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https://mathoverflow.net/questions/347932 | -2 |
>
> Given sum of elements of each row of a positive definite square matrix $M$ of order $n$ all of whose entries are non-negative, when is it possible to find the sum of all elements of the matrix $M^{-1}$?
> Are there any papers or research done in this direction?
>
>
>
To fully explain my question ,
Let $M$... | https://mathoverflow.net/users/140263 | When is it possible to find the sum of all elements of inverse of a matrix? | If the condition on $M$ (in order for the sum of all entries of $M^{-1}$ to be uniquely determined) is to be expressed only in terms of the row sums of $M$ (with the additional restrictions that all entries of $M$ be non-negative and all eigenvalues of $M$ be positive), then the answer is the following:
>
> The su... | 5 | https://mathoverflow.net/users/36721 | 347962 | 147,315 |
https://mathoverflow.net/questions/347852 | 1 | I saw a result in notes on by Olivier Debarre (Rational Curves on Hypersurfaces, Lecture notes for the II Latin American School of Algebraic Geometry and Applications 1-12 of June 2015 in Cabo Frio, Brazil) that if $ Z $ is a hypersurface in $ \mathbb{P}^{n}\_{\mathbb{C}} $, of degree less than or equal to $ n $, then ... | https://mathoverflow.net/users/113893 | Does anyone know a reference in the literature regrading a proof that every projective hypersurface with vanishing canonical divisor is uniruled | Not the most important thing, but your title does not match your question. The canonical divisor (if it exists!) of these objects is not vanishing. It is actually better to talk about the *canonical sheaf* than the *canonical divisor* and probably what you had in mind was that the canonical sheaf does not have non-zero... | 2 | https://mathoverflow.net/users/10076 | 347964 | 147,317 |
https://mathoverflow.net/questions/346964 | 5 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). We call $H$ *proper* if $E\neq\emptyset, \emptyset \notin E$ and for no $e\_1\neq e\_2\in E$ we have $e\_1\subseteq e\_2$. A *matching* is a set $M$ of pairwise disjoint edges (members of $E$), and $T\subseteq V$ is said to be a *transversal* if... | https://mathoverflow.net/users/8628 | Sizes of matchings and transversals in hypergraphs | Yes. For fixed $\beta$ we construct the hypergraph $G\_\beta$ with edges $\{e\_x\}\_{x\in \beta}$, any two edges $e\_x,e\_y$ have unique common vertex $v\_{x,y}$, and there are no other vertices. In this hypergraph the maximal matching has size 1 and the minimal transversal has size $\beta$, since any set consisting of... | 1 | https://mathoverflow.net/users/4312 | 347965 | 147,318 |
https://mathoverflow.net/questions/347969 | 0 | Let $\mu$ be a locally finite Borel measure on a Polish space $X$. A $\mu$-continuity set $A$ is such that $\mu(\partial A) = 0$, such sets are important when working with Portmanteau's theorem. A few years ago I remember seeing that for a fixed $x \in X$, all but at most countably many balls with centre $x$ in $X$ are... | https://mathoverflow.net/users/149688 | At most countably many balls are $\mu$-continuity sets in a Polish space | 1) Locally finite measure on the separable metric space $X$ is $\sigma$-finite. Indeed, fix a dense countable set $Z\subset X$ and call a ball with center in $Z$ and rational radius good, if it has finite measure. Let $Y\subset X$ be a union of all good balls. Clearly $\mu$ is $\sigma$-finite on $Y$. Assume that $Y\ne ... | 0 | https://mathoverflow.net/users/4312 | 347973 | 147,319 |
https://mathoverflow.net/questions/347693 | 7 | Take the integers $\mathbb{Z}$ and the addition
\begin{align\*}
+: \mathbb{Z} \times \mathbb{Z} &\to \mathbb{Z}
\\
(a,b) &\mapsto a+b.
\end{align\*}
Using the Stone-Čech compactification $\beta\mathbb{Z}$ in two steps, $A$ can be "continuously" extended to
\begin{equation\*}
+: \beta\mathbb{Z} \times \mathbb{Z} \to... | https://mathoverflow.net/users/18191 | Partitioning $\beta \mathbb{Z} \setminus \mathbb{Z}$ | Yes, this is possible.
First of all, let me suggest a way of thinking about the $+$ operation. If you're familiar with the idea of taking limits along an ultrafilter, then given $p,q \in \mathbb Z^\*$, you can think of $p+q$ as
$$p+q = q\text{-lim}\_{n \in \mathbb Z} \ (p+n).$$
I tend to look at this dynamically: we... | 6 | https://mathoverflow.net/users/70618 | 347974 | 147,320 |
https://mathoverflow.net/questions/347953 | 4 | Let $X$ be a separable Banach space. Is this property equivalent to the approximation property?
There exists a chain $X\_n$ of finite-dimensional subspaces of $X$, each being a range of some projection $P\_n$, such that $\bigcup X\_n$ is dense in $X$ and for every $x\in X$ we have $P\_nx \to x$ as $n\to \infty$?
Do... | https://mathoverflow.net/users/148734 | Approximation property of a Banach space in terms of finite-rank projections | It seems to me that the property which you describe is equivalent to the $\pi$-property. An interesting example, showing that it does not follow from the approximation property was discovered by Charles Read (1958-2015). Unfortunately Charles never published this nice paper, but you can find it on my web page: <http://... | 6 | https://mathoverflow.net/users/37822 | 347976 | 147,321 |
https://mathoverflow.net/questions/347884 | 2 | I'd like to learn about the period-doubling route to chaos of the logistic family $f\_\lambda(x)= \lambda x (1-x)$ and got interested in the fine properties of the bifurcation diagram of this family as we vary $\lambda$.
On wikipedia <https://en.wikipedia.org/wiki/Logistic_map> they claim that for most parameters $\l... | https://mathoverflow.net/users/91098 | Fine structure of bifurcation diagram of logistic family | The questions you are asking are fundamental to the theory of one-dimensional dynamical systems. I would suggest starting with an introductory textbook, such as *An Introduction to Chaotic Dynamical Systems* by Devaney. Books with more in-depth results include *Iterated Maps of the Interval as Dynamical Systems* by Col... | 2 | https://mathoverflow.net/users/6514 | 347982 | 147,322 |
https://mathoverflow.net/questions/347992 | 1 | Let $Z=XY$ where $X$, $Y$ are random variables with support of non-trivial measure. For what distributions of $X$ and $Y$ can $Z$ be guaranteed to be Gaussian?
| https://mathoverflow.net/users/17243 | Which distributions of $X$ and $Y$ yield a Gaussian $Z=XY$? | It was shown in [this paper (see formula (2) there)](http://arxiv.org/abs/1803.09838v1) that any normal random variable (r.v.) $Z$ is multiplicatively infinitely divisible; that is, for each natural $k$ there exist iid r.v.'s $W\_1,\dots,W\_k$ such that $Z$ equals $W\_1\cdots W\_k$ in distribution; the distribution of ... | 4 | https://mathoverflow.net/users/36721 | 348000 | 147,327 |
https://mathoverflow.net/questions/347997 | 3 | I wondered if it were possible to clarify the basic first lemma in *A note on the Intersection of Veronese Surfaces*.
Setup: $X\_1$ and $X\_2$ are two Veronese surfaces in $\mathbb{P}^5$ intersecting along a zero-dimensional scheme $W$.
Statement: Viewing $W$ as a subscheme of $X\_1\cong \mathbb{P}^2$, $h^1(I\_{W/\... | https://mathoverflow.net/users/16356 | Veronese surfaces in $\mathbb{P}^5$ intersecting finitely | The restriction of the resolution of $\mathcal{O}\_{X\_2}$ to $X\_1$ computes $\mathrm{Tor}\_i(\mathcal{O}\_{X\_1},\mathcal{O}\_{X\_2})$. If the intersection $X\_1 \cap X\_2$ is dimensionally transverse, higher $\mathrm{Tor}\_i$ vanish, so the restriction of the resolution is a resolution of $W$.
EDIT. As @DCT observ... | 3 | https://mathoverflow.net/users/4428 | 348017 | 147,330 |
https://mathoverflow.net/questions/347910 | 5 | In a family of smooth projective curves over a reduced complex scheme of finite type the list of isomorphism classes of automorphism groups of the fibers is finite. This follows from the Hurwitz's bound and the constancy of the genus on each connected component of the base.
Can one prove a similar result for families... | https://mathoverflow.net/users/nan | Can the automorphism group vary too much in families of complex projective varieties? | In the case of a family of minimal smooth projective varieties *of general type*, we can repeat exactly the same argument as in the case of curves of genus $\geq 2$.
In fact, by a result of Hacon, McKernan and Xu **[HMcKX13]** we know that, if $X$ is any such a variety, we have $$|\mathrm{Aut}(X)| \leq c \cdot \mathr... | 6 | https://mathoverflow.net/users/7460 | 348028 | 147,334 |
https://mathoverflow.net/questions/347308 | 9 | **Statement.** Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$ in *analtytic topology*. It is *well known* that there exists a rank $k+1$ complex vector bundle $V$ over $\mathbb CP^n$ such that $X$ is isomorphic as a *projective variety* to the projectivisation $\mathb... | https://mathoverflow.net/users/13441 | $\mathbb CP^k$ bundles over $\mathbb CP^n$ are projectivisations of vector bundles. Any reference? | I would like to give a final answer to this question (it took me some time to get it, thanks to Angelo and Donu) which is, in fact, the following statement.
**Statement'**. Let $Y$ be a projective variety with $H^2(Y,\mathbb {\cal O}^\*)=0$. Then any holomorphic $\mathbb P^m$-bundle over it is a projectivisation of a... | 0 | https://mathoverflow.net/users/13441 | 348049 | 147,341 |
https://mathoverflow.net/questions/348021 | 2 | We know that a set may be of zero harmonic measure without its Lebesgue measure being zero (see Armitage and Gardiner, classical potential theory, pg 178).
Now, consider the following problem. Let $ V $ be a bounded open set in $ \mathbb{R}^{m} $, $ m\geq 2 $, and $W$ the interior of the closure of $V$. Let $E$ be a... | https://mathoverflow.net/users/100746 | A set of zero harmonic measure | It is easy to construct such an example in $R^n$ for $n\geq 3$ (see my comment).
In $R^2$ let $V$ be a rectangle with removed segments:
$$\{ x+iy:|x|<1,0<y<2\}\backslash \left((0,i]\cup\cup\_{n=2}^\infty ((1/n,1/n+i]\cup(-1/n,-1/n+i])\right),$$
and $E=(0,i)$.
Similar example will work in $R^n$ if you want the $n-1$ L... | 1 | https://mathoverflow.net/users/25510 | 348052 | 147,342 |
https://mathoverflow.net/questions/348045 | 6 | let me say that I am not a set theorist, but I have to settle up some things in category theory and I need your help.
What I'd like to do is, in some way, use axiom of choice for proper classes.
I say that axiom of choice holds for a set $X$ if there exist a function $f:X \to \bigcup X$ such that $f(x) \in x$.
