parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/253995 | 9 | Given an $m$ by $n$ matrix $A$ I'm familiar with the standard method to compute a basis for the null space of $A$ by computing a QR factorization of $A^T$. If $A$ is large and sparse, we can use sparse-QR techniques with pivoting, but the resulting $Q$ matrix (and portion used for the null space basis) may be quite den... | https://mathoverflow.net/users/23064 | How can one construct a sparse null space basis using recursive LU decomposition? | *On the LUQ decomposition*
--------------------------
The algorithm implemented in `luq` (see reference given below) computes bases for the left/right null spaces of a sparse matrix $A$. Unfortunately, as far as I can tell, there seems to be no thorough discussion of this particular algorithm in the literature. In pl... | 5 | https://mathoverflow.net/users/64449 | 254095 | 115,004 |
https://mathoverflow.net/questions/254086 | 10 | I recently came across this elegant translation of etale $\mathbb{G}\_m$-torsors into line bundles:
Let $\text{Spec }R$ be a $\mathbb{G}\_m$-torsor over $\text{Spec }A$ for the etale topology, where $\mathbb{G}\_m = \text{Spec }A[x,x^{-1}]$, then the action of $\mathbb{G}\_m$ on $\text{Spec }R$ is given by a homomorp... | https://mathoverflow.net/users/88840 | $\mathbb{G}_m$-torsors and line bundles | If $G$ is a commutative monoid (for your question we will want $G = \mathbb{Z}$), then $A[G]$-comodules identify with $G$-graded $A$-modules. A reference is Demazure, Gabriel, *Introduction to Algebraic Geometry and Algebraic Groups*, II, §2, no 2, Example 1.
Indeed, if $M$ is an $A$-module with an $A[G]$-module stru... | 12 | https://mathoverflow.net/users/98306 | 254109 | 115,006 |
https://mathoverflow.net/questions/254107 | 0 | Let $S$ be the semiring in $\mathbb{R}[x,y]$ generated by the nonnegative real numbers $\mathbb{R}\_+$ and the polynomials $x, y, x + y - 1, 1 - (x+y)$, i.e.
$$S = \left\{ \sum\_{i,j,k,\ell \ge 0 } a\_{i,j,k,\ell} x^{i} y^{j} (x + y - 1)^{k} (1 - (x+y))^{\ell} : \, a\_{i,j,k,\ell} \ge 0 \,\, \forall i,j,k,\ell \,\right... | https://mathoverflow.net/users/nan | The semiring generated by $\mathbb{R}_+$ and the polynomials $x, y, x + y - 1, 1 - (x+y)$ | The question can be reformulated as: *Does $\pi^{-1}(0)$ lie in $S$?* Indeed, if so, then for all $s\in S$, $t\in \mathbb R[x,y]$ with $\pi(s)=\pi(t)$ we have $t-s\in S$, so $t=s+(t-s)\in S$ as well.
And this seems to be true. Actually, every $t\in \pi^{-1}(0)$ is a linear combination of the polynomials $m\_{ij}=x^iy... | 3 | https://mathoverflow.net/users/17581 | 254110 | 115,007 |
https://mathoverflow.net/questions/254119 | 11 | I have a Polish space $X$ and a subset $A \subset X$.
I know that $A$ is completely metrizable (in its induced topology) if and only if $A$ is a $G\_\delta$-set in $X$.
This means: If I want to show that $A$ *is* completly metrizable then it suffices to find a sequence $U\_1,U\_2,\ldots$ of open sets such that $A=\... | https://mathoverflow.net/users/58628 | How to show that something is not completely metrizable | There is an equivalent condition for complete metrizability in terms of a game known as the Choquet game. The game is described, for example, in the book *Classical Descriptive Set Theory* by Kechris, who calls it the strong Choquet game.
The game goes like this. The rounds are labeled with natural numbers. On round ... | 9 | https://mathoverflow.net/users/5442 | 254127 | 115,015 |
https://mathoverflow.net/questions/254101 | 6 | Suppose $k$ is an algebraically closed field of characteristic $p>0$. There is an $\infty$-category of motivic spectra over $k$, denoted $\mathcal{S}pt(k)$. As in algebraic topology, there are motivic Eilenberg-Maclane spectra $\mathbf{EM}(A)$ for each abelian group $A$. I have a few related questions:
1) Do $\mathbf... | https://mathoverflow.net/users/98793 | Localization, Slice Tower, and Motivic Spectra | The stable motivic category is a presentable symmetric monoidal ∞-category so smashing with *any* motivic spectrum preserves all homotopy colimits. On the other hand smashing with a spectrum need not to preserve the slice filtration exactly for the reasons that Tom Bachmann mentioned in the comments: the single slice c... | 1 | https://mathoverflow.net/users/43054 | 254131 | 115,016 |
https://mathoverflow.net/questions/254090 | 4 | Let $A\subset\mathbb{R}^p$ and $B\subset\mathbb{R}^q$, it’s not difficult to show that $$m^\*(A\times B)\leq m^\*(A)\cdot m^\*(B)$$, where $m^\*()$stands for the outter measure in Lebesgue meaning.
If A and B are measurable, then "=" holds. My question is whether "=" holds for all of/ none of/ some of the non-measura... | https://mathoverflow.net/users/69999 | Product of two non-measurable sets | For simplicity, assume $p = q = 1$ and $A, B \subseteq [0, 1]$. Let $\mu\_k, \mu\_k^{\star}$ denote the $k$-dimensional Lebesgue measure, outer measure respectively. Let $\{U(n) : n \geq 1\}$ be a sequence of decreasing open sets each containing $A \times B$ such that $\mu\_2(\bigcap\_n U(n)) = \mu\_2^{\star}(A \times ... | 4 | https://mathoverflow.net/users/100811 | 254134 | 115,017 |
https://mathoverflow.net/questions/254130 | 4 | Suppose we have a positive integer $n\geq 4$. We call a graph $G=(V,E)$ an *$K^{(n)}\_{n-1}$-graph* if
1. there are $n$ subsets $S\_1,\ldots, S\_n$, each consisting of $n-1$ points, and each are cliques;
2. for positive integers $i<j\leq n$ we have $|S\_i\cap S\_j| = 1$, and
3. for all $v\in V$ there are positive int... | https://mathoverflow.net/users/8628 | A graph consisting of cliques pairwise intersecting in 1 point | The answer is yes (in a certain sense; see below). This proof seems a bit long... so perhaps someone can come up with an easier argument!
The assumption that the sets form cliques in a graph seems to be irrelevant here. In fact, this problem isn't really a "graph problem" at all, so let's rephrase it in terms of hype... | 8 | https://mathoverflow.net/users/39146 | 254136 | 115,018 |
https://mathoverflow.net/questions/248611 | 2 | Is there any way to compute/express $\sum\limits^m\_{i=0}\{\frac{q\*i}{m}\}(\frac{i}{m})^n$ ? Here $q,m,n$ are natural numbers, one can assume $gcd(q,m)=1$. Furthermore, $n$ can be treated as a parameter, i.e. I will be quite happy to know the formula for each particular $n$. At least for say $0<n<5$.
Maybe in this g... | https://mathoverflow.net/users/2900 | the sum of fractional parts times the ordinary powers | As Gerry already commented, your formulas are equivalent to a reciprocity theorem of Tom Apostol; see, e.g., the bibliography in [this paper](https://arxiv.org/abs/1008.0038) (Apostol's paper is too old to be on the arXiv): Your sums are of the form $s\_n(a,b) + s\_n(b,a)$ where
$$
s\_n(a,b) = \sum\_{m=1}^b \left\{\fr... | 3 | https://mathoverflow.net/users/3193 | 254137 | 115,019 |
https://mathoverflow.net/questions/254142 | 7 | My question arises from thinking about how we can obtain class forcing extensions from truncations of set forcing extensions in the presence of a (strongly) inaccessible cardinal in the following sense. Let $\kappa$ be an inaccessible cardinal and let $\mathbb P$ be a forcing such that $\mathbb P \subseteq V\_{\kappa}$... | https://mathoverflow.net/users/100285 | Class forcing as set forcing followed by truncation | No, there can be no such forcing. The point is that such a forcing should have size $\kappa$ and by my proof given in [Singularizing forcing of "small" cardinality?](https://mathoverflow.net/questions/142086/singularizing-forcing-of-small-cardinality), we have $\Vdash\_{\mathbb{P}} |\kappa|=cf(\kappa)$, so $\mathbb{P}$... | 6 | https://mathoverflow.net/users/11115 | 254148 | 115,021 |
https://mathoverflow.net/questions/254140 | 8 | Call a real number $\lambda>1$ **special** if it is a root of a polynomial $f(x)$ such that
1. $f(x)$ is monic with integer coefficients,
2. all roots of $f(x)$ are distinct,
3. for all $z\in\mathbb{C}$, if $f(z)=0$ and $z\neq \lambda$ then $|z|<1$.
For example, the Golden Ratio $\frac{1+\sqrt{5}}{2}$ is special.
