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https://mathoverflow.net/questions/432410 | 2 | $\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let
\begin{equation\*}
K:=K\_{n,t}(a):=\inf\_{x\in\R^n}(\|a-x\|\_2+t\|x\|\_1),
\end{equation\*}
\begin{equation\*}
M:=M\_{n,t}(a):=\min(\|a\|\_2,t\|a\|\_1),
\end{equation\*}
and (for nonzero $a$)
\begin{equation\*}
R:=R\_{n,t}(a):=\frac KM,... | https://mathoverflow.net/users/36721 | On the infimal convolution of two norms on $\mathbb R^n$ | $\newcommand\ka\kappa\renewcommand{\R}{\mathbb R}\newcommand{\de}{\delta}\newcommand\ep\varepsilon$Take any nonzero $a=(a\_1,\dots,a\_n)\in\R^n$. We have
\begin{equation\*}
K=\inf\_{x\in\R^n}\ka(x),\quad \ka(x):=\ka\_a(x):=(\|a-x\|\_2+t\|x\|\_1). \tag{1}\label{1}
\end{equation\*}
Since the norms $\|\cdot\|\_p$ ar... | 2 | https://mathoverflow.net/users/36721 | 432411 | 174,980 |
https://mathoverflow.net/questions/432400 | 17 | I am reading the three texts on condensed mathematics by Scholze and Clausen. I am also interested in paper *"A $p$-adic 6-functor formalism in rigid-analytic geometry"* by Lucas Mann.
To advance in the texts I will have to learn about derived categories and later about $\infty$-categories. In these texts the authors... | https://mathoverflow.net/users/130868 | Derived categories and $\infty$-categories necessary for condensed mathematics | There are several questions (implicit) here.
1. In the texts as they are written, how much knowledge on derived categories (as triangulated categories, or as stable $\infty$-categories) is assumed?
2. Does the development of condensed mathematics, and/or its use in applications, require knowledge of derived ($\infty$... | 29 | https://mathoverflow.net/users/6074 | 432423 | 174,982 |
https://mathoverflow.net/questions/432306 | 2 | I have a matrix $A$ as follows:
$$
A=\begin{pmatrix}
0 & \boldsymbol{W} \\
\boldsymbol{W}^{\dagger} & \boldsymbol{H}
\end{pmatrix}
$$
where $H$ and $W$ are a random Hermitian $N\times N$ matrix and an $N$-component vector of independently distributed complex variables, respectively. The matrix elements have zero mean a... | https://mathoverflow.net/users/482984 | Resolvent (Green's function) of this random matrix | *This is a small varation on Pastur's derivation of the semicircle law.*
We seek the average $\langle G(z)\rangle$ of the Green's function
\begin{equation}
G(z)=(z-A)^{-1}=z^{-1}\textstyle{\sum\_{p=0}^{\infty}}(A/z)^{p}.
\end{equation}
Gaussian averages of $A^{p}$ consist of sums of all pairwise contractions. For $N\... | 3 | https://mathoverflow.net/users/11260 | 432427 | 174,983 |
https://mathoverflow.net/questions/432420 | 1 | When computing the dimension of moduli space for complete intersections of type $(a,b)$ in $\mathbb{P}^n$, what do we need to consider? In general we have the following part:
$$|\mathcal{O}\_{\mathbb{P}^n}(a)|+|\mathcal{O}\_{\mathbb{P}^n}(b)|-\text{PGL}(n+1)$$
then what should one eliminate again? Here some concrete ex... | https://mathoverflow.net/users/nan | The (expected) dimension of moduli space for complete intersection | First you want to compute $h^0(\mathcal{O}\_X(a) \oplus\mathcal{O}\_X(b))$, if $X$ is your complete intersection. You can do it by writing down the Koszul complex for $X$ and tensoring it with $\mathcal{O}\_X(a) \oplus\mathcal{O}\_X(b)$.
Using the properties of the Euler characteristic, you can find the desired dimensi... | 2 | https://mathoverflow.net/users/52811 | 432439 | 174,988 |
https://mathoverflow.net/questions/432437 | 2 | Investigating further questions around this question: [Example of sequence of graphs which satisfy the Riemann hypothesis?](https://mathoverflow.net/questions/432283/example-of-sequence-of-graphs-which-satisfy-the-riemann-hypothesis) leads to the partition function $Z$ of the Ising model of the graph defined here: [Why... | https://mathoverflow.net/users/165920 | What are the coefficients of this partition function in the following Ising model? | In general, the partition function of the Ising model is usually nicer when defining $x = \tanh(\beta)$ instead of $t = \exp(- \beta)$. One explanation for this is that the Ising model partition function is the same as the partition function for the random cluster model and the partition function of the Ising model can... | 1 | https://mathoverflow.net/users/143779 | 432443 | 174,991 |
https://mathoverflow.net/questions/432445 | -4 | A famous theorem of Whitehead essentially states that spaces are determined by their homotopy groups. Is this true for spectra too?, i.e,
$$
\text{question: is a spectrum $E$ determined by its homotopy groups $\pi\_\*E$?}
$$
| https://mathoverflow.net/users/173315 | Are spectra determined by their homotopy groups? | Your statement of Whitehead's theorem is highly misleading. It is very easy to have inequivalent based spaces $X$ and $Y$ with $\pi\_\*(X)\simeq\pi\_\*(Y)$, and the same is true for spectra. For example, the spectra $KU$ and $\bigvee\_{n\in\mathbb{Z}}\Sigma^{2n}H\mathbb{Z}$ are inequivalent but have isomorphic homotopy... | 15 | https://mathoverflow.net/users/10366 | 432446 | 174,992 |
https://mathoverflow.net/questions/432399 | 15 | Let $G$ be an infinite group with a finite generating set $S$. For $n \geq 1$, let $p\_n$ be the probability that a random word in $S \cup S^{-1}$ of length at most $n$ represents the identity. Is it possible for $p\_n$ to not go to $0$ as $n$ goes to $\infty$?
| https://mathoverflow.net/users/492896 | Probability that a random element of a group is trivial | The answer is "no", and it has nothing to do with free groups, cogrowth, or Schreier graphs. We are talking here about the return probabilities of the simple random walk on $G$ (i.e., the one whose step distribution is equidistributed on the set $S\cup S^{-1}$). The reason is the following simple property differentiati... | 11 | https://mathoverflow.net/users/8588 | 432456 | 174,995 |
https://mathoverflow.net/questions/432453 | 14 | Let $f=\sum\_{n\ge 1}a\_nq^n$ be a normalized Hecke eigenform which is not of CM-type, of weight $k\ge 2$ for the congruence subgroup $\Gamma\_0(N)$. Let $a\in\mathbb{Z}$ and define
$$
\pi\_f(x,a):=\#\{p\le x\; :\; a\_p=a \}
$$
I am trying to understand the proof of Theorem 5.1 from K. Murty's paper "Modular Forms an... | https://mathoverflow.net/users/477704 | A question on a paper of K. Murty | There is a flaw in Murty's argument. Once one corrects this flaw, the bound \eqref{1} weakens to $\ll x(\log\log x)^3/(\log x)^2$. Fortunately, [Thorner and Zaman - A Chebotarev variant of the Brun–Titchmarsh theorem and bounds for the Lang–Trotter conjectures](https://arxiv.org/abs/1606.09238) (Sections 1 and 9) fixes... | 24 | https://mathoverflow.net/users/111215 | 432465 | 174,999 |
https://mathoverflow.net/questions/432425 | 4 | Suppose $X$ and $Y$ are smooth affine surfaces over $\mathbb C$. Suppose there is a biholomorphism $f: X\to Y$. Does it follow that $X$ and $Y$ are isomorphic as affine surfaces (i.e. there exists an algebraic isomorphism $g: X\to Y$)?
What if we additionally know that $X$ and $Y$ are rational surfaces?
| https://mathoverflow.net/users/13441 | Biholomorphic but not isomorphic complex affine surfaces? | Let $\overline{C}$ be a complex projective curve of genus $g>0$. Let $C\subset \overline{C}$ be the open affine complement of one (closed) point $p$. The composition $\text{Pic}^0(\overline{C})\to \text{Pic}(\overline{C})\to \text{Pic}(C)$ is an isomorphism. Let $L$ be any nontrivial (geometric) rank $1$ vector bundle ... | 6 | https://mathoverflow.net/users/13265 | 432474 | 175,001 |
https://mathoverflow.net/questions/432470 | 31 | I came across a [post](https://physics.stackexchange.com/a/14944/1046) by Ron Maimon on physics.SE that makes what seems to me to be a very interesting conjecture I've never seen before about what it would take to settle every question of arithmetic. First I'll try to be more precise: a *question of arithmetic* is a fi... | https://mathoverflow.net/users/290 | Do we expect that sufficiently large computable ordinals settle every question of arithmetic? | The question of whether a computable linear order is well-founded is $\Pi^1\_1$-complete, so this is true in a sense:
>
> There is a computable function $F$ such that, for every sentence $\varphi$ in the language of arithmetic with Godel number $\ulcorner\varphi\urcorner$, $F(\ulcorner\varphi\urcorner)$ is an index... | 24 | https://mathoverflow.net/users/8133 | 432478 | 175,004 |
https://mathoverflow.net/questions/432480 | 0 | Consider the inequalities
$$\frac{(2A-1)^2}{4A^2}xy\leq \Big(\frac{x+y}2\Big)^2\leq\frac{(2A-1)^2}{4(A-1)^2}xy$$
$$x,y\geq0$$ where $A>10^9$.
Is the set of integer solutions to $x,y$ finite?
| https://mathoverflow.net/users/10035 | Is set of integer solutions to these inequalities finite? | No. If you consider the inequalities in $\mathbb R^2$, the set of solutions is invariant under scaling and contains an open set. Hence it contains arbitrarily large balls and so infinitely many integer points.
| 2 | https://mathoverflow.net/users/11054 | 432482 | 175,006 |
https://mathoverflow.net/questions/432314 | 2 | Given $H\_1$ and $H\_2$ i.i.d. $\mathit{GUE}$ matrices, what is the single eigenvalue distribution of $H\_1 H\_2 H\_1$ in the large $N$ limit? This matrix is Hermitian, and so its eigenvalues are still real.
---
As some background, I'm practicing moment methods to find the distribution of single eigenvalues of ra... | https://mathoverflow.net/users/153549 | Eigenvalues of $H_1 H_2 H_1$, where $H_1$, $H_2$ independent $\mathit{GUE}$ | A follow-up answer by [Bob Hanlon](https://mathematica.stackexchange.com/a/274735/69425) on Mathematica Stack Exchange further simplifies Mathematica's output of the probability density, found via the Fourier transform technique detailed in the update in my question:
$$
\begin{aligned}
f\_{\lambda}(x) =&
\,\frac{1}{... | 0 | https://mathoverflow.net/users/153549 | 432483 | 175,007 |
https://mathoverflow.net/questions/432418 | 4 | Let $\operatorname{Part}(n)$ be the set of integer partitions of $n$.
A partition $p \in \operatorname{Part}(n)$ has $k$ summands and $d$ distinct summand $n\_i$, with $d \leq k$ and $d$ frequencies $f\_i$ such that $\sum\_i^d f\_i \cdot n\_i = n$.
