less elementary than the modular ones.
To answer your se... | 6 | https://mathoverflow.net/users/114143 | 291986 | 128,606 |
https://mathoverflow.net/questions/291989 | 3 | Let $p,q$ be a pair of coprime positive integers. Let $S(N,p,q)$ be the number of integer solutions $(x,y)$ of $N=p x^2+q y^2$ such that $x$ and $y$ are coprime. If $(p,q)=(1,1)$, it follows from Fermat's two square theorem that $S(N,1,1)$ has no upper bound as $N\to\infty$. What will happens if $(p,q)\neq(1,1)$, i.e. ... | https://mathoverflow.net/users/97252 | The number of coprime integer solutions of: $ N=p x^2+q y^2$ | Let $F$ be a positive definite binary quadratic form. For a positive number $Z$ put $N\_F(Z)$ for the number of pairs of integers $(x,y)$ such that $F(x,y) \leq Z$, and let $R\_F(Z)$ denote the number of integers $n \leq Z$ such that the equation $F(x,y) = n$ has solutions in integers $x$ and $y$. It is easy to see fro... | 2 | https://mathoverflow.net/users/10898 | 291993 | 128,607 |
https://mathoverflow.net/questions/291992 | 5 | A lot of results are available for the following chain-rule problem:
(**CRP1**) Let $f\colon \mathbb R \to \mathbb R$ be a $C^1$/Lipschitz function and let $g \colon \mathbb R^d \to \mathbb R$ be a weakly differentiable function (e.g. $W\_{\rm loc}^{1,p}$ or $BV\_{\rm loc}$). Then the function $f \circ g$ is weakly ... | https://mathoverflow.net/users/111164 | Chain-rule and change of variables in BV/Sobolev | Your formula can be wrong even if $f$ and $g$ are both Lipschitz. For criteria when such a result holds (and related results) see e.g. Leoni, Giovanni, Morini, Massimiliano: Necessary and sufficient conditions for the chain rule in $W^{1,1}\_{loc}(ℝ^N;ℝ^d)$ and $BV\_{loc}(ℝ^N;ℝ^d)$. J. Eur. Math. Soc. (JEMS) 9 (2007). ... | 4 | https://mathoverflow.net/users/90407 | 292006 | 128,613 |
https://mathoverflow.net/questions/292008 | 3 | Take $p\in (1,\infty)\setminus \{2\}$. Let $X$ be a subspace of $\ell\_p^n$ and let $U\colon X\to \ell\_p^m$ ($m\geqslant n$) be a linear isometry. Is it possible to extend $U$ to a (non-surjective) linear isometry $\hat{U}\colon \ell\_p^n\to \ell\_p^m$?
For contractions this is not necessarily true however the count... | https://mathoverflow.net/users/106520 | Extending linear isometries from subspaces of $\ell_p^n$ | No, take $m=n=2$ and let $X$ be the $x$-axis. For every line $Y$ through the origin there is an isometry from $X$ to $Y$, and if $p\neq 2$ they obviously don't extend to isometries from $\ell^n\_p$ to itself.
| 4 | https://mathoverflow.net/users/23141 | 292009 | 128,614 |
https://mathoverflow.net/questions/291999 | 0 | Let $\alpha\ge 1$ be an even integer, and $k$ an intger s.t, $1\le k\le \alpha$. Set $\alpha'=\alpha/2$,
$$
A=\mathrm{Card}(\{n : 0\le n\le k-1, k\mid\alpha'(4n+1)\}),\quad B=\mathrm{Card}(\{n : 0\le n\le k-1, k\mid\alpha'(4n+3)\})
$$
So my question is: Can we find $A$ and $B$ explicitly in terms of $\alpha'$ and $k$?... | https://mathoverflow.net/users/120314 | Is $\mathrm{Card}(\{n : 0\le n\le k-1, k\mid\alpha'(4n+1)\})=\mathrm{Card}(\{n : 0\le n\le k-1, k\mid\alpha'(4n+3)\})$? | This is a straightforward exercise in elementary number theory, but I felt like doing it.
For any integer $m$, the relation $k\mid\alpha'm$ is equivalent to $\frac{k}{\gcd(k,\alpha')}\mid m$. $A$ is the number of such $m$'s with the additional properties $m\equiv 1\pmod{4}$ and $1\leq m<4k+1$, while $B$ is the numbe... | 2 | https://mathoverflow.net/users/11919 | 292010 | 128,615 |
https://mathoverflow.net/questions/292012 | 2 | While working on some problem (not relevant here), it turned out to be convenient to be able to enclose arbitrary compact subsets in "nicer" compact subsets, hence the question:
>
> if $(M,g)$ is a Riemannian manifold and $K \subseteq M$ is compact, does there exist a connected open subset $U \subseteq M$ such that... | https://mathoverflow.net/users/54780 | Riemannian manifolds: every compact subset is contained in a connected relatively compact open subset | You will need to assume $K$ lies in a connected component of $M$.
Then for every point $x\in K$ there is a small open balls in $M$ whose closure is compact (you can just do this in charts). Since $K$ is compact it is covered by finitely many of such balls. Now choose a path that goes through these (finitely many) ope... | 3 | https://mathoverflow.net/users/12156 | 292013 | 128,616 |
https://mathoverflow.net/questions/291797 | 3 | Let $X$ be a non-singular hyperelliptic curve (over $\mathbb{C}$) and $\pi:X \to \mathbb{P}^1$ be a $2:1$ covering. Let $\sigma:X \to X$ be the hyperelliptic involution and $E$ be a locally free sheaf on $X$ such that $
\sigma^\* E \cong E$. Then, does there exist a locally free sheaf $F$ on $\mathbb{P}^1$ such that $\... | https://mathoverflow.net/users/32151 | Descent of coherent sheaves on finite coverings | Assume $\sigma^\*E \cong E$. Then $E$ is a pullback if and only if the action of $\sigma$ on the fiber of $E$ at each ramification point of $\pi$ is trivial.
In the counterexample of nfdc23, the $\sigma$ acts by $-1$ on the fiber at $x$.
| 2 | https://mathoverflow.net/users/4428 | 292025 | 128,619 |
https://mathoverflow.net/questions/292023 | 6 | A function $f: \mathbb{R} \to \mathbb{R}$ is called a Darboux function if and only if it maps every connected subset of $\mathbb{R}$ to a connected set.
As an example :
We know that (a.k.a., the Intermediate Value Property of continuous functions) all continuous functions are Darboux functions, further all the funct... | https://mathoverflow.net/users/109471 | Is the sum of a Darboux function and a polynomial necessarily a Darboux function? | No, the fact that $f$ is Darboux does not even imply that $x\mapsto f(x)+x$ is Darboux.
Here is one way to construct a counterexample. Fix a bijection $\psi\colon\mathbb{R}/\mathbb{Q}\to\mathbb{R}$ (which exists because both have the same cardinality). Let $g\colon\mathbb{R}\to\mathbb{R}$ be the composition of $\psi$... | 15 | https://mathoverflow.net/users/17064 | 292026 | 128,620 |
https://mathoverflow.net/questions/291896 | 2 | Given $n$, the number of vertices, what is the number of labeled triangle-free graphs on $n$ vertices?
| https://mathoverflow.net/users/119603 | Counting labeled triangle-free graphs on $n$ vertices | There shouldn't be any sensible exact formula, as Ira Gessel says.
But there are very good asymptotics and a structural description.
An old result of Erdős, Kleitman and Rothschild is that almost all triangle-free graphs are bipartite. Proemel, Schickinger and Steger refined this to show that almost all triangle-free... | 8 | https://mathoverflow.net/users/36212 | 292027 | 128,621 |
https://mathoverflow.net/questions/292022 | 2 | Given a finite family $A:=(A\_j)\_{j\in J}$ of operators $A\_j \in\mathcal{L}(H)$,
>
> It is true that for all $L\in \mathcal{L}(H)$, we have
> $$\big\|\sum\_{j\in J}A\_j^\*LA\_j\big\|\le \big\|\sum\_{j\in J}A\_j^\*A\_j\big\|\ \|L\|?$$
>
>
>
If $L\in \mathcal{L}(H)^+$, one has
$$\big\|\sum\_{j\in J}A\_j^\*LA... | https://mathoverflow.net/users/113054 | $\big\|\sum_{j\in J}A_j^*LA_j\big\|\le \big\|\sum_{j\in J}A_j^*A_j\big\|\ \|L\|?,\;\forall \,L\in \mathcal{L}(H)$ | Actually yes: it follows from the analogous inequality for any $x$ and $y$, and Cauchy-Schwarz. One has
$$ \big(\sum\_{j\in J}A\_j^\*LA\_jx, \ y\big) = \sum\_{j\in J}(A\_j^\*LA\_jx,y) = \sum\_{j\in J}(LA\_jx,A\_jy)\le \sum\_{j\in J}\|L\|\ \|A\_jx\|\|A\_jy\|$$
$$ \le\|L\|\sqrt{ \sum\_{j\in J}\|A\_jx\|^2 \sum\_{j\in J... | 5 | https://mathoverflow.net/users/6101 | 292028 | 128,622 |
https://mathoverflow.net/questions/292015 | 5 | Conjecture: Let $f:{\mathbb C}^n\rightarrow{\mathbb C}$ be an entire function in $n$ complex variables. Assume
that for every $x\in{\mathbb R}^n$ there exists a $y\_x\in{\mathbb R}^n$ such that
$f(x+iy\_x)\equiv f(x\_1+iy\_{x,1},\ldots,x\_n+iy\_{x,n})=0$. Then $f\equiv 0$. (We don't assume continuity
of $x\mapsto y\_... | https://mathoverflow.net/users/10901 | zeros of holomorphic function in n variables | The conjecture is obviously false even for $n=2$. Check $f(z,w)=(w-z^2)(z-(w+1)^2)$. Write $z=x+iy$ and $w=u+iv$. Given $x$ and $u$, I can make first term zero unless $u>x^2$. I can make the second term zero unless $x>(u+1)^2$. Since both inequalities can not be true, we are done.
| 8 | https://mathoverflow.net/users/1811 | 292030 | 128,623 |
https://mathoverflow.net/questions/291961 | 9 | Are the [Atiyah-Bott-Shapiro Orientation](https://ncatlab.org/nlab/show/K-orientation#UniversalAtiyahBottShapiroOrientation) and the [Anderson-Brown-Peterson Splitting](https://projecteuclid.org/euclid.bams/1183527786) compatible in any sense?