... | https://mathoverflow.net/users/38253 | Polynomials with all but one root inside the unit disc | This is a well-known topic. Such numbers are called Pisot numbers. Salem (1945) proved that they form a closed subset $S$ of $(1,+\infty)$, and Siegel (1944) proved that its lowest element is the positive root of $X^3-X-1$. Therefore the answer to your question is **No**.
The Golden ratio is however the smallest amon... | 13 | https://mathoverflow.net/users/8799 | 254154 | 115,023 |
https://mathoverflow.net/questions/254155 | 3 | Let $(M,g)$ be a Riemannian manifold of dimension $n$. (In the case I am interested in, $M$ is a complex symmetric domain, but I do not think that this is relevant for the question.)
Let $N$ be a submanifold of $M$. I would like to say that $N$ is "much curved". For the time being, let me just say that the second fun... | https://mathoverflow.net/users/48866 | Curvature and intersection of submanifolds | If you assume that the codimension of $N$ is large,
then your embeddings are called "free" (see the Gromov's book on h-principle). In this case there are no geodesics and all the curves in $N$ have non-vanishing normal curvature.
For small codimension, the statement does not hold, say consider the paraboloid $z=xy$... | 4 | https://mathoverflow.net/users/1441 | 254157 | 115,025 |
https://mathoverflow.net/questions/254164 | 1 | Is every finite quasi-simple group generated by 2 elements ?
Recall that G is quasi-simple if G is perfect and G/Z(G) is simple.
Edit: If the answer is yes, do we know of a larger class of finite groups (beyond the finite quasi-simple groups) having the 2-generation property ?
Edited: Yes, there is another class.... | https://mathoverflow.net/users/46323 | Is every finite quasi-simple group generated by 2 elements? | It turns out the answer is Yes.
This is because G/Z(G) is simple and hence generated by 2 elemnts>
Moreover, if 2 elements generate a perfect group modulo its center, then they evidently generate the group since
G = < a, b> Z(G) and G = [G, G].
imply G = < a, b>. That is, G is generated by a and b.
Moreover, G ... | 1 | https://mathoverflow.net/users/46323 | 254171 | 115,027 |
https://mathoverflow.net/questions/253848 | 4 | This question is a follow up to [Are $G$-limits of a slender group $G$ also slender?](https://mathoverflow.net/questions/253686/are-g-limits-of-a-slender-group-g-in-the-space-of-marked-groups-also-slender) The MO user @Ycor gave an excellent answer to this question, in which they show that $\mathbb{Z}\wr\mathbb{Z}$ emb... | https://mathoverflow.net/users/38698 | Are $G$-limits of a slender group $G$ in the space of marked groups small? | Thanks to Ycor's answer and an other [answer due to Anton Klyatchko](https://mathoverflow.net/questions/146812/minimal-number-of-generators-of-subgroups-of-noetherian-groups), the answer is **no, limits of slender groups are not small.**
In that Anton Klyatchkos's answer, [Obraztsov's embedding theorem](http://www.ma... | 2 | https://mathoverflow.net/users/38698 | 254172 | 115,028 |
https://mathoverflow.net/questions/254166 | 3 | Think of the three-sphere as given by $\lbrace|z|^2+|w|^2=1, \;z,w\in \mathbb{C}^2\rbrace$. We can regard it in terms of Hopf coordinates
\begin{align\*}
z&= \cos(\theta/2)e^{i(\phi+\psi)}\\
w&= \sin(\theta/2)e^{i\psi}
\end{align\*}
where $0\leq\theta\leq \pi$ and $0\leq \phi,\psi<2\pi$. Now I want to consider the orbi... | https://mathoverflow.net/users/41130 | Orbifold of the three-sphere (and lens spaces) | The quotient of $S^3$ by your action is again $S^3$. You can see that by dividing the sphere into regions where (say) $|w| \leq 1/2$ and $|w| \geq 1/2$. The quotient of each of these is a solid torus, and you can check that they are glued together to give a 3-sphere. A sanity check is that the fixed point set is the ci... | 2 | https://mathoverflow.net/users/3460 | 254189 | 115,034 |
https://mathoverflow.net/questions/254173 | 0 | I encountered the following problem. Since this is somewhat not related to what I normally do, I wanted to know what the best estimates in this field are.
Let $A \in \mathbb{R}^{n \times n}$ be a self-adjoint matrix and $B:=A + \sum\_{i=1}^{k} w\_i C\_i$ where $w\_i \in \mathbb{R}$ and $C\_i$ an orthonormal system of... | https://mathoverflow.net/users/100835 | Perturbation theory for matrices | By orthonormal, I suppose you mean $\text{tr}(C\_i C\_j) = 0$ for $i \ne j$,
$\text{tr}(C\_i^2) = 1$?
The estimate you gave is tight, in the sense that it is an equality if, for example, all $w\_i = \epsilon > 0$ sufficiently small and the matrices
$A$ and $C\_i$ all share the same eigenvector for the lowest eigenv... | 1 | https://mathoverflow.net/users/13650 | 254190 | 115,035 |
https://mathoverflow.net/questions/254053 | 4 | Let $Gpd$ denote the category of groupoids and functors. Let $Gpd\_{con}$ denote the subcategory spanned by connected groupoids, i.e for every $x,y\in Ob(Gpd\_{con})$, there is at least one morphism $x\rightarrow y$.
Does the canonical inclusion $i:Gpd\_{con}\rightarrow Gpd$ have a left adjoint? If so, does it preser... | https://mathoverflow.net/users/84563 | Left adjoint to inclusion of Connected Groupoids into Groupoids | No, it does not. If it did, then $\mathrm{Gpd}\_{\mathrm{con}}$, like any reflective subcategory, would be closed under limits in $\mathrm{Gpd}$. But it is not closed under equalizers. For instance, let $X$ be the contractible groupoid with two objects $x,y$, and $G$ any nontrivial group regarded as a connected groupoi... | 3 | https://mathoverflow.net/users/49 | 254200 | 115,038 |
https://mathoverflow.net/questions/254182 | 5 | What I mean by classical: For the case of $GL\_2$, the answer to my question would be that the automorphic forms are either Maas forms or modular forms. For $GSp(2n)$ these are the Siegel modular forms.
Are there any "classical" objects which appeared in mathematics before this automorphic perspective which align wit... | https://mathoverflow.net/users/47195 | "Classical" description of automorphic forms on unitary groups | These go by the name of "Hermitian modular forms". They occur very frequently in papers of Shimura (e.g. his monograph *Arithmeticity on the theory of automorphic forms*) and in other more recent works. For instance, [this paper by Bouganis](https://www.math.uni-bielefeld.de/documenta/vol-20/37.pdf) describes in detail... | 10 | https://mathoverflow.net/users/2481 | 254205 | 115,040 |
https://mathoverflow.net/questions/254211 | 0 | **Problem.** Does every compact countable space contain a non-trivial convergent sequence?
This question concerns non-Hausdorff compact spaces. An example of such space is any infinite set $X$ endowed with the Zariski topology $\tau=\{\emptyset\}\cup\{X\setminus F:F$ is finite$\}$. Observe that this space is compact,... | https://mathoverflow.net/users/61536 | Does every compact countable space contain a non-trivial convergent sequence? | Let $X=(x\_i)$ be a sequence of all elements of our space without repetitions. If it does not converge to $x\_1$, then there exists a neighbrhood $U\_1$ of $x\_1$ and a subsequence $X\_1$ of $X$ avoiding $U\_1$. If $X\_1$ does not converge to $x\_2$, then there exists a neighbrhood $U\_2$ of $x\_2$ and a subsequence $X... | 5 | https://mathoverflow.net/users/17581 | 254213 | 115,043 |
https://mathoverflow.net/questions/254215 | 1 | Can a non-amenable connected Lie group have a faithful finite-dimensional unitary representation?
| https://mathoverflow.net/users/50457 | Faithful representations of non-amenable Lie groups | You have an extension $1\rightarrow R\rightarrow G\rightarrow S\rightarrow 1$ where $R$ is a solvable subgroup and $S$ semisimple by using the Levi decomposition.
Let $f:G\rightarrow U(n)$ be a faithful representation. The restriction of $f$ to $S$ is also faithful. Consider the Iwasawa decomposition of $S=KAN$ where... | 3 | https://mathoverflow.net/users/80891 | 254216 | 115,044 |
https://mathoverflow.net/questions/254214 | 1 | **Problem.** Assume that a compact space $X$ can be written as the union $X=K\cup D$ of a compact metrizable subspace $K$ and a discrete subspace $D$.
Does $D$ contain a non-trivial convergent sequence in $X$?
As shown by Ilya Bogdanov ([Does every compact countable space contain a non-trivial convergent sequence?](h... | https://mathoverflow.net/users/61536 | Convergent sequences in compact spaces | Oh, sorry! I wrote this question and after some thinking found a (relatively simple) answer.