My ... | https://mathoverflow.net/users/140013 | Very large axiom of choice | If you have ZFC in the ambient theory, including the axiom of choice, then indeed the axiom of choice holds in every Grothendieck-Zermelo universe (also sometimes known as Grothendieck universes). A Grothendieck-Zermelo universe is a rank-initial segment $V\_\kappa$ of the cumulative hierarchy, where $\kappa$ is an ina... | 8 | https://mathoverflow.net/users/1946 | 348053 | 147,343 |
https://mathoverflow.net/questions/348031 | 5 | Let $\mathbb{F}$ be a local field and let $\pi$ be a principal series representation of $GL\_2(\mathbb{F})$ that is $\pi=Ind\_B^{GL\_2}(\chi\_1\otimes\chi\_2)$ for two characters $\chi\_1$ and $\chi\_2$ of the maximal torus. Then I came to know from the book **"Automorphic Forms on Adele group"** by **Gelbert** that du... | https://mathoverflow.net/users/140336 | Conductor of Principal series representation | Here is the story as I know it. The more general setup is for *induced representations of Langlands type*. These are certain generic representations of $\mathrm{GL}\_n(F)$, where $F$ is a nonarchimedean local field. The construction of such representations is the following.
Suppose that $n = n\_1 + \cdots + n\_r$ wit... | 6 | https://mathoverflow.net/users/3803 | 348065 | 147,347 |
https://mathoverflow.net/questions/348071 | 4 | Working with relations in a purely set theoretic manner i.e. as just sets of ordered pairs, we see for any relation $R$ there exists unique inclusion minimal sets $A$ and $B$ such that $R\subseteq A\times B$ so that if we write $\text{dom}(R)=A$ and $\text{rng}(R)=B$ with $\text{fld}(R)=A\cup B$ then $\text{fld}(R)$ by... | https://mathoverflow.net/users/38626 | Characterizing relations by forbidden induced subsets | I am going to rephrase the question in terms of first order relational structures. I believe the answer will be sufficiently close to what you are looking for.
Let $L$ be the first order language containing only a binary relation symbol $R$. Call a class $\mathcal{P}$ of $L$-structures **universal** if there is a col... | 4 | https://mathoverflow.net/users/38253 | 348078 | 147,353 |
https://mathoverflow.net/questions/201251 | 4 | Basically, I wonder whether a theory similar to geometric group theory has been or could be developed for rings and semirings.
One direction would be the following. Consider $\mathbb{N}$ (with the French convention, i.e. including $0$), and let $n\in\mathbb{N}$.
Consider all algorithms based on the following operatio... | https://mathoverflow.net/users/4961 | Geometry of rings and semi-rings | (1) As already mentioned in the comments by Emil Jeřábek, your questions about $\mathbb{N}$ are studied under the name of straight-line programs or addition-multiplication chains. They are closely related to major complexity questions, e.g. if you cannot compute $n!$ using only $O((\log n)^c)$ such operations for any c... | 5 | https://mathoverflow.net/users/38434 | 348095 | 147,361 |
https://mathoverflow.net/questions/347697 | 8 | I would like to know if it is possible to calculate in closed-form, or well what work can be done about it, the definite integral $$\int\_0^1\int\_0^1\int\_0^1\frac{3dxdydz}{3-z(x+\sqrt{xy}+y)},\tag{1}$$
where I was inspired in a well-known integral representation for the Apéry constant involving the volume $x\cdot y\c... | https://mathoverflow.net/users/142929 | On the closed-form of $\int_0^1\int_0^1\int_0^1\frac{dxdydz}{1-\frac{z}{3}(x+\sqrt{xy}+y)}$ | Let me expand the integrand in powers of $z$ and integrate over $z$,
$$I=\int\_0^1 dx\int\_0^1 dy\int\_0^1 dz\;\frac{3}{3-z(x+\sqrt{xy}+y)}$$
$$\qquad\qquad=\sum\_{n=0}^\infty\,\frac{3^{-n}}{n+1} \int\_0^1 dx\int\_0^1 dy\;(x+\sqrt{xy}+y)^n.$$
The integral over $x$ and $y$ is an element $c\_{n}\in\mathbb{Q}$,
$$I=\sum\_... | 13 | https://mathoverflow.net/users/11260 | 348102 | 147,362 |
https://mathoverflow.net/questions/344294 | 3 | Does there exist a definition of Chern character (or Chern classes) for a coherent sheaf $\mathscr{F}$ on a singular variety $X$? In this case I might not be able to find a projective resolution for $\mathscr{F}$.
It seems that Vakil define it in his notes <https://math.stanford.edu/~vakil/245/245class19.pdf> on pag... | https://mathoverflow.net/users/122284 | Chern character of coherent sheaf on singular variety | Let me summarize what is known about Chern classes and the Chern character on singular varieties, expanding on the comments to the question.
1. On a normal variety the first Chern class is easily defined by removing the singular locus (which is of codimension at least two) and closing up the first Chern class on the ... | 3 | https://mathoverflow.net/users/111491 | 348107 | 147,364 |
https://mathoverflow.net/questions/346802 | 3 | I am looking for a *reference* for the following basic fact:
>
> Let $R$ be a noetherian ring, let $M$ be an artinian $R$-module, let $N$ be a finitely generated $R$-module, and let $i\in\mathbb{N}$. Then, $Tor\_i^R(M,N)$ is artinian.
>
>
>
(I know that it is easy to prove. I guess nevertheless that this is wr... | https://mathoverflow.net/users/11025 | Artinian Tor modules (Reference request) | This follows immediately from the slightly more general Lemma 2.2 in M. Brodmann, S. Fumasoli, R. Tajarod, *Local cohomology over homogeneous rings with one-dimensional local base ring,* Proc. Amer. Math. Soc. 131 (2003), 2977-2985, which considers additionally a change of rings.
| 1 | https://mathoverflow.net/users/11025 | 348121 | 147,369 |
https://mathoverflow.net/questions/348110 | 16 | It is [well-known](http://brauer.maths.qmul.ac.uk/Atlas/v3/clas/U42) that there is an isomorphism between $PSp(4,3)$ (the symplectic group of dimension $4$ over $\mathbb F\_3$) and $PSU(4,2^2)$ (the unitary group defined by $4\times4$ unitary matrices over $\mathbb F\_4$).
**Question:** Can the exceptional isomorphis... | https://mathoverflow.net/users/125498 | Geometric interpretation of the exceptional isomorphism $PSp(4,3)=PSU(4,2^2)$ | I believe that a finite geometric proof is given by Jean Dieudonné here:
*Dieudonné, Jean*, [**Les isomorphismes exceptionnels entre les groupes classiques finis**](http://dx.doi.org/10.4153/CJM-1954-029-0), Can. J. Math. 6, 305-315 (1954). [ZBL0055.01904](https://zbmath.org/?q=an:0055.01904).
If you go to Section ... | 14 | https://mathoverflow.net/users/801 | 348124 | 147,371 |
https://mathoverflow.net/questions/348118 | 3 | Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta\_g$ be the Laplacian and $L=\Delta\_g-\partial\_t$ the heat operator. Let $0<\alpha<1$, $0<t\_0<T$.
Let $$u\in C^{2,1}(M\times[0,T))\cap C^\infty(M\times(0,T))$$ solve $$Lu=f$$ where $$f\in C^0(M\times[0,T))\cap C^\infty(M\times(0,T))... | https://mathoverflow.net/users/90076 | Reference request: Schauder estimate in the space variable for parabolic equations | For the heat equation in $\mathbb R^n$ the estimate holds. The main thing is the smoothness of the volume potential $g=\Gamma\*f$, where $\Gamma$ is the fundamental solution. It is obtained in O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva, *Linear and Quasi-linear Equations of Parabolic Type*, ch.4, $\S2$. ... | 1 | https://mathoverflow.net/users/14551 | 348136 | 147,375 |
https://mathoverflow.net/questions/348100 | 4 | Assume that $X$ and $Y$ are compactly generated weak Hausdorff spaces (CGWH spaces for short). Assume that they are also well-pointed (so the inclusions of the base points are Hurewicz cofibrations).
Is then the mapping space $Y^X$ of pointed maps from $X$ to $Y$ again well-pointed (with base point the constant map fro... | https://mathoverflow.net/users/70808 | Cofibrations and mapping spaces in compactly generated weak Hausdorff spaces | If the inclusion $A\to X$ is a cofibration, then $A$ is a $G\_\delta$ subset of $X$ (i.e. the intersection of a countable family of open subsets), as one can deduce easily from the fact that $\{0\}$ is $G\_\delta$ in $[0,1]$. If $T$ is uncountable and discrete then the basepoint in $[0,1]^T$ is not $G\_\delta$. Thus, t... | 4 | https://mathoverflow.net/users/10366 | 348139 | 147,377 |
https://mathoverflow.net/questions/348112 | 10 | Is there a nice theorem about the algebra of invariants $\mathbb{C}[\mathfrak{gl}\_n]^{SO\_n \times SO\_n}$, where the action is by left and right multiplication? I'm hoping for something along the lines of the Chevalley restriction theorem, with a list of generators for the algebra being a nice bonus if possible (the ... | https://mathoverflow.net/users/44191 | Invariants for $SO_n \backslash \mathfrak{gl}_n / SO_n$ | Let $X$ denote the $n \times n$ complex matrices and let $D$ be the diagonal $n \times n$ complex matrices. We first claim that a generic matrix in $X$ factors as $U\Sigma V$ with $U$ and $V \in SO(n)$ and $\Sigma \in D$. Proof: Let $\mu$ be the multication map $SO(n) \times D \times SO(n) \to X$. Then it is not bad to... | 6 | https://mathoverflow.net/users/297 | 348146 | 147,381 |
https://mathoverflow.net/questions/348150 | 2 | Let $\mathbb{P}=\langle P, \leq \rangle$ be a p.o.
* Two elements $p$ and $q$ of it are called
compatible if there is an $r \in \mathbb{P}$ such that $r\leq p$ and $r \leq q$; otherwise they
are called incompatible.
* A subset $D$ of $\mathbb{P}$ is said to be **dense** in $\mathbb{P}$ if for each $p\in \mathbb{P}$ t... | https://mathoverflow.net/users/138770 | About product of Baire spaces and forcing | The way you set this up, it might not be dense, since you only have that $p'$ forces that $f$ is a function from $\omega$ to the ordinals. Perhaps other incompatible conditions force that $f$ is not a function, or empty, or is whatever, in such a way that $f(\check n)$ is not meaningful.
But you can fix things by ar... | 1 | https://mathoverflow.net/users/1946 | 348155 | 147,384 |
https://mathoverflow.net/questions/346114 | 4 | Let $S$ be a Noetherian affine scheme. Let $S'\to S$ be a flat surjective morphism of affine schemes. Let $X\to S$ be a morphism such that $X\_{S'}\to S'$ is projective. Is $X\to S$ projective? It is not difficult to see that this holds when $S'\to S$ is a field extension.
The analogous statement for proper morphisms... | https://mathoverflow.net/users/nan | Projective after fpqc base change | Hironaka's example is a locally projective birational morphism $f \colon \tilde X \to X = \mathbf P^3$ where $\tilde X$ is not projective; in particular $f$ is not projective. This already shows that the property of being projective is not Zariski-local on the target, so in particular not fppf-local.