Notice that $\sum\_i^d f\_i = k$.
The probability of $n\_i$ withi... | https://mathoverflow.net/users/42854 | Maximal entropy of integer partitions of $n$ | To summarize and make more complete what has already been figured out:
**Claim:** Let $T\_i = {i+1 \choose 2}$ for all $i$. Let $j$ be the integer such that $T\_j \leq n < T\_{j+1}$. Then $H\_{max}(n) = \log(j)$.
*Proof:* Write $T\_j = 1 + \dots + (j-1) + j$. By increasing the last summand, we obtain a partition of... | 1 | https://mathoverflow.net/users/29697 | 432485 | 175,009 |
https://mathoverflow.net/questions/432407 | 18 | $\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-dimensional complex representation of $\Sp\_{2k}(\FF\_p)$. If you read most descriptions of the Weil representation, they hit a... | https://mathoverflow.net/users/297 | Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$? | Just a comment, here are some original references for the construction of the Weil representation.
[1] B. Bolt, T. G. Room and G. E. Wall, On the Clifford collineation, transform and
similarity groups. I, J. Austral. Math. Soc., 2 (1961-62), 60-79. [DOI](https://doi.org/10.1017/S1446788700026379)
[2] I. M. Isaacs, ... | 4 | https://mathoverflow.net/users/38068 | 432489 | 175,010 |
https://mathoverflow.net/questions/432488 | 3 | For a symmetric Gaussian random matrix $G=\{G\}\_{1\le i,j \le n}$ with iid $E[G\_{ij}]=0$ and $E[G\_{ij}^2]=1/n$ (it is normalized), ordering its eigenvalues $\lambda\_1\le \lambda\_2\le\cdots \lambda\_n$.
Is there any results about the asymptotic result for the smallest gap $\delta=\min\_{1\le i,j \le n}\{|\lambda\... | https://mathoverflow.net/users/168083 | Asymptotic results for smallest gap of Gaussian random matrix | For a complex Hermitian matrix (GUE ensemble) the probability distribution of the smallest eigenvalue spacing $\delta\_{\rm min}$ is such that the rescaled minimal spacing $x=\delta\_{\rm min}n^{4/3}$ has for large $n$ the asymptotic distribution
$$P(x)=3x^2e^{-x^3},$$
see [Closest Spacing of Eigenvalues](https://arxiv... | 3 | https://mathoverflow.net/users/11260 | 432493 | 175,012 |
https://mathoverflow.net/questions/431288 | 7 | In Arhangel'skii's book "Topological function spaces" there is a part where the author uses that, if $\kappa>\omega$ is a cardinal number, then the space $$\Sigma\_\*(\kappa):=\left\{x\in \mathbb{R}^{\kappa} : \forall \varepsilon>0\left(\left|\left\{\alpha<\kappa : |x\_\alpha|\geq \varepsilon\right\}\right|<\omega\righ... | https://mathoverflow.net/users/146942 | $\Sigma_*$-product is not $\sigma$-countably compact | Let $X =\Sigma\_\*(\omega) = \{ f \in {\mathbb{R}}^\omega : \forall \epsilon > 0, \{ x \in \omega : |f(x)| < \epsilon\} \mbox{ is finite.} \} $. Since $X$ is homeomorphic to a closed subset of $\Sigma\_\*(\kappa)$ and is metrizable, it is enough to show that $X$ is not $\sigma$-compact.
[The idea is this: Suppose $X ... | 2 | https://mathoverflow.net/users/89233 | 432494 | 175,013 |
https://mathoverflow.net/questions/431670 | 3 | Let $K/k$ be an extension of fields, not necessarily algebraic; let $G$ and $H$ be split, reductive groups over $K$; and let $f : H \to G$ be an embedding of groups.
Do there exist split, reductive groups $G'$ and $H'$ over $k$, an embedding $f' : H' \to G'$ of groups, and isomorphisms $G'\_K \cong G$ and $H'\_K \con... | https://mathoverflow.net/users/2383 | Embeddings of reductive groups over algebraically closed fields | In positive characteristic the answer to the question is negative. The reason for that is that there is exists a semisimple groups $H'/k$ admitting a family of finite dimensional representations $\rho\_t:H'\to GL(n,k)$, $t\in\mathbb A^1$, whose members are pairwise non-isomorphic. This family then defines a representat... | 3 | https://mathoverflow.net/users/89948 | 432495 | 175,014 |
https://mathoverflow.net/questions/432512 | 4 | Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective over an affine scheme and $M$ is an étale sheaf, we know that the Čech cohomology $\smash{\check{\mathrm{H}}}^\bullet(X,M)$... | https://mathoverflow.net/users/131975 | Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$? |
>
> Is there a Cech-like way of describing the (hyper)cohomology H∙(X,M∙) or, even better, the complex Rf∗M∙ for some map f?
>
>
>
Yes, the [Verdier hypercovering theorem](https://ncatlab.org/nlab/show/hypercover#DescentAndCohomology) allows one to compute sheaf cohomology on any site in terms of hypercovers.
... | 6 | https://mathoverflow.net/users/402 | 432516 | 175,022 |
https://mathoverflow.net/questions/432466 | 3 | In paper [1] Brezis and Merle prove theorem 3 by using the following fact. Let $w\_n=u\_n-v\_n$, $\Delta w\_n=0$ on $\Omega$ (a bounded domain in $\mathbb{R^2}$) and $w\_n^+$ is bounded in $L^{\infty}\_\mathrm{loc}(\Omega)$. Then by Harnack's principle either
1. a subsequence $w\_{n\_k}$ is bounded in $L\_\mathrm{loc... | https://mathoverflow.net/users/492965 | A detail in one step in a theorem from a paper of Brezis and Merle | This follows from the mean value theorem. Assume that (up to a subsequence) $w\_n(x\_n) \geq -B$ with $(x\_n) \in K$ (a compact subset of $\Omega$). If $x\_n \to x\_0 \in K$ and $B(x\_n,r) \in \Omega$ for every $n$, then $\int\_{B(x\_n,r)} w\_n \geq -B$ and then $$\int\_{B(x\_n,r)} w\_n^- \leq B+Ar^N, \quad \int\_{B(x\... | 4 | https://mathoverflow.net/users/150653 | 432523 | 175,024 |
https://mathoverflow.net/questions/432520 | 4 | Let $\mu$ be a nonatomic probability measure on a Banach space $X$. Is it true that for $\mu$ a.e. $x \in X$, the function $g\_x: (0, \infty) \to \mathbb R$ given by
$$g\_x (r) := \mu(B\_r (x))$$
is continuous in $r$?
*Note: Here $B\_r (x)$ denotes the open ball of radius $r$ around $x$.*
| https://mathoverflow.net/users/173490 | Are nonatomic probability measures on a Banach space nicely shrinking a.e? | No, consider $X=\mathbf{R}^2$ with the $\ell\_\infty$ norm, and let $\mu$ be a non-atomic probability measure giving mass $\frac 1 2$ to both segments $I\_0=[-1,1]\times\{0\}$ and $I\_1=[-1,1]\times\{2\}$. Then for every $x$ in the support of $\mu$ (so in particular for a.e. $x$), $g\_x$ is discontinuous at $2$: $g\_x(... | 8 | https://mathoverflow.net/users/10265 | 432524 | 175,025 |
https://mathoverflow.net/questions/432344 | 15 | If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift operator of summation. These, of course, are not real proofs, since the series do not converge, but one might try to generalize the ... | https://mathoverflow.net/users/113200 | Generalizations of summation methods of divergence series | $\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\SSS}{\mathcal S}\newcommand{\CC}{\mathcal C}\newcommand{\sh}{\operatorname{sh}}$First of all, as was noted in the previous [comment](https://mathoverflow.net/questions/432344/generalizations-of-summation-methods-of-divergence-seri... | 10 | https://mathoverflow.net/users/36721 | 432535 | 175,028 |
https://mathoverflow.net/questions/432543 | 5 | A matrix $A\in\textbf{Mat}\_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}\_{k\to\infty}A^k(v) = 0$. Assume that $\mathcal{A}, \mathcal{B}$ be maximal (under inclusion) among those subsets of $\textbf{Mat}\_n(\mathbb{R})$ with only asymptotically nilpotent matrices and which are close... | https://mathoverflow.net/users/100140 | Asymptotically nilpotent matrices | The answer to your first question is no: they're not all conjugate.
Indeed, let $A$ be the set of all upper triangular matrices of absolute value $<1$ on the diagonal. Then $A$ consists of asymptotically nilpotent matrices (clear) and is maximal for this property (1).
Let $B\_0$ be the set of all $d\times d$ matric... | 8 | https://mathoverflow.net/users/14094 | 432546 | 175,032 |
https://mathoverflow.net/questions/432552 | 3 | Let $f\colon \Omega \to \mathbb{R}$ be a Lipschitz function on an open subset $\Omega\subset \mathbb{R}^n$.
By [the Rademacher theorem](https://en.wikipedia.org/wiki/Rademacher%27s_theorem) $f$ has first derivative almost everywhere. We denote it $\nabla f$.
On the other hand $f$ has a derivative in the sense of di... | https://mathoverflow.net/users/16183 | Do two ways to differentiate Lipschitz functions coincide? | $\newcommand{\g}{\nabla f}\newcommand{\tg}{\widetilde{\nabla f}}\newcommand{\R}{\mathbb R}
\newcommand{\vpi}{\varphi}\newcommand{\Om}{\Omega}$The answer is yes.
For $\Om=\R^n$, this follows from the identity
\begin{equation\*}
\int\_{\R^n}\frac{f(x+tu)-f(x)}t\,\vpi(x)\,dx=\int\_{\R^n}\frac{\vpi(x-tu)-\vpi(x)}t\,f(x)... | 6 | https://mathoverflow.net/users/36721 | 432555 | 175,033 |
https://mathoverflow.net/questions/432510 | 1 | In this [article](https://www.emis.de/journals/JTNB/2007-2/article03.pdf) (Theorem 1.2) there is a proof for Robin's inequality for odd numbers,
$\sigma(n)/n< e^{\gamma}\log(\log(n))$ where $\gamma$ is the Euler-Mascheroni constant and $\sigma(n)$ is the divisor function. The proof is not hard but uses Euler's Totient ... | https://mathoverflow.net/users/98050 | Robin's inequality for odd numbers | Choie et. al. had more concerns than just odd $n;$ if we ask when their argument kicks in, it is simpler in appearance.