Consider the set $\mathcal P$ of pairs $(A,I)$ where $A$ is a non-empty closed subset of the compact metrizable space $K$ and $I$ is an infinite subset of $D$ such that every open neighborhood $U$ of $A$ in $X=K\cup D$ contai... | 2 | https://mathoverflow.net/users/61536 | 254217 | 115,045 |
https://mathoverflow.net/questions/254143 | 3 | I am trying to calculate the packing density of cylindrical bottles in a box, assuming that the bottles are randomly dumped in the box.
I have read on the packing density of spheres here <https://en.wikipedia.org/wiki/Random_close_pack>
but have found nothing on cylinders.
For example, if I randomly dump bottles wi... | https://mathoverflow.net/users/100817 | Random packing density of cylinders in a volume | An overview of results for the random packing densities of basic 3D objects, including cylinders, is given in [this article](http://link.springer.com/article/10.1007/s11434-009-0650-0) (2010). Here is the plot for cylinders as a function of aspect ratio (height over diameter):
 $A\_{\neg\neg}$ via the nucleus which maps an open $U$ of $A$ to $\neg \neg U$, which is the interior of the closure of $U$. This quotient frame can be viewed as a ... | https://mathoverflow.net/users/82487 | Products of double-negation sublocales (and probability distributions on them) | For your first question, if $X$ and $Y$ are two boolean locale then $X \times Y$ is boolean only if $X$ or $Y$ is discrete. So unless $\neg \neg A$ or $\neg \neg B$ are discrete, $\neg \neg A \times \neg \neg B$ and $\neg \neg (A \times B)$ cannot be isomorphic because one is boolean and the other is not.
For you sec... | 3 | https://mathoverflow.net/users/22131 | 254238 | 115,054 |
https://mathoverflow.net/questions/254237 | 9 | Let's say I have a sequence of random variables $X\_n$ such that $$\mathbf E X\_n^k = \mathbf E X^k+O(a\_k/\sqrt{n})\quad\text{for all }k\in\mathbb N,\tag{$\ast$}$$ where $X$ is a random variable of standard (zero mean, unit variance) Gaussian distribution and $a\_k$ are some constants which typically grow with $k$ (fo... | https://mathoverflow.net/users/89934 | Distance between distributions and distance of moments | The natural thing to compare in this context seems to be the [moment generating functions](https://en.wikipedia.org/wiki/Moment-generating_function) of $X\_n$ and $X$. In particular, consider:
\begin{align\*}
\mathbf{E} \exp(t X\_n) - \mathbf{E} \exp(t X) &= \sum\_{k=0}^{\infty} \frac{t^k}{k!} \left( \mathbf{E} X\_n^k ... | 5 | https://mathoverflow.net/users/64449 | 254242 | 115,055 |
https://mathoverflow.net/questions/156633 | 8 | I've just come across [this popular article](https://web.archive.org/web/20140904103408/http://www.simonsfoundation.org/quanta/20140130-perfecting-the-art-of-sensible-nonsense/) about [a breakthrough](https://eprint.iacr.org/2013/451.pdf) (which can be purchased [here](https://ieeexplore.ieee.org/document/6686139)), pu... | https://mathoverflow.net/users/2821 | Is this obfuscation scheme unbreakable? | *"Is this obfuscation scheme unbreakable?"*
"Well.. no." said people a couple of years later.
1. On GGHRSW13 specifically: [Cryptanalyses of Candidate Branching Program Obfuscators](https://eprint.iacr.org/2016/998)
See also (concurrent, similar flavor):
2. [Cryptanalysis of Indistinguishability Obfuscations of... | 9 | https://mathoverflow.net/users/16024 | 254244 | 115,056 |
https://mathoverflow.net/questions/254240 | 10 | Sorry for the naive question. Let $X\_1$, $X\_2$ and $Y$ be three projective varieties over an algebraically closed field of characteristic zero. If we have $X\_1\times Y\cong X\_2\times Y$, do we automatically get $X\_1\cong X\_2$? If not, do we have counter examples? If necessary, we could put stronger conditions (fo... | https://mathoverflow.net/users/24965 | Do we have "cancellation law" for products of varieties | You may look at T. Fujita's paper "Cancellation Problem of Complete Varieties". He proved that if $X\_1$ and $Y$ are "Picard independent" (look at Proposition 3 therein for a definition) then cancellation holds for any $X\_2$. In the same paper (Remark 8), the author cites T. Shioda "Some remarks on abelian varieties" ... | 14 | https://mathoverflow.net/users/31724 | 254245 | 115,057 |
https://mathoverflow.net/questions/254183 | 5 | In Ralf Schmidt's appendix to "Jacquet-Langlands-Shimizu correspondence for theta lifts to $\mathrm{GSp}(2)$ and its inner forms" by Narita and Okazaki , he computes the representations of $\mathrm{GSp}(1,1)$ by examining the tables in his book and discarding the representations whose Weil-Deligne representations fall ... | https://mathoverflow.net/users/6084 | Irrelevant parabolics and inner forms of GSp(4) | I've never really got my head around ``relevant $L$-parameters'', but the condition on the $L$-parameter (i.e. the associated WD-rep) which ensures transfer to a given inner form should look something like the condition for being in the discrete series. Roughly speaking, representations whose $L$-parameters have fairly... | 4 | https://mathoverflow.net/users/12055 | 254251 | 115,058 |
https://mathoverflow.net/questions/253578 | 9 | Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{Ext}^1(E, E)$? I am primary interested in two situations:
* $\mathcal M$ is the moduli space of semi-stable coherent ... | https://mathoverflow.net/users/43639 | Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$ | **Theorem 2.6** (page 9) from [Hartshorne *Lectures on Deformation Theory*](https://math.berkeley.edu/~robin/math274root.pdf) (it seems that Hartshorne uses the same notation both for an affine scheme $D$ and its function algebra):
Let $X$ be a scheme over $k$, and let $\mathcal F$ be a coherent sheaf on $X$. We defin... | 6 | https://mathoverflow.net/users/43639 | 254256 | 115,061 |
https://mathoverflow.net/questions/254219 | 32 | Suppose you have a closed $m$-dimensional manifold $M$, which embeds in $\mathbb{R}^{n+1}$ for some $n$. Can it have a closed submanifold $N$ (of dimension strictly smaller than $m$) which does *not* embed in $\mathbb{R}^n$? (By which I mean there is no embedding, not just that the restriction/projection of the first o... | https://mathoverflow.net/users/91903 | Manifold embedded in $R^{n+1}$ with a submanifold that doesn't embed in $R^n$ | Here's another way to get examples, in codimension one and in low dimensions. There are lots of oriented closed 3-manifolds that don't embed in 4-space, for example any 3-manifold $M$ with $H\_1(M) \cong \mathbb{Z}\_{2n}$. But a theorem of Hirsch says that $M$ embeds in $\mathbb{R}^5$. By a standard transversality argu... | 23 | https://mathoverflow.net/users/3460 | 254257 | 115,062 |
https://mathoverflow.net/questions/253864 | 27 | This is my first question on this wonderful site. The following question about Arakelov geometry is gonna be quite long and wide; to be clear one of that kind of questions that are usually ignored. The point is that I haven't found any help in the literature and I feel lost.
---
Let's fix an **arithmetic surface*... | https://mathoverflow.net/users/100660 | Analogies between classical geometry on complex surfaces and Arakelov geometry | These are indeed good questions, and while there is a very good corpus of answers to them, the analogy is not perfect.
*0. The non-archimedean analogy*
First of all, I would like to go back to the relative situation of a
surface $\mathcal X\to B$ fibered over a germ of curve $(B,b)$.
Then any local function $f$, ... | 13 | https://mathoverflow.net/users/10696 | 254260 | 115,064 |
https://mathoverflow.net/questions/254199 | 1 | Let $X$ be a smooth projective algebraic variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of complementary codimension in $X$. Let $n$ denote their intersection product, and let $Z$ denote the union of the zero dimensional components of the intersection (with multiplicities). ... | https://mathoverflow.net/users/100848 | Zero dimensional components of an intersection | It's not true scheme-theoretically, at least. Let $X=\mathbb P^4$ with coordinates $w,x,y,z,\Omega$, let $A$ be the $xy\Omega$-plane, and $B$ the union of the $wx\Omega$- and $yz\Omega$-planes. Then $A\cap B$ is a fat point of length $3$, whereas the intersection number is $2$.
| 2 | https://mathoverflow.net/users/391 | 254277 | 115,072 |
https://mathoverflow.net/questions/254250 | 27 | Reading "H. Matsumura - Commutative Ring theory" I had the impression that the definitions were all made to mean something in algebraic geometry afterwards. I wonder what was commutative algebra before the modern algebraic geometry (which ideas were developed, how sophisticated they were, etc).