But you want the... | 4 | https://mathoverflow.net/users/82179 | 348162 | 147,387 |
https://mathoverflow.net/questions/348022 | 8 | Given is an integer $n\ge 2$ and two rows of $n$ positive real numbers each, with each number not more than $n-0.5$, such that the numbers in each row sum to $n$. Is it always possible to choose some numbers such that from each row, at least one but not all numbers are chosen, and the sum of the chosen numbers is not l... | https://mathoverflow.net/users/83212 | Two rows of bounded numbers | The answer is Yes, and it follows from the following claim:
>
> **Claim.** There exist two non-empty subsets $A\subsetneq \{a\_1,\dots a\_n\}$ and $B\subsetneq \{b\_1,\dots b\_n\}$ such that $|sum(A)-sum(B)|\leq 0.5$.
>
>
>
Indeed, replacing the set $A$ with $B$ in the sum $a\_1+\dots+a\_n$ results in the sum... | 6 | https://mathoverflow.net/users/7076 | 348166 | 147,389 |
https://mathoverflow.net/questions/348163 | 10 | A set $S \subseteq \mathbb{Z}/p\mathbb{Z}$ is called a Sidon set if given $a, b, c, d \in S$ and $a+ b = c+ d$, then $\{a, b\} = \{c,d\}$. I was interested in knowing about the largest possible Sidon set $\mathbb{Z}/p\mathbb{Z}$ has (Here $p$ is prime). What are known about the largest possible Sidon sets contained in ... | https://mathoverflow.net/users/84272 | Sidon sets of $\mathbb{Z}/p\mathbb{Z}$ | I am not sure of your level of knowledge of Sidon sets, but a good reference is O' Bryant's [survey](https://arxiv.org/pdf/math/0407117.pdf) (see section 4.3 in particular) which contains several constructions in the integers. If you are already aware of the three constructions then as far as I know, we do not know muc... | 10 | https://mathoverflow.net/users/50426 | 348172 | 147,392 |
https://mathoverflow.net/questions/343897 | 5 | I have just learned that a general framework in constrained optimization is called "proximal gradient optimization". It is [interesting](https://archive.siam.org/books/mo25/mo25_ch6.pdf) that the $\ell\_0$ "norm" is also associated with a proximal operator. Hence, one can apply iterative hard thresholding algorithm to ... | https://mathoverflow.net/users/120253 | If $\ell_0$ regularization can be done via the proximal operator, why are people still using LASSO? | The application of proximal gradient to this exact problem is considered in (equations (2) and (35) in) the Uncertainty in Artificial Intelligence (UAI) 2019 paper [Fast Proximal Gradient Descent for A Class of Non-convex and Non-smooth Sparse Learning Problems, Yingzhen Yang and Jiahui Yu](http://auai.org/uai2019/proc... | 5 | https://mathoverflow.net/users/75420 | 348178 | 147,395 |
https://mathoverflow.net/questions/348167 | 3 | Let $b\_1,b\_2,\dots$ be positive real numbers such that
$$s\_1<\infty\quad\text{and}\quad z\_1<\infty,
$$
where
$$s\_k:=\sum\_{j=k}^\infty b\_j\quad\text{and}\quad z\_k:=\sum\_{j=k}^\infty\frac{b\_j}{\sqrt{s\_j}}
$$
for natural $k$. Does it then necessarily follow that
$$\limsup\_{k\to\infty}\frac{z\_k}{\sqrt{s\_... | https://mathoverflow.net/users/36721 | Comparing the tails of two related convergent series | In fact $z\_k$ (or better any partial sum for it) is a lower Riemann sum for the function ${1\over \sqrt{x}}$ wrto the infinitesimal decreasing sequence $s\_k>s\_{k+1}>\dots s\_{m} $, so $z\_k< \int\_0^{s\_k}{1\over \sqrt{x}}dx=2\sqrt{s\_k}$, and the constant $2$ is sharp.
| 2 | https://mathoverflow.net/users/6101 | 348189 | 147,402 |
https://mathoverflow.net/questions/348226 | 1 | I'm looking for functions $g: \mathbb{R}\_+ \to \mathbb{R}$ such that the hitting time
$$\tau := \inf \{t \geq 0 : B\_t \nleq g(t) \} $$
has an explicit density with respect to the Lebesgue measure, where $B\_t$ is a standard Brownian motion. What are possible examples?
The case where, for some $R> 0$, $g(t) \equ... | https://mathoverflow.net/users/141749 | Explicit densities for Brownian motion hitting times | See e.g. [ANALYTIC CROSSING PROBABILITIES FOR CERTAIN
BARRIERS BY BROWNIAN MOTION](https://arxiv.org/pdf/0704.2826.pdf), especially Example 2 (page 9) there for a square-root barrier, Example 3 (page 9) there for a square-root-times-log-factor barrier, and further references therein.
| 2 | https://mathoverflow.net/users/36721 | 348232 | 147,415 |
https://mathoverflow.net/questions/348067 | 13 | Let $\mathcal{C}$ be an ordinary 1-category and suppose that there exists some object $X \in \mathcal{C}$ such that the following conditions are satisfied,
(1) For every $C \in \mathcal{C}$ we have $\operatorname{Hom}(X,C)\neq \emptyset$ , $\operatorname{Hom}(C,X)\neq \emptyset$.
(2) $\operatorname{Hom}(X,X)=\*$.
... | https://mathoverflow.net/users/141150 | About contractibility of certain categories | Let $\mathcal{C}$ be the category with two objects $X$ and $Y$, generated by morphisms $\alpha\_1,\alpha\_2:X\to Y$ and $\beta\_1,\beta\_2:Y\to X$ subject to relations $\beta\_i\alpha\_j=\text{id}\_X$ for all $i,j$.
So the only non-identity morphisms other than $\alpha\_1,\alpha\_2,\beta\_1,\beta\_2$ are
the composit... | 10 | https://mathoverflow.net/users/22989 | 348242 | 147,418 |
https://mathoverflow.net/questions/348229 | 14 | Let $M$ be a matroid with set of basis $\mathcal{B}$. The *basis graph* of $M$ is a graph with set of vertices $\mathcal{B}$ and edges $(B,B')$ always that $B$ and $B'$ differ (as sets) by exactly one element.
It is easy to see that this graph can be thought as the $1$-skeleton of the basis polytope of $M$.
My que... | https://mathoverflow.net/users/147861 | Does the basis graph of a matroid determine it? | I think this question is answered in the paper [A Graphical Representation of Matroids](https://doi.org/10.1137/0125060) by C. A. Holzmann, P. G. Norton, and M. D. Tobey. From the abstract:
"A base graph of a matroid is the graph whose points are the bases of the matroid. Two bases are adjacent if they differ by exac... | 12 | https://mathoverflow.net/users/51668 | 348245 | 147,421 |
https://mathoverflow.net/questions/347548 | 11 | A real matrix $M$ is called totally positive if all of its minors are positive; these matrices have been extensively studied, and there are generalizations to other Lie types, for example by Lusztig.
I am interested in the following $q$-analog: a matrix $M$ with entries in $\mathbb{R}[q]$ is $q$-totally positive if a... | https://mathoverflow.net/users/33089 | $q$-analogs of total positivity | This is a wonderful question but unfortunately I don't think that there is a definite answer in the literature just yet. Let's look at two somewhat recent lines of research in this direction:
1) A. Sokal outlines a project of understanding what he calls coefficientwise total positivity in [this research proposal](htt... | 8 | https://mathoverflow.net/users/2384 | 348248 | 147,422 |
https://mathoverflow.net/questions/348247 | 4 | Consider $\nabla$ a connection in a vector bundle above a smooth manifold $M$.Consider a local frame $\sigma=(\sigma\_1, \sigma\_2,...,\sigma\_m )$ on a contractible open set $U\subset M$ and calculate the curvature matrix $\Omega$ with respect to this frame.Take $\theta\_k$ a frame of $2$ forms in $TU.$ Look at the ma... | https://mathoverflow.net/users/70498 | A consequence of Ambrose-Singer theorem on holonomy | Your first question is a bit ambiguous. Are you asking whether, for *each* $p\in U$, the matrices $S\_k(p)$ span the Lie algebra of $\mathrm{Hol}^0\_p(\nabla)$ or are you asking whether, after taking the span of the union of the images of $S\_k(p)$ over all $p\in U$, the result is the Lie algebra of $\mathrm{Hol}^0\_p(... | 9 | https://mathoverflow.net/users/13972 | 348259 | 147,426 |
https://mathoverflow.net/questions/347869 | 0 | I have two matrices,
$$A= \left[ \begin{matrix} 5 & 10 & 15 & \cdots \\ 17 & 28 & 39 & \cdots \\ 35 & 52 & 69 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{matrix} \right]$$
$$B= \left[ \begin{matrix} 6 & 13 & 20 & \cdots \\ 18 & 31 & 44 & \cdots \\ 36 & 55 & 74 & \cdots \\ \vdots & \vdots & \vdots & \ddots \... | https://mathoverflow.net/users/49311 | Sum elements in two dimensional arithmetic progression | It seems highly likely that the answer is yes and also quite possible that that it is difficult or even practically impossible to prove.
One could consider the question: can **every** $n \gt 1$ be written as a sum of two numbers missing from $A$ and $B?$ It isn't much different because every $n \gt 38$ can be written... | 3 | https://mathoverflow.net/users/8008 | 348262 | 147,427 |
https://mathoverflow.net/questions/348227 | 5 | Define the Steenrod algebra $A^\ast$ to be the algebra of all stable mod $p$ cohomology operations. Without actually computing $A^\ast$, is it possible to see that $A^\ast$ acts faithfully on $H^\ast(BC\_p; \mathbb F\_p)$?
My question is closely related to [this one](https://mathoverflow.net/questions/83096/is-there-... | https://mathoverflow.net/users/2362 | Why does the Steenrod algebra act faithfully on $H^\ast(BC_p)$? | As is commented by @Connor Malin, the action of the Steenrod algebra on $H^\*B\mathbb{Z}/p$ is not faithful. Consider the case $p=2$. $Sq^3Sq^1$ acts trivially on $H^\*(B \mathbb{Z}/2)$, since $Sq^{2n+1}x^{2m}=0$ by the Cartan formula. As a matter of fact, the computation of the Hopf ring structure of $H\_\*K(\mathbb{Z... | 11 | https://mathoverflow.net/users/43326 | 348264 | 147,428 |
https://mathoverflow.net/questions/348228 | 10 | Here are the exact definitions of the terms:
Let $G$ be a topological group.
Then $G$ has the small index property if every subgroup of countable (including finite) index is open in $G$. Furthermore, $G$ has automatic continuity if every group homomorphism from $G$ to any separable topological group is continuous.
... | https://mathoverflow.net/users/55865 | Is there a topological group with the small index property that does not have automatic continuity? | Here is a counterexample. Consider $\mathbb{R}$ with addition. We define a topology on this group by giving the cosets of all countable index subgroups as a sub-basis.