Small prime $p,$
$$s(n)=s(pn)/s(p)< \frac{p}{1+p}\left(e^{\gamma}\log(\log(pn))+\frac{0.64821365}{\log(\log(pn))}\right) \; ?< ? \; e^{\gamma}\log(\log(n))$$
or
$$ \frac{p}{1+p}\left( \frac{\log(\... | 3 | https://mathoverflow.net/users/3324 | 432562 | 175,035 |
https://mathoverflow.net/questions/432558 | 7 | I am interested in the following situation: I have two codimension-2 knots $K\_1$ and $K\_2$ in $S^n$ and they are not isotopic. Furthermore, $K\_1$ is not isotopic to the mirror image of $K\_2$ and vice versa. Could it be that $K\_1$ and $K\_2$ become isotopic after connect summing (away from the knots) another $n$-ma... | https://mathoverflow.net/users/419791 | Small knots becoming isotopic after connect sum | I believe can happen in dimension $4$, and probably in all higher dimensions. Take two inequivalent knots $K, K'$ with the same exterior (Cappell-Shaneson; Gordon). Then $K'$ is obtained from $K$ by a "Gluck twist", in other words trivialize the normal bundle of $K$ (there's only one way to do this), remove it, and glu... | 11 | https://mathoverflow.net/users/3460 | 432564 | 175,036 |
https://mathoverflow.net/questions/430956 | 2 | **Setup.** Let $K$ be an algebraically closed field of characteristic zero, and let $A/K$ be a simple abelian variety of dimension $n$. Let $\{ x\_1,x\_2,\dots,x\_{m^{2n}}\}$ denote the $m$-torsion points of $A$.
**Question.** Does there exist an irreducible divisor $D$ on $A$ such that the collection of divisors $\{... | https://mathoverflow.net/users/56667 | Intersection of translate of divisors on abelian variety | I think one can perhaps make a dimension count argument to show that a generic divisor and its torsion translates, as in your question, are in general position. I'll assume $A$ is an abelian surface.
Suppose $L$ is a very ample line bundle on $A$ and let $\mathbb{P}=\mathbb{P}(H^0(A,L))\cong \mathbb{P}^N$ be its comp... | 1 | https://mathoverflow.net/users/484855 | 432581 | 175,041 |
https://mathoverflow.net/questions/432261 | 6 | Let $c>1$, $c\not\in\mathbb{Z}$ and consider the sum
$$
\sum\_{n\leq x} \tau(\lfloor n^c \rfloor),
$$
where $\tau(n)$ is the number of divisors of $n$. I'm almost certain I've seen an evaluation of this sum for an appropriate range of $c$, but do not know where. Does anyone happen to know where this sum has been studie... | https://mathoverflow.net/users/307675 | Mean value of the divisor function over Piatetski-Shapiro sequences | Since asking my question, I have stumbled upon the answer myself, so I post it here in case some future person finds this post.
It appears that the only paper that explicitly considers the problem above is the paper "On the number of divisors of $\lfloor n^c \rfloor$" of D. I. Tolev from 1990. The paper is 2 pages lo... | 1 | https://mathoverflow.net/users/307675 | 432584 | 175,043 |
https://mathoverflow.net/questions/432571 | 6 | This question is on a point in D.R. Adams paper "A Sharp Inequality of J. Moser for Higher Order Derivatives". Precisely the lemma says:
Given $a(s,t)$ be a non negative measureable function on $(-\infty,\infty)\times [0,\infty)$ such that
$$
a(s,t)\leq 1\;\text{ when }\;0<s<t\label{1}\tag{1}
$$ and
$$
\sup\_{t>0}\l... | https://mathoverflow.net/users/493046 | Doubts in first lemma in the paper of Adams regarding sharp Moser inequality | For the second, rewrite $F(t) \leq \lambda$ as
$$t - \lambda \leq \left(\int\_{\mathbb{R}} a(s,\,t)\phi(s)\,ds\right)^q,$$
which reduces the problem to showing that
$$\int\_{\mathbb{R}} a(s,\,t)\phi(s)\,ds \leq (b^q + t)^{1/q}(1-L^p(t))^{1/p} + bL(t).$$
To verify this inequality write the left side as
$$\int\_{-\infty}... | 4 | https://mathoverflow.net/users/16659 | 432586 | 175,044 |
https://mathoverflow.net/questions/432579 | 3 | Let $X$ be an abelian variety defined over a number field $K$. We know that the Neron--Tate height machine associates to a class in the Picard group of $X$ a unique quadratic function which is zero at the identity of $X$. And it is known that modulo torsion this association homomorphism is injective. Consider the homom... | https://mathoverflow.net/users/70360 | Are there any quadratic functions on an abelian variety not from the height machine? | The source has countable dimension over $\mathbb R$, since $A$ has countably many divisors defined over a finite extension of $K$, while the target, being the space of quadratic functions on a countably-infinite-dimensional vector space, has uncountable dimension over $\mathbb R$, so the map can never be surjective.
| 1 | https://mathoverflow.net/users/18060 | 432589 | 175,046 |
https://mathoverflow.net/questions/432595 | 2 | The [De Bruijn–Erdős theorem](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory)) states that when all finite subgraphs of a graph $G$ can be colored with $n$ colors, the same is true for the whole graph.
There is a natural notion of coloring for [hypergraphs](https://en.wikipedia.org/... | https://mathoverflow.net/users/8628 | De Bruijn–Erdős theorem for hypergraphs | The space $X=\{1,\dots,n\}^V$ of all colorings (proper or not) of $H=(V,E)$ with $n$ colors is compact in the product topology. Given a finite set $F \subset E$, the set $K\_{F}$ of proper colorings of $(V,F)$ is a closed set in $X$.
For any finite collection
$F\_1,\dots,F\_k$ of finite subsets of $E$, the intersecti... | 5 | https://mathoverflow.net/users/7691 | 432596 | 175,048 |
https://mathoverflow.net/questions/430851 | 7 | It is well-known and easy to check that a continuous map between topological spaces is an embedding if and only if it has the LLP with respect to $A \to \*$ and $B \to \*$ where $A$ is the two-point codiscrete space and $B$ is the Sierpiński space.
Are closed embeddings also characterized by a left lifting property?
... | https://mathoverflow.net/users/12547 | Are closed embeddings characterized by a left lifting property in the category of topological spaces? | As it turns out, there is a weak factorization system $(\mathcal{L}, \mathcal{R})$ where $\mathcal{L}$ is the class of closed embeddings and $\mathcal{R}$ is the class of all maps with the RLP with respect to $\mathcal{L}$. Unfortunately, my argument does not provide a very concrete description of $\mathcal{R}$ and I w... | 2 | https://mathoverflow.net/users/12547 | 432604 | 175,049 |
https://mathoverflow.net/questions/432615 | 1 | Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e\_\* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module structure is given by
$r\cdot s=r^{p^e}s, \forall r \in R, s\in F^e\_\* R$. Assume that $F^1\_\* R$ is a finitely generated ... | https://mathoverflow.net/users/386496 | When is $R$ a direct summand of Frobenius pushforwards? | If there exists one $e > 0$ so that $R \to F^e\_\* R$ splits, then by composing splittings one sees that $R \to F^{ne}\_\* R$ splits for all $n > 0$. Ie, if $\phi : F^e\_\* R \to R$ is a splitting (sends $F^e\_\* 1 \mapsto 1$), then $\phi \circ (F^e\_\* \phi) : F^{2e}\_\* R \to R$ also sends $F^{2e}\_\* 1 \mapsto 1$.
... | 4 | https://mathoverflow.net/users/3521 | 432617 | 175,051 |
https://mathoverflow.net/questions/310706 | 8 | $\newcommand{\C}{\mathbb{C}} \newcommand{\U}{\mathbb{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}}$
Let $F$ be a number field.
Let $\chi\colon \mathbb{A}\_F^\times/F^\times \to \C^\times$ be a Hecke character with local components $\chi\_v$ for each place $v$ of $F$.
When $v$... | https://mathoverflow.net/users/40821 | Are there partially algebraic Hecke characters? | Yes, such partially algebraic characters exist. This is proved in Section 5.6 (Section 5.5 in the published version) of my [paper](https://hal.inria.fr/hal-03795267) with Pascal Molin *Computing groups of Hecke characters*.
Let me repeat the construction here: assume $F$ is a quadratic extension of another number fie... | 1 | https://mathoverflow.net/users/40821 | 432623 | 175,053 |
https://mathoverflow.net/questions/432621 | 1 | Suppose $G$ is a locally free sheaf on $P^{d}$. $F\_{1}$,$F\_{2}$ are two subsheaves of
$G$ and they concide on a dense open subscheme of $P^{d}$ .If the quotients of $G$ corresponding to these two subsheaves are torsion free,can we conclude that $F\_{1}=F\_{2}$
everywhere?
| https://mathoverflow.net/users/493073 | Subsheaves of a locally free sheaf on $P^{d}$ | Let $F\_3 = F\_1 + F\_2$. Then $F\_3$ coincides with $F\_1$ and $F\_2$ on the same dense open. Therefore $F\_3/F\_1$ and $F\_3/F\_2$ are torsion. But they are also torsion free, as subsheaves of $G/F\_1$ and $G/F\_2$, so they are zero, and $F\_1 = F\_3 = F\_2$.
| 4 | https://mathoverflow.net/users/3847 | 432625 | 175,055 |
https://mathoverflow.net/questions/432607 | 3 | Consider the category of simplicial presheaves $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ endowed with the projective model structure, i.e. weak equivalences and fibrations are point-wise.
>
> I need a functorial cofibrant replacement
> $Q:\mathsf{sSet}^{\mathcal{C}^{\text{op}}} \to
> \mathsf{sSet}^{\mathcal{C}^{\t... | https://mathoverflow.net/users/493091 | A projective-cofibrant replacement in $\mathsf{sSet}^{\mathcal{C}^{\text{op}}}$ such that $\operatorname{ev}_0Q(S) \cong \operatorname{ev}_0S$ |
>
> Before someone answers I add a related less general question. Does there exist a cofibrant replacement (not necessarily functorial) of the point ∗∈sSetCop with the property that ev0Q(∗)≅∗∈SetCop ?
>
>
>
No. Cofibrations in the projective model structure on simplicial presheaves on C are retracts of transfini... | 3 | https://mathoverflow.net/users/402 | 432632 | 175,058 |
https://mathoverflow.net/questions/432403 | 1 | Let $G$ be a symmetric Gaussian random matrix with iid $E[G\_{ij}]=0$ and $E[G\_{ij}^2]=\frac{1}{n}$, and ordering its eigenvalues $\lambda\_1\le \lambda\_2\le \dots \le \lambda\_n$ corresponding eigenvectors $v\_1,\dots, v\_n$. Define $X\_t=\{X^i\_t\}$ is a vector on $R^n$ for time $t\ge 0$ and $X\_0$ is distributed u... | https://mathoverflow.net/users/168083 | How to prove that upper bound of the hitting time holds with high probability? | I don't think this upper bound holds.
Notice that the probability distribution of $h(0)$ is a Gaussian of mean 0 and width $1/\sqrt n$, so the probability that $|h(0)|<\alpha$ is $\alpha\sqrt n$ for $0<\alpha<1/\sqrt n$.