Moreover, whether the su... | https://mathoverflow.net/users/74069 | What was commutative algebra before (modern) algebraic geometry? | here is an excerpt from Zariski's talk at the Icm several decades ago:
"The arithmetic trend in algebraic geometry is not in itself a radical departure from the past. This trend goes back to Dedekind and Weber who have developed, in their classical memoir, an arithmetic theory of fields of algebraic functions of one ... | 26 | https://mathoverflow.net/users/9449 | 254281 | 115,075 |
https://mathoverflow.net/questions/254289 | 6 | $A$ and $B$ are two players, each have exactly one turn. $A$ goes first. $A$ keeps on choosing a random number uniformly distributed over $(0,1)$ and add the values. If at one point it exceeds $1$, $A$ loses. If $A$ thinks his cumulative sum is very close to $1$, hence there is a risk of losing, he stops. Then $B$ star... | https://mathoverflow.net/users/100886 | An Interesting Two Players' Game Involving Cumulative Sum of Uniform Distribution | The optimal strategy for A is to play until she reaches 0.570557. This is a root of $3e^x-x-2xe^x=e$. To see this, one can compute that A's probability of winning if she stops at $\alpha$ is $f(\alpha)$, where $f(x)=1-(1-x)e^x$. Her probability of winning if she carries on for one more turn and then stops is $g(\alpha)... | 11 | https://mathoverflow.net/users/11054 | 254291 | 115,078 |
https://mathoverflow.net/questions/254288 | 7 | Let $G$ be a split semisimple algebraic group scheme over $\mathbf{Z}$ (I'm mostly interested in the case $G = Sp\_4$).
Let $V$ be an irreducible representation of the generic fibre $G\_{\mathbf{Q}}$, of some highest weight $\lambda$, and $v$ a choice of highest-weight vector. Then there is a notion of an "admissible... | https://mathoverflow.net/users/2481 | Integral lattices in Lie group representations | Let $\mathfrak{n}\_-$ be the Lie algebra of the derived subgroup $N\_-$ of $B\_{-}$ where $B\_{-}=TN\_-$ is a Borel subgroup of $G$ opposite to a Borel subgroup $B\_+\subset P$ (here $T$ is a split maximal torus of $G$ defined over $\mathbb{Z}$).
If $V\_\mathbb{Z}$ is a minimal admissible $\mathbb{Z}$-lattice then the... | 6 | https://mathoverflow.net/users/24386 | 254294 | 115,080 |
https://mathoverflow.net/questions/254286 | 3 | I am learning about local limit theorems. The following example is probably why we don't have a "convergence in density/pmf."
Ex: $X\_1,X\_2,\ldots$ is a sequence of independent RVs with mean $a$ and variance $\sigma^2$, then the distribution function $F\_n(x)$ of the normalized sum $Z\_n=\frac{\sum\_{i=1}^n (X\_i-a)... | https://mathoverflow.net/users/95756 | Is there a notion of Convergence in PDF/PMF | You should look at the old book
>
> Gnedenko and Kolmogorov: *Limit Distributions for Sums of Independent Random Variables*.
>
>
>
where they have examples of local limit theorems. I assume many more things were discovered in the meantime. Here is an example of theorem that answers your question.
**Theorem... | 6 | https://mathoverflow.net/users/20302 | 254297 | 115,081 |
https://mathoverflow.net/questions/89842 | 13 | In ${\bf M}\_n(\mathbb R)$, let us consider the usual operator norm
$$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$
where $\|x\|$ is the Euclidian norm.
The closed unit ball $B$ is the set of *contractions* (in the terminology used by operator theorists). It is a convex compact subset of ${\bf M}\_n(\mathbb R)$. By Krein-Milman ... | https://mathoverflow.net/users/8799 | What is the "positive part" of the unit ball in $M_n(R)$ ? | I'm a bit late in answering this. But in case there is still interest, please have a look at:
Saunderson, Parrilo, Willsky. [Semidefinite descriptions of the convex hull of rotation matrices](http://epubs.siam.org/doi/pdf/10.1137/14096339X), SIAM J. Optimization, 25(3), 1314-1343, 2015.
This paper provides explicit... | 11 | https://mathoverflow.net/users/8430 | 254305 | 115,083 |
https://mathoverflow.net/questions/254265 | 5 | According to Wikipedia
<https://en.wikipedia.org/wiki/Algebraic_matroid>
"For fields of characteristic zero (such as the real numbers) linear and algebraic matroids coincide"
I cannot find the two references it provides. There is a theorem that if you are algebraic over F then you are linear over an extension of F. ... | https://mathoverflow.net/users/100874 | Over the complex numbers, is there an example of an algebraic but non-representable matroid? | Algebraic over $\mathbb{C}$ implies realizable over $\mathbb{C}$. If your matroid is algebraic over $\mathbb{C}$, then it is realizable over some $\mathbb{C} \subset K$, so it is realizable over the algebraic closure $\overline{K}$, which explicitly means that a list of equalities and inequalities over $\overline{K}$ h... | 3 | https://mathoverflow.net/users/297 | 254313 | 115,087 |
https://mathoverflow.net/questions/251807 | 2 | "There are 536 class of quartic forms Q (header) [in 8 boolean variables] providing bent functions of the form Q+f where f is a cubic functions."
[Philippe Langevin, 2008.](http://langevin.univ-tln.fr/project/quartics/quartics.html "Classification of Boolean Quartics Forms in eight Variables")
What is the current pr... | https://mathoverflow.net/users/12911 | How to enumerate the extended affine equivalence classes of bent functions of degree 4 in 8 variables? | I asked the question to Philippe Langevin by e-mail, here's his answer, hopefully it will help--I am not very au fait with the technicalities of this topic. Here is the link to his [projects page](http://langevin.univ-tln.fr/project/) referred to below.
>
> I guess you speak about the affine classification of bent ... | 1 | https://mathoverflow.net/users/17773 | 254322 | 115,089 |
https://mathoverflow.net/questions/254328 | 2 | Given a Primal LP (P) and it Dual LP (D) we know that the optimal solutions to P ($x\_{opt}$) and D $(y\_{opt})$ satisfy complementary slackness condition, i.e. under optimal solutions either a constraint is tight in P or the corresponding variable in D is zero. If we know that a Dual feasible point, $y$, is approximat... | https://mathoverflow.net/users/90465 | Complementary slackness for approximately optimal Dual solution | Yes, we do have "approximate complementary slackness" in the following sense.
Consider the standard (primal) linear programming problem:
max $c^T x$ s.t. $A x \le b$, $x \ge 0$
If $x^\*$ and $y^\*$ are primal and dual feasible solutions, we have
$$ c^T x^\* \le y^\* A x^\* \le y^\* b$$
with equality iff $x^\*$... | 2 | https://mathoverflow.net/users/13650 | 254331 | 115,092 |
https://mathoverflow.net/questions/254335 | 0 | The simplest case of a well known theorem of Penner states that given a pair of filling curves, a positive twist about one curve together with a negative twist about the other curve is a pseudo-anosov mapping class. Suppose I have an explicit pair of curves that fill a closed surface, is there a reasonable way to expli... | https://mathoverflow.net/users/100917 | Constructing Invariant Lamination of a Pseudo-Anosov Given By Dehn Twists | Your question is explicitly answered in Section 6 of Thurston's Bulletin article, freely available [here](https://projecteuclid.org/euclid.bams/1183554722). In the end it boils down to a calculation in $\mathrm{SL}(2, \mathbb{Z})$.
| 1 | https://mathoverflow.net/users/1650 | 254338 | 115,093 |
https://mathoverflow.net/questions/254342 | -2 | $$f:N \rightarrow B,\space B\subset N $$ and $B$ is finite, $S$ is the sequence constructed by $f(1),f(2)\cdots f(i)\cdots $.
Now, if $f$ is a computable function,is $S$ eventually periodic?
Update: Secondly, if the computable function is computable in p time, is the $S$ eventually periodic? Or, under which computat... | https://mathoverflow.net/users/14024 | If the set of the output of a computable function is finite, is the sequence periodic eventually? | Regarding the 2nd question, the set of output sequences of an [autonomous finite automaton](https://books.google.com/books?id=s58EXPW6B6sC&pg=PA215&lpg=PA215&dq=output%20of%20finite%20automaton%20periodic&source=bl&ots=Liwgupg94l&sig=IKEczqfQqDhR6QZ7-CfG4YV57FE&hl=en&sa=X&ved=0ahUKEwjUz8Ody53QAhWIi1QKHeEwB8UQ6AEIIzAB) ... | 3 | https://mathoverflow.net/users/4600 | 254346 | 115,095 |
https://mathoverflow.net/questions/254315 | 15 | I would like to know if for $A,B\in SO(3)$ the inequality
$$
\|AB-BA\|\_F\leq \|A-I\|\_F\|B-I\|\_F
$$
holds, where $\|\cdot\|\_F$ denotes the Frobenius norm and $I$ the identity matrix. Using the identity
$$
AB-BA=(A-I)(B-I)-(B-I)(A-I)
$$
one can show the inequality with a factor $2$. However the inequality seems to be... | https://mathoverflow.net/users/100908 | matrix inequality with orthogonal matrices | I confirm Peter's suggestion that the best constant is $\frac1{\sqrt2}$. For let $\omega$ be this best constant. Then the inequality amounts to writing
$$3-{\rm Tr}(A^tB^tAB)\le2\omega^2(3-{\rm Tr}\,A)(3-{\rm Tr}\,B).$$
Let me parametrize $A$ by the angle of rotation $\theta$ and the axis of rotation $u$, a unit vector... | 9 | https://mathoverflow.net/users/8799 | 254359 | 115,102 |
https://mathoverflow.net/questions/254362 | 6 | Let $D$ be a divisor in $\mathbb P^2\_{\mathbb C}$ and let $X= \mathbb P^2\_{\mathbb C} - D$.