The first guess is that the ABS-Orientation is related to the projections... | https://mathoverflow.net/users/21985 | Atiyah Bott-Shapiro orientation Vs Anderson-Brown-Peterson Splitting | According to Michael Hopkins, Mark Hovey, *[Spin cobordism determines real K-theory](https://ncatlab.org/nlab/files/HopkinsHoveyCobordismK.pdf),* Mathematische Zeitschrift 210.1 (1992): 181-196, 4th page of the pdf file, the Atiyah-Bott-Shapiro Orientation is just one of the Anderson-Brown-Peterson Splitting map $\pi ^... | 8 | https://mathoverflow.net/users/43326 | 292037 | 128,626 |
https://mathoverflow.net/questions/292032 | 3 | I'am wondering whether there exists a non-discrete hyperbolic totally disconnected locally compact group such that the boundary is a finite-dimensional sphere. If the answer is positive, could you please provide few examples?
| https://mathoverflow.net/users/120328 | Examples of (non-discrete) hyperbolic totally disconnected locally compact groups whose boundaries are spheres | Non-discrete is not a reasonable assumption: for any hyperbolic group take the direct product with a compact group. Every hyperbolic group locally compact group $G$ has a unique maximal compact normal subgroup $W(G)$, which for $G$ non-elementary is the kernel of the $G$-action on the boundary $\partial G$. A reasonabl... | 6 | https://mathoverflow.net/users/14094 | 292046 | 128,630 |
https://mathoverflow.net/questions/292049 | 2 | Let $W = aI\_{n\times n} + bJ\_{n\times n}$, where $I$ is an identity matix, $J$ is the matrix of all ones, $a,b\in\mathbb{R}$ and a+b>0. Also, let $A = \mathbf{P} - \mathbf{p}\mathbf{p}^{T}$, where $\mathbf{p} = (p\_{1},\ldots,p\_{n})^{T}$ and $\mathbf{P} = \rm{diag}(\mathbf{p})$ with $\sum\_{i=1}^{n}p\_i = 1$ and $0<... | https://mathoverflow.net/users/120111 | Proving symmetry of trace function of special matrix | To check the symmetry of the trace under a given permutation of the $p\_i$'s, take the corresponding permutation matrix $S$ and insert in the trace noting that $S^TS=I$ and $SWS^T=W$:
$${\rm tr}\,(AW)^K={\rm tr}\,(AS^T SWS^TS)^K={\rm tr}\,(\tilde{A}W)^K$$
with $\tilde{A}=SAS^T$ the matrix $A$ after the desired perm... | 2 | https://mathoverflow.net/users/11260 | 292052 | 128,633 |
https://mathoverflow.net/questions/291740 | 5 | A metric space $X$ is defined to be an *absolute Lipschitz extensor for compacta* if each Lipschitz map $f:K\to K$ defined on a compact subset $K\subset X$ extends to a Lipschitz map $\bar f: X\to X$.
**Question.** Is each Banach space an absolute Lipschitz extensors for compacta?
I admit that the answer to this qu... | https://mathoverflow.net/users/61536 | Which Banach spaces are absolute Lipschitz extensors for compacta? | Here is one way you can prove that a space $X$ is not an absolute extensor for compacta: Find sequences $(E\_n)$ and $(F\_n)$ of finite dimensional subspaces of $X$ and a constant $C$ so that for every $n$ there is a linear isomorphism $T\_n$ from $E\_n$ onto $F\_n$ s.t. $\|T\_n\|=1$, $\|T\_n^{-1}\|\le C$, each $F\_n$ ... | 6 | https://mathoverflow.net/users/2554 | 292058 | 128,635 |
https://mathoverflow.net/questions/291866 | 1 | I'm not very familiar with differential geometry and am coming from a general relativity background, so would appreciate help with a question from that context. If this question could be posed in a more abstract sense, that would also be of tremendous help!
Suppose I have a curved space-time with metric $g\_{\mu \nu}... | https://mathoverflow.net/users/99341 | Invariance of a vector under parallel transport along an infinitesimal orthogonal loop | In dimension $3$, the curvature tensor can always be written with respect to an orthonormal frame as something like
$$
R\_{ijkl} = R\_{ik}\delta\_{jl} + R\_{jl}\delta\_{ik} - R\_{il}\delta\_{jk} - R\_{jk}\delta\_{il} - \frac{1}{2}S(\delta\_{ik}\delta\_{jl} - \delta\_{il}\delta\_{jk})
$$
If you let $v$, $a$, $b$ be the ... | 6 | https://mathoverflow.net/users/613 | 292062 | 128,638 |
https://mathoverflow.net/questions/292047 | 1 | In <https://www.sciencedirect.com/science/article/pii/0001870891900378> section 6 a cotilting module T over an algebra A is said to be strong in case $\hat{add(T)}$ coincides with the subcategory of modules having finite injective dimension.
Here $\hat{add(T)}$ is just the full subcategory of all module $M$ such that... | https://mathoverflow.net/users/61949 | Question on strong cotilting modules | I believe I had this exact same question some time ago. Auslander and Reiten attribute this result to a paper of Auslander and Green. I still have the latter paper, but can no longer find where this result is in that paper. From my old notes, it looks like I concluded that this proposition is misstated in Auslander and... | 1 | https://mathoverflow.net/users/11791 | 292063 | 128,639 |
https://mathoverflow.net/questions/292077 | 9 | For me, a fusion category (over $\mathbb{C}$) is just a tensor $F$ (the associator, with $6$ simple-object labels and $4$ fusion space indices) and a tensor $d$ (the quantum dimensions, with one simple-object label), such that the $F$-tensor fulfils the pentagon equation. (This seems to be the most useful language for ... | https://mathoverflow.net/users/115363 | What do "pivotal" and "spherical" mean for (unitary) fusion categories on the level of the $F$-symbols? | Any unitary fusion category has a canonical spherical structure. See Example 2.12 of [this paper](https://arxiv.org/pdf/1611.04620.pdf) and the references therein. It also follows from a more general result Prop 8.23 in [this paper](https://arxiv.org/abs/math/0203060). So you don't need to do anything in the unitary ca... | 12 | https://mathoverflow.net/users/22 | 292085 | 128,646 |
https://mathoverflow.net/questions/291853 | 2 | Consider the natural bijection between $(0,1]$ and the infinite $\{0,1\}$-sequences with infinitely many ones (written in base $2$).
Define also the sequence $(x\_n)$ by $x\_n=1$ if $n$ is odd and $x\_n=0$ otherwise. Hence, for each $\omega \in (0,1]$, we can consider the subsequence $((x \upharpoonright \omega)\_n)... | https://mathoverflow.net/users/32898 | Meager set of subsequences | The answer is yes.
Denote here by $\mathbb{N}$ the set of positive (nonzero) integers. First consider the application :
$$f : (y\_k)\_{k \in \mathbb{N}} \longrightarrow \sum\_{k \in \mathbb{N}} 2^{-(y\_1 + \ldots + y\_k)}.$$
This is a continuous bijection from the Baire space $\mathbb{N}^\mathbb{N}$ to $(0, 1]$ (to a... | 3 | https://mathoverflow.net/users/120363 | 292096 | 128,650 |
https://mathoverflow.net/questions/292073 | 12 | Let $R$ be a DVR with uniformizer $\pi$, fraction field $K$ and residue field $k$. Let $X/K$ be a Severi-Brauer variety and $\mathscr X/R$ a smooth, projective model of it. Is it true that $\mathscr X\_k/k$ is also a Severi-Brauer variety? What if we only assume that $\mathscr X/R$ is smooth and proper?
One way to tr... | https://mathoverflow.net/users/58001 | Smooth projective models of Severi-Brauer varieties over a DVR are also Severi-Brauer varieties | I wrote up some notes on this in 2004. There have been some developments since then that I will indicate below.