This subbasis is actually a basis as the intersection of finitely many subgroups of countable index is another subgroup of countable index. Moreover a... | 11 | https://mathoverflow.net/users/149865 | 348271 | 147,431 |
https://mathoverflow.net/questions/339378 | 7 | Let $E$ and $F$ be Banach spaces, and let $\mathfrak L\_{co}(E,F)$ denote the space of bounded linear operators $E \to F$ equipped with the topology of uniform convergence on the absolutely convex compact subsets of $E$.
Defant and Floret [DF93, §5.5] point out that there is a natural linear map
$$ D\_F : F' \mathop... | https://mathoverflow.net/users/120251 | Do the operators in $B(E,F)$ separate points on the projective tensor product $F' \mathop{\tilde\otimes_\pi} E$? | This question was settled in 2012 by Petr Hájek and Richard J. Smith [HS12]. They prove the following beautiful result (reformulated here to match the notation from the question).
>
> **Theorem** (cf. [HS12, Theorem 2.5])**.** Let $F$ be a Banach space with the AP. Then the following conditions are equivalent:
>
>... | 2 | https://mathoverflow.net/users/120251 | 348301 | 147,440 |
https://mathoverflow.net/questions/335464 | 0 | ***Note:** Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a [(very elementary) research paper from 2018](http://www.gioenia.unict.it/bollettino/bollettino2018/Full_Papers/Arithmetic%20Rings%... | https://mathoverflow.net/users/142698 | Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)? | The first identity holds in an arithmetical ring since the identity holds iff it holds locally and an arithmetical ring is locally a chained ring and its easy to see that the identity holds in a chained ring.
| 0 | https://mathoverflow.net/users/106601 | 348316 | 147,446 |
https://mathoverflow.net/questions/348317 | 0 | There are many algorithms to estimate the density of probability distributions. I am looking for one that is not sensitive to the initial condition. For instance, Expectation–maximization algorithm starts by initializing the parameters theta to some random values and then it goes from there to other steps. So for diffe... | https://mathoverflow.net/users/103418 | a probability density algorithm that is not sensitive to the initial condition | The result obtained by a local numerical optimization solver applied to a non-convex optimization problem is sensitive to the starting value used by the optimizer. Expectation-maximization as applied to find a local optimum of a non-convex maximum likelihood estimation problem is one such example of a local optimizatio... | 2 | https://mathoverflow.net/users/75420 | 348319 | 147,447 |
https://mathoverflow.net/questions/348272 | 6 | Let $A$ be a commutative noetherian ring of finite Krull dimension.
Is the set
$W = \{p \in Spec(A) \mid A\_p \mbox{ is equidimensional}\}$
a dense open subset of $Spec(A)$?
(I guess dense is trivial, because it contains all minimal primes).
And if yes, can I conclude that this $W$ is covered by open sets $D(f)$... | https://mathoverflow.net/users/3759 | Equidimensional locus of a noetherian ring | This is true if $A$ is catenary. We will use that a catenary Noetherian *local* ring $A$ has a dimension function $d\_A \colon \operatorname{Spec} A \to \mathbf Z$ [Stacks, Tags [02I8](https://stacks.math.columbia.edu/tag/02I8) and [0ECF](https://stacks.math.columbia.edu/tag/0ECF)]; such a function is well-defined up t... | 1 | https://mathoverflow.net/users/82179 | 348328 | 147,449 |
https://mathoverflow.net/questions/325279 | 5 | Erdos and Selfridge open their paper ["The Product of Consecutive Integers is Never a Power"](https://projecteuclid.org/download/pdf_1/euclid.ijm/1256050816) (1974) with the theorem
>
> $\text{Theorem 1:}$ *The product of two or more consecutive positive integers is never a power.*
>
>
>
**Question:** While I ... | https://mathoverflow.net/users/123309 | Extension of Erdos-Selfridge Theorem | It's not hard to find number fields where there are examples. E.g., the equation $(y-3)(y-2)(y-1)=y^2$, equivalently $y^3-7y^2+11y-6=0$, has a root $\alpha$, an algebraic integer, with $5<\alpha<6$, so if we let $x=\alpha-4$, then $x$ is a positive algebraic integer with $(x+1)(x+2)(x+3)=(x+4)^2$.
| 3 | https://mathoverflow.net/users/3684 | 348331 | 147,451 |
https://mathoverflow.net/questions/348310 | 2 | Let $L\subset L'\in S^3$ be two links such that $L$ has one less number of components than $L'$. Further, $L$ is hyperbolic. Under what conditions is the link $L'$ hyperbolic. To be more specific $L, L'$ are shown in [here](https://i.stack.imgur.com/dbTGW.png).
| https://mathoverflow.net/users/149888 | Hyperbolic links | There is not enough information in your picture to give a definitive answer. If you have more details about the link $L$ then an answer will probably factor through Thurston's characterisation of hyperbolic links. [Purcell's book](http://users.monash.edu/~jpurcell/hypknottheory.html) is an introduction to the subject, ... | 3 | https://mathoverflow.net/users/1650 | 348336 | 147,453 |
https://mathoverflow.net/questions/348335 | 0 | I am trying to understand a paper on options titled "OPTION PRICING: A SIMPLIFIED APPROACH" [1]. In it option price is calculated as the expected payoff from the possible states of stock prices by binomial distribution approach.
I am stuck at one step.
What does the following sentences exactly mean?
>
> Now, ... | https://mathoverflow.net/users/149902 | What is a complementary binomial distribution function $\Phi[a; n,p]$? | The complementary binomial distribution is defined by
$$\Phi(a;n,p)=\sum \_{j=a}^n \binom{n}{j} p^j(1-p)^{n-j},\;\;0<p<1,\;\;0\leq a\leq n.$$
see [The derivation of diffusion-jump modes for power plant projects under risk](http://www.worldresearchlibrary.org/up_proc/pdf/1124-15114160424-7.pdf). A more correct terminolo... | 2 | https://mathoverflow.net/users/11260 | 348341 | 147,456 |
https://mathoverflow.net/questions/348339 | 1 | The comments in the post
[Almost sure identity](https://mathoverflow.net/questions/348332/almost-sure-identity?noredirect=1#comment872443_348332)
has led me to the following question:
Suppose that $(\Omega,\mathcal{F},P)$ is a probability space.
Denote by $\mathcal{B}(\mathbb R)$ the Borel sigma-algebra on $\ma... | https://mathoverflow.net/users/70540 | Measurable selection | Yes! This actually follows from standard measure theory.
Let $\mathcal{R}$ denote the collection of rectangles $U\times V$ where $U\in\mathcal{F}$ and $V\in\mathcal{B}(\mathbb{R})$. The finite disjoint unions of elements of $\mathcal{R}$ form an algebra $\mathcal{A}$ on $\Omega\times\mathbb{R}$ which in turn generate... | 4 | https://mathoverflow.net/users/23297 | 348342 | 147,457 |
https://mathoverflow.net/questions/348270 | 3 | Assume that we have the following equation to solve
$$\sum\_{\ell=1}^L A\_\ell X\_{\ell} B\_{\ell} =0$$ over complex matrices
where each $A\_{\ell}$ is a given $m\times n$ matrix, each $B\_{\ell}$ is a given $n\times t$ matrix, and we are solving for the $n\times n$ matrices $X\_\ell$. $\{A\_{\ell}\}$ satisfy
$$\sum\_... | https://mathoverflow.net/users/124751 | Conditions for a certain matrix equation to have a full rank solution | We ignore the supplementary condition $\sum\_lA\_lA\_l^\*=I$. Adding conditions 24 hours after the first post is making fun of the world.
We assume that $n,m,t$ are fixed positive inytegers s.t.$m,t\geq n,Ln^2>mt$ and that the $(A\_l[i,j]),(B\_l[i,j])$ are $d=Ln(m+t)$ complex parameters (they are elements of a transc... | 1 | https://mathoverflow.net/users/9091 | 348344 | 147,459 |
https://mathoverflow.net/questions/348323 | 2 | Given a function, $c(x,y):\mathbb{R}\times \mathbb{R}\to \mathbb{R}$, what are sufficient conditions for this to be the covariance of some (centered) Gaussian random field $X:\mathbb{R}\to \mathbb{R}$,
$$
c(x,y) = \mathbb{E}[X(x)X(y)]
$$
Obviously, we would like $c$ to be symmetric, $c(x,y) =c(y,x)$, but what is also n... | https://mathoverflow.net/users/51335 | Sufficient conditions to be a covariance | Polya's criterion says that if $f:\mathbb{R}\to \mathbb{R}$ is even, convex on $[0,\infty)$, with $f(0)>0$ and zero limit at infinity, then $c(x,y) = f(\vert x-y\vert)$ is a positive definite kernel, hence the existence of the Gaussian random field. It would apply to your function for $p$ less than $1$ for example (to ... | 5 | https://mathoverflow.net/users/112954 | 348351 | 147,462 |
https://mathoverflow.net/questions/346844 | 4 | Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let $\textrm{mod}\_\mathcal{H}$ be the monoidal abelian category of finite-dimensional modules over $\mathcal{H}$. Fix $X\in\textrm{Obj}(\textrm{mod}\_\mathcal{H})$. We know that the functor
$$(-\otimes X):\textrm{mod}\_\mathcal{H}\rightarrow\textrm{mod}\_\mathcal{... | https://mathoverflow.net/users/137269 | Subfunctor of internal Hom | I am not an expert, but I think that this result is an instance of a more general phaenomenon. If this does not apply to your situation, feel free to ignore my answer or to downvote it.
It looks to me that your framework is a special case of the one described and developed in the following paper.
>
> **Hopf monad... | 1 | https://mathoverflow.net/users/104432 | 348353 | 147,463 |
https://mathoverflow.net/questions/348346 | -7 | Wanting to learn a bit about Ramsey's theory, I read the corresponding article on Wikipedia and stumbled upon this:
"Le théorème de van der Waerden[2] : pour tous entiers c et n, il existe un entier[3] W tel que si l'ensemble {1, 2, … , W} est coloré avec c couleurs, il contient une progression arithmétique monochrom... | https://mathoverflow.net/users/13625 | Is Green-Tao's theorem a consequence of Van der Waerden theorem? | The primes are a density zero subset of the natural numbers. Therefore, van der Waerden's theorem or its great strengthening, Szemerédi's theorem is not directly applicable. In fact the idea of Green and Tao is to "make Szemerédi's theorem work for primes" by proving a "transference theorem" from the natural numbers to... | 13 | https://mathoverflow.net/users/11919 | 348361 | 147,465 |
https://mathoverflow.net/questions/348327 | 8 | Let $T^2$ be a compact smooth surface and let $p\in T^2$. Suppose that $T^2$ admits a symmetric $\left(0,2\right)$-tensor which is a flat Riemannian metric restricted to $T^2-\{p\}$. Is it true that $\chi(T)=0$?
| https://mathoverflow.net/users/117091 | Flat metric on compact surface minus a point | Here are explicit examples when $M$ is compact, connected, and $\chi(M)\le0$.