Under what condition on $D$ is the fundamental group of $X$ infinite?
| https://mathoverflow.net/users/100945 | Fundamental groups of complements of divisors in $\mathbb P^2$ | I'd leave this as a comment, but I don't have enough reputation. Consider the long exact sequence in homology of the pair $(\mathbb{P}^2, \mathbb{P}^2-D)$. Since $H\_1(\mathbb{P}^2,\mathbb{Z}) = 0$ and $H\_2(\mathbb{P}^2, \mathbb{Z}) = \mathbb{Z}$, this has a segment
$$\mathbb{Z}\rightarrow H\_2(\mathbb{P}^2, \mathbb{P... | 11 | https://mathoverflow.net/users/98320 | 254366 | 115,104 |
https://mathoverflow.net/questions/254364 | 26 | Other than reading MathSciNet regularly, is there a way to get notified when one of my papers gets a review on MathSciNet?
| https://mathoverflow.net/users/955 | MathSciNet review alert? | While you could do as @Geoff Robinson suggests and look at MathSciNet regularly, in early 2017, we plan to have email alerts enabled on MathSciNet. The target is February 2017.
Other new features will be rolled out in January and February. The first demos will be at the Joint Mathematics Meetings in Atlanta.
Edwa... | 72 | https://mathoverflow.net/users/49409 | 254367 | 115,105 |
https://mathoverflow.net/questions/254060 | 5 | I learned that the sphere has the smallest total mean curvature among all convex solids with a given surface area. This actually implies the sphere also has the smallest total mean curvature among all convex solids with a given volume. The first result can be proved by using Steiner symmetrization, and the latter resul... | https://mathoverflow.net/users/51546 | Lower bound for total mean curvature among mean convex set in $3D$ with fixed volume | Even with an area constraint, the minimization of total mean curvature for mean-convex surfaces seems to be open. You should look at this recent article <http://link.springer.com/article/10.1007/s12220-015-9646-y>
| 3 | https://mathoverflow.net/users/100951 | 254369 | 115,106 |
https://mathoverflow.net/questions/254341 | 10 | A slight variant $\tilde Mot\_{num}(k,\mathbb{Q})$ of the category of pure motives $Mot\_{num}(k,\mathbb{Q})$ is a Tannakian category equivalent to a category of representations of some algebraic group $GMot\_k$ (the Motivic galois group):
$$
Mot\_{num}(k,\mathbb{Q}) \simeq Rep(GMot\_k)
\,.
$$
[nLab](https://ncatla... | https://mathoverflow.net/users/83957 | Derived version of equivalence between motives and representations of Motivic galois groups? | Let $k$ be a field and $\operatorname{DM}\_{gm}(k)\_{\mathbb Q}$ the ∞-category of rational geometric motives over $k$. A mixed Weil cohomology theory induces a symmetric monoidal exact functor
$$
R: \operatorname{DM}\_{gm}(k)\_{\mathbb Q} \to D\_c(\mathbb Q),
$$
where $D\_c(\mathbb Q)$ are the compact=dualizable o... | 11 | https://mathoverflow.net/users/20233 | 254383 | 115,110 |
https://mathoverflow.net/questions/254394 | 6 | It is well known that any homotopy type can be obtained as the classifying space of a ($1$-)category. The classifying space of a category $\mathcal{C}$ can be interpreted in at least two ways:
1. We can view $\mathcal{C}$ as an object in the $(\infty,1)$-category of $(\infty,1)$-categories, and localise $\mathcal{C}... | https://mathoverflow.net/users/80483 | Understanding model independently the equivalence of two ways of obtaining homotopy types from categories | Here is an argument, which is basically Denis Nardin's comment.
To have a model independent proof you need model independent definitions of the hocolim and of the localization. You can define them via adjunctions, but from a pragmatic point of view I am not sure this is so helpful. Ultimately to do any kind of calcu... | 8 | https://mathoverflow.net/users/184 | 254395 | 115,114 |
https://mathoverflow.net/questions/254386 | 2 | Do there exist functions $g(\epsilon)$ and $h(\epsilon)$ defined for $\epsilon>0$ such that $g(\epsilon)\to 0$ and $h(\epsilon)\to 0$ as $\epsilon\to 0^+$ with the following property:
If $X$ and $Y$ are random variables with joint pmf $p(X,Y)$ satisfying
$$\mathbb{I}(X;Y)=\mathbb{E}\_{X,Y}\left(\log\frac{p(X,Y)}{p(X)... | https://mathoverflow.net/users/68835 | Does small expected value of a random variable show the small probability for its tail? | So you can recover what you want directly from the Pinsker inequality.
Set $g(\epsilon)=\sqrt[4]\epsilon$. Let $p\_{ij}=\mathbb P(X=i,Y=j)$, $p\_i=\mathbb P(X=i)$ and $q\_j=\mathbb P(Y=j)$.
Then
$$
\mathbb P(\Lambda\_\epsilon)=\sum\_{(i,j)\in\Lambda\_\epsilon}p\_{i,j}.
$$
If $(i,j)\in\Lambda\_\epsilon$, then $p\... | 2 | https://mathoverflow.net/users/11054 | 254412 | 115,119 |
https://mathoverflow.net/questions/254411 | -2 |
>
> A $k$-regular graph is a graph with all vertices having degree k.
> A graph $X$ is called vertex-transitive if it's automorphism group acts transitively on the vertex set.
>
>
> We know that all the vertex-transitive graphs are regular graphs but my question is that whether the reverse of this statement true w... | https://mathoverflow.net/users/92161 | Is connected k-regular graphs are always vertex-transitive? | No. For example, the [Frucht graph](https://en.wikipedia.org/wiki/Frucht_graph) is 3 regular but not vertex transitive.
| 4 | https://mathoverflow.net/users/97414 | 254413 | 115,120 |
https://mathoverflow.net/questions/254337 | 8 | Let $\mathcal{O}$ be a finite-dimensional, paracompact, Hausdorff, smooth (and compact, if it helps) orbifold. Is there an isomorphism between the real Čech cohomology and singular cohomology **of the underlying space** $|\mathcal{O}|$?
(NB. The answers of Simon Rose and David C refer to an earlier version of the que... | https://mathoverflow.net/users/93960 | Is the Čech cohomology of an orbifold isomorphic to its singular cohomology? | It's a classical result that if $X$ is paracompact and locally contractible, then singular cohomology and Cech cohomology of $X$ coincide, with coefficients in any abelian group. A reference is Spanier's textbook. This applies in particular to the underlying space of an orbifold, in which case a small neighborhood of a... | 3 | https://mathoverflow.net/users/1310 | 254414 | 115,121 |
https://mathoverflow.net/questions/253963 | 2 | Inspired by [this question](https://mathoverflow.net/questions/250287/g-cocycle-split-to-a-coboundary-in-j-via-a-group-extension), let us take a nontrivial 3-cocycle $\omega\_3^G(g\_a, g\_b, g\_c) \in H^3(G,\mathbb{R}/\mathbb{Z})$ in the cohomology group of $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwor... | https://mathoverflow.net/users/44768 | Trivialize a cup-product 3-cocycle of $G$ in a larger group $J$ | The group $J$ has to be of order divisible by 16, due to the fact that if the order of the group $J$ is $8n$ where $n$ is an odd number, then the inflation of the cocycle will not be trivial (for example due to the fact that the quotient map $r:J\to G$ will split).
There is no one minimal $J$. The reason for this is ... | 3 | https://mathoverflow.net/users/41644 | 254448 | 115,132 |
https://mathoverflow.net/questions/254403 | 0 | [I asked this question first on MSE but there was no activity.](https://math.stackexchange.com/questions/2004494/estimating-pointwise-multiplication-conjugated-by-a-fourier-multiplier)
Let $m(D)$ be a Fourier multiplier and $f$ a known function. I'm trying to estimate the operator
$$Tu=m^{-1}(D)(f(x)m(D)u)$$
in say $... | https://mathoverflow.net/users/70155 | Estimating pointwise multiplication conjugated by a Fourier multiplier | Let's just do some naive things.