Since $\lambda\_2-\lambda\_1\simeq n^{-2/3}$, the hitting time $T\_\epsilon$ for $h(0)=\alpha$ ... | 1 | https://mathoverflow.net/users/11260 | 432637 | 175,060 |
https://mathoverflow.net/questions/432635 | 1 | Consider a smooth projective variety of ample $\omega\_X$, how can I quickly see that
$$\textbf{Coh}(X)=\{\mathcal{F}^{\bullet}\mid\text{Hom}(\omega\_X^{\otimes i},\mathcal{F}^\bullet[n])=0\text{ for } n \neq0\text{ and }i\ll0\}\subset D^b(X)$$
This is used in the proof of [Bondal—Orlov reconstruction theorem](https://... | https://mathoverflow.net/users/nan | A characterization on coherent sheaves inside $D^b(X)$ | First, one has
$$
\mathrm{Hom}(\omega\_X^{\otimes i}, F[n]) \cong \mathbb{H}^n(X, F \otimes \omega\_X^{\otimes -i}).
$$
Next, there is the hypercohomology spectral sequence
$$
H^q(X, \mathcal{H}^p(F) \otimes \omega\_X^{\otimes -i})
\Rightarrow
\mathbb{H}^{p+q}(X, F \otimes \omega\_X^{\otimes -i}).
$$
Now, since $\ome... | 4 | https://mathoverflow.net/users/4428 | 432644 | 175,063 |
https://mathoverflow.net/questions/432636 | 10 | According to [this source](https://math.mit.edu/%7Epoonen/papers/sampler.pdf) (p. 10), determining whether a simplicial complex is a simplicial sphere (the *sphere recognition problem*) is undecidable.
According to [this source](https://www2.mathematik.tu-darmstadt.de/%7Epfetsch/apropo/steinitz_problem.html), determi... | https://mathoverflow.net/users/108884 | Determining whether a lattice is the face lattice of a polytope - NP hard or undecidable? | There's no contradiction:
1. I don't know the correct complexity, but I recall hearing several times that it is at least as hard as NP.
2. It is not difficult to show (using Tarski's algorithm, as indicated in the comments) that recognition of polytopal face lattices is decidable. Given a face lattice, you need to ch... | 14 | https://mathoverflow.net/users/75344 | 432656 | 175,066 |
https://mathoverflow.net/questions/432382 | 15 | Sequence A93637 of the OEIS (<https://oeis.org/A093637>) starting as $1,1,2,4,9,20,49,117,297,746,1947,\ldots$ is defined by
the coefficients $a\_0,a\_1,\ldots$ of the unique formal power series
defined by the equality
$$A(x)=\prod\_{n=0}^\infty \frac{1}{1-a\_nx^{n+1}}=\sum\_{n=0}^\infty a\_n x^n\ .$$
Experimentally, $... | https://mathoverflow.net/users/4556 | Convergency radius of the generating series for A93637 | This is pretty simple, really. Note that we can obtain our power series in the following way. Define on (formal) power series with positive coefficients the transform
$$
T\sum\_{k=0}^\infty b\_kx^k=\text{Expansion of }\prod\_{k=0}^\infty\frac 1{1-b\_k x^{k+1}}.
$$
Start with $f\_0(x)=1$ and iterate $f\_n(x)=Tf\_{n-1}(x... | 16 | https://mathoverflow.net/users/1131 | 432664 | 175,069 |
https://mathoverflow.net/questions/432657 | 6 | **Question.** Is there any structure theorem for the class of monoids $H$ with the property that $xy = x$ or $xy = y$ for all $x, y \in H$? Or does this look hopeless for some good reasons?
A monoid with the above property is idempotent and need not be commutative. Examples include (i) the unitization of a left (resp... | https://mathoverflow.net/users/16537 | Structure theorem for a class of idempotent monoids (where $xy = x$ or $xy = y$ for all $x, y$) | The semigroups in the question seem to have first been introduced by Redei in his book Algebra. Vol 1. The book was originally published in Hungarian in 1954, and then German in 1959. The English edition is 1967. In the English version they are called breakable semigroups. The main result in the English edition about t... | 7 | https://mathoverflow.net/users/15934 | 432691 | 175,077 |
https://mathoverflow.net/questions/432669 | 11 | A field $K$ is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety over $K$ has a $K$-point. Let $L$ be the maximal totally real subfield of $\overline{\mathbb Q}$. A few places claim that $L(i)$ is PAC, but I can't find any proofs of this. Does anyone have a reference, or know why this is ... | https://mathoverflow.net/users/140821 | PAC and totally real fields | The reason why $L(i)$ is PAC is the following theorem.
>
> **Theorem.** Let $K$ be a global field. Let $S$ be a finite set of places of $K$. Let $X$ be a smooth, geometrically integral $K$-variety. For each $v\in S$, let $\Omega\_v\subset X(K\_v)$ be open (for the $v$-adic topology) and nonempty. Then there exist:
... | 12 | https://mathoverflow.net/users/7666 | 432699 | 175,079 |
https://mathoverflow.net/questions/432320 | 3 | It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed orientable hyperbolic 3-manifold with volume less than or equal to that of the Weeks manifold, then $N$ is homeomorphic to... | https://mathoverflow.net/users/492828 | Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds? | The answer given in the comment by Ryan Budney should be enough, but let me give you a couple of theorems if it is still not so clear.
Suppose you have a finite volume hyperbolic complete orientable manifold $M$. Suppose that $M$ has cusps (hence, it is not compact).
Theorem [Thick-thin decomposition]: $M$ is diffe... | 2 | https://mathoverflow.net/users/128408 | 432700 | 175,080 |
https://mathoverflow.net/questions/432639 | 4 | There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the **strong positivity** even for the basic semigroups like the Neumann laplacian semigroup.
Here is a simplified mathematical problem:
$$
\begin{cases}\dfrac{\partial u... | https://mathoverflow.net/users/61629 | Strong positivity of Neumann Laplacian | As other users have indicated in the comments, for sufficiently smooth domains one can get it by combining, for instance, elliptic regularity with Hopf's boundary point lemma (and then go from the elliptic to the parabolic case by, for instance, a semigroup argument).
However, the same result remains true in much mor... | 7 | https://mathoverflow.net/users/102946 | 432713 | 175,085 |
https://mathoverflow.net/questions/432675 | 3 | Let $(X,\mu,f)$ be a two-sided full shift system. Assume that there is $t \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ and $x \in X$, we can define $T(x)=f^{n+m(x)}(x)$, where $m(x) \leq t; $ $m(x) \in \mathbb{N}$ and that depends on $x$. I also assume that $T$ is measure preserving.
It is well-known that $... | https://mathoverflow.net/users/127839 | Entropy of $f^{m(x)+n}$ of full shift | If I understand right, your function $m$ (for a fixed $n$) takes on only finitely many values, which are all measurable sets. You can then define the partition $\{m^{-1}(i)\}$ and the associated finite $\sigma$-algebra $\mathcal{B}$.
If we define $\mathcal{A}$ to be any finite sub-$\sigma$-algebra of your original $\... | 3 | https://mathoverflow.net/users/116357 | 432723 | 175,088 |
https://mathoverflow.net/questions/432726 | 8 | What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3\_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as one might argue that maybe for sufficiently large $n$, the resultant manifold have trivial mapping class group. But can s... | https://mathoverflow.net/users/33064 | On trivial mapping class group of 3-manifolds | Dave Gabai [proved that the mapping class group of a closed hyperbolic 3-manifold is isomorphic to its isometry group](https://mathscinet.ams.org/mathscinet-getitem?mr=1354958). For a hyperbolic knot $K$ without any symmetries, for large enough $n$, $S^3\_{1/n}(K)$ will have a very short geodesic core by Thurston’s [hy... | 18 | https://mathoverflow.net/users/1345 | 432729 | 175,091 |
https://mathoverflow.net/questions/330622 | 3 | Recall that a space is:
* "Lindelof", if every open cover has a countable subcover.
* "Linearly Lindelof", if every open cover which is linearly ordered by $\subseteq$ has a countable subcover.
* "weakly Lindelof" if every open cover has a countable subcollection whose union is dense in the space.
Of course every L... | https://mathoverflow.net/users/11647 | Is there a linearly Lindelof space which is not weakly Lindelof? | (I hope the following comments are still worthwhile @SantiSpadaro)
We say that a space $X$ is dually-$\mathcal{P}$ if for every open neighbourhood assignment $\phi$ of $X$, there is a subspace $Y$ of $X$ with property $\mathcal{P}$ such that $\phi(Y)=X$. The class of dually-$\mathcal{P}$ spaces is denoted by $\mathca... | 1 | https://mathoverflow.net/users/476373 | 432742 | 175,094 |
https://mathoverflow.net/questions/432719 | 4 | Consider classical bond percolation on $\mathbb{Z}^d$. Each edge is included with probability $p$ and deleted with probability $1-p$. As is well known, there is a $p\_c(d) \in (0,1)$ such that $p>p\_c$ means there is an infinite connected component of the resulting graph (with probability one), and $p<p\_c$ means there... | https://mathoverflow.net/users/5678 | Percolation: at what length scale do we see it? | As Ofer said in his comment, this is should be equivalent to estimating the correlation length. Of course there are a few different ways to define the correlation length, but usually proving these ways are equivalent can be done in a completely quantitative way. E.g. in his scaling relation paper <https://projecteuclid... | 3 | https://mathoverflow.net/users/41827 | 432743 | 175,095 |
https://mathoverflow.net/questions/432720 | 1 | Let $A$ be an invertible, symmetric and tridiagonal matrix of size $n \times n$. Assume that $A\_{i,i}=a \neq 0$ for $i=1\dotsc n$ and all the elements in the sub- and super-diagonal of $A$ are $b \neq 0$. I would like to simplify the following Kronecker product: $e^{-A} \otimes e^{A}$.
I know that, given the Kroneck... | https://mathoverflow.net/users/478084 | Kronecker product: Is it possible to simplify this product $e^{-A} \otimes e^{A}$ where $A$ is an invertible and symmetric matrix | One can diagonalize your $A = VDV^{-1}$ explicitly; the closed formulas are [here](https://de.wikipedia.org/wiki/Tridiagonal-Toeplitz-Matrix) for instance.
Once you have those matrices, you can write the orthogonal eigendecomposition
$$
\exp(-A) \otimes \exp(A) = (V\otimes V) \,\, (\exp(-D)\otimes \exp(D)) \, \,(V\ot... | 6 | https://mathoverflow.net/users/1898 | 432744 | 175,096 |
https://mathoverflow.net/questions/432740 | 2 | Suppose $X$ is a prime Fano threefold of index 1 such that $H = -K\_X$ is ample. There is a full classification of the derived category of such threefolds depending on the genus of $X$; in the case that $g \geq 6$ we have
$$
D^b(X) = \langle \operatorname{Ku}(X), \mathcal{E}, \mathscr{O}\rangle
$$
where $\operatorname{... | https://mathoverflow.net/users/458355 | Right adjoint of subcollection of semi-orthogonal decomposition | First, mutations are exact, hence commute with shifts. Second,
$$
\mathrm{RHom}(\mathcal{O},\mathcal{O}(H)) =
H^\bullet(X, \mathcal{O}(H)) =
H^\bullet(\mathbb{P}^{g+1}, \mathcal{O}(H)),
$$
hence the evaluation morphism on $X$ is the restriction of the analogous evaluation morphism on $\mathbb{P}^{g+1}$, where its con... | 4 | https://mathoverflow.net/users/4428 | 432746 | 175,098 |
https://mathoverflow.net/questions/432683 | 1 | Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every second ball from the highest-numbered box that has more than one ball.