*Orientable case:* Let $M$ be the 1-point compactification of the hyperelliptic Riemann surface defined in the affine plane $\mathbb{C}^2$ by
$$
y^2 = x^{2g+1}-1.
$$
This is a smooth Riemann surface of genus $g\ge1$ and hence $\chi(M) = 2-2... | 18 | https://mathoverflow.net/users/13972 | 348364 | 147,466 |
https://mathoverflow.net/questions/348241 | 10 | In the paper by Bhatt and Scholze on prismatic cohomology (<https://arxiv.org/pdf/1905.08229.pdf>), it is stated that the de Rham comparison theorem for prismatic cohomology can be lifted to an equivalence of cdga's. This confuses me, because I don't see why the de Rham complex of a smooth and proper scheme over a fiel... | https://mathoverflow.net/users/115052 | Is the de Rham complex in characteristic $p$ a CDGA? | For a smooth and proper scheme $X$ over a field $k$ of characteristic $p$, the $E\_\infty$-algebra $R\Gamma(X, DR\_{X/k})$ over $k$ is not represented by a $k$-cdga. Indeed, if it were, then most of the Steenrod operations on its cohomology groups would have to vanish, which is simply not true. You get a counterexample... | 8 | https://mathoverflow.net/users/149917 | 348367 | 147,467 |
https://mathoverflow.net/questions/347970 | 3 | There are general algorithms for finding all of the integer points on an elliptic curve given in Weietstrass form, but this special case is likely much easier. There’s a simple point of order 2, and the polynomial on the RHS generates a quadratic rather than a cubic number field (in fact it’s an imaginary quadratic fie... | https://mathoverflow.net/users/25 | Integer points on $y^2=x^3+Dx$, $D\in\mathbb{N}$ squarefree | I think that you're right that the problem is *easier* than the general case, but still it's not *easy*. I don't think there was an effective method known before Baker's work, and even now, you'll need to use linear forms in logs of some sort to get an effective upper bound of the form $|x|\le C(D)$. Even for $D$ prime... | 6 | https://mathoverflow.net/users/11926 | 348381 | 147,472 |
https://mathoverflow.net/questions/348371 | 8 | Let $U$ be a $d\times d$ unitary matrix, and $U\_{i,j}$ be its matrix elements. I am interested in the following quantity
$$\int dU \max\_j |U\_{1,j}|^2 \ , $$
where $dU$ is the uniform Haar measure over ${\rm SU}(d)$.
Please let me know if you have any idea for calculating this integral for general $d$.
| https://mathoverflow.net/users/149918 | Average of the maximum matrix element over the Haar measure | The answer to the question as stated (maximum of row elements) has been solved in [Extreme statistics of complex random and quantum chaotic states](https://arxiv.org/abs/0708.0176), see also this [MO posting](https://mathoverflow.net/a/298776/11260):
$$\int dU \max\_j |U\_{1,j}|^2 =\frac{1}{d}\sum\_{j=1}^d \frac{1}{... | 11 | https://mathoverflow.net/users/11260 | 348387 | 147,475 |
https://mathoverflow.net/questions/348386 | 4 | Let $A$ be a Banach algebra. Is there a Banach algebra $B$ which contains $A$ but the spectrum of each elements of $B$ has empty interior(as a subset of $\mathbb{C}$)?
The motivation comes from the fact that the spectrum of elements in a smaller algebra possibly loses its interior when we compute its spectrum in a la... | https://mathoverflow.net/users/36688 | Removing the interior of spectrums | The answer is *no*, in general. Here is a counterexample:
Let $A$ be the algebra of bounded linear operators on $\ell^2(\mathbb{N})$, and let $a \in A$ be the left shift on $\ell^2(\mathbb{N})$. Then the spectrum of $a$ is the closed unit disk $\overline{D}$, and the point spectrum of the operator $a$ is the open uni... | 9 | https://mathoverflow.net/users/102946 | 348392 | 147,477 |
https://mathoverflow.net/questions/348404 | -1 | Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense subset.
>
>
> >
> > Question: Suppose that the set of $f(x)$ for $x\in B$ is closed and convex set. Is the set of $f(x... | https://mathoverflow.net/users/127839 | Closed on generic set implies closed set whole set | Obviously $f(B) \subseteq f(A)$. And since $B$ is dense in $A$, we have $A \subseteq \overline{B}$, and so $f(A) \subseteq f(\overline{B})$. But by the continuity of $f$, we have $f(\overline{B}) \subseteq \overline{f(B)}$. Since $f(B)$ is closed by assumption, this shows that $f(A) \subseteq f(B)$. So in fact we have ... | 1 | https://mathoverflow.net/users/4832 | 348406 | 147,485 |
https://mathoverflow.net/questions/348408 | 5 | The question is basically the one outlined in the title. Let $\mathcal{T}$ be a triangulated category containing infinite direct sums (e.g. $D\_{qc}(X)$ for some separated, finite type over a field $k$, scheme $X$) and consider the subcategory $\mathcal{E}$ generated by an object $E$ of $\mathcal{T}$. Here by subcatego... | https://mathoverflow.net/users/91572 | Can homotopy colimits recover cohomology sheaves? | No.
Let $j:\mathbb{A}^2\_k\smallsetminus\{0\}\to \mathbb{A}^2\_k$ be the canonical open embedding. Then the derived pushforward $Rj\_\*$ is fully faithful and colimit-preserving. In particular, the subcategory of $D\_{qc}(\mathbb{A}^2)$ generated under colimits by $\mathscr{A}:=Rj\_\*\mathcal{O}$ is contained in thi... | 7 | https://mathoverflow.net/users/43054 | 348409 | 147,487 |
https://mathoverflow.net/questions/348369 | 6 | Fix a forcing notion $\mathbb{P}$. Say that a formula $\varphi(x)$ with parameters is *$\mathbb{P}$-enforceable* if there is some countable set $\mathcal{D}$ of dense sets in $\mathbb{P}$ such that for every $\mathcal{D}$-generic filter $G\subseteq\mathbb{P}$ there is some $p\in G$ such that for every $\mathcal{D}$-gen... | https://mathoverflow.net/users/8133 | When does "sufficient genericity" actually suffice? | Assume $\text{AD}^{L(\mathbb R)}$. Let $A$ be projective. Since $A$ is ${}^\infty$Borel in $L(\mathbb R)$, there is a set of ordinals $S$ and a formula $\psi$ such that $\omega\_1$ is strongly inaccessible in $L[S]$ and for all reals $x$, $x\in A$ if and only if $L[S,x]\vDash \psi(S,x).$ Let $\mathcal D$ be the set of ... | 7 | https://mathoverflow.net/users/102684 | 348415 | 147,490 |
https://mathoverflow.net/questions/347713 | 1 | Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi\_{\{x>0\}}(X(t,x)), &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
where $\chi$ denotes the [indicator function of a set](https://en.wikipedia.org/wiki/Indicator_function).
* What is the unique [regular Lagrangian flow](http:/... | https://mathoverflow.net/users/122620 | Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$ | The flow $X:[0,T]\times \mathbb{R}\to\mathbb{R}$ defined by $X(t,x)= x+t\chi\_{\mathbb{R}\_+}(x)$, for all $(t,x)\in [0,T]\times \mathbb{R}$, is a regular Lagrangian flow solution to $(\star)$ in the sense of ***Definition (4)*** of the linked paper (for a.e. initial datum $x\in \mathbb{R}$, in fact for all, one has $(... | 1 | https://mathoverflow.net/users/6101 | 348420 | 147,492 |
https://mathoverflow.net/questions/348413 | 4 | **EDIT:** The well known [Jordan curve theorem](https://en.wikipedia.org/wiki/Jordan_curve_theorem) says: let $C\subset S^2$ be a closed simple curve on the 2-sphere. Then its complement $S^2\backslash C$ consists of two connected components, both homeomorphic to discs (in fact it is known that the closure of each comp... | https://mathoverflow.net/users/16183 | Closed simple curves in $\mathbb{R}\mathbb{P}^2$ | If C is null-homologous, then the complement of C has two components: a disk and a Möbius strip (as one sees since the preimage of C in the 2-sphere is 2 disjoint Jordan curves).
If C is not null-homologous, then the complement of C is a single disk.
| 5 | https://mathoverflow.net/users/105095 | 348421 | 147,493 |
https://mathoverflow.net/questions/348349 | 12 | Let $G$ be a reductive algebraic group (over some alg. closed field $k$ of char 0), and $H$ a subgroup such that $(G, H)$ is *spherical* (i.e., the Borel $B$ of $G$ has an open orbit on $G/H$). Then $k[G/H]$ is multiplicity-free as a $G$-module. I'm looking for a nice description of the irreducible $G$-representations ... | https://mathoverflow.net/users/2481 | For a spherical pair $(G, H)$, which $G$-representations appear in $k[G/H]$? | Let $X\_0=B/B\_{x\_0}=Bx\_0$ be the open $B$-orbit in $X=G/H$. Then every character of $B$ which is trivial on $B\_{x\_0}$ yields a $B$-semiinvariant $f\_\lambda$ on $X\_0$. In general, it does not extend to a regular function on $X$. If it does, this means that $V\_\lambda$ occurs in $k[X]$. So your second problems as... | 6 | https://mathoverflow.net/users/89948 | 348424 | 147,495 |
https://mathoverflow.net/questions/348388 | 4 | I'm interested in whether the finitely-generated discrete Heisenberg group admits a notion of "convex set". Below a formalization of what I need from the convex sets, in particular they should all be finite, should be arbitrarily large, the family should be translation-invariant, closed under intersections and satisfy ... | https://mathoverflow.net/users/123634 | Convex sets on the discrete Heisenberg group | I claim that the answer is yes for the Heisenberg group, and more generally for every finitely generated torsion-free 2-step nilpotent group $\Gamma$.
Indeed, embed $\Gamma$ in its real Malcev closure $G$, and view $G=\mathfrak{g}$ as a Lie algebra, by identification through the exponential map. Recall that $G$ is un... | 2 | https://mathoverflow.net/users/14094 | 348428 | 147,498 |
https://mathoverflow.net/questions/348389 | 6 | Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of [comonoids](https://ncatlab.org/nlab/show/comonoid) of monoids?
This category has (at least) two different descriptions, namely it is also th... | https://mathoverflow.net/users/98306 | Comonoids in the category of monoids | George Bergman characterizes representable endofunctors of monoids in 10.6 of <https://math.berkeley.edu/~gbergman/245/3.2.pdf> by classifying the comonoids in the category of monoids.
| 4 | https://mathoverflow.net/users/15934 | 348429 | 147,499 |
https://mathoverflow.net/questions/348418 | 11 | Let $\mathcal C$ be a category. Say that a class of objects $\mathcal S \subseteq \mathcal C$ is *weakly cogenerating* if the functors $Hom\_{\mathcal C}(-,S)$ are jointly conservative, for $S \in \mathcal S$. That is, a map $X \to Y$ in $\mathcal C$ is an isomorphism if and only if it induces bijections $Hom\_C(Y,S) \... | https://mathoverflow.net/users/2362 | Do spaces admit a weak cogenerating set? | For any infinite set $X$ let $S\_X$ be the group of bijections $\sigma \colon X\to X$ such that $\{x : \sigma(x)\neq x\}$ is finite. This still has signature homomorphism, and the alternating subgroup $A\_X$ is simple, and has the same cardinality as $X$. Now let $\mathcal{G}$ be a set of groups, and put $\kappa = \max... | 6 | https://mathoverflow.net/users/10366 | 348433 | 147,502 |
https://mathoverflow.net/questions/348400 | 2 | A fusion ring $\mathcal{F}$ is given by a finite set $B = \{b\_1,b\_2, \dots, b\_r \}$ such that $b\_i b\_j = \sum\_k n\_{i,j}^k b\_k$ with $n\_{i,j}^k \in \mathbb{Z}\_{\ge 0}$, satisfying axioms slightly augmenting the group axioms (see the details [here](https://mathoverflow.net/q/344079/34538)).