1. Your $m(D)$ is not bounded on any $H^s$, due to the exponential growth. In fact, your $m(D)$ is not a bounded map from $H^{s\_1} \to H^{s\_2}$ for any pair $(s\_1,s\_2)$. (I assume you mean the $L^2$ Sobolev space by $H^s$.) [This is mostly a comment on your assertion that you wan... | 1 | https://mathoverflow.net/users/3948 | 254450 | 115,133 |
https://mathoverflow.net/questions/253131 | 3 | Let $k$ be a field. For each $a \in k^\times$ and each $b \in k$, let $g\_{a, b}: k \to k$ be an affine-linear map given by $g\_{a, b}(x) = a \cdot x + b$. The transformations $\{g\_{a, b}, \text{ }a \in k^\times, b \in k\}$ form a group $G(k)$ with respect to the composition operation.
**Question.** What is the clas... | https://mathoverflow.net/users/98682 | Classification of finite-dimensional continuous irreps of affine group up to isomophism? | Assume that $V$ is a finite dimensional irreducible complex representation of the group $G=\{g\_{a,b}\}$. Let $H=\{g\_{1,b}\}$. Then the subgroup $H$ is commutative. As a result (for this we do not even need continuity), the subgroup $H$ fixes a flag inside $V$. Let $W\subseteq V$ be the subspaces spanned by all vector... | 3 | https://mathoverflow.net/users/41644 | 254452 | 115,134 |
https://mathoverflow.net/questions/254440 | 25 | Consider a polynomial $P(X)\in\mathbb Z[X]$. Is it true that $P(N)$ divides $N!$ for infinitely many integer $N$?
This question is motivated by the special case where $P(X) = X^2 + 1$ that appeared in a math olympiad.
I was wondering if anyone can point me to references of this question.
| https://mathoverflow.net/users/88670 | Integral polynomials dividing N! | In general this is an open problem. A closely related problem (essentially equivalent) is to ask for the values $P(n)$ for $n$ of size $X$ to be $X$ smooth (i.e. composed only of primes below $X$). This is known for quadratic polynomials, but already open for general cubic irreducible polynomials. For some special poly... | 16 | https://mathoverflow.net/users/38624 | 254471 | 115,138 |
https://mathoverflow.net/questions/253197 | 5 | Let $G$ be a linear algebraic semisimple group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$, and let $(g,X)\in G\times\mathfrak{g}^{\operatorname{reg}}$ be such that $\mathrm{Ad}\_gX=X$ (where $X\in\mathfrak{g}^{\operatorname{reg}}$ means that $Z\_{\mathfrak{g}}(X)$ has minimal dimension).
>
> **Conjecture.** ... | https://mathoverflow.net/users/90299 | A strong relationship between $\mathrm{ad}(X)$ and $1-\mathrm{Ad}_g$ when $\mathrm{Ad}_gX=X$ | Consider the Killing form on $\mathfrak g$. Since $Ad\_g$ is orthogonal, it is easy to see that $im(1-Ad\_g)^\perp=ker(1-Ad\_g)$. Since $ad\_g$ is anti-selfadjoint, $im(ad(x))^\perp=ker(ad(x))$. Therefore, your two statements are equivalent. Since you know the second holds, so does the first.
| 6 | https://mathoverflow.net/users/2653 | 254474 | 115,140 |
https://mathoverflow.net/questions/254468 | 4 | The classifying space $BDiff(S^1 \times S^1)$ of the diffeomorphism group of the torus is a 2-type with $\pi\_1 = GL(2, \mathbb{Z})$ and $\pi\_2 = \mathbb{Z} \times \mathbb{Z}$ and all higher $\pi\_i$ = 0.
Presumably the action of $\pi\_1$ on $\pi\_2$ is the standard action of $GL(2, \mathbb{Z})$ on $\mathbb{Z} \time... | https://mathoverflow.net/users/401 | What is the first Postnikov invariant of $BDiff(S^1 \times S^1)$? | It is zero, because $BDiff(S^1 \times S^1) \to BGL\_2(\mathbb{Z})$ is split by the standard action of $GL\_2(\mathbb{Z})$ on $S^1 \times S^1 = \mathbb{R}^2/\mathbb{Z}^2$. In other words
$$BDiff(S^1 \times S^1) \simeq B(S^1 \times S^1 \rtimes GL\_2(\mathbb{Z})).$$
| 4 | https://mathoverflow.net/users/318 | 254477 | 115,141 |
https://mathoverflow.net/questions/254478 | 3 | Let $\text{char}(K) = p > 0$, and let $E/K$ be an elliptic curve with $j(E) \notin \overline{\mathbb{F}}\_p$.
Does it follow that $\text{End}(E) = \mathbb{Z}$?
| https://mathoverflow.net/users/nan | $\text{char}(K) = p > 0$, $E/K$ elliptic curve with $j(E) \notin \overline{\mathbb{F}}_p$, follows that $\text{End}(E) = \mathbb{Z}$? | Yes. This seems to be due to Deuring. There are various proofs. You'll find a proof, for example, in Mumford's *Abelian Varieties*, page 217.
| 4 | https://mathoverflow.net/users/11926 | 254480 | 115,143 |
https://mathoverflow.net/questions/254473 | 2 | The following looks like a strengthening of the approximation property, but I don't know, maybe this is equivalent. I would be grateful if somebody could explain this.
Let $X$ be a Banach space with the approximation property, and $K$ a compact set in $X$.
>
> Is it true that for each $\varepsilon>0$ there exist... | https://mathoverflow.net/users/18943 | A variant of the approximation property? | Since every separable closed subspace of $X$ is the closed linear span of a compact set, your property is equivalent to the HAP (hereditary approximation property).
Approximate $K$ by a finite rank bounded linear operator $T$ on the closed linear span of $K$. A small perturbation argument shows that WLOG $T$ ranges i... | 2 | https://mathoverflow.net/users/2554 | 254486 | 115,144 |
https://mathoverflow.net/questions/254372 | 14 | In my understanding the introduction of a paper is a very important section because it not only gives a concise summary of the results, but it is also the only section most people will read at all. Therefore I often ask myself how to arrange it in such a way that, for example,
* the results are easy to grasp,
* the ... | https://mathoverflow.net/users/98306 | Arrangement of the introduction of a paper | To some extent this depends on the nature of your result, but the principle I generally follow is this:
>
> State your result as soon as possible, subject to the constraint that when readers arrive at the statement of your result, they will have some idea of what question you are answering and why.
>
>
>
For e... | 6 | https://mathoverflow.net/users/3106 | 254491 | 115,146 |
https://mathoverflow.net/questions/254490 | 3 | I have trouble in going through a proof in a paper for quite a while. To simplify notation and make it readable to a larger audience, let me just present the simplest case:
Let $\Omega$ be a bounded smooth star-shaped domain in $\mathbb{R}^3$ with $H \ge 0$ everywhere on $\partial \Omega$. Here $H$ is the mean curvat... | https://mathoverflow.net/users/51546 | How to evolve a star-shaped mean convex set to a strictly mean convex set? | Represent $\partial \Omega$ as an embedding $F:\mathbb S^2 \to \mathbb R^3$ and consider the mean curvature flow $\{F\_t :\mathbb S^2 \to \mathbb R^3: t\in [0,T)\}$, which is a family of embeddings so that
$$\frac{\partial F}{\partial t} = H\_t v\_t,.$$
where $v\_t$ is the inward unit normal of the embedding $F\_t... | 5 | https://mathoverflow.net/users/41094 | 254499 | 115,148 |
https://mathoverflow.net/questions/254496 | 3 | To formulate my question I need the construction of the algebra $J^n\_M(K)$ of jets of degree $n$ on a compact set $K$ of a smooth manifold $M$. I'll describe it for the simplest case of $M={\mathbb R}$, $K=[0,1]$ and $n=1$. This is done in two steps.
1. First, we need the construction of the *algebra of polynomials... | https://mathoverflow.net/users/18943 | Does the Banach algebra of jets have the approximation property? | Is $x\_0+\tau x\_0' \mapsto (x\_0'(\cdot), x\_0(0))$ a linear homeomorphism from $J^1$ to $C([0,1])\times \mathbb R$? Wouldn't that imply that $J^1$ inherits the approximation property from $C^0([0,1])$?
| 3 | https://mathoverflow.net/users/101022 | 254514 | 115,155 |
https://mathoverflow.net/questions/254517 | 6 | Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and fix a Weierstrass equation for $E$ having coefficients in $\mathbb{Z}$. Are there infinitely many primes $p \in \mathbb{Z}$ such that the reduced curve $E/\mathbb{F}\_p$ has Hasse invariant $1$?
| https://mathoverflow.net/users/101031 | Infinitely many primes $p \in \mathbb{Z}$ where reduced curve $E/\mathbb{F}_p$ has Hasse invariant $1$? | Yes. In fact, Elkies and Serre [1] have independent proofs that for every $\epsilon>0$ there is a constant $C\_\epsilon>0$ such that
$$
{\#\{p\le X : E/\mathbb{F}\_p \text{ is ordinary}\}} \ge C\_\epsilon
X^{3/4-\epsilon}.