Denote the smooth, proper morphism as follows, $$\pi:\mathcal{X}\to \text{Spec}\ R.$$ Since $\pi$ is flat and proper, also the fiber product morphism, $$\mathbb{P}^n\times\_{\text{Spec}\ R}\mathcal{X}\to \t... | 16 | https://mathoverflow.net/users/13265 | 292099 | 128,651 |
https://mathoverflow.net/questions/292074 | 2 | Consider characters of $PGL(2)$. The conductor of a local component $\pi\_p$ of an automorphic representation is defined as the smallest index such that there is nontrivial fixed vector by the congruence subgroup
$$K\_0(p^r) = \left\{
M \in GL(2, \mathcal{O}\_{F\_p}) \ : \ M \equiv \left(
\begin{array}{cc}
\star & \sta... | https://mathoverflow.net/users/43737 | Conductor of characters | The lower-right corner is supposed to be a $1$, not a star (unless the central character is trivial, which it isn't in your case unless $\chi\_0$ is quadratic).
Regardless, as LSpice notes, these representations do not have a conductor in that sense, unless the conductor of $\chi\_0$ is $1$, because they are not inv... | 4 | https://mathoverflow.net/users/18060 | 292100 | 128,652 |
https://mathoverflow.net/questions/292108 | 5 | Let $G$ be a semi-simple group over an algebraically closed field of characteristic zero. In which cases there is a unique wonderful compactification of $G$ (modulo isomorphism)?
For instance, is the wonderful compactification of $SL(n,\mathbb{C})$ unique?
| https://mathoverflow.net/users/nan | Uniqueness of the wonderful compactification of a semi-simple group | I assume you mean the variety $G$ considered as a $G \times G$ variety via the action $(g,h) \cdot x = gxh^{-1}$, which is the standard interpretation in the literature. The variety $G$ is *spherical* as a $G \times G$ variety, meaning that it contains a dense $B \times B$-orbit (where $B$ is a Borel subgroup of $G$). ... | 7 | https://mathoverflow.net/users/16002 | 292112 | 128,654 |
https://mathoverflow.net/questions/292114 | 2 | Let $f\left(z\right)=\sum\_{n=0}^{\infty}a\_{n}z^{n}$ be a power series with $0$s and $1$s as its coefficients ($a\_{n}\in\left\{0,1\right\}$ for all $n$) with a radius of convergence of $1$. I call such holomorphic functions “digital" functions. I've been investigating the boundary values and limiting behaviors of dig... | https://mathoverflow.net/users/120369 | Boundary behavior of power series vs. boundedness of partial sums | I think this question is odd because boundedness of $S\_N(\xi)$ is a very weak condition whereas existence of an analytic continuation to some open neighborhood of $\xi$ (and even much less nice behavior) is a very strong condition.
A "random" Taylor series has no analytic continuation beyond its circle of convergen... | 3 | https://mathoverflow.net/users/9833 | 292126 | 128,657 |
https://mathoverflow.net/questions/292084 | 0 | I have a doubt in proof of Lemma $4.7$ of [this paper.](https://arxiv.org/pdf/1511.09020.pdf)
---
**Lemma:** Let $A$ be a closed operator on a complex Banach space $E$ and assume that $0$ is an eigenvalue of $A$ and a pole of the resolvent $R(\cdot, A).$ Denote by $P$ the corresponding spectral projection.
$(i)... | https://mathoverflow.net/users/119514 | Doubt in proof of $\lim_{n \to \infty} [\lambda R(\lambda, A)]^n = P.$ | Here are a few details which might be helpful to understand the argument in the paper better:
(1) First note that $P$ is also the spectral projection for the spectral value $\frac{1}{\lambda}$ of the operator $R(\lambda,A)$; this result can, for instance, be found in [Engel/Nagel: One-Parameter Semigroups for Linear ... | 3 | https://mathoverflow.net/users/102946 | 292127 | 128,658 |
https://mathoverflow.net/questions/292117 | 5 | Let $(V,\pi)$ be an irreducible, admissible, supercuspidal representation of $G = \operatorname{GL}\_n(F)$ for $F$ a $p$-adic field. Let $B = TU$ be the usual Borel subgroup, maximal torus, and unipotent radical of $G$. Let $f\_{v^{\ast},v}(g) = \langle v^{\ast}, \pi(g)v \rangle$ be a matrix coefficient for $v \in V$ a... | https://mathoverflow.net/users/38145 | Generic supercuspidal representations of $\operatorname{GL}_n$ can be defined by integrals over $U$ | The answer to both questions is yes.
1. All irreducible supercuspidal representations of ${\rm GL}(N,F)$ are generic. See e.g. I. M. Gelfand and D. A. Kajdan,
Representations of the group ${\rm GL}(n,K)$ where $K$ is a local field,
Lie groups and their representations.
2. All ingredients to prove 2. are in
Paskuna... | 9 | https://mathoverflow.net/users/4767 | 292137 | 128,660 |
https://mathoverflow.net/questions/292128 | 0 | Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Consider the Green function $G(x,y)=\sum\_{n\geq 0}\mu^{\*n}(x^{-1}y)$, where $\mu^{\*n}$ is the $n$th convolution power of $\mu$ and $x,y\in \Gamma$.
Denote by $e$ the neutral element of $\Gamma$.
The Martin kernel is then d... | https://mathoverflow.net/users/111917 | Harmonicity of the Martin kernels | There are explicit examples of radially symmetric random walks on free groups for which the Martin boundary contains potentials, see Theorems 2 and 4 in [Cartwright and Sawyer](https://mathscinet.ams.org/mathscinet-getitem?mr=1088396).
Concerning your second question, there are finitely generated groups $G$ for which... | 1 | https://mathoverflow.net/users/8588 | 292139 | 128,661 |
https://mathoverflow.net/questions/292138 | 0 | Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and let $Q$ be the uniform distribution on $(\Omega, \mathcal{F})$ such that $q = dQ / d\mu$ exists. Then the KL-divergence for some probability measure $P$ with $p = dP / \mu$ is given by
$$
D\_{\text{KL}}(P || Q) = \int\_\Omega p \log \frac{p}{q} d\mu.
$$i.e. we... | https://mathoverflow.net/users/120389 | Upper bound for KL divergence on compact space | First, the definition of the KL divergence has nothing to do with compactness as it is defined entirely in terms of the density of one measure with respect to the other one.
Second, the KL divergence is **not** bounded even for finite spaces. Take $\Omega$ to be a two point set, $P$ be the distribution with the weigh... | 0 | https://mathoverflow.net/users/8588 | 292140 | 128,662 |
https://mathoverflow.net/questions/292145 | 4 | Consider the projective space $\mathbb{P}^N$ parametrizing morphisms $f:\mathbb{P}^1\rightarrow\mathbb{P}^n$, $f(x,y) = [f\_0(x,y):\dots:f\_n(x,y)]$ of degree $d$.
Let $Z\_i\subset\mathbb{P}^N$ be the locus of morphisms such that $f\_0,\dots,f\_n$ have $i$ common zeros, for $i = 1,\dots, d$. Then $Z\_d\subset Z\_{d-1... | https://mathoverflow.net/users/nan | Blowing-up projective spaces of parametrized rational curves | One modular interpretation of $X\_1$ mimics the modular interpretation of the space of complete collineations. Let $V$ be a $k$-vector space of finite dimension.
**Definition 1.** For every $k$-scheme $T$, for every invertible $\mathcal{O}\_T$-module $\mathcal{L}$, a $(V,\mathcal{L})$-**system** on $T$ is a homomorp... | 3 | https://mathoverflow.net/users/13265 | 292161 | 128,666 |
https://mathoverflow.net/questions/292119 | 4 | In a field, if we require that all polynomials have at least one root, then it's algebraically closed and all polynomials factor completely. In a *ring*, the same requirement implies that it's an ACF, because linear polynomials need a root and this ensures multiplicative inverses. This raises the question, is there som... | https://mathoverflow.net/users/97603 | Ring where nonlinear polynomials have at least one root | Rings in which every monic polynomial has a root, are called *absolutely integrally closed*.
Such rings are not necessarily a field: For example the ring of all algebraic integers (that is the ring all roots of monic polynomials with rational integral coefficients) is absolutely integrally closed, but no field.
For... | 9 | https://mathoverflow.net/users/18571 | 292162 | 128,667 |
https://mathoverflow.net/questions/292094 | 5 | I have several questions related to ordinal analysis.
According to [[1](https://ncatlab.org/nlab/show/ordinal+analysis)], here are the proof-theoretic ordinal of some well-known theories (using $|T|$ do denotate the proof-theoretic ordinal of $T$):
* $ |\text{ATR}\_0|=\Gamma\_0,|\text{ATR}|=\Gamma\_{\varepsilon\_0}... | https://mathoverflow.net/users/78441 | A question about ordinal analysis | First of all, note that we don't (yet) have ordinal analyses of subsystems of second order arithmetic beyond $\Pi^1\_2$-CA$\_0$.
Still, we can say something about the pattern you indicate using known results about these systems. A good reference is the book by Stephen G. Simpson: [Subsystems of Second Order Arithmeti... | 6 | https://mathoverflow.net/users/2004 | 292169 | 128,669 |
https://mathoverflow.net/questions/292168 | 3 | Let $\Lambda \stackrel{F}{\to} \Omega \stackrel{G}{\leftarrow} \Gamma$ be a diagram of groupoids and functors and $\Gamma \times\_\Omega \Lambda$ the homotopy pullback. We will regard all these groupoids as spaces and compute the cohomology with coefficients in some field.