The fusion ring $... | https://mathoverflow.net/users/34538 | Is there a noncommutative simple fusion ring? | The fusion ring of one of the even parts of Extended Haagerup will work (see [Appendix A of our paper](https://arxiv.org/abs/0909.4099)). It has rank 8, and you can see that $AB \neq BA$. Surely there's less complicated examples though.
| 2 | https://mathoverflow.net/users/22 | 348434 | 147,503 |
https://mathoverflow.net/questions/348257 | 2 | Let $\mathscr C$ be a category so that every morphism is '*invertible*' only up to equivalence and so that it makes sense to say two morphisms are '*homotopic to each other*'. Probably this is called $(2,1)$-category.
(The category in my mind could be such that objects are topological spaces and morphisms are homotop... | https://mathoverflow.net/users/69190 | Transitive closure in category setting | If every morphism is a equivalence, then you really want a $(2,0)$-category, also known as a 2-groupoid. However, I see no reason to assume every morphism is an equivalence here.
This is not a generalization or a categorification of a transitive closure, in any case: it is literally the transitive closure of the rel... | 2 | https://mathoverflow.net/users/43000 | 348435 | 147,504 |
https://mathoverflow.net/questions/348440 | 9 | I'm looking for a reference for a description of the outer automorphism groups of $\operatorname{SL}(2,\mathbb{F}\_q)$ for $q = p^n$.
I'm sure such a thing must exist somewhere, but I'm having trouble locating a reference.
| https://mathoverflow.net/users/88840 | What is the outer automorphism group of $\operatorname{SL}(2,\mathbb{F}_q)$? | "Each automorphism $\sigma$ of $G$ can be written $\sigma = g f d i$, with $i$, $d$, $f$, and $g$ being inner, diagonal, field, and graph automorphisms, respectively" (Steinberg - [Automorphisms of finite linear groups](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/automorphisms-of-fin... | 5 | https://mathoverflow.net/users/2383 | 348441 | 147,507 |
https://mathoverflow.net/questions/348448 | 54 | The Prime Number Theorem relates primes to the important constant $e$.
Here I report my following surprising discovery which relates primes to $\pi$.
**Conjecture** (December 15, 2019). Let $s(n)$ be the sum of all primes $p\le n$ with $p\equiv1\pmod4$, and let $s\_\*(n)$ be the sum of those $x\_py\_p$ with $p\le n... | https://mathoverflow.net/users/124654 | A mysterious connection between primes and $\pi$ | Here is a proof of the conjecture. We shall use Hecke's theorem that the angles of the lattice points $(x\_p,y\_p)$ are asymptotically equidistributed in $[\pi/4,\pi/2]$, cf. [this MO post](https://mathoverflow.net/questions/133410/hecke-equidistribution/133447).
Let $t\_p\in[\pi/4,\pi/2]$ be the angle of the lattice... | 76 | https://mathoverflow.net/users/11919 | 348452 | 147,510 |
https://mathoverflow.net/questions/348200 | 19 | Let $X$ be a positive-dimensional, smooth, connected projective variety (say over $\Bbb{C}$), and let $\sigma $ be an involution of $X$ with a finite number of fixed points; then this number is even. The proof I have is somewhat artificial: blow up the fixed points, take the quotient variety, observe that the image of ... | https://mathoverflow.net/users/40297 | Number of fixed points of an involution | Indeed, the number of fixed points is divisible by $2^{\dim X}$. This is actually an old result of Conner and Floyd, *Periodic maps which preserve a complex structure*, Bull. Amer. Math. Soc. 70, no. 4 (1964), 574-579. As @SashaP mentions, Atiyah and Bott observed that it is also a consequence of the holomorphic Lefsch... | 9 | https://mathoverflow.net/users/40297 | 348459 | 147,513 |
https://mathoverflow.net/questions/348279 | 3 | Consider a binary vector $a\_0, a\_1,\,\dots\,, a\_n$ and an equation
$$\sum\_{i=0}^n a\_i \cdot (-1)^i {n \choose i} = 0.$$
You can satisfy this trivially when
1) all $a\_i$ are 0, or
2) all $a\_i$ are 1, or
3) $n$ is odd and $a$ satisfies $a\_i = a\_{n-i}$ for all $i$, because the summands for $i$ and $n-i$... | https://mathoverflow.net/users/149563 | Non-trivial alternating sums of binomial coefficients | (Reposting my comment as an answer, as requested by the OP.)
If you have a solution with $a\_i \in \{−1,1\}$, then you also have a solution with $a\_i \in \{0,1\}$ simply by replacing each $a\_i$ with $(a\_i+1)/2$ (using the fact that setting all $a\_i=1$ is also a valid solution). In other words, your question had a... | 1 | https://mathoverflow.net/users/12858 | 348462 | 147,515 |
https://mathoverflow.net/questions/348362 | 4 | I'm looking for readable references on calculating the growth rates of surface groups.
There's some approach done briefly in page 159 of de la Harpe's "Topics in Geometric Group Theory", who cites:
>
> Cannon 1980, The growth of the closed surface groups and the compact hyperbolic Coxeter groups, Cannon
>
>
>
... | https://mathoverflow.net/users/149916 | Growth rates of surface groups | Just to make the method as concrete as possible, I'll compute the growth rate for the fundamental group $G$ of a surface of genus two. The Cayley graph of $G$ is the 1-skeleton of a tiling of the hyperbolic plane by regular octagons, with eight octagons meeting at each vertex.
Given an element $g\in G$, let $|g|$ den... | 9 | https://mathoverflow.net/users/6514 | 348473 | 147,519 |
https://mathoverflow.net/questions/348313 | 10 | Let $R$ be a subring of $\Bbb C$ closed under complex conjugation and let $P$ be an $n\times n$ positive semidefinite matrix with entries in $R$. I'm curious if it is always possible to factor $P$ as $P=A^\ast A$ where $A$ is $m\times n$ with entries in $R$.
If $P$ is $1\times 1$ and $R=\Bbb Z$, then Lagrange's four... | https://mathoverflow.net/users/33377 | Do matrices of the form $A^\ast A$ where $A$ has entries in $R\subset\Bbb C$ account for all positive semidefinite matrices with entries in $R$? | $\def\ZZ{\mathbb{Z}}\def\QQ{\mathbb{Q}}\def\RR{\mathbb{R}}\def\CC{\mathbb{C}}$I did some searching and thinking, and this is what I have come up with. Disclaimer: I couldn't actually find full copies of the papers I'm going to cite by Mordell and Ko, so this is based on reading summaries of them in other papers.
--... | 5 | https://mathoverflow.net/users/297 | 348485 | 147,522 |
https://mathoverflow.net/questions/341570 | 12 | A [new preprint by Terry Tao](https://arxiv.org/abs/1909.03562) has recently appeared and has established some interesting results regarding the topic of Collatz conjecture. I will not cite the precise result, but rather an equivalent formulation which Tao notes in his Remark 1.4:
>
> For any $\delta>0$ there exist... | https://mathoverflow.net/users/30186 | Explicit bounds from Tao's result on Collatz conjecture | In v2 of the arXiv preprint, there is more detail, including fairly explicit dependence on $\delta$.
| 2 | https://mathoverflow.net/users/806 | 348495 | 147,527 |
https://mathoverflow.net/questions/348496 | 10 | That is, is there a Kähler manifold $X$ on which there is no map
$$
\tau:X\to X
$$
such that
$$
d\tau\circ I=-I\circ d\tau
$$
and
$$
\tau\circ \tau=\mathrm{Id}\_X?
$$
| https://mathoverflow.net/users/123207 | Is there a Kähler manifold with no anti-holomophic involution? | The moduli space $\mathcal{M}\_1^\mathbb{R}$ of real algebraic curves of genus $1$ equals the real part of the moduli space $\mathcal{M}\_1=\mathbb{C}$. In particular, the general elliptic curve has *no* anti-holomorphic involution, as suggested by P. Achinger in his comment.
Furthermore, similar statements hold also... | 11 | https://mathoverflow.net/users/7460 | 348506 | 147,531 |
https://mathoverflow.net/questions/348507 | 18 | ([This question](https://math.stackexchange.com/questions/3258252) is originally from Math.SE where it was suggested that I ask the question here)
Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning that $B\_r(x)\subseteq S$ for some $r>0$.
Is it necessarily the cas... | https://mathoverflow.net/users/95685 | Fundamental group of punctured simply connected subset of $\mathbb{R}^2$ | Here is a proof that $\pi\_1(S\setminus\{x\})\cong\mathbb Z$. It is along similar lines to Jeremy Brazas' answer, and also makes use of Lemma 13 from the paper [*The fundamental groups of subsets of closed surfaces inject into their first shape groups*](https://arxiv.org/abs/math/0512343v1https://).
>
> **Lemma 13*... | 6 | https://mathoverflow.net/users/1004 | 348528 | 147,538 |
https://mathoverflow.net/questions/348208 | 2 | By definition, a group is FC if all its conjugacy classes are finite.
Has anything been published about a generalization of the FC property for topological groups?
| https://mathoverflow.net/users/128442 | Topological analogue of an FC group? | In 1963, Usakov characterised topological FC groups:
<https://mathscinet.ams.org/mathscinet-getitem?mr=165031>
| 1 | https://mathoverflow.net/users/150006 | 348545 | 147,543 |
https://mathoverflow.net/questions/348480 | 6 | The [nLab](https://ncatlab.org/nlab/show/semicartesian+monoidal+category) states that a [semicartesian monoidal category](https://ncatlab.org/nlab/show/semicartesian+monoidal+category) equipped with natural transformations $\delta\_x : x \to x \otimes x$ such that $\pi\_1 \circ \delta\_x = 1\_x$ and $\pi\_2 \circ \delt... | https://mathoverflow.net/users/66017 | A semicartesian monoidal category with diagonals is cartesian: proof? | As pointed out by Dylan Wilson, the nLab postulates that $\delta$ be a monoidal natural transformation. This requires that the functor $Gx = x \otimes x$ be a (lax/oplax) monoidal functor. Let us assume that the category is symmetric monoidal, then $G$ is strong (in the sense of non-lax) monoidal. Then we have an isomo... | 4 | https://mathoverflow.net/users/66017 | 348548 | 147,544 |
https://mathoverflow.net/questions/348532 | 3 | Consider the following integral
$$
I\_\delta(\lambda)=\int\_0^\delta e^{i\lambda \exp(-x^{-2})}dx.
$$
Here, $\phi(x)=\exp(-x^{-2})$ is the phase function. I would like to study the rate of decay of $I(\lambda)$ as $\lambda\to \infty$.