I conjecture that for $n=2^k$ ($k>0$) we have
$... | https://mathoverflow.net/users/231922 | Number of steps required to get one ball in each box for $n=2^k$ | The reason is that for $n=2^k$ the following recursion holds:
$a(n)=1+2a(n/2)+(n/2-1)n/2$.
To see why this holds, notice that first you split the $n$ balls into two groups of $n/2$, one group in the first box, the other in the second.
After that, you do $a(n/2)$ steps for the balls in the second box.
Then, ... | 2 | https://mathoverflow.net/users/955 | 432747 | 175,099 |
https://mathoverflow.net/questions/432759 | 1 | Let $N$ be a large integer and $I = [aN, bN]$ for some $0 < a < b < 1$. Denote by $\chi\_I(x) = 1$ if $x \in I$, $0$ otherwise. I was wondering if there exists a smooth function $w$ with the property that $w (x) = \chi\_I(x)$ if $x \in I$ or $\operatorname{dist}(x,I)>1/2$ and
$$
\int\_{\mathbb{R}} |w^{(n)}(x)| dx \leq ... | https://mathoverflow.net/users/84272 | Existence of a smooth function that approximates a characteristic function of an interval with certain property | Consider $\rho$ be a $C^\infty$ function supported in $(-1/8,1/8)$ with integral 1 and set
$
w=\chi\_I\ast \rho,
$
so that, for $n\ge 1$, we have
$$
w^{(n)}(x)=\bigl(\chi\_I\ast \rho^{(n)}\bigr)(x)=
\bigl(\chi'\_I \ast \rho^{(n-1)}\bigr)(x)=\rho^{(n-1)}(x-aN)-
\rho^{(n-1)}(x-bN)
$$
and $\Vert w^{(n)}\Vert\_{L^1}\le ... | 3 | https://mathoverflow.net/users/21907 | 432763 | 175,103 |
https://mathoverflow.net/questions/432761 | 4 | A **Frobenius algebra** is a vector space that is both an algebra and a coalgebra in a compatible way. (See [here](https://ncatlab.org/nlab/show/Frobenius+algebra#AsAssociativeAlgebraWithLinearForm) for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a Frobenius algebra? What is an ... | https://mathoverflow.net/users/491434 | A subalgebra of a Frobenius algebra that is not again a Frobenius algebra? | Any finite dimensional $k$-algebra $A$ is also a subalgebra of its trivial extension $$T(A)=A \oplus \rm{Hom}\_k(A,k),$$ which is Frobenius. In this way you get examples where the Frobenius algebra is not semi-simple.
The (symmetric) $k$-bilinear form on $T(A)$ is given by $$\langle (a,f),(b,g) \rangle = g(a)+f(b),$$... | 6 | https://mathoverflow.net/users/18756 | 432771 | 175,105 |
https://mathoverflow.net/questions/302244 | 10 | The property of well-orderability is upward absolute for transitive models of ZF: by Replacement in the smaller class, specifically Mostowski collapse, this is equivalent to the upward absoluteness of von Neumann ordinals, which holds, by Foundation in the larger class, since the property of being a von Neumann ordinal... | https://mathoverflow.net/users/15819 | Absoluteness of well-orderability | Upward absoluteness of well-foundedness fails for transitive models of Zermelo set theory (i.e. with full Separation but no Replacement). That is, you can find $M \subseteq N$ both transitive models of Zermelo so that there's a linear order $L \in M$ which $M$ thinks is a well-order but $N$ sees is ill-founded.
This ... | 9 | https://mathoverflow.net/users/64676 | 432772 | 175,106 |
https://mathoverflow.net/questions/432769 | 5 | In 2002, by using Floer theory, Froyshov defined the $h$-invariant for intergal homology 3-spheres, which is a surjective group homomorphism $\Theta^3\_{\Bbb Z}\to \Bbb Z$, where $\Theta^3\_{\Bbb Z}$ is the group of homology cobordism classes of integral homology 3-spheres with connected sum as operation. (<https://arx... | https://mathoverflow.net/users/164671 | Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres? | No, not yet. The state of the art in computing instanton homology for Seifert spaces is in [this paper](https://arxiv.org/abs/2010.03800). This is like knowing how to compute $\widehat{HF}(\Sigma)$, which is not enough: you also want to understand $\widehat{HF}\_{red}(\Sigma)$. More or less equivalently, you want compa... | 4 | https://mathoverflow.net/users/40804 | 432782 | 175,108 |
https://mathoverflow.net/questions/432778 | 2 | Is there any condition over a number field $K$ for an unramified quadratic extension of $K$ to admit an embedding into an unramified cyclic extension of degree 4 of $K$?
| https://mathoverflow.net/users/483989 | A question about unramified quadratic extension of number field | Ah... perhaps an amplification of my comment and @Aurel's would be useful to you:
First, this kind of thing is not really in Hilbert's "Zahlbericht", because he only treats a sub-class of extensions... which was already a novelty, etc.
But by the 1920's, Takagi and Artin had clarified/proved the reasonable general ... | 4 | https://mathoverflow.net/users/15629 | 432805 | 175,115 |
https://mathoverflow.net/questions/431371 | 1 | The theory of theta functions can be interpreted as automorphic representations on metaplectic groups (2-fold covering groups of $\mathrm{Sp}\_{2}$, or $\mathrm{GL}\_2$), and there's also a notion of $n$-fold covering groups which are studied by Brylinski-Deligne, Kubota, Weissman, and many people. Patterson and Bump-H... | https://mathoverflow.net/users/95471 | Automorphic representations on non-cyclic covering groups | Let $\tilde{G}$ be a central extension of $G$ by a finite group $A$. Let $V$ be a complex representation of $G$. Then $V$ decomposes as a direct sum over all characters $\chi:A\to \mathbb{C}^\times$.
$$
V=\bigoplus\_{\chi} V\_\chi,
$$
where
$$
V\_\chi = \{v\in V \mid av=\chi(a)v \ \ \ \forall\ a\in A\}.$$
The action ... | 3 | https://mathoverflow.net/users/425 | 432807 | 175,116 |
https://mathoverflow.net/questions/432794 | 4 | I'm trying to understand the details of the almost toric mutation process as explained in Section 8.4 in <https://arxiv.org/pdf/2110.08643.pdf>. More specifically, given an almost toric fibration $f: (M,\omega) \rightarrow B$ from a symplectic manifold $(M,\omega)$ to a base $B$, the process of mutation briefly involve... | https://mathoverflow.net/users/92483 | Almost toric mutations | Mutation doesn't even change the integral affine base, which is why it doesn't change the symplectic manifold. All you're doing is changing the way the integral affine base is drawn. If you're given an integral affine manifold, the way you get a picture is by making some branch cuts to pick a fundamental domain in the ... | 5 | https://mathoverflow.net/users/10839 | 432816 | 175,119 |
https://mathoverflow.net/questions/431952 | 2 | Let $X$ be an n-dimensional polyhedral space with, say, $n\geq 3.$ Let also $p\in X$ be a vertex on a triangulation $\tau$ of $X,$ so a vertex on the polyhedral space.
The tangent cone (as a metric space) of $X$ at $p$ is given by the limit $$\lim\_{\lambda \to \infty} (X,\lambda d\_X, p),$$ where $d\_X$ is the dista... | https://mathoverflow.net/users/100597 | Tangent cone on polyhedral spaces | Your question isn't completely clear, but I think you mean that $p$ is the vertex of a simplex in the triangulation of $X$. In that case, take all the simplices containing $p$, viewed as simplices embedded in $\mathbb R^3$, and note how they are glued together inside $X$. Now take the *cone* over each simplex *based at... | 3 | https://mathoverflow.net/users/153128 | 432840 | 175,125 |
https://mathoverflow.net/questions/432837 | 5 | It seems to be considered a classical fact that one cannot have a spherical polyhedral/cone-metric on the 2-sphere with precisely one conical point. However, I've never actually seen it proven anywhere in full generality. I realise that it's not too hard to prove using the holonomy and developing map, but I would prefe... | https://mathoverflow.net/users/153128 | Nonexistence of sphere with one conical point [reference request] | The proof is very simple. Let $f$ be the developing map (take an isometry of some small disk on your surface to a region in the plane with constant curvature metric, and then perform analytic continuation along all paths not passing through singularity). The monodromy representation is in the group of isometries of you... | 6 | https://mathoverflow.net/users/25510 | 432841 | 175,126 |
https://mathoverflow.net/questions/432798 | 9 | **Question.** Let $M^{n+1}$ be a closed manifold without boundary. Which closed submanifolds $\Sigma^n \subset M^{n+1}$ (of codimension one) are leaves of a foliation of $M$ minus some finite collection of points? Does one know *a priori* the number of points one is forced to remove from $M$?
I suspect the answer mig... | https://mathoverflow.net/users/103792 | Which submanifolds are leaves of a foliation? | If the normal bundle of $\Sigma$ in $M$ is orientable, then there always exists such a foliation. The idea is that one can construct a smooth function $f$ on $M$ such that $\Sigma$ is the set of zeros of $f$ and $\mathrm{d}f$ does not vanish on a tubular neighborhood of $\Sigma$. Then, since, by Theorem 6.2, Chapter II... | 11 | https://mathoverflow.net/users/13972 | 432846 | 175,129 |
https://mathoverflow.net/questions/432191 | 3 | Consider mappings $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ with differential
\begin{align}
\mathsf{d} f= \begin{pmatrix}
\cos\psi(x) &\cos\phi(y) \\
\sin \psi(x)& \sin\phi(y)
\end{pmatrix},
\end{align}
being $\psi(x)$ and $\phi(y)$ arbitrary functions satisfying $0<\psi(x)-\phi(y)<\pi$. (Here $x$ and $y$ are c... | https://mathoverflow.net/users/171439 | Shrinking a disk with fixed differential | Here are a few comments that you might find useful, though they don't completely solve the problem. First, using symmetries of the problem, you can easily reduce to the case that $f$ is mapping the interior of the unit circle $x^2+y^2<1$ diffeomorphically onto the interior of a circle $u^2+v^2 < r^2$ for some $r$. Seco... | 3 | https://mathoverflow.net/users/13972 | 432857 | 175,132 |
https://mathoverflow.net/questions/432851 | 13 | This aim of this question is to determine whether there exists a proof (or some counterexample) to the following statement : "If $R$ is a subring of Dedekind domain $S$, such that $S$ has a power basis as an $R$-module, then $R$ is itself a Dedekind domain".