There should be a map
$$ C^\*(\Gamma) \stack... | https://mathoverflow.net/users/119240 | Cohomology of a homotopy pullback of groupoids | No, consider the multiplication by $2$ map on the integers. The homotopy fiber is two points. But the derived tensor product (for $\mathbb Q$ coefficients) is $\mathbb Q \otimes^{L}\_{\mathbb Q[x]} \mathbb Q[x] = \mathbb Q$.
| 6 | https://mathoverflow.net/users/52918 | 292174 | 128,671 |
https://mathoverflow.net/questions/292163 | 0 | Suppose $w$ is a solution of
$$\frac{d^2}{dx^2}w+\{u(x)+k^2\}w=0$$
with asymptotic condition
$$\lim\_{x\rightarrow \infty}w(x)e^{ikx}=1$$
and $u\in L^1\_1(\mathbb{R})=\{f:\int\_\mathbb{R}(1+|x|)|f|dx<\infty\}$, and $k>0$
Is the solution unique?
| https://mathoverflow.net/users/62005 | Does asymptotic behavior guarantee uniqueness? | Under your assumption on the potential, there is indeed such a unique solution,
(I assume you mean $x\to+\infty$ in your boundary condition. This is proved by reducing your differential equation to an integral one, which is solved by the method of successive approximations. See, for example,
V. A. Marchenko, Sturm-Liou... | 1 | https://mathoverflow.net/users/25510 | 292177 | 128,672 |
https://mathoverflow.net/questions/285746 | 19 | This is a [repost from MSE](https://math.stackexchange.com/q/2503975/72031) as I haven't got anything so far there.
---
Ramanujan gave the following series evaluation $$1+9\left(\frac{1}{4}\right)^{4}+17\left(\frac{1\cdot 5}{4\cdot 8}\right)^{4}+25\left(\frac{1\cdot 5\cdot 9}{4\cdot 8\cdot 12}\right)^{4}+\cdots=... | https://mathoverflow.net/users/15540 | Ramanujan's series $1+\sum_{n=1}^{\infty}(8n+1)\left(\frac{1\cdot 5\cdots (4n-3)}{4\cdot 8\cdots (4n)}\right)^{4}$ | There is a constant $C$ such that
$$\sum\_{n=0}^{\infty} \frac{(\frac14)\_n^3(\frac14 - k)\_n}{(1)\_n^3(1+k)\_n} (8n+1) = C \frac{\Gamma(\frac12+k) \Gamma(1+k)}{\Gamma^2(\frac34+k)}$$
Proof: WZ-method + a Carlson's theorem (see [this paper](https://arxiv.org/abs/math/9306213)).
Then, taking $k=1/4$ we see that th... | 20 | https://mathoverflow.net/users/120265 | 292180 | 128,673 |
https://mathoverflow.net/questions/292111 | 8 | Consider a classical space $M\_k(N)$ of elliptic modular forms of weight $k$ for $\Gamma\_0(N)$. The definition of an unramified Hecke operator $T\_{p^m}$ in terms of double cosets is the disjoint union of double cosets
$$\Gamma\_0(N) \begin{pmatrix} a&b \\ c&d \end{pmatrix} \Gamma\_0(N),$$
where $a, b, c, d \in \mathb... | https://mathoverflow.net/users/6518 | Origin of definitions of ramified Hecke operators | These operators certainly appeared in the 1970 paper by Atkin and Lehner:
>
> Atkin, A. O. L.; Lehner, J. *Hecke operators on $\Gamma\_0(m)$*. Math. Ann. 185
> (1970), 134–160.
>
>
>
I don't know for sure that this is the *first* time these operators appeared in print, but the *Mathematical Reviews* entry for... | 7 | https://mathoverflow.net/users/2481 | 292192 | 128,678 |
https://mathoverflow.net/questions/292176 | 6 | I'm having a difficult time finding any theory on an inverse problem I've come up against. Let's say I have an unknown function $f:[0,1] \rightarrow \mathbb{R}$, and I know $\int\_{a}^{b} f$ for some collection $A$ of pairs $(a,b)\in[0,1]^2$. I'm looking for pointers to any material that discusses condtions on $f$ and ... | https://mathoverflow.net/users/120414 | Reference Request - Recovering a function from its definite integrals (inverse problem) | Here is how you make an inverse problem of this problem: Choose a space $X$ for the function $f$ you are looking for (e.g. $L^2(0,1)$ to work in Hilbert spaces, but other spaces may be more suitable, depending on your needs).
I assume that you only have finitely many definite integrals (since I assume that this is a ... | 4 | https://mathoverflow.net/users/9652 | 292203 | 128,686 |
https://mathoverflow.net/questions/292206 | 9 | A residue formula
=================
I have strong evindence to believe that the following identity holds:
$$
\frac{n!}{2\pi i}\oint\_{|z-1|=\epsilon} \frac{z^{a-1} \mathrm{d}z}{(z^d-1)^{n+1}} = d^{-n-1}\prod\_{j=1}^{n}\left(a-jd\right),\quad a,d,n+1 \in \mathbb{Z}\_{\ge 1}.
$$
>
> Have you seen this formula befor... | https://mathoverflow.net/users/45609 | A natural residue formula | Let us consider the function $w=z^d$ around $z=1$. It is holomorphic and locally invertible with a holomorphic inverse. So we can think of $z$ as $w^{1/d}$ where the latter symbol means a branch of the $d$-th root function taking $1$ at $1$. With this notation, the integral can be written as
$$\frac{n!}{2\pi i}\oint\_{... | 20 | https://mathoverflow.net/users/11919 | 292207 | 128,688 |
https://mathoverflow.net/questions/292157 | 5 | $\require{AMScd}\def\D{\mathbb{D}}\def\prepull{\vcenter{\lrcorner}}$
It is well known that for a category $\cal C$ the existence of finite limits is equivalent to the existence of finite products and equalizers, or to the existence of a terminal object and pullbacks.
What about (pre)derivators? Is it true that, for e... | https://mathoverflow.net/users/7952 | Equivalence between existence of limits and products+equalizers, for derivators | $\require{AMScd}\def\D{\mathbb{D}}\def\prepull{\vcenter{\lrcorner}}$
The result you mention from the paper by Ponto-Shulman does not say exactly what you stated. The setting is that of derivators (so you already have all the homotopy Kan extensions indexed by small cats in some fixed category of diagrams). In that sett... | 4 | https://mathoverflow.net/users/24891 | 292208 | 128,689 |
https://mathoverflow.net/questions/292213 | 35 | Here is a question I asked myself years ago. Since it is not really in my field, I hope to find some (partial) answers here... Since it was unclear, I precise that I am looking for an answer in ZFC, so using the axiom of choice if needed.
The chromatic number of a graph $G$ is the minimal cardinal of a partition of $... | https://mathoverflow.net/users/120363 | Chromatic number of a topological space | The chromatic number $\chi(X)$ of a topological space $X$ is related to the separation dimension $t(X)$ introduced and studied by [Steinke](https://link.springer.com/content/pdf/10.1007%2FBF01192781.pdf).
The separation dimension $t(X)$ is defined inductively:
$\bullet$ $t(\emptyset)=-1$
$\bullet$ $t(X)=0$ for ... | 31 | https://mathoverflow.net/users/61536 | 292228 | 128,695 |
https://mathoverflow.net/questions/292225 | 4 | Send me back to Mathematics Stack Exchange if this question is not research level.
At Terence Tao's [blog post](https://terrytao.wordpress.com/2017/07/06/correlations-of-the-von-mangoldt-and-higher-divisor-functions-i-long-shift-ranges/) there is the expression:
$$\sum\limits\_{n \leq X} \Lambda(n)\Lambda(n+h) \ \ ... | https://mathoverflow.net/users/25104 | Do prime gaps that are a power of "h" have the same density? | Yes - conjecturally.
A special case of the [Hardy-Littlewood prime-tuple conjecture](https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture) states that, for any even $\Delta$,
$$\lim\_{x \to \infty} \frac{\sum\_{1 \le n\le x} \Lambda(n)\Lambda(n+\Delta)}{x} =C(\Delta),$$
where
$$C(\Delta)... | 12 | https://mathoverflow.net/users/31469 | 292230 | 128,696 |
https://mathoverflow.net/questions/292040 | 14 | At the end of his 1956 paper [On Manifolds Homeomorphic to the 7-Sphere](http://www.maths.ed.ac.uk/~aar/papers/exotic.pdf), Milnor shows that either
1. There exists a closed topological 8-manifold with no smooth structure; or
2. The first Pontryagin class $p\_1$ of an open smooth 8-manifold is not a topological invar... | https://mathoverflow.net/users/109769 | Why is the first integral Pontryagin class a homeomorphism invariant? | The topological invariance of the first Pontryagin class is proved in the paper
* B.L. Sharma. Topologically invariant integral characteristic classes. Topology Appl. 21 (1985), no. 2, 135–146. [(link to Elsevier website)](https://www.sciencedirect.com/science/article/pii/0166864185901002)
In low degrees, Sharma co... | 7 | https://mathoverflow.net/users/50846 | 292236 | 128,697 |
https://mathoverflow.net/questions/292219 | 4 | I can't quite figure this out. Perhaps it's simple.