In Stein's *Harmonic Analysis*, the case where the phase function has finite order ... | https://mathoverflow.net/users/111012 | Decay of oscillatory integral for non-analytic phase function | Let me fix $\delta=1$ for simplicity. Let us use a Van der Corput method. We have for $\epsilon\in (0,1)$ to be chosen later, with $\phi(x)= e^{-x^{-2}}, $ noting that $\phi'(x)=\phi(x) 2 x^{-3}$
$$
I(\lambda)=\underbrace{\int\_0^{\epsilon} e^{i\lambda \phi(x)} dx}\_{O(\epsilon)}+\underbrace{\int\_{\epsilon}^1
\frac{d}... | 3 | https://mathoverflow.net/users/21907 | 348552 | 147,545 |
https://mathoverflow.net/questions/346060 | 5 | The ordinal $\tau\_1$ corresponds to $\lambda^{\textit{it}}$ (the supremum of all ordinals writable by *iterated ITTMs*) — see Definition 3.1 in the paper [“ITTMs with Feedback” [Robert S. Lubarsky]](https://www.semanticscholar.org/paper/ITTMs-with-Feedback-Lubarsky/980d560b8f991113f5d132b4d161a78e712a067c). According t... | https://mathoverflow.net/users/122796 | Which ordinal is larger, the supremum of ordinals writable by iterated Infinite Time Turing Machines or the smallest $\Sigma_2^1$-reflecting ordinal? | The quickest answer is to say $\zeta^{it}$ is the smaller because it can be computed inside any transitive $ZF^-$ model; if $L\_\gamma$ is the least such model, then $\gamma < \sigma$ - the latter ordinal also being the smallest $\Sigma^1\_2$-reflecting ordinal. (Indeed it can be computed inside the least $L\_\tau$ whe... | 7 | https://mathoverflow.net/users/6942 | 348561 | 147,550 |
https://mathoverflow.net/questions/348567 | 1 | Let $A$ be a non-negatively graded algebra such that $A\_0 = k$. We say that $A$ is Koszul if $k$ has a projective resolution by projective modules such that the i-th piece is generated in degree $i$. Let us now replace the condition $A\_0 = k$ with $S: = A\_0$ being semi-simple. Then, $A$ can be considered as an augme... | https://mathoverflow.net/users/91572 | Augmented algebras over semisimple ring | This is the set-up of *Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang*, [**Koszul duality patterns in representation theory**](http://dx.doi.org/10.1090/S0894-0347-96-00192-0), J. Am. Math. Soc. 9, No. 2, 473-527 (1996). [ZBL0864.17006](https://zbmath.org/?q=an:0864.17006).
It has been cited 888 times acco... | 4 | https://mathoverflow.net/users/18756 | 348576 | 147,553 |
https://mathoverflow.net/questions/348547 | 5 | Disclaimer: this a [cross post from MSE](https://math.stackexchange.com/questions/3421905/is-the-wikipedia-depiction-of-the-ergosphere-of-a-kerr-black-hole-a-cassini-oval), where this question was asked on November 4th 2019 and has so far received no upvote, no comment and no answer whatsoever.
Glancing at <https://e... | https://mathoverflow.net/users/13625 | Is the Wikipedia depiction of the ergosphere of a Kerr black hole a Cassini oval? | It's not a Cassini oval.
To see this, recall that we're talking about the outer static limit of the black hole, whose (Boyer-Lindquist) $r$ coordinate in function of $\theta$ is given by $r = M + \sqrt{M^2 - a^2\,\cos^2\theta}$ (where, as usual, $M$ is the black hole mass and $a$ its angular momentum per unit mass). ... | 8 | https://mathoverflow.net/users/17064 | 348579 | 147,555 |
https://mathoverflow.net/questions/348564 | 9 | Let $K$ be a finite extension of $\mathbb{Q}\_p$ with absolute Galois group $G\_K$. Let $A$ be an abelian variety defined over $K$. The (geometric) Tamagawa number is defined as the order of the quotient $$c(A/K)=A(K)/A\_0(K)$$ where $A\_0(K)$ denotes the $K$-points of $A$ that are sent to the identity component of the... | https://mathoverflow.net/users/45982 | Tamagawa numbers | Denote by $\Phi$ the quotient of $\mathcal A^\vee$, the special fiber of the smooth (but not necessarily proper) model of the dual abelian variety $A^\vee$, by the connected component of $0$ of $\mathcal A^\vee$. Then it is indeed true that the *Tamagawa number* $c\_p$ as defined in *$L$-functions and Tamagawa number o... | 11 | https://mathoverflow.net/users/2284 | 348586 | 147,559 |
https://mathoverflow.net/questions/348599 | 18 | Let $M$ be a smooth closed manifold. Let $f\colon M\to M$ be a homeomorphism.
>
> Does there exist a sequence of diffeomorphisms $f\_i\colon M\to M$ which conveges to $f$ uniformly, i.e. in $C^0$-topology:
> $$\sup\_{x\in M}dist(f(x),f\_i(x))\to 0,\, \sup\_{x\in M}dist(f^{-1}(x),f^{-1}\_i(x))\to 0 \mbox{ as } i\to... | https://mathoverflow.net/users/16183 | Approximation of homeomorphism by diffeomorphism | No. The space of homeomorphisms of a compact manifold is locally contractible:
A. V. Černavskiı̆. Local contractibility of the group of homeomorphisms of a manifold. Mat.
Sb. (N.S.), 79 (121):307–356, 1969.
So if there were such a sequence then for large enough $i$ the diffeomorphism $f\_i$ would be topologically i... | 26 | https://mathoverflow.net/users/318 | 348603 | 147,564 |
https://mathoverflow.net/questions/348288 | 3 | I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x\_0$, where $A$ generates an **analytic** $C\_0$-semigroup on a Banach space $X$. I'm looking for some ... | https://mathoverflow.net/users/146543 | Inequality for initial data | I fear that the question is posed in too general terms. As stated, the answer is **No**, because of the following example.
Consider the Hilbert space $X=L^2(0,1)$ and the operator $S\_t\in{\cal L}(X)$ defined by
$$(S\_ta)(x)=\left\{\begin{array}{lr}
0, & x\in(0,t), \\ a(x-t), & x\in(t,1).
\end{array}\right.$$
This de... | 2 | https://mathoverflow.net/users/8799 | 348605 | 147,565 |
https://mathoverflow.net/questions/348625 | 2 | What is the largest area/volume of any $(d-1)$-dimensional flat object that fits into the $d$-dimensional unit hypercube? For instance, for $d=2$, the answer is $\sqrt 2$, as this is the length of the (1-dimensional) diagonal. But how does this quantity grow as a function of $d$?
| https://mathoverflow.net/users/41364 | Maximal area/volume of (d-1)-dimensional object in d-dimensional hypercube | The maximal area of a $d-1$ dimensional slice through a $d$-dimensional hypercube is $\sqrt 2$ in any dimension, this was proven by K.M. Ball, [Cube slicing in $\mathbb{R}^n$](https://www.jstor.org/stable/2046239) (1986).
| 4 | https://mathoverflow.net/users/11260 | 348630 | 147,573 |
https://mathoverflow.net/questions/348619 | 3 | Suppose $\Gamma\_1(V\_1, E\_1)$ and $\Gamma\_2(V\_2, E\_2)$ are simple graphs with countably many vertices. And suppose $A\_1$ and $A\_2$ are initially empty sets. Suppose two players play the following game: each turn, the first player choses either to add a vertex from $V\_1$ to $A\_1$ or a vertex from $V\_2$ to $A\_... | https://mathoverflow.net/users/110691 | Is following function a metric on the set of isomorphism classes of graphs with countably many vertices? | To prove that this is a metric, consider the following theorem.
**Theorem.** If the second player can survive for $n$ steps in the $(\Gamma\_1,\Gamma\_2)$ game, and for $m$ steps in the $(\Gamma\_2,\Gamma\_3)$ game, then he can survive for $\min(n,m)$ steps in the $(\Gamma\_1,\Gamma\_3)$ game.
**Proof.** The idea ... | 3 | https://mathoverflow.net/users/1946 | 348632 | 147,575 |
https://mathoverflow.net/questions/348638 | 4 | For $w$ a permutation, the associated **(ordinary) Schubert polynomial** $\mathfrak{S}\_w(\textbf{x})$ is a multivariate polynomial that represents for the cohomology class of the Schubert variety $X\_w$ in the flag variety.
There are many references for this fact. See, e.g., this [text](https://books.google.com/books?... | https://mathoverflow.net/users/7717 | Transition equations for double Schubert polynomials | On pg. 5 of <https://arxiv.org/abs/1909.13777>, Allen Knutson suggests that Lascoux proves a transition formula for double Schubert polynomials in the paper "Transition on Grothendieck polynomials", Physics and Combinatorics 2000, pp. 164-179, available online (if you pay for it) at <https://www.worldscientific.com/doi... | 4 | https://mathoverflow.net/users/25028 | 348639 | 147,576 |
https://mathoverflow.net/questions/348636 | 6 | I know there are already a couple of questions on this on the site, but I haven't seen an answer to this particular form...
We know, from the Fundamental Theorem of Algebra, that the complex algebraic numbers contain a unique maximal ordered subfield, namely the real algebraic numbers, and the complex algebraic numbe... | https://mathoverflow.net/users/36212 | Fundamental Theorem of Algebra, via algebra | Despite the extensive discussion in the comments, I'm not sure I completely understand the question, but I think the following fact may settle the question in the negative.
Work of Läuchli implies that there is a model of ZF in which $\mathbb Q$ has an algebraic closure with no real-closed subfield. For more details,... | 10 | https://mathoverflow.net/users/3106 | 348665 | 147,584 |
https://mathoverflow.net/questions/348651 | 3 | Suppose that $D$ is a bounded open convex subset of $\mathbb{R}^2$ with analytic boundary. You can parametrize the boundary of $D$ using the angles of the support lines at each point, but it isn't obvious that the parametrization you get is a real analytic function of the angles. Is there a quick reference to show that... | https://mathoverflow.net/users/54756 | Convex sets with analytic boundary, using angles to parametrize boundary | Such a parametrization will not be in general real analytic, because the angle of the support line may be varying too slowly at some points. E.g., let the boundary of the convex set $D$ be
$$C:=\{(x,y)\in\mathbb R^2\colon x^4/2+(y-1)^2=1\}.
$$
Then the points on $C$ in a neighborhood of the point $(0,0)\in C$ will be... | 8 | https://mathoverflow.net/users/36721 | 348670 | 147,586 |
https://mathoverflow.net/questions/348650 | 6 | Let $S$ be a (compact, connected) Riemann surface of genus $g$ and $f: S\to \mathbb CP^1$ be a degree $d$ meromorphic function. Then the Riemann-Hurwitz formula tells us that the number of ramifications of $f$ counted with multiplicity is $2(d-1)+2g-2$.