$$
But if you just want to know that there are infinitely many ordinary primes, there is an elem... | 9 | https://mathoverflow.net/users/11926 | 254520 | 115,157 |
https://mathoverflow.net/questions/251694 | 0 | Can anyone point out (if any) references which contain any possible links between association schemes and matroids?
| https://mathoverflow.net/users/nan | Any existing relations between association schemes and matroids | Check out Section 6.4 in [Network Coding](https://books.google.at/books?id=_JyPg18-XRMC&pg=PT193&lpg=PT193&dq=%22network%20coding%22%20al%20agha&source=bl&ots=8Lh4I-LaGz&sig=fjG_kAYkUG63j_TsEviUphuNDmY&hl=en&sa=X&ved=0ahUKEwio2Pyd5qPQAhUH7hoKHdHbALAQ6AEIMzAD#v=onepage&q=%22association%20scheme%22&f=false) and Section 2... | 0 | https://mathoverflow.net/users/94968 | 254534 | 115,162 |
https://mathoverflow.net/questions/254188 | 6 | Setup:
-------
Suppose $X\_t$ solves the SDE
$$
dX\_t = \mu(t,X\_t)dt +\sigma(t,X\_t)dZ\_t,
$$
where $Z\_t$ is a Lévy process on $\mathbb{R}^d$, $g(t,s,x):[0,T]\times[0,1]\times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is $C^{1,2,2}$ and $S\_t$ is an SDE solving
$$
dS\_t = a(t,S\_t,X\_t)dt + b(t,S\_t,X\_t)dY\_t,
$$
wh... | https://mathoverflow.net/users/94001 | SDE for conditioning on subfiltration | You can obtain an S*P*DE (or, rather, an S*I*DE - stochastic integro-differential equation) for the conditional density of $S$ given the history of $X$. From here you can calculate the quantity you're looking for easily.
In the continuous case (i.e. $Z$ and $Y$ are Wiener processes), there are plenty of references, l... | 4 | https://mathoverflow.net/users/100941 | 254538 | 115,163 |
https://mathoverflow.net/questions/254532 | 4 | I'm looking for the tightest upper bound on the number of different binary matrices $A \in {\{-1,1\}^{m \times n}}$ for which $\mathrm{rank}(A)\leq r$. I'm interested in the regime $1 \ll r \ll m \leq n$.
I wonder if there is a way to get a better bound than $2^{mr+nr-r^2} m !$, which is the best bound I could derive... | https://mathoverflow.net/users/44790 | Upper bound on the number of binary matrices with small rank | The following observation goes back to Komlos's work in the 1960's deriving bounds on the probability that a $(0,1)$ matrix is singular:
Any such $A$ must have a collection of $r$ rows that contain the remainder in their span. By paying a $\binom{m}{r}$ multiplicative factor, we can assume that they are the first $r$... | 4 | https://mathoverflow.net/users/405 | 254541 | 115,165 |
https://mathoverflow.net/questions/253954 | 7 | I have attached [pg. 275](https://i.stack.imgur.com/nZRSh.png) and [pg. 276](https://i.stack.imgur.com/YY7nI.png) of [BS91]. My concern is with the claim (2.7) on pg. 276. To prove this claim, I require the following additional assumption, which is not made by the authors:
**Assumption**: $u^{\rho}$ is upper semicont... | https://mathoverflow.net/users/35874 | Discontinuity of solutions to approximation schemes in the Barles-Souganidis framework | After checking the arguments with @Jeff, I am willing to conclude that the issue in the question above is an extremely minor error in the original paper [BS91]. Both Jeff and I came up with "fixes" for the case of $u^{\rho}$ not continuous. Both are detailed below.
**Update**: I had the chance to speak to Professor B... | 2 | https://mathoverflow.net/users/35874 | 254542 | 115,166 |
https://mathoverflow.net/questions/254535 | 1 | Let $\mathbb{F}$ be a finite field and $S\subset\mathbb{F}\_{\leq d}[x,y]$, a set of bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$. Assume the linear span of $S$ is all of $\mathbb{F}\_{\leq d}[x,y]$. Let $L$ be the set of one-dimensional lines in $\mathbb{F}^2$. Suppose the following pro... | https://mathoverflow.net/users/90531 | question about sets of polynomials with a special agreement guarantee | Still, a required estimate is hopeless. Set $q=|\mathbb F|$.
Take into $S$ all polynomials of the form $k(\ell x+my+n)^d$ with $k,\ell,m,n\in\mathbb F$ (there are in fact $\Omega(q^3)$ such polynomials). They satisfy all the requirements. Indeed, if $ux+vy+w=0$ is an equation of some affine line, then $k(\ell x+my+n)... | 3 | https://mathoverflow.net/users/17581 | 254547 | 115,169 |
https://mathoverflow.net/questions/254518 | 9 | Before formulating my question, let me briefly sum up what I know about the topic (feel free to correct me if something I claimed is false!). This is for you good to see what my state of knowledge is, and maybe some people could learn something.
This is what I know
===================
An interesting problem in logi... | https://mathoverflow.net/users/nan | What exactly is a judgement? | I highly recommend reading Martin-Löf's paper referenced by Ulrik Buchholtz in the comments to your question. Apart from that, here are a couple of point that might help, some of which were already made by Andreas Blass in his answer.
A judgement is an act of knowing, or asserting a piece of knowledge about a mathema... | 10 | https://mathoverflow.net/users/1176 | 254548 | 115,170 |
https://mathoverflow.net/questions/254552 | 18 | Let $k$ be a field, $\overline{k}$ its algebraic closure, and $A$ an abelian variety defined over $k$, of dimension $g$. For each integer $m \ge 1$, let $A\_m$ denote the group of elements $a \in A(\overline{k})$ such that $ma = 0$. let $l$ be a prime number different from the characteristic of $k$, and let $T\_l(A)$ d... | https://mathoverflow.net/users/nan | On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main result? | I don't have any contribution for the intuition beyond the fact that, I can't construct something outside the image of (1) so I hope it's surjective.
Here is a sketch of the central idea of Tate's proof. Consider $A = A' \times A''$ and try to find an endomorphism of $A$ from an endomorphism of its Tate module. The e... | 23 | https://mathoverflow.net/users/2290 | 254555 | 115,173 |
https://mathoverflow.net/questions/254571 | 10 | I've read many times that moving coframes where a convenient tool for computations in Riemannian Geometry, especially on surfaces, but never really used it. Lately to get a better feel of the method, I've tried to reprove classical results on surfaces using moving coframes and the Cartan structure equations :
>
> *... | https://mathoverflow.net/users/8887 | Bochner's formula on surfaces using moving coframes | The straightforward way to do this is to note that we have, for some $u\_1$ and $u\_2$,
$$
\mathrm{d}u = u\_1\,\eta^1 + u\_2\,\eta^2
$$
Taking the exterior derivative of this and using Cartan's Lemma then shows that,
for some $u\_{11}$, $u\_{12}$, and $u\_{22}$,
$$
\begin{aligned}
\mathrm{d}u\_1 &= \phantom{-}u\_2\,\om... | 13 | https://mathoverflow.net/users/13972 | 254578 | 115,175 |
https://mathoverflow.net/questions/254114 | 26 | Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}\_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably a well-known question, but I found nothing in Google:
$$\text{Why not a scheme $\mathrm{Sub}\_X(E)$ parametrizing ar... | https://mathoverflow.net/users/43639 | Why there is a Quot-scheme, not a Sub-scheme? | For standard universal properties, you need the scheme to behave well under base change, which in these cases would mean tensor products. Tensor product is right exact, so a quotient remain a quotient, not left exact, so a sub may not remain a sub.
| 21 | https://mathoverflow.net/users/9502 | 254579 | 115,176 |
https://mathoverflow.net/questions/254564 | 4 | I'm reading the book "Principles of Algebraic Geometry". After proving the statement
*Every meromorphic function on an algebraic variety $V \subset \mathbb{P}^n$ is rational*
Griffith and Harris say that it is not hard to see(p 170):
1. Any meromorphic differential form on a smooth variety is algebraic.
2. Any ho... | https://mathoverflow.net/users/40042 | Application of the G.A.G.A. principle | I'll address the question about rational differential forms, since that one is a bit more subtle (for reasons indicated below). As you note, the global nature of meromorphicity and compactness need to be used in some way (i.e., if we merely analytify the sheaf of rational functions we do **not** get the sheaf of meromo... | 7 | https://mathoverflow.net/users/81332 | 254580 | 115,177 |
https://mathoverflow.net/questions/254589 | 1 | The [infinite Ramsey theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem#Infinite_Ramsey_theorem) implies that, if we color the $n$-element subsets of $N:=\{0,1,2,\ldots\}$ in a finite number of colors, then there will exist an *infinite* subset $A\subseteq N$ such that all $n$-element subsets of $A$ will be of t... | https://mathoverflow.net/users/61113 | Infinite Ramsey theorem for strings (instead of sets)? | Color each vector $(x\_i)$ in any $k$ such that $x\_k=\max\limits\_{1\leq i\leq n} x\_i$. Then there is no dominating monochromatic set: if it had color $k$, then there is no $S$-dominating bector for $k\notin S$.
| 3 | https://mathoverflow.net/users/17581 | 254590 | 115,179 |
https://mathoverflow.net/questions/253887 | 4 | I was trying to understand [this interesting question](https://mathoverflow.net/questions/250287/g-cocycle-split-to-a-coboundary-in-j-via-a-group-extension) by example.