The context is the following: I have recently been working ... | https://mathoverflow.net/users/493301 | Can a Dedekind domain have a power basis over a ring that isn't a Dedekind domain? | If $R\subseteq S$ with $S$ Dedekind and free as an $R$-module, then $R$ is Dedekind because every $R$-ideal $I$ is projective (hence invertible, if non-zero). For $I\otimes\_RS$ $\cong$ $IS$ is projective over $S$, hence over $R$. But $I\otimes\_RS$ $\cong$ $I^{\oplus n}$ - or, more generally, $I\otimes\_RS$ $\cong$ $\... | 13 | https://mathoverflow.net/users/31923 | 432860 | 175,134 |
https://mathoverflow.net/questions/432864 | 1 | I'm looking for a reference for the following:
Suppose that $G$ is a finite group, that $M$ is a smooth $G$-manifold, and that $A\subseteq M$ is a closed $G$-invariant subspace of $M$ such that the action on $M\setminus A$ is free. Suppose also that $G$ acts on $\mathbb{R}^n$. If $f\colon M\rightarrow \mathbb{R}^n$ i... | https://mathoverflow.net/users/489804 | Relative equivariant Thom transversality | See Prop 2.2 of ON THE GROUPS JO(G), Chung-Nim Lee and Arthur Wasserman, Memoirs of the American Mathematical Society Number 159
| 2 | https://mathoverflow.net/users/121316 | 432868 | 175,136 |
https://mathoverflow.net/questions/432869 | 1 | $\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\CC}{\mathcal C}$[This previous question](https://mathoverflow.net/q/432344/36721) introduced the following notion of a summability space.
Let $\N:=\{1,2,\dots\}$. Let $T$ be the shift operator on $\R^\N$ defined by the formula $... | https://mathoverflow.net/users/36721 | On summation methods of divergent series | $\renewcommand{\R}{\mathbb R}\renewcommand{\N}{\mathbb N}\renewcommand{\si}{\sigma}\newcommand\ep\varepsilon$Take indeed any sequence $b\in\R^\N$ such that
\begin{equation\*}
b\_n\to l \tag{0}\label{0}
\end{equation\*}
for some real $l\ne0$.
To obtain a contradiction, suppose that $b$ is good, so that $b\in S$ for s... | 2 | https://mathoverflow.net/users/36721 | 432870 | 175,137 |
https://mathoverflow.net/questions/432853 | 5 | This is my first, and probably my last, (for a while) posting on MO. I am very much a student and I don't claim to be a research mathematician, at all, but I have seen that sometimes "regular" MSE users ask questions here if they feel their question is too obscure to receive a good answer on MSE.
I think my question ... | https://mathoverflow.net/users/320040 | Why is the category of strong braided functors from the braid category to a braided monoidal $M$ equivalent to the subcategory of *strict* functors? | It is indeed true that any strong braided monoidal functor from the braid category to a strict braided monoidal category is equivalent to a strict braided monoidal functor, however that comes out of the proof.
MacLane's argument is the following.
(1) There is an equivalence
$$Hom\_{BM}(\mathfrak{B}, M) \simeq Hom... | 4 | https://mathoverflow.net/users/184 | 432872 | 175,138 |
https://mathoverflow.net/questions/432814 | 2 | This is copied from [math.SE](https://math.stackexchange.com/questions/4556836/overall-idea-of-estimating-major-arcs-in-warings-problem) after a kind comment's suggestion as I am sure people here are very well knowledged in this method :)
I am currently reading Vaughan's "The Hardy-Littlewood Method", and in particul... | https://mathoverflow.net/users/493261 | Overall idea of estimating major arcs in Waring's problem | The following explanation not only accounts for the treatments in the major arcs of Waring's problem, but also the major arcs in a general situation.
Suppose $F(\alpha)$ is some exponential sum that we wish to extract arithmetical information from. Then our task will be to estimate
$$g(n)=\int\_0^1F(\alpha)e(-n\alp... | 3 | https://mathoverflow.net/users/449628 | 432875 | 175,139 |
https://mathoverflow.net/questions/432824 | 3 | Let $X\neq\emptyset$ be a set. A family ${\cal S}\subseteq {\cal P}(X)$ has *property $\mathbf{B}$* if there is $T\subseteq X$ such that for all $S\in{\cal S}$ we have $S\cap T\neq \emptyset$ and $S\not\subseteq T$. Moreover, ${\cal S}$ is said to be *linear* if $|S\_1 \cap S\_2| \leq 1$ for all $S\_1\neq S\_2\in{\cal ... | https://mathoverflow.net/users/8628 | Property $\mathbf{B}$ for maximal linear set systems on $\omega$ with finite members | **Yes.** Partition $\omega$ into two disjoint infinite subsets $T\_1$ and $T\_2$. Recursively construct a $3$-uniform linear hypergraph (Steiner triple system) $\mathcal S\subseteq\binom\omega3$ so that each element of $\binom\omega2$ is contained in a unique element of $\mathcal S$, and each element of $\mathcal S$ me... | 5 | https://mathoverflow.net/users/43266 | 432876 | 175,140 |
https://mathoverflow.net/questions/432904 | -1 | **Definition:** Let $G$ be a group. For $g\in G$ and a subset $F\subseteq G$ fix
the notation $gF:=\{gf\mid f\in G\}$. A sequence $(F\_{i})\_{i\in\mathbb{N}}\subseteq G$
is called a *Følner sequence* if
\begin{eqnarray}
\nonumber
\lim\_{i\rightarrow\infty}\frac{\#(gF\_{i}\triangle F\_{i})}{\#F\_{i}}=0
\end{eqnarray}
fo... | https://mathoverflow.net/users/64444 | Følner sequences of the integers | No. We have an estimate of the form $|f(k) - f(k-1)| \le C / \sqrt{|k|+1}$. Therefore $|f(k) - f(k-1)| \le \epsilon$ except at $O(\epsilon^{-2})$ values, so $\sum\_{k \in F\_i} |f(k) - f(k-1)| / |F\_i| \le O(\epsilon) + O(\epsilon^{-2} / |F\_i|)$. Putting $\epsilon = 1/|F\_i|^{1/3}$, we get $\sum\_{k \in F\_i} |f(k) - ... | 7 | https://mathoverflow.net/users/20598 | 432906 | 175,145 |
https://mathoverflow.net/questions/432918 | 1 | Consider a function $f:\mathbb{R}\_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is *increasing* in each argument, *continuous*, additively (or multiplicatively) *separable*, that is, it can be written in the form $$f(x,y)=a(x)+b(y)$$ for functions of on... | https://mathoverflow.net/users/30484 | Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable? | If we allow $f$ to be discontinuous, then the answer is **yes**: $f$ need not be differentiable.
We can choose $a(x)$ and $b(y)$ so that their ranges $A$ and $B$ are Cantor-like sets which have the following property:
$$ \text{if $\alpha, \alpha' \in A$ and $\beta, \beta' \in B$, and $\alpha + \beta = \alpha' + \beta... | 3 | https://mathoverflow.net/users/108637 | 432921 | 175,150 |
https://mathoverflow.net/questions/421036 | 1 | For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^\*) = \overline{ \sigma(A)}$ the complex conjugation of the spectrum of the matrix $\sigma(A)$. The results also has immediate generalisations to operator... | https://mathoverflow.net/users/143779 | Spectrum invariant under (generalised) transpose as operator on trace class operators | It is not true that $\tilde A$ maps trace class operators to trace class operators in general. For a counterexample, consider the maps $A:X\mapsto \mathrm{Tr}(X) \vert 1\rangle \langle 1 \vert$. Then $\tilde A$ should send $\vert 1\rangle \langle 1 \vert$ to $\mathrm{Id}\_{\mathcal H}$, which is not trace class.
What... | 3 | https://mathoverflow.net/users/10265 | 432922 | 175,151 |
https://mathoverflow.net/questions/432919 | 2 | In [this review](http://home.iscte-iul.pt/%7Ejaats/myweb/papers/new_kuras.pdf) of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $K r$,
$$
r = K r \int\_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta
$$
where $g(\omega)$ is some unknown function (though is... | https://mathoverflow.net/users/90619 | Power series expansion of the order parameter in the Kuramoto model | Presumably (because this is physics) $g$ is analytic in a neighborhood of the origin, so we can Taylor expand.
I think the really important part in deriving this equation for $r$ is the fact that the contribution from the first-order derivative term in $g$ is zero.
Just underneath equation (10), they assume that the ... | 2 | https://mathoverflow.net/users/49417 | 432925 | 175,152 |
https://mathoverflow.net/questions/432883 | 2 | Condition (a) of lemma 3.4 in the paper [“Countable ranks at the first and second projective levels”](https://arxiv.org/abs/2207.08754) [M. Carl, P. Schlicht, P. Welch] is
>
> $\alpha^{+L} = \omega\_1,$
>
>
>
where $\alpha$ denotes any infinite countable ordinal and $\omega\_1 = \omega\_1^V$. I am unable to ex... | https://mathoverflow.net/users/122796 | What is the meaning of $\alpha^{+L}$ for $\alpha$ an infinite countable ordinal? | The meaning of $$\alpha^{+L}$$ for $\alpha$ an infinite ordinal (countable or not) is just "The *cardinal* successor of $\alpha$ as seen by $L$." If that isn't clear, you may prefer the following phrasing:
>
> "The unique ordinal $\beta$ such that $L\models$ "$\beta$ is the smallest cardinal greater than $\alpha$".... | 9 | https://mathoverflow.net/users/8133 | 432929 | 175,153 |
https://mathoverflow.net/questions/432936 | 5 | I am interested in learning about standard monomial theory and Seshadri's program. I find the topic interesting, but I could not yet find a resource which kind of "dumbs it down" enough (a kind of introduction to a layman etc.). Could an expert please point me to some not so difficult to read introductions to SMT, if t... | https://mathoverflow.net/users/81645 | References on standard monomial theory | Seshadri wrote a book, "Introduction to the Theory of Standard Monomials" (<https://doi.org/10.1007/978-981-10-1813-8>), which is very easy-going, especially in the beginning. But perhaps it does not cover exactly what you're interested in?
| 6 | https://mathoverflow.net/users/25028 | 432937 | 175,154 |
https://mathoverflow.net/questions/432933 | 6 | For an integer $n \geq 2$, define $f\_n(\alpha\_0, \alpha\_1, \ldots, \alpha\_{n-1}) = \prod\limits\_{0 \leq i < j < n}\sin^2\left(\alpha\_i - \alpha\_j\right)$ and $$M\_n = \max\limits\_{(\alpha\_0, \alpha\_1, \ldots, \alpha\_{n-1}) \in \mathbb R^n}\{f\_n(\alpha\_0, \alpha\_1, \ldots, \alpha\_{n-1})\}.$$
I *think* t... | https://mathoverflow.net/users/22733 | Maximizing $\prod_{i < j} \sin^2(\alpha_i - \alpha_j)$ | Denote $z\_k=e^{2i\alpha\_k}$, then you want to maximize $\prod\_{j<k}|z\_j-z\_k|$, i. e., the product of all sides and diagonals of an inscribed to the unit circle $n$—gon $A\_1\ldots A\_n$. For any given $j=1,2,\ldots, n-1$ the product of $A\_kA\_{k+j}$ over all $k=1,2,\ldots,n$ (indices are cyclic mod $n$) is maximi... | 8 | https://mathoverflow.net/users/4312 | 432941 | 175,156 |
https://mathoverflow.net/questions/432926 | 2 | Let $A\subseteq [0,1]^d$, $d\geq 2$, a set with Hausdorff dimension $\operatorname{dim}\_{\mathcal{H}}A=s$. What is the minimum $s$ (if any) which guarantee that $A$ has non-empty intersections with a positive fraction of lines passing through the origin?