Suppose we consider instead a map $\varphi: S\to \mathbb CP^1$ ... | https://mathoverflow.net/users/13441 | Riemann-Hurwitz for real maps | This is not possible, just using the singularities and the homological degree of the map.
Let $f\_1$ be a generic degree $3$ holomorphic map $\mathbb P^1 \to \mathbb P^1$. It has four quadratic singularities. Let $E$ be an elliptic curve, and $f\_2$ a degree $2$ map $E \to \mathbb P^1$, with four simple ramification ... | 4 | https://mathoverflow.net/users/18060 | 348673 | 147,588 |
https://mathoverflow.net/questions/348676 | 0 | Let $H$ be a Hilbert space. I am interested in isometries $f\colon H\to L^2(X,\mu)$ where $\mu$ is a probability measure on some measure space $X=(X,\mathcal F)$ where $\mathcal F$ is a $\sigma$-algebra on $X$. My question is whether there exists a canonical choice of $(X,\mu,f)$ that depends only on $H$ and not "arbit... | https://mathoverflow.net/users/29961 | Canonical embedding of Hilbert space in $L^2$ space | This is impossible, at least in the case when $X=H$, as in your "finite-dimensional" example.
Indeed, suppose that $H$ is infinite dimensional. If you want the measure $\mu$ not to depend on the choice of an orthonormal basis, you have to make $\mu$ spherically invariant. But such a probability measure can only be $... | 3 | https://mathoverflow.net/users/36721 | 348680 | 147,592 |
https://mathoverflow.net/questions/348541 | 1 | Let $\mathbb{N}$ be the set of positive integers. Given a set $A\subseteq \mathbb{N}$ we let the *(upper) density* of $A$ be defined by $$\mu^+(A) = \lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
If $\alpha\in\mathbb{R}$, we say $q\in\mathbb{N}$ is *good for approximating $\alpha$* if there is $p\in\mathbb{... | https://mathoverflow.net/users/8628 | Density of the set of numbers that are "good approximators" to a given real in the sense of Dirichlet's approximation theorem | If $\alpha$ is irrational, then the sequence $\alpha,2\alpha,\ldots$ is equidistributed modulo 1 (Weyl's theorem). Thus the inequality $|q\alpha-p|<1/q$ holds for $q$ of density $0$. If $\alpha=a/b$ ($a,b$ are coprime) is rational, then the density of your numbers equals $1/b$.
| 3 | https://mathoverflow.net/users/4312 | 348690 | 147,594 |
https://mathoverflow.net/questions/348681 | 3 | $P$ means polynomial complexity.
$S\_p$ is the class of all $P$\_random set, and $S\_{pc}$ is the class of all $P$ noncomputable sets, is $S\_p \bigcap S\_{pc}$ empty? If not empty, any example?
what is the result, if we replace $P$ complexity with $NP$?
Moreover, $S$ is the class of all random sets, and $S\_c$ is ... | https://mathoverflow.net/users/14024 | Relationship between P-noncomputable and P-random sets | No, the intersection is not empty.
To see this, notice that we can easily find sets $A\subset\mathbb{N}$ that are random with respect to much stronger notions of complexity. And indeed, there will be uncountably many such sets $A$. Not all these can be computable. So we find $A$ that are both random and undecidable,... | 4 | https://mathoverflow.net/users/1946 | 348692 | 147,596 |
https://mathoverflow.net/questions/348461 | 2 | Let $k$ be an algebraically closed field. Consider a smooth group scheme $G$ over $k$. It is well known that the category $\textbf{Rep}\_{G}$ is semisimple if and only if one of the following assertions holds.
1. $G$ is reductive and $\mathrm{char}(k)=0$.
2. $G$ is an algebraic torus and $\mathrm{char}(k) = p >0$.
... | https://mathoverflow.net/users/147687 | Global homological dimension of reductive groups | In positive characteristic the only connected groups of finite homological dimension are the tori.
We need the following result from Jantzen, Representations of algebraic groups. [J, I 5.13], [J, I 4.6 b].
Let $H$ be a flat subgroup scheme of $G$, such that $G/H$ is an affine scheme. Then $\mathrm {ind}\_H^G$ is ex... | 6 | https://mathoverflow.net/users/4794 | 348699 | 147,599 |
https://mathoverflow.net/questions/348646 | 1 | We got reduction graph to planar bounded treewidth graph,
but this is unlikely to be true.
Let $H$, the planarizing gadget, be planar graph with four
distinguished vertices $u,u',v,v'$ on the outer faces.
Take graph $G$ drawn on the plane. Add new vertex $S$,
adjacent to all vertices of $G$. So far the diameter is
... | https://mathoverflow.net/users/12481 | Reduction graph to planar bounded treewidth graph | If gadgets are applied to pairs of crossing edges as described then I'm not sure such gadgets exist as they probably preserve the presence of forbidden minors.
If gadgets are applied around crossing points then two vertices at distance $d$ in $G$ are not necessarily at distance $O(d)$ in $G'$, as a path of length $d$... | 2 | https://mathoverflow.net/users/25485 | 348709 | 147,602 |
https://mathoverflow.net/questions/348287 | 3 | Let $X$ be a smooth variety over $\mathbb{C}$, and let $\omega \in \operatorname{Pic}(X)\_\mathbb{R}$ be an ample class.
I would like to know if any $\mu\_\omega$-semistable sheaf $E \in \operatorname{Coh}(X)$ admits a Jordan-Hölder sequence, i.e. a finite sequence
$$0 = E\_0 \subset E\_1 \subset \dots \subset E\_{n-1}... | https://mathoverflow.net/users/111897 | Jordan–Hölder sequence for $\mu$-semi stable sheaves | So I talked to my advisor, and one of the misconceptions I had is about $\mu\_\omega$-semistability: A torsion-free sheaf $E$ is called $\mu\_\omega$-semistable, if for all subsheaves $0 \neq E' \subset E$ with $\operatorname{rk}(E') < \operatorname{rk}(E)$, one has $\mu\_{\omega}(E') \leq \mu\_\omega(E)$. Similar for ... | 1 | https://mathoverflow.net/users/111897 | 348713 | 147,606 |
https://mathoverflow.net/questions/348686 | 4 | This question came up when thinking about an [older question](https://mathoverflow.net/questions/348541/density-of-the-set-of-numbers-that-are-good-approximators-to-a-given-real-in-t) that hasn't been answered as of now.
Let $\mathbb{N}$ be the set of positive integers. If $\alpha\in\mathbb{R}$, we say $q\in\mathbb{N... | https://mathoverflow.net/users/8628 | On the set of good approximators in the sense of Dirichlet's theorem | The answer is yes. For an irrational number $\alpha$, the inequality says that the fractional part $\{q\alpha\}$ lies in $(0,1/q)\cup(1-1/q,1)$. In particular, $\{q\alpha\}\not\in[1/3,2/3]$ when $q\geq 3$. However, it is easy to show by Dirichlet's approximation theorem that the fractional parts $\{q\alpha\}$ are dense... | 4 | https://mathoverflow.net/users/11919 | 348726 | 147,611 |
https://mathoverflow.net/questions/348648 | 5 | Consider the tensor product $G \otimes\_{\mathbb{Z}} H$ of two abelian groups $G$ and $H$. If $G$ and $H$ are topological groups, we can give $G \otimes\_{\mathbb{Z}} H$ a topology as follows. For any $k$, consider the function.
\begin{align}
\phi\_k: G^k \times H^k = G \times \dots \times G \times H \times \dots \ti... | https://mathoverflow.net/users/144100 | Is the tensor product of compactly generated Hausdorff abelian groups again Hausdorff? | This is not true. Our example will be $G = H = S^1$ with its standard topology.
Let $\mu \subset S^1$ be the subgroup of roots of unity. The group $S^1$ is divisible and $\mu$ consists entirely of torsion, and so
$$
\mu \otimes S^1 = 0.
$$
This lets us conclude that the natural map of groups
$$
S^1 \otimes S^1 \to (S... | 3 | https://mathoverflow.net/users/360 | 348727 | 147,612 |
https://mathoverflow.net/questions/348610 | 2 | The idea of this post arises when I've considered simple variants of the known as *Firoozbakht's conjecture* (see this corresponding [Wikipedia](https://en.wikipedia.org/wiki/Firoozbakht%27s_conjecture) or [1]), and comparisons by trial and error with means (if you are interested feel free to do yourself comparisons mo... | https://mathoverflow.net/users/142929 | On an attempt to create interesting variants of Firoozbakht's conjecture, evoking combinations of different generalized means | Yes, (2) holds for all large enough $n$.
According to Wikipedia (see also [this question](https://mathoverflow.net/q/259997/18698)) $\frac{p\_n}n=\log n+\log\log n-1+\frac{\log\log n}{\log n}(1+o(1))$, so
$$\frac{n+p\_{n+1}}{n+1}=\log(n+1)+\log\log(n+1)+\frac{\log\log n}{\log n}(1+o(1))$$
Since $\log\log(n+1)\approx\lo... | 1 | https://mathoverflow.net/users/18698 | 348737 | 147,614 |
https://mathoverflow.net/questions/348655 | 3 | Let $X$ be a finite set and let $f$ and $g$ be permutations with $f\ne g\ne f^{-1}$.
Can you show that the proportion of subsets $S$ with $|S\cap f(S)|=|S\cap g(S) |$ is at most $3/4$?
In other words
$$\frac{|\{S\subseteq X\mid|S\cap f(S)|=|S\cap g(S) | \} |}{|\{S\subseteq X\}|}\leq\frac{3}{4} \cdot 2^{|X|}$$
| https://mathoverflow.net/users/45242 | On a certain proportion concerning sets and permutations | True, of course (except the correct exception was given by Robert Israel, not by the OP), but nearly trivial.
Let us consider all vectors $\delta=(\delta\_1,\dots,\delta\_n)$ consisting of $0$ and $1$. The question is just how often it can happen that $F(\delta)=\langle\delta,A\delta\rangle=0$ where the matrix $A=P-... | 6 | https://mathoverflow.net/users/1131 | 348746 | 147,616 |
https://mathoverflow.net/questions/348748 | 5 |
>
> The boundary of any convex open set $X$ is $\mathbb R^n$ is a rectifiable hypersurface.
>
>
>
To see this, intuitively, simply take a sphere $S\_d$ with diameter $d\in(0,+\infty]$ that contains $X$. The nearest point projection from $S\_d$ to $\partial X$ is one-to-one onto.
Although the rectifiability re... | https://mathoverflow.net/users/109527 | Reference for the rectifiablity of the boundary hypersurface of convex open set | Any convex function is Lipschitz continuous so the boundary of a convex set is locally a graph of a Lipschitz function and therefore it is rectifiable.
For a proof of Lipschitz continuity of convex functions, see for example Theorem 2.31 in:
**B. Dacorogna,** *Direct methods in the calculus of variations.* Second e... | 4 | https://mathoverflow.net/users/121665 | 348749 | 147,617 |
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