If $S$ is an affine scheme over $\mathbb{C}$ and $s$ and $t$ are $\mathbb{C}$ points in the same connected component, is there a finite type connected $\mathbb{C}$ scheme mapping to $S$ such that $s$ and $t$ are contained in the image of this map?
| https://mathoverflow.net/users/nan | Connected components of infinite type schemes | That is false, and the suggestions by Ariyan Javanpeykar and Joe Berner lead to a counterexample. Let $R\_0$ be $\mathbb{C}[t]$. Let $\Sigma$ be the multiplicative closed set of all elements $u(t)\in \mathbb{C}[t]$ such that $u(0)u(1)\neq 0$. Let $R$ be the fraction ring $\Sigma^{-1}R\_0$. This is $\mathbb{C}$-algebra ... | 5 | https://mathoverflow.net/users/13265 | 292237 | 128,698 |
https://mathoverflow.net/questions/287845 | 3 | The definition of a "connected Shimura datum" (as in Milne's notes) is a pair $(G, X)$, where $G$ is a reductive algebraic group and $X$ is a $G(\mathbb{R})$-conjugacy class of morphisms
$$
x: \mathbb{S}^1 \to G\_\mathbb{R},
$$
where $\mathbb{S}^1$ is the norm one subtorus of $\text{Res}\_{\mathbb{C}/\mathbb{R}}$ satis... | https://mathoverflow.net/users/118227 | What is wrong with this modification of the definition of Shimura datum? | Any reasonable interpetation of the first axiom (on the weights of the action on the adjoint representation) will force the homomorphisms $\mu\_n \to G$ to canonically factor through $\mathbb G\_m$, and so a generalized Shimura variety is just a Shimura variety.
The reason is that any representation of $\mu\_\infty$ ... | 3 | https://mathoverflow.net/users/18060 | 292245 | 128,701 |
https://mathoverflow.net/questions/292229 | 2 | Let $\Omega = [0,1]^d$ and consider $f : \Omega \to R$ Lipschitz continuous with constant 1.
Consider the regular decomposition of $\Omega$ into $d$-dimensional cubes $\Omega\_i$, $i=1 ... k^d$ with $k$ subdivision intervals along each dimension.
Consider the best approximation (in the sense of $ \| \cdot \|\_\inf... | https://mathoverflow.net/users/44552 | Approximation error of 1-Lipschitz function on cubical mesh | It's easy if $d = 1$, right? So the first case of interest is $d = 2$. Here is an example which shows why I think no nontrivial bound is possible.
(Of course you can always let $\hat{f}$ be a constant function which takes a value halfway between the max and min values of $f$; then $\|f - \hat{f}\|\_\infty \leq \frac{... | 1 | https://mathoverflow.net/users/23141 | 292248 | 128,702 |
https://mathoverflow.net/questions/292233 | 3 | The Grötzsch graph is triangle-free and has chromatic number 4. At 11 vertices it is the (unique) smallest graph with these properties.
What is the smallest number of vertices needed for a triangle-free graph with chromatic number 5? The Mycielskian of the Grötzsch graph has 23 vertices, so it's not larger than that.... | https://mathoverflow.net/users/6043 | Smallest triangle-free graph with chromatic number 5 | 22 vertices, there are 80 of them.
Jensen and Royle, Small graphs with chromatic number 5 : a computer search
Journal of Graph Theory, 1995.
| 8 | https://mathoverflow.net/users/1492 | 292255 | 128,706 |
https://mathoverflow.net/questions/292273 | 1 |
>
> For $A= (A\_1,\cdots,A\_d)\in {\cal L}(E)^d$ such that $A\_iA\_j=A\_jA\_i$ for all $i,j$. Why
> $$\sum\_{f\in F(n,d)} A\_{f}^\*A\_{f}=\displaystyle\sum\_{|\alpha|=n}\frac{n!}{\alpha!}{A^\*}^{\alpha}A^{\alpha}\,?$$
> Note that $F(n,d)$ denotes the set of all functions from $\{1,\cdots,n\}$ into $\{1,\cdots,d\}$ ... | https://mathoverflow.net/users/113054 | Why $\sum_{f\in F(n,d)} A_{f}^*A_{f}=\sum_{|\alpha|=n}\frac{n!}{\alpha!}{A^*}^{\alpha}A^{\alpha}?$ | More generally for any $L\in\mathcal{L}(H)$
$$\sum\_{f\in [d]^n} A\_{f}^\*LA\_{f}=\displaystyle\sum\_{|\alpha|=n}{n\choose \alpha}\ {A^\*}^{\alpha}LA^{\alpha}\ .$$
It is just an instance of the expansion of the $n$-power of the sum of $d$ commuting objects in a ring, $$(X\_1+\dots +X\_d)^n=\sum\_{\alpha\in\mathbb{N}^d... | 2 | https://mathoverflow.net/users/6101 | 292276 | 128,710 |
https://mathoverflow.net/questions/292254 | 3 | Say the [KL divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) between two distributions $A$ and $B$ is $\varepsilon$. Can we give bounds, or a precise computation, of the KL divergence between $A^k$ and $B^k$ (the product distributions)?
| https://mathoverflow.net/users/120465 | Kullback–Leibler divergence of product distributions | Transforming usul's comment into a proper answer: if the KL divergence between $A$ and $B$ is $\varepsilon$, the KL divergence between $A^k$ and $B^k$ is $k\varepsilon$. This follows directly from the chain rule (Theorem 5.3 of [this PDF](http://www.math.tau.ac.il/~mansour/advanced-agt+ml/scribe5-lower-bound-MAB.pdf), ... | 3 | https://mathoverflow.net/users/120465 | 292277 | 128,711 |
https://mathoverflow.net/questions/291988 | 0 | I have a cloud of points, and I want to compute its 'diversity'. Variance is not appropriate, because a cloud clustering around few points can still have a large variance.
To that end, I see the cloud of points as the realizations of a random variable.
I want to compute the topological entropy of a random variable,... | https://mathoverflow.net/users/26207 | How to compute the entropy of a random variable with values in a metric space? | 1. Estimate the probability density, using [Kernel density estimation](https://en.wikipedia.org/wiki/Kernel_density_estimation)
2. Compute the differential entropy.
| 0 | https://mathoverflow.net/users/26207 | 292290 | 128,714 |
https://mathoverflow.net/questions/292019 | 1 | People found there were only existing 15 types of pentagonal tiling after one hundred years' work, see [Pentagonal tiling](https://en.wikipedia.org/wiki/Pentagonal_tiling#cite_note-FOOTNOTESchattschneider1978-8).
These 15 types of pentagonal was named by finding date except type 9 and type 10:
type 10 pentagonal wa... | https://mathoverflow.net/users/120323 | Order question about pentagonal tiling type 9 and type 10 | The "numbers" of types 9 up to 13 were probably fixed by Doris Schattschneider "Tiling the plane with congruent pentagons" (1978). She tried to explain the possible cause(s) for the omissions of Kershner: in types 1 up to 8, each tile either edge-to-edge, and/or each tile is adjacent to at most 6 other tiles. But eithe... | 1 | https://mathoverflow.net/users/38477 | 292300 | 128,716 |
https://mathoverflow.net/questions/292292 | 3 | The integral converges as it is easily seen to be upper bounded by $\sqrt{\pi/2}$.
However, [Laplace's method](https://en.wikipedia.org/wiki/Laplace%27s_method) does not seem to work out as the maxima of the function $S(x) = -a\sqrt{1-e^{-x}}-x^2/2$ is located at the end point $0$. [This question](https://mathoverflow... | https://mathoverflow.net/users/64194 | Finding a tight upper bound of $\int_0^\infty e^{-a\sqrt{1-e^{-x}}-x^2/2} dx$ as a function of $a$, $a>0$ | Here again, like in the previous question the Laplace method [DLMF](https://dlmf.nist.gov/2.3#E15) applies. With $p(x)=\sqrt{1-\exp(-x)}$ and $q(x)=\exp(-x^2/2)$, we have the expansion
$$ p(x)=x^{1/2}-\frac1 4 x^{3/2}+\frac 5{96}x^{5/2}+\cdots$$
which is of the form
$$p(x)\sim p(0)+\sum\_{s=0}^{\infty}p\_{s}t^{s+\mu}$$... | 2 | https://mathoverflow.net/users/46744 | 292305 | 128,721 |
https://mathoverflow.net/questions/275741 | 10 | Say that a complete theory $T$ is a limit of stable theories if for every $\phi \in T$ there is a stable completion of $\{\phi\}$. (Equivalently, $T$ is the ultraproduct of stable theories.)
**Question:** Is every simple theory a limit of stable theories? (We allow that theories of finite structures are stable.)