Thank you in advance for any suggestion.
| https://mathoverflow.net/users/169603 | Hausdorff dimension and non-empty intersections with lines | This is true if the dimension of $A$ is strictly larger than $d-1$; on the other hand taking $A = \{ x^d = 0 \}$ shows that $s = d-1$ is not enough. To prove the first claim we use the co-area formula.
*Remark.*
Generally, when working with Lipschitz functions for example, one has to be careful with the application o... | 3 | https://mathoverflow.net/users/103792 | 432942 | 175,157 |
https://mathoverflow.net/questions/425787 | 3 | Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and
$$
E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}.
$$
Then we have the following result which is
>
> [Theorem:](https://math.stackexchange.com/a/4483272/1019043) If $X= \mathbb R^d$ and $f$ is convex, then the Hausdorff dimension of $... | https://mathoverflow.net/users/99469 | Hausdorff dimension of the non-differentiability set a convex function | I just stumbled across your question. I have no idea how the proofs of these results go—and I am inclined to believe that they would indeed also prove the version you seek—but here's a way to deduce the local result from that on $\mathbf{R}^d$.
Let $X \subset \mathbf{R}^d$ be an open, convex set, $f: X \to \mathbf{R}... | 3 | https://mathoverflow.net/users/103792 | 432950 | 175,159 |
https://mathoverflow.net/questions/432945 | 3 | An antisymmetric relation is defined as a binary relation $R$ on a set $S$ such that $(xRy \land yRx) \rightarrow x=y$, for all $x,y$ in $S$. Certainly, they can't be defined in first-order logic without equality. However, what is an axiomatization of the equality-free theory of antisymmetric relations? I conjecture th... | https://mathoverflow.net/users/43439 | What is an axiomatization of the equality-free theory of antisymmetric relations? | Your proposed sentence is not strong enough. Consider, for example, the "distance-$<17$" relation on $\mathbb{R}$ with the usual metric.
The issue is that we need "transitivity within reflexivity regions:" if $x=y$, then $x$ and $y$ must be related to the same objects. The following pair of sentences on their own do ... | 9 | https://mathoverflow.net/users/8133 | 432952 | 175,160 |
https://mathoverflow.net/questions/432974 | 7 | $\DeclareMathOperator\MLS{MLS}$Recall that the **median operation**, on the power set $2^Y$ of subsets of a set $Y$, is the ternary law $m(A,B,C)$ mapping a triple of subsets to the set of elements belonging to at least two of them. If we view $2^Y$ as the Boolean algebra $(\mathbf{Z}/2\mathbf{Z})^Y$, this is just $m(A... | https://mathoverflow.net/users/14094 | Free median algebras and maximal linked systems | The answer is yes. The reason is rather simple: since the median operation is a special case of a *majority* operation, that is, an operation satisfying the identities
$\forall x,y,\ m(x,x,y) = m(x,y,x) = m(y,x,x) = x,$
every median subalgebra of $2^{2^X}$ is determined by its binary projections, by the Baker-Pixle... | 7 | https://mathoverflow.net/users/2363 | 432978 | 175,168 |
https://mathoverflow.net/questions/432928 | 1 | Let $S\_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S\_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\operatorname{Diff}(S\_g) \to \operatorname{Aut}(H\_1(S\_g))=GL\_{2n}(\mathbb Z)$? It is called symplectic group if we restrict... | https://mathoverflow.net/users/91826 | Name for extension of the symplectic group | I think it is sometimes written $\operatorname{GSp}\_{2g}(\mathbb{Z})$, and called the group of symplectic similitudes
| 8 | https://mathoverflow.net/users/318 | 432980 | 175,169 |
https://mathoverflow.net/questions/432994 | 1 | At the end of the proof of lemma 10, lemma 8 is cited. In order to use it and finish the contradiction, we need to show $n$ is not a multiple of $3.$ However, I don't see any contradiction in having $n \equiv \pm 2 \mod 8$ and $n \equiv 0 \mod 3.$ I also asked on MSE but not even a halfpenny of thoughts were given.
P... | https://mathoverflow.net/users/127521 | Elementary proof of cannonball problem: why can't $n$ be a multiple of $3$? | Lemma 8 is used to conclude that the second factor on the left is $-1$. Note that this factor is
$$
\left( \frac{5}{u\_{2^s}}\right)
$$
meaning you apply Lemma 8 for $m=2^s$ and **not** for $n$. It is obvious that $m$ is not a multiple of $3$.
| 3 | https://mathoverflow.net/users/11552 | 432996 | 175,172 |
https://mathoverflow.net/questions/432897 | 6 | By a *Tarski plane* (resp. *plane*) I understand a mathematical structure $(X,B,\equiv)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and the 4-ary congruence relation ${\equiv}\subseteq X^2\times X^2$ satisfying the [Tarski's axioms](https://en.wikipedia.org/wiki/Tarski%27s_axioms) minus th... | https://mathoverflow.net/users/61536 | The algebraic structure of a line in a (Tarski) plane | This question is addressed by W. Schwabhäuser on p. 156 of his paper *Metamathematical methods in foundations of geometry*. Logic, Methodology and Philosophy of Science (Proc. 1964 Internat. Congr.) North-Holland, Amsterdam, 1965, pp. 152–165. If the Tarski plane (as you have defined it) is hyperbolic (i.e. not Euclide... | 6 | https://mathoverflow.net/users/18939 | 433000 | 175,173 |
https://mathoverflow.net/questions/433005 | 2 | Let $m,n$ be an integer. Denote by ${\rm O}\_n(m)$ be the multiplicative order of $m$ modulo $n$. I want to know what is the possible values of $\frac{m^{{\rm O}\_n(m)}-1}{n}$. Is it true that for fixed $m$, for any integer $N$, we can find $n=n(m,N)$ such that $N$ is a factor of $\frac{m^{{\rm O}\_n(m)}-1}{n}$?
Than... | https://mathoverflow.net/users/45092 | The multiplicative order of $m$ modulo $n$ | If $m$ and $N$ are not coprime, then $n=n(m,N)$ does not exist. Indeed, $m^{{\rm O}\_n(m)}\equiv 1\pmod{nN}$ implies that $m$ and $N$ are coprime.
If $m$ and $N$ are coprime, then $n=n(m,N)$ exists. To see this, we shall use [Zsigmondy's theorem](https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem): for any $\ell>6$ ... | 4 | https://mathoverflow.net/users/11919 | 433012 | 175,179 |
https://mathoverflow.net/questions/433016 | 7 | Let $\pi\_1, \pi\_2$ be two $k$-dimensional subspaces of $\mathbb R^n$. Using elements of the orthogonal group $O(n)$, how much can we simplify $\pi\_1, \pi\_2$? Certainly there always exists $A \in O(n)$ such that $A \cdot \pi\_1$ is the span of the first $k$ canonical bases, but this doesn't say anything about $A \cd... | https://mathoverflow.net/users/156792 | Fundamental domain for two Grassmannians | Your argument about the dimension of the quotient doesn't take into account that there may be elements of the orthogonal group that don't do anything to the pair $(\pi\_1,\pi\_2)$. For example, if $2k<n-1$, then the span of the two planes has codimension at least $2$ in $\mathbb{R}^n$, so there will be rotations that a... | 11 | https://mathoverflow.net/users/13972 | 433025 | 175,181 |
https://mathoverflow.net/questions/433014 | 3 | I need to compute a Groebner basis of a polynomial system with parameters.
The only recent results I found is Groebner cover:
<https://www.sciencedirect.com/science/article/pii/S0747717110000970>
Are there any more advanced algorithms for the study of parametric polynomial systems?
| https://mathoverflow.net/users/493428 | Groebner basis with parameters | Mathscinet mentions some 30 papers citing the paper you mention, among which the following looks like potentially relevant:
[Kapur, Deepak (1-NM-C); Sun, Yao (PRC-ASBJ-MML); Wang, Dingkang (PRC-ASBJ-MML)
An efficient algorithm for computing a comprehensive Gröbner system of a parametric polynomial system. (English su... | 7 | https://mathoverflow.net/users/1306 | 433026 | 175,182 |
https://mathoverflow.net/questions/432981 | 2 | Let $A$ be a full triangulated subcategory of $B$, $u:A\rightarrow B$ the corresponding embedding. Let $f:B\rightarrow A$ be a triangulated functor
satisfying:
1. $f\circ u = id$
2. Let $b \in B $, if $f(b)=0$ then $b=0$.
**Question:** do we have $K\_{0} (A)= K\_{0}(B)$ ?
| https://mathoverflow.net/users/165456 | Grothendieck group of triangulated categories | Let $A$ be a triangulated category, and let $B=A\times A$, with $A$ regarded as a full triangulated subcategory of $B$ via the embedding $u(X)=(X,0)$, and let $f:B\to A$ be the functor $f(X,Y)=X\oplus Y$.
Then $f\circ u=\text{id}\_A$, and $K\_0(B)\cong K\_0(A)\oplus K\_0(A)$, which might not be isomorphic to $K\_0(A)... | 9 | https://mathoverflow.net/users/22989 | 433027 | 175,183 |
https://mathoverflow.net/questions/433024 | 0 | let ILP be an integer linear program with constraints-matrix $\boldsymbol{\mathrm{M}}\in\mathbb{Z}^{m\times n}$ and cost vector $\boldsymbol{\mathrm{c}}\in\mathbb{Z}^n$,
${\boldsymbol{\mathrm{x}}^\*}\in\mathbb{Z}^n,\,{\boldsymbol{\mathrm{x}}^\*}^T\boldsymbol{\mathrm{c}}\in\mathbb{Z}\le\boldsymbol{\mathrm{x}}^T\boldsy... | https://mathoverflow.net/users/31310 | Benefit of adding a trivial constraint to ILPs | This question is addressed in a few OR StackExchange questions:
* <https://or.stackexchange.com/questions/419/feeding-known-lower-bounds-to-solvers>
* <https://or.stackexchange.com/questions/3777/how-to-exploit-known-solution-in-milp>
* <https://or.stackexchange.com/questions/5331/objective-integrality-cuts>
The su... | 3 | https://mathoverflow.net/users/141766 | 433028 | 175,184 |
https://mathoverflow.net/questions/432962 | 1 | Given a matrix $M$ that consists of a set of $4K$ binary row vectors (each vector entry is 0 or 1) each of length $K$. Moreover, it is known/promised that no subset of rows in matrix add to an all 1 vector. For a given integer $X$, the only the following operations permitted on the rows of the matrix:
1. Modular addi... | https://mathoverflow.net/users/493386 | A query about modular arithmetic on a matrix | The question is equivalent to finding an integer vector $x$ such that
$$xM = \iota\_K,$$
where $\iota\_k$ is the all-1 vector of length $K$.
By [Rouché–Capelli theorem](https://en.wikipedia.org/wiki/Rouch%C3%A9%E2%80%93Capelli_theorem), this equation has a solution modulo prime $X$ iff the rank of $M$ equals the rank... | 3 | https://mathoverflow.net/users/7076 | 433035 | 175,186 |
https://mathoverflow.net/questions/433021 | 11 | For $f: \mathbb R \to \mathbb R$ a measurable function, we say $g: \mathbb R \to \mathbb R$ is a *modification* of $f$ if $f = g$ a.e.