... | https://mathoverflow.net/users/26705 | limits of stable theories | No, not every simple theory is a limit of stable theories. For example, let $K$ be a pseudoalgebraically closed field with a small, nontrivial Galois group not isomorphic to $\widehat{\mathbb{Z}}$. Possibly after replacing $K$ with a finite algebraic extension, there will be some natural number $n$ coprime to the chara... | 6 | https://mathoverflow.net/users/5147 | 292308 | 128,723 |
https://mathoverflow.net/questions/291808 | 4 | The (2-dimensional) Weisfeiler-Leman algorithm is a method for partitioning the ordered pairs of vertices of a graph in a canonical way which gives rise to a powerful graph invariant (see for instance Section 4 of this [paper](https://arxiv.org/pdf/0801.2322.pdf)). If the partitions arising from two graphs are NOT "equ... | https://mathoverflow.net/users/18606 | Smallest pair of non-isomorphic graphs equivalent under the Weisfeiler-Leman algorithm | The 15-vertex "strongly regular" tournament is the smallest non-Schurian
example of an association scheme; it is [unique](http://math.shinshu-u.ac.jp/~hanaki/as/) as an assoc. scheme (there is no Schurian scheme with these parameters - but see PS below!). The smallest case of two non-isomorphic association schemes with... | 5 | https://mathoverflow.net/users/11100 | 292315 | 128,726 |
https://mathoverflow.net/questions/292318 | 10 | Let $S\in\mathcal S\_N$ be a $N\times N$ symmetric matrix over the reals, and introduce the (normalised) gaussian measure
$$
\mathrm d\mu(S):=2^{-\frac 12N}\pi^{-\frac14N(N+1)}\exp\left[-\frac12\operatorname{tr}(S^2)\right]\mathrm dS
$$
where
$$
\mathrm dS:=\prod\_{i\le j\le N}\mathrm dS\_{ij}
$$
I am interested in t... | https://mathoverflow.net/users/106114 | Gaussian integrals over the space of symmetric matrices | [A recursion formula for the moments of the Gaussian orthogonal ensemble](https://projecteuclid.org/euclid.aihp/1249391383), M. Ledoux (2009).
The desired recursion formula for the moment $b\_p^N\equiv E\,[\,{\rm tr}\,(S\_N^{2p})]$ is
I notice a difference in nor... | 11 | https://mathoverflow.net/users/11260 | 292321 | 128,730 |
https://mathoverflow.net/questions/292249 | 1 | Assume That $A$ is a commutative complex Banach algebra. Let $G$ be the connected component of invertible elements containing the identity.
>
> Is there an smooth embedded curve $c:(-\epsilon, \epsilon) \to A$ with $c(0)=0$ and a neighborhood $W$ of $0$ and a continuous (or Frechet differentiable or holomorphic\* )... | https://mathoverflow.net/users/36688 | Local branch of logarithm in commutative Banach algebras | My main reference here is [Lorch - The Theory of Analytic Functions in Normed Abelian Vector Rings (1942)](https://www.ams.org/tran/1943-054-03/S0002-9947-1943-0009090-0/S0002-9947-1943-0009090-0.pdf) (freely available from AMS).
Basically, in this paper Lorch shows that the classical function theory of a single comp... | 3 | https://mathoverflow.net/users/1849 | 292330 | 128,734 |
https://mathoverflow.net/questions/292332 | 1 | I have been working on characterizing the asymptotic behavior for large $n$ for the following sum:
$$\Gamma(1+t)\sum\_{k=1}^n \frac{\binom{n}{k} (-1)^{k-1}}{k^t}$$
where $t(>0)$ is a given constant. The case for $t=1$ can be somewhat found here:
<http://www.math.wvu.edu/~gould/Vol.4.PDF>
in equation $8.25$. Usi... | https://mathoverflow.net/users/120498 | An asymptotic characterization for a series | Mathematica does the job for concrete values of $t\in \mathbb{N}$:
```
Series[Gamma[1 + 2]*Sum[Binomial[n, k]*(-1)^(k - 1)/k^2, {k, 1, n}], {n, Infinity, 2}]
```
>
> $$\left(\log ^2(n)+2 \gamma \log (n)+\gamma ^2+\frac{\pi ^2}{6}\right)+\frac{\log (n)+\gamma -1}{n}+\frac{-2 \log (n)-2 \gamma +9}{12 n^2}+O\left(\... | 0 | https://mathoverflow.net/users/35959 | 292335 | 128,735 |
https://mathoverflow.net/questions/292241 | 9 | Let $k$ be a field, $X$ a smooth projective variety over $k$, $\overline{X} := X\times\_k {k}^{\rm sep}$ for a separable closure ${k}^{\rm sep}$ of $k$, $\ell$ a prime with $\ell\in k^{\times}$.
>
> Are the Galois cohomology groups $$H^i(\text{Gal}({k}^{\rm sep}/k),H^j\_{\rm ét}(\overline{X},\mathbf{Z}\_{\ell}))$$
... | https://mathoverflow.net/users/nan | Finiteness of Galois cohomology | I'm not sure what's going on with this question, but let me drop a few lines to summarize what the official position should be.
First off, I can't think about any situation these groups may occur beyond that of a Hochschild-Serre spectral sequence, so I hope group cohomology here is meant to be continuous group cohom... | 7 | https://mathoverflow.net/users/nan | 292337 | 128,737 |
https://mathoverflow.net/questions/292018 | 1 | Let $\omega=(\omega\_1,\ldots,\omega\_{m})$ be an $m$-tuple of real numbers. Let $|\omega|\_{m}:=\sup\limits\_{1 \leq j \leq m}|\omega\_j|\_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\mathbb{Z}^m$, i. e. $|\theta|\_{1}$ is the distance from $\theta$ to nearest integer.
We say that an $m$-tuple $\om... | https://mathoverflow.net/users/85336 | Simultaneous Diophantine Condition and Growth Rate of Convergents Denominators | **Theorem.** Suppose that an $m$-tuple $\omega$ satisfies the Diophantine condition of order $\nu \geq 0$. Then there exists a sequence $\{q\_{k}\}$ of natural numbers, $k=1,2,\ldots$, and a constant $\hat{C}=\hat{C}(\omega)>0$ such that
* $|\omega q\_{k}|\_{m} \leq \hat{C} \cdot
\left(\frac{1}{q\_{k+1}}\right)^{1/m... | 0 | https://mathoverflow.net/users/85336 | 292340 | 128,739 |
https://mathoverflow.net/questions/292270 | 1 | Let $\Delta$ be a set of simple roots for a reduced root system, and let $(W,S)$ be the associated Coxeter system, where $W$ is the Weyl group and $S$ is the set of simple reflections corresponding to the elements of $\Delta$.
For each $\theta \subseteq \Delta$, there is a unique element $w\_0 \in W$ such that $w\_0... | https://mathoverflow.net/users/38145 | Reduced decomposition for Weyl group elements which support a Bessel function | I'd be extremely surprised if such tables or database existed, mainly because the number of possible reduced decompositions for a Weyl group element tenda to grow very large as the rank increases. Even for the well-studied symmetric groups, this poses a serious problem. (It also makes tables of Kazhdan-Lusztig polynomi... | 2 | https://mathoverflow.net/users/4231 | 292349 | 128,743 |
https://mathoverflow.net/questions/292352 | 1 | We know that accelerated proximal gradient descent method can be applied to solve the following convex programming problem:
$$\min{f(x)+g(x)}$$
where $f$ is smooth and convex, and $g$ is a non-smooth convex function such as $|x|$, whose proximal operator is easy to calculate.
However, is there an algorithm that can... | https://mathoverflow.net/users/88212 | Optimization of non-smooth convex function in a polytope | You could dualize the $h$ to get a saddle point problem. To be specific: Write $h(x) = H(Ax)$ with $H(y) = I\_{\cdot\leq b}(y)$ and write $H(Ax) = \sup\_y (Ax)^Ty - H^\*(y)$. The resulting saddle point problem (min over $x$ max over $y$) could be solved by several primal-dual methods, e.g. the one by Condat (similar to... | 0 | https://mathoverflow.net/users/9652 | 292360 | 128,747 |
https://mathoverflow.net/questions/292324 | 4 | A theorem by Kollár asserts that if $X$ and $Y$ are projective varieties with $X$ smooth, and $f : X \to Y$ is a surjective map, then the higher direct images $R^if\_\*\omega\_X$ vanish for $i$ greater than the generic fiber dimension. I'd like to know if one can weaken the assumption that $X$ is smooth, for example, t... | https://mathoverflow.net/users/120495 | Extension of Kollár's vanishing theorem to singular varieties? | If $X$ has rational singularities, then this vanishing (and the torsion-freeness as well) follows almost trivially. Let $g:Z\to X$ be a resolution of singularities. Then
$Rg\_\*\omega\_Z\simeq \omega\_X$ and hence $R^if\_\*\omega\_X\simeq R^i(f\circ g)\_\*\omega\_Z=0$ for $i>\dim X-\dim Y$ by Kollár's theorem (I suppo... | 4 | https://mathoverflow.net/users/10076 | 292368 | 128,749 |
https://mathoverflow.net/questions/292369 | 2 | Consider the following iterative procedure for solving the $p$-Laplace equation $\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$ with fixed Dirichlet boundary data:
1. $u\_0$ is our initial guess, for example a harmonic function.
2. $u\_k$ solves the equation $\nabla \cdot (|\nabla u\_{k-1}|^{p-2} \nabla u\_k) = 0$.
... | https://mathoverflow.net/users/1445 | Iterative method for $p$-Laplacian | I know this method under the name *lagged diffusivity*. I learned it from the paper
>
> Vogel, Curtis R., and Mary E. Oman. "Iterative methods for total variation denoising." SIAM Journal on Scientific Computing 17.1 (1996): 227-238.
>
>
>
and the paper
>
> Chan, Tony F., and Pep Mulet. "On the convergenc... | 1 | https://mathoverflow.net/users/9652 | 292373 | 128,751 |
https://mathoverflow.net/questions/292351 | 8 | This question is a follow-up of an [old question](https://mathoverflow.net/questions/231770/integral-quaternary-forms-and-theta-functions) posted on MathOverflow.
**Motivation:** The exact equations of modular curves $X\_0^{+}(p)=X\_0(p)/w\_p$($p>13$ is a prime number,$w\_p$ is the Fricke involution) are intriguing o... | https://mathoverflow.net/users/18286 | Basis problem of modular forms arising from quaternion algebra | Hecke conjectured that $\theta\_I$ form a basis for the space $S\_2(p)$, but this was found to fail for $p=37$ by Eichler. In fact, Gross realized that whenever you get vanishing central $L$-values you get a linear relation among theta series. This happens in $S\_2(37)$ since there is an elliptic curve with root number... | 8 | https://mathoverflow.net/users/6518 | 292377 | 128,753 |
https://mathoverflow.net/questions/292371 | 4 | Let "*Injective Replacement*" be the following schema:
If $\phi(x,y)$ is a formula in which only x,y occur free, and only free, then:
$\small \forall
A \ [\forall x \in
A \exists y (\phi(x,y)) \wedge \forall x,y,z,u (\phi(x,y) \wedge \phi(z,u) \rightarrow (x=z \leftrightarrow y=u)) \rightarrow \exists B \ \forall... | https://mathoverflow.net/users/95347 | What would be the effect of replacing Separation by Injective Replacement? | Replacing separation with injective replacement is equivalent to adding the usual replacement axiom, and so your theory $Z^{inj}$ is the same as ZF. From this, you can easily answer all your questions.
The first thing is to note [thanks to Emil in the comments below] is that because you have stated your injective re... | 9 | https://mathoverflow.net/users/1946 | 292384 | 128,755 |
https://mathoverflow.net/questions/292366 | 5 | Let $P=(p(x,y))\_{x, y\in N}$ be the transition matrix over countable states $N$.
Consider the generating Green function $G(x, y|t)=\sum\_{0}^{\infty} p^n(x, y) t^n$, where $p^n(x,y)$ is the $(x,y)$-entry of the matrix $P^n$.
The spectral radius is given by $\rho(P)=\limsup\_{n\to\infty} p^n(x,y)^{1/n}$. It is clea... | https://mathoverflow.net/users/37302 | Divergence of Green function of random walks at spectral radius | Concerning random walk on non-amenable groups, the Green function converges at the spectral radius: this is a result of Guivarc'h, quoted
in Wolgang Woess's book (Random walks on infinite graphs and groups), chapter IIB (in particular Theorem 7.8).
For a bit more explicit statement, one can laso look at the introduc... | 5 | https://mathoverflow.net/users/120527 | 292396 | 128,762 |
https://mathoverflow.net/questions/292392 | 0 | Let $A \in \mathbb R^{n\times n}$ and assume that all eigenvalues lie in the left open halfplane.
Is it true that the logarithmic norm $\mu\_2 (A):= \lambda\_{\max}
\left(\frac{A + A^T}{2}\right)<0?$
| https://mathoverflow.net/users/nan | Does stability imply that the logarithmic norm is negative? | No. Take
$$A=\left(\begin{array}{cc}a&b\\ 0&a\end{array}\right),$$
where $a$ is negative but $b$ is positive and large.
The largest eigenvalue of $(A^T+A)/2$
is $a/2+b/2$.
| 2 | https://mathoverflow.net/users/25510 | 292399 | 128,763 |
https://mathoverflow.net/questions/292383 | 5 | We call a map $f:{\mathbb Z}\times {\mathbb Z} \to {\mathbb Z}$ a *Fibonacci grid* if for all $x,y \in {\mathbb Z}$ we have that $f(x,y)$ is the sum of the neighboring values, that is, $$f(x,y) = f(x-1,y)+f(x+1,y) + f(x,y-1) + f(x,y+1).$$
Are there uncountably many Fibonacci grids?
| https://mathoverflow.net/users/8628 | Fibonacci grids | Yes.
You may make two adjacent rows using any two strings of integers that you like:
>
> ...$\ 3\ \ 1\ \ \ 4\ \ \ 1\ \ \ 5\ \ \ 9\ \ \ 2\ \ \ 6\ \ \ 5\ \ \ 3\ \ \ 5\ \ \ 8\ $...
>
>
> ...$\ 2\ \ 7\ \ \ 1\ \ \ 8\ \ \ 2\ \ \ 8\ \ \ 1\ \ \ 8\ \ \ 2\ \ \ 8\ \ \ 4\ \ \ 5\ $...
>
>
>
Then fill in the rows above ... | 11 | https://mathoverflow.net/users/70618 | 292400 | 128,764 |
https://mathoverflow.net/questions/292319 | 17 |
>
> Is there a closed, smooth, orientable manifold which is not spin${}^c$ but has a finite cover which is spin${}^c$?
>
>
>
Such examples exist when spin${}^c$ is replaced by spin: an Enriques surface is not spin but it is double covered by a K3 surface which is spin.
Every orientable manifold of dimension at... | https://mathoverflow.net/users/21564 | An orientable non-spin${}^c$ manifold with a spin${}^c$ covering space | This is probably overkill, but I couldn't resist advertising a preprint that Diarmuid Crowley and I recently posted to the arXiv: <https://arxiv.org/abs/1802.01296>
In the final section we discuss examples due to Teichner of closed $6$-manifolds with extraordinary cohomological properties. These examples are construc... | 7 | https://mathoverflow.net/users/8103 | 292401 | 128,765 |
https://mathoverflow.net/questions/292376 | 2 | Is there an elliptic operator $D$ on $C^{\infty}(S^2)$ whose principal symbol is not identical to thats of Laplacian but it satisfies $\int\_{S^2} fDf =\int\_{S^2} f\Delta (f)$ for all $f\in C^{\infty}(S^2)$?
| https://mathoverflow.net/users/36688 | Realization of symbol of Laplace operator via certain integral | No: the operator must be of even order, and therefore the symbol of the adjoint operator $D^\*$ is the same as the symbol of $D$. Hence we can assume without loss of generality that $D$ is symmetric. By applying the functional identity to the family of functions $f+t g$ for $t\in(-\epsilon,\epsilon)$ one easily finds t... | 5 | https://mathoverflow.net/users/4572 | 292404 | 128,766 |
https://mathoverflow.net/questions/292365 | 8 | Let $\mathcal{C}$ be the $E\_k$-monoidal $\infty$-category of left modules over a fixed connective $E\_{k+1}$-ring spectrum $A$. Suppose that $M$ is an object of $\mathcal{C}$ which is an $E\_0$-algebra, in other words $M$ admits a unit map $A\to M$. There is an adjunction $U\colon Alg\_{E\_k}(\mathcal{C})\leftrightarr... | https://mathoverflow.net/users/11546 | Cofiber of the inclusion of an $E_0$-algebra $M$ into the free $E_k$-algebra generated by it | The following is an expansion of Dylan's suggestion to use an arity filtration. The reader's digest version is: there's a filtration (which I'm going to construct below) whose associated graded is the free $E\_k$-algebra on the cofiber $M/A$, this free algebra has a decomposition into homotopy colimits of diagrams of t... | 3 | https://mathoverflow.net/users/360 | 292408 | 128,767 |
https://mathoverflow.net/questions/292107 | 2 | I'm studying a paper (see citation below) on numerical analysis, and came across this estimate. I am unable to figure out what was done in the final step, and I am not certain if this was just a typo in the paper.
**Preliminary information**: $\Delta{t}$ is the time step, $n$ is the iteration count from $0$ to final ... | https://mathoverflow.net/users/15045 | Resolving an inequality in the final step of derivation of an *a priori* energy estimate | It looks to me like a discrete version of Gronwall's inequality.
If you have a sequence of numbers satisfying
$$ E\_n \leq k \Delta t + (1 + \ell \Delta t) E\_{n-1} $$
You can rewrite
$$ A\_n = (1 + \ell \Delta t)^{-n} E\_n $$
to get
$$ A\_n \leq \frac{k \Delta t}{(1 + \ell \Delta t)^n} + A\_{n-1} $$
wh... | 2 | https://mathoverflow.net/users/3948 | 292410 | 128,768 |
https://mathoverflow.net/questions/285789 | 1 | I am looking for a reference for the following statement:
For every integer $d$, there exist an integer $k$, such that for all polynomials $P\_1, \ldots, P\_k$ of degree $d$ there exist integers $N, q, a$, such that all integers $n>N$ satisfying $n\equiv a\pmod{q}$ can be represented as $n=P\_1(x\_1)+\dots+P\_k(x\_k)... | https://mathoverflow.net/users/37555 | Reference Request: Waring's problem for different polynomials | Morally, any first attack on Waring's problem (e.g. Chapter 2 of [Vaughan](http://dx.doi.org/10.1017/CBO9780511470929)) works here, but to be rigorous one might modify the argument on the major arcs. If you want to quote an explicit result I suggest [Birch](http://www.jstor.org/stable/2414232); Theorem 1 in Section 7 n... | 3 | https://mathoverflow.net/users/116794 | 292415 | 128,769 |
https://mathoverflow.net/questions/292362 | 1 | I ran into this problem in my research:
Let $y\_0$ be the root of
$$-(y+a)e^{y^2}\mathit{erfc}(y)+\frac{b}{\sqrt{\pi}}=0$$
on interval $[-a,\infty)$, while $a>0$ and $0<b<1$.