Suppose $f$ Is a measurable function that is differentiable a.e.
We say that a modification $g$ of $f$ is *maximally differentiable* if whenever $h$ is another modification of $f$, ... | https://mathoverflow.net/users/173490 | Does every differentiable a.e. function admit a maximally differentiable representative? | $\DeclareMathOperator\*\appliminf{app-liminf}\DeclareMathOperator\*\applimsup{app-limsup}\DeclareMathOperator\*\applim{app-lim}\DeclareMathOperator\*\essliminf{ess liminf}\DeclareMathOperator\*\esslimsup{ess limsup}$The answer is indeed yes. Further, the assumption that $f$ be differentiable a.e. is unnecessary. The ma... | 12 | https://mathoverflow.net/users/173490 | 433040 | 175,188 |
https://mathoverflow.net/questions/432951 | 10 | Certain categories of mathematical structures have had synthetic axiom systems developed for them. One particularly well known such category is the category of sets and functions $\mathit{Set}$, which was axiomatised by William Lawvere as the [Elementary Theory of the Category of Sets](https://www.ncbi.nlm.nih.gov/pmc/... | https://mathoverflow.net/users/483446 | Axioms for the category of groups | As requested, here is an answer summarizing axioms for the category of groups that were [given](https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/une-caracterisation-de-la-categorie-des-groupes/BDD4B7EA2FD122F7E0F1C81236162FE4) by Pierre Leroux, and which I learned from an MSE answer of Arn... | 10 | https://mathoverflow.net/users/43000 | 433048 | 175,191 |
https://mathoverflow.net/questions/433041 | 5 | Given a measurable subset $A$ of $[0, 1]$, a sequence of functions $f\_n: [0, 1] \to \mathbb R$ is said to be *equi-Lebesgue continuous* on $A$ if for every $x \in A$, and $\varepsilon > 0$, there exists some $\delta > 0$ such that for all $0 < r < \delta$, we have
$$\frac{1}{2r} \int\_{B\_r (x)} \lvert f\_n (x) - f\... | https://mathoverflow.net/users/173490 | Arzelà–Ascoli for equi-Lebesgue continuous functions | $\newcommand\ep\varepsilon\newcommand\ze\zeta\newcommand{\al}{\alpha}\newcommand{\be}{\beta}\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}$The answer is yes.
Indeed, take any real $\be>0$. Let
\begin{equation\*}
\al:=\be/2,\quad\ep:=\be^2/48,\quad\ze:=\eta:=\be/4.
\end{equation\*}
Write $B\_x(r):=[0,1]\cup(x-r,x... | 6 | https://mathoverflow.net/users/36721 | 433056 | 175,192 |
https://mathoverflow.net/questions/432982 | -2 | A matrix $$\begin{bmatrix}w &x \\\ y &z\end{bmatrix}\in\mathbb Z^{2\times 2}$$ is unimodular if $$|wz-xy|=1$$ holds.
>
> Is there a parametrization of such matrices with $2wy>(wz+xy)$ and $2xz>(wz+xy)$ with $wz-xy=1$?
>
>
>
| https://mathoverflow.net/users/10035 | On a criterion for unimodular matrix | The OP clarified in a comment that the variables $w,x,y,z$ were meant to be nonnegative. Under this restriction, the pair of inequalities
$$2wy>wz+xy\qquad\text{and}\qquad 2xz>wz+xy$$
has no solution. Indeed, these inequalities feature nonnegative numbers on both sides, hence multiplying them yields
$$4wxyz>(wz+xy)^2.$... | 3 | https://mathoverflow.net/users/11919 | 433061 | 175,193 |
https://mathoverflow.net/questions/432561 | 5 | I would like to enter the world of derivators. We can find a little history here and there about the limitations of triangulated categories and the motivation to enhance them, but also to compute homotopy limits and colimits and others. There is also a relation with the theory of higher categories. History also says th... | https://mathoverflow.net/users/429204 | Axioms of derivators | With the comments having clarified the question a bit, let me just say that the a priori motivation for these particular axioms was just whatever was going through Grothendieck’s and Heller’s heads when they wanted to build something that looks like a 2-functor of categories of diagrams and Kan extensions between them.... | 5 | https://mathoverflow.net/users/43000 | 433073 | 175,197 |
https://mathoverflow.net/questions/433062 | 40 | Does there exist a group $G$ such that
1. for any finite $K$ there is a monomorphism $K \to G$
2. for any $H$ with property 1 there is a monomorphism $G \to H$
If yes, is it the only one?
| https://mathoverflow.net/users/148161 | Is there a smallest group containing all finite groups? | No. To show that it doesn't exist it is enough to produce two groups $G,H$ which contain isomorphic copies of all finite groups, but such that no group containing isomorphic copies of all finite groups embeds into both $G$ and $H$.
Let $(G\_n)$ be an enumeration of all finite groups. Let $G=\bigoplus G\_n$ be the res... | 49 | https://mathoverflow.net/users/14094 | 433074 | 175,198 |
https://mathoverflow.net/questions/432932 | 16 | In conjecture 6.1.14 of this [article](https://arxiv.org/abs/2210.01404), Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived categories of) smooth representations of $G:=\mathrm{GL}\_{2}(F)$ on $\mathcal{O}$-modules,... | https://mathoverflow.net/users/85392 | How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands? | Briefly (I will elaborate below): One expects that their fully faithful functor from (roughly) $p$-adic representations of $G(\mathbb Q\_p)$ to (roughly) coherent sheaves on the Emerton--Gee stack extends to an equivalence between (roughly) $p$-adic sheaves on $\mathrm{Bun}\_G$ and (roughly) coherent sheaves on the Eme... | 8 | https://mathoverflow.net/users/6074 | 433083 | 175,202 |
https://mathoverflow.net/questions/433087 | 3 | Austin-Braam approach uses the multicomplexes of de Rham complex on critical submanifolds to describe Bott-Morse theory.
For more details, see the follows:
<https://link.springer.com/chapter/10.1007/978-3-0348-9217-9_8>
Could we construct the similar approaches for the Floer type theory? For example, if the one-par... | https://mathoverflow.net/users/120948 | Is there an analogy of Austin-Braam approach to Bott-Morse type Hamiltonian Floer homology? | First, Austin and Braam *did* already apply their machine to a Floer-type theory: it was just instanton homology and not Hamiltonian Floer homology [here](https://www.sciencedirect.com/science/article/pii/0040938395000046). Their machine uses $\Bbb R$ coefficients, but you can work over $\Bbb Z$ by using an appropriate... | 3 | https://mathoverflow.net/users/40804 | 433089 | 175,204 |
https://mathoverflow.net/questions/433054 | 1 | Rick Miranda's "The basic theory of elliptic surfaces" the Example (I.5.1) see page 7 on a pencil of plane curves contains an argument that I do not understand yet.
Let $C\_1$ be a smooth cubic curve in $\mathbb{P^2}$ and let $C\_2$ be any other cubic. By intersection theory and Bezout's theorem the intersection numb... | https://mathoverflow.net/users/108274 | Blow-up of a pencil of cubic curves (from Miranda's basic theory of elliptic surfaces) | As explained in abx's comment, the canonical divisor of your surface is given by $$K\_X=-3 \pi^\*L + \sum E\_i,$$ and this is precisely the class of $-\widetilde{C}\_1$ (note that $C\_1$ is a curve in $\mathbb{P}^2$, so I put the tilde to specify that we are considering its strict transform in $X$).
Now, $\widetilde{... | 2 | https://mathoverflow.net/users/7460 | 433090 | 175,205 |
https://mathoverflow.net/questions/433063 | 18 | Can one color the positive integers with finitely many colors, so that no two different numbers of the same color add to a square?
Some easy to prove remarks:
1. at least 4 colors are needed, since the sum of any two in $\{386, 2114, 3970, 10430\}$ is square;
2. if $N$ colors suffice for each finite subset, then $N... | https://mathoverflow.net/users/2480 | Can the positive integers be colored so that elements of same color never add to a square? | No. See the paper below, which handles more polynomials than just perfect squares.
[On the number of monochromatic solutions of $x+y = z^2$](https://doi.org/10.1017/S0963548305007169). Ayman Khalfalah and Endre Szemerédi. Combinatorics, Probability and Computing (2006) 15, 213–227.
| 17 | https://mathoverflow.net/users/129185 | 433095 | 175,207 |
https://mathoverflow.net/questions/433098 | 1 | Is there a profinite group $G$ with a locally finite subgroup $H$ such that $\overline H$, the closure of $H$, is not torsion?
| https://mathoverflow.net/users/84700 | Looking for an example of profinite groups | You can take $G=\prod\_{k>0}\mathbb{Z}/2^k$ and $H=\bigoplus\_{k>0}\mathbb{Z}/2^k$. Then every finitely generated subgroup of $H$ is finite, and $\overline{H}=G$, but the element $(1,1,1,\dotsc)\in G$ is not torsion.
| 7 | https://mathoverflow.net/users/10366 | 433100 | 175,208 |
https://mathoverflow.net/questions/433097 | 2 | *Note: We define the signum function, $\text{sgn}$ by $\text{sgn}(x) = 1$ if $x \geq 0$, and $-1$ otherwise.*
Suppose $f: [0, \infty) \to \mathbb R$ is continuous and of locally bounded variation, with $f(0) = 0$. Is it true that the integral
$$\int\_0^t f \, d\, \text{sgn(f)} := f(t) \, \text{sgn}(f(t)) - \int\_0^... | https://mathoverflow.net/users/173490 | On a certain deterministic integral related to Tanaka’s formula | $\newcommand\sgn{\operatorname{sgn}}$Yes, this is true. Indeed, fix any real $t>0$. Let
\begin{equation}
S\_+:=\{s\in(0,t)\colon f(s)\ge0\},\quad S\_-:=\{s\in(0,t)\colon f(s)<0\}.
\end{equation}
Then
\begin{equation}
I:=\int\_0^t\sgn(f(s))\,df(s)=I\_+ - I\_-,
\end{equation}
where
\begin{equation}
I\_-:=\int\_{S\_-}... | 2 | https://mathoverflow.net/users/36721 | 433106 | 175,210 |
https://mathoverflow.net/questions/433105 | 1 | Let $K$ be a number field, $\mathcal{O}\_K$ its ring of integers and $S$ a
subset of the real places. Let $\mathfrak{m} \subset \mathcal{O}\_K$ an ideal. The ideal
theoretic
*ray class group* of $\mathfrak{m} $ and $S$ is the quotient group
$$ I^{\mathfrak{m}}/P^{\mathfrak{m}} $$
where $ I^{\mathfrak{m}} $ is the g... | https://mathoverflow.net/users/108274 | The $ 1 \operatorname{ mod } \mathfrak{m}$ congruence relation in ray $ P^{\mathfrak{m}}$ of the ideal theoretic ray class group | Gauß had studied classes of binary quadratic forms with arbitrary discriminant. Dedekind realized that the class groups of forms with fundamental discriminant are ideal class groups (in the strict sense) of quadratic number fields. In order to find something similar for forms with discriminant $\Delta = df^2$, where $d... | 4 | https://mathoverflow.net/users/3503 | 433118 | 175,214 |
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