How can I show
$$y\_0\leq \frac{a(b-2)+\sqrt{a^2b^2+2b(1-b)}}{2(1-b)}?$$
It is [already known](https://mathoverflow.net/questions/285... | https://mathoverflow.net/users/116983 | Root problem involving error function | Your conjecture is correct.
Indeed, let
\begin{equation\*}
h(y):=\sqrt\pi\,(y+a)e^{y^2}\text{erfc}(y)=\frac{f(y)}{g(y)},\quad
f(y):=\sqrt\pi\,\text{erfc}(y),\quad g:=f/h.
\end{equation\*}
Then the "derivative ratio"
\begin{equation\*}
\rho(y):=\frac{f'(y)}{g'(y)}=\frac{2 (a + y)^2}{1 + 2 a y + 2 y^2}
\end{equat... | 3 | https://mathoverflow.net/users/36721 | 292429 | 128,772 |
https://mathoverflow.net/questions/292350 | 5 | I'm currently working on some universal algebra using proper classes (in MK class theory), and I repeatedly run into situations where I want to collect together some proper classes as the members of a new algebraic structure.
For example, the construction given [here](https://mathoverflow.net/questions/286791/name-f... | https://mathoverflow.net/users/92164 | An axiom for collecting proper classes | Really, what you want to have is a set-theoretic structure on top of the universe, with classes and meta-classes and hyper-classes and so on, in a set-theoretic realm continuing to build into ranks above the ordinals.
One can make this precise by talking about classes coding these meta-class structures. The process i... | 6 | https://mathoverflow.net/users/1946 | 292433 | 128,773 |
https://mathoverflow.net/questions/292416 | 1 | We say that an $m$-tuple $\omega=(\omega\_1,\ldots,\omega\_m)$ satisfies *the Diophantine condition* of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p\_1,\ldots,p\_m$ the inequality
$$\max\limits\_{1 \leq j \leq m}|\omega\_j q - p\_{j}| \geq C \left(\frac{1}{q}\right)^{\fra... | https://mathoverflow.net/users/85336 | Does the set of Diophantine $m$-tuples has full measure? | I'm pretty sure that plenty of those kind of questions are covered in Cassels' book.
The modern approach to this kind of problems follows from dynamics on homogeneous spaces via Dani's correspondence, and in-particular this issue is tightly related to the question of divergence under the geodesic flow (or more genera... | 3 | https://mathoverflow.net/users/8857 | 292435 | 128,774 |
https://mathoverflow.net/questions/292343 | 4 | The concentration of measure on $ [0, 1]^n $ equipped with uniform probability measure $\mu\_{\infty}$,
states that for any $A \subset [0, 1]^n $ with $ \mu\_{\infty}(A) \geq \frac{1}{2} $,
we have:
$$ 1 - \mu\_{\infty}(A\_{\epsilon}) \leq e^{- \pi \epsilon^2 }, \epsilon > 0, $$
where $A\_{\epsilon} = \{ x \in [0, 1]... | https://mathoverflow.net/users/105627 | On the 1/2 assumption on concentration of measure for continuous cube | By the Tsirel’son--Ibragimov--Sudakov argument, reviewed on the first page in [Bobkov](http://www-users.math.umn.edu/~bobko001/papers/2010_JMS-165_Conc.on.the.cube.pdf),
pushing the measure forward from the cube to the canonical Gaussian on $\mathbb R^n$ and using the Gaussian isoperimetric inequality,
we have
\begi... | 6 | https://mathoverflow.net/users/36721 | 292439 | 128,777 |
https://mathoverflow.net/questions/291842 | 15 | It is well known that a topological space is more or less the same as an $\infty$-groupoid. I'm wondering if there is an analogous construction which starts with a manifold endowed with Morse theory (either a function or a gradient-like vector field) and leads to an $(\infty, 1)$-category. The idea should be that morph... | https://mathoverflow.net/users/33286 | Constructing ($\infty, 1)$-category from Morse theory on a manifold | This problem has a long history going back at least to the paper of Cohen, Jones and Segal (available here: [http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0883-04.pdf](http://www.kurims.kyoto-u.ac.jp/%7Ekyodo/kokyuroku/contents/pdf/0883-04.pdf)).
In terms of your first definition,
the answer is that i... | 6 | https://mathoverflow.net/users/8032 | 292441 | 128,778 |
https://mathoverflow.net/questions/283454 | 6 | A triangulation of a surface is called a Fisk triangulation if the degree of all but two vertices is even, and these two exceptional vertices of odd degree are neighbors.
I would like to know what conditions must be satisfied by the degree sequence of such graphs.
I've found several results about the degree sequences o... | https://mathoverflow.net/users/955 | Eberhard-type theorems for Fisk triangulations? | Günter Rote has just shown me a (5,6,..,6,7) triangulation of the Klein-bottle where 5 and 7 are neighbors. He has also found several similar higher genus triangulations.
| 1 | https://mathoverflow.net/users/955 | 292446 | 128,779 |
https://mathoverflow.net/questions/292413 | 3 | It's very well known that if $A$ is a Noetherian normal domain then every prime ideal $\mathfrak{p}\in \mathsf{Spec} \, A$, associated to a principal ideal $aA\subset A$, has height equal to $1$. Moreover, in this setup one can prove that $\mathfrak{p}A\_\mathfrak{p}$ is a principal ideal.
My question is the followi... | https://mathoverflow.net/users/80084 | Primes associated to a principal ideal | Geometrically, the question you ask is the following:
**Question.** Let $X$ be an integral Noetherian scheme, and let $D \subseteq X$ be a Cartier divisor. Then can $D$ have embedded points?
Indeed, your question is the case $X = \operatorname{Spec}(A)$ and $D = V(a)$ for some element $a \in A$.
**Answer.** The a... | 3 | https://mathoverflow.net/users/82179 | 292447 | 128,780 |
https://mathoverflow.net/questions/292443 | 5 | I have been trying for some time to get a grip on how large Mahlo cardinals are, but am finding the definition rather unsuggestive.
Let $\kappa$ be the smallest Mahlo cardinal. By definition, the set of inaccessible cardinals smaller than $\kappa$ is stationary in $\kappa$. Using this, I can prove (I think) that the... | https://mathoverflow.net/users/94232 | Is a Mahlo cardinal also a stationary limit of m-inaccessible cardinals? | Yes. Erin Carmody gives a good account of this in her dissertation.
* Erin Carmody, Force to change large cardinal strength, [arXiv:1506.03432](https://arxiv.org/abs/1506.03432), 2015.
If you see the material leading up to her theorem 11, she first develops the degrees of of inaccessibility beyond $\alpha$-inacces... | 8 | https://mathoverflow.net/users/1946 | 292449 | 128,781 |
https://mathoverflow.net/questions/235548 | 2 | In the following, $T$ is a bounded operator on a Banach space $X$.
* $T$ is called "power bounded" if $\sup\_{n\in \mathbb N}\|T^n\|<\infty$;
* $T$ is called "mean ergodic" if the Cesàro sums $\frac{1}{n}\sum\_{k=1}^n T^k$ converge strongly as $n\to \infty$;
* $T$ is called a "Ritt operator" if its spectrum is contai... | https://mathoverflow.net/users/26039 | Are Ritt operators mean ergodic? | There exists a Ritt operator $T$ on $\ell^\infty$ which is not mean ergodic.
Indeed, let $T$ be the "multiplication operator" given by $Tx = \big((1-\frac{1}{n})x\_n\big)\_{n \in \mathbb{N}}$ for each $x = (x\_n)\_{n \in \mathbb{N}} \in \ell^\infty$.
The spectrum of $T$ is given by $\sigma(T) = \{1-\frac{1}{n}: \;... | 2 | https://mathoverflow.net/users/102946 | 292455 | 128,784 |
https://mathoverflow.net/questions/292420 | 9 | Let $X$ be a smooth projective complex analytic space, $i,p\ge 0$ integers, $\mathbf{Z}(p)\_{\mathcal{D}}$ the Deligne complex of $X$, $H^i\_{\mathcal{D}}(X,\mathbf{Z}(p))$ its hypercohomology.
What properties does the subgroup of torsion elements of $H^i\_{\mathcal{D}}(X,\mathbf{Z}(p))$ have?
* For instance, is i... | https://mathoverflow.net/users/nan | Torsion in Deligne cohomology | The following is not an answer, since the last comment to your question already answers most of it.
It is, rather, a "better question".
What you really want to ask is: let $f: X\to\text{Spec}(k)$ be a proper, geometrically connected, geometrically reduced scheme over a perfect field.
For integers $p,q$, $p\ge 0$,... | 4 | https://mathoverflow.net/users/nan | 292456 | 128,785 |
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