Let me follow their previous discussion and ask: Let a generic nontrivial 2-cocycle $\omega\_2^G(g\_1,g\_2) \in H^2(G,\mathbb{R}/\mathbb{Z})$ in the ... | https://mathoverflow.net/users/44768 | $SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$? | Let $G$ be a group, and let $$1\to A\to J\to G\to 1$$ be an extension of groups with an abelian kernel. Choose a set-theoretical lifting $s:G\to J$ of the quotient map $p:J\to G$. Now define a function $\beta:G^2\to A$ by the formula $$\alpha(g,h) = s(g)s(h)s(gh)^{-1}.$$
This formula defines a two cocycle, and this is ... | 3 | https://mathoverflow.net/users/41644 | 254606 | 115,180 |
https://mathoverflow.net/questions/249907 | 8 | For a finite group $G$, the prime graph of $G$ is an undirected graph such that its vertices are all prime divisors of $\vert G\vert$ and two distinct vertices $p$ and $q$ are adjacent when there is an element in $G$ of order $pq$.
Which finite groups does have a complete prime graph? i.e I am looking for finite gro... | https://mathoverflow.net/users/97247 | Finite groups whose prime graphs are complete | To understand the "minimal troublemakers" in the case of solvable groups, I would start with a solvable group $G$ such that every proper section of $G$ has a complete prime graph but $G$ does not. Recall that a section of $G$ is a group $X/Y$ where $Y \lhd X$ and $X$ is subgroup of $G$.
Such a group $G$ must be a $\{... | 3 | https://mathoverflow.net/users/14450 | 254608 | 115,181 |
https://mathoverflow.net/questions/254605 | 2 | **Background:**
A pointed object $X$ in a category $C$ with terminal object $\*$ is a map $\*\rightarrow X$. Such objects with basepoint-preserving maps form their own category of pointed objects $C^{\*/}$. There is a canonical forgetful functor $U:C^{\*/}\rightarrow C$ that forgets the basepoint. Furthermore, this h... | https://mathoverflow.net/users/84563 | "Maybe Monad" for multi-pointed objects? | Let $\mathcal{C}$ be a category with coproducts and $S \in \mathcal{C}$. Then we have the slice category $S/\mathcal{C}$. The objects of it are morphisms $S \to X$, where $X$ is an object of $\mathcal{C}$. This generalizes your construction. There is a forgetful functor $S/\mathcal{C} \to \mathcal{S}$ mapping $(S \to X... | 6 | https://mathoverflow.net/users/98306 | 254621 | 115,185 |
https://mathoverflow.net/questions/254617 | 5 | Let $E/F$ be a quadratic extension of number fields and $v$ is a place of $F$.
Let $\chi\_1,\chi\_2$ be the unramified characters of $F\_v^{\times}$.
If $B(\chi\_1,\chi\_2)$ is the unramified principal series representation of $GL\_2(F\_v)$, what is the $BC(\pi)$, the base change of $\pi$ to $GL\_2(E\_v)$?
I supp... | https://mathoverflow.net/users/29422 | Explicit formula of base change for GL(n) | The answer to your final question should be yes.
Let me assume that $v$ is inert in $E$, so that $E\_v / F\_v$ is a honest quadratic extension.
Assuming that $B(\chi\_1, \chi\_2)$ is irreducible, it corresponds via local Langlands (a theorem for $\mathrm{GL}\_2(K)$ and any $p$-adic field $K$) to the 2-dimensional rep... | 5 | https://mathoverflow.net/users/101091 | 254622 | 115,186 |
https://mathoverflow.net/questions/254612 | 10 | For which fields $k$ does the following hold for all $n \geq 1$? Let $(a\_1,\ldots,a\_n) \in k^n$. Then there exists a polynomial $f(x\_1,\ldots,x\_n) \in k[x\_1,\ldots,x\_n]$ such that $f(a\_1,\ldots,a\_n) = 0$ but $f(b\_1,\ldots,b\_n) \neq 0$ for all $(b\_1,\ldots,b\_n) \neq (a\_1,\ldots,a\_n)$?
This is clearly imp... | https://mathoverflow.net/users/101081 | Fields for which there exist multivariable polynomials vanishing at single specified point | If $k$ is not algebraically closed, such a polynomial always exists (the opposite is also true and is mentioned in the post).
We may assume that $a\_i=0$ for all $i$. Take an irreducible polynomial $g(x)$ of degree $d>1$, then for the homogeneous form $G(x,y)=y^dg(x/y)$ we have $G(x,y)=0$ if only if $x=y=0$. This so... | 21 | https://mathoverflow.net/users/4312 | 254625 | 115,188 |
https://mathoverflow.net/questions/254495 | 1 | Let $(\epsilon\_t)\_t$ be a sequence of iid random variables, distributed according to the density $f:\mathbb{R}\to (0,\infty)$ and
$$
x\_t = q( \theta^\star, x\_1,x\_2, \ldots, x\_{t-1}) + \epsilon\_t \,.
$$
Assume that the derivative of $q$ with respect to $\theta$ is bounded from below, i.e. for all $\theta, x$
$$
... | https://mathoverflow.net/users/82510 | convergence of Bayesian posterior with non iid data | Probably not true just with the assumptions you have there. It seems to accommodate a model in which $\theta$ is the starting point of a random walk. That would only require $q = \theta + x\_1 + ... + x\_t$. It is known and easy to believe that in this case you cannot estimate $\theta $ consistently, and I would think ... | 1 | https://mathoverflow.net/users/nan | 254629 | 115,189 |
https://mathoverflow.net/questions/254632 | 2 | I would like to compute
$$\int\_X \exp\left(-\frac{1}{2}(Au)^2\right)\mathrm d\mu\_0(u)$$
with a linear and continuous operator on a Banach space $A:X\to \mathbb R$ (in my case $X=C([0,1])$) and $\mu\_0$ a centred Gaussian measure with covariance operator $Q$ on $X$. Is there any hope this can be calculated explicitly?... | https://mathoverflow.net/users/88505 | Normalization of Gaussian w.r.t. Gaussian in a Banach space | Assume $\mu\_0$ is centered. The covariance **form** (not operator) of $\mu\_0$ is a bilinear form $Q$ on the dual $X^\*$. $\newcommand{\bR}{\mathbb{R}}$ Regard the continuous linear functionals $\alpha:X\to\bR$ as random variables on the probability space $(X,\mu\_0)$. Then
$\newcommand{\bE}{\mathbb{E}}$
$$Q: X^\*\... | 4 | https://mathoverflow.net/users/20302 | 254636 | 115,190 |
https://mathoverflow.net/questions/250017 | 11 | Given rational numbers $a\_1,\ldots, a\_k$ and $u\_0, \ldots, u\_k$, let $(u\_n)\_{n \geq k}$ be the linear recurrence defined by
$$u\_n := a\_1 u\_{n-1} + \cdots + a\_k u\_{n-k}, \text{ for } n \geq k .$$
Obviously, $u\_n \in \mathbb{Q}$ for any $n \geq 0$.
My question is: *Is there an effective way to decide if it ... | https://mathoverflow.net/users/nan | Are the terms of a linear recurrence integral? | The problem is effectively decidable. To test whether $u\_n$ is eventually integral, first use the recurrence relation for $u\_n$ to construct relatively prime polynomials $A,B\in \mathbb{Z}[x]$ such that the rational function $A/B$ has power series expansion $\sum\_nu\_nx^n$. (Here *relatively prime* will always mean ... | 13 | https://mathoverflow.net/users/5229 | 254638 | 115,191 |
https://mathoverflow.net/questions/228990 | 22 | Suppose that $M$ is a closed Riemannian manifold with bounded geometry, i.e., curvature between $-1$ and $1$ and injectivity radius at least $1$. Since $M$ is a smooth manifold, it has a triangulation. Does it necessarily have a triangulation that is "nice" with respect to the metric?
For instance, is there an $\epsi... | https://mathoverflow.net/users/25051 | Does a Riemannian manifold have a triangulation with quantitative bounds? | A series of papers has been published (or rather is being published) on this topic, staring with Stability of Delaunay-type structures for manifolds (see <http://dl.acm.org/citation.cfm?id=2261284>) by Boissonnat, Dyer, and Ghosh. I think the paper in the series you might find most useful is Delaunay triangulation of m... | 5 | https://mathoverflow.net/users/101109 | 254643 | 115,192 |
https://mathoverflow.net/questions/254635 | 5 | I was thinking in solving the following problem for the general case :
\*\*) Given a list of pairs $((n\_i, A\_i))\_{i=1}^k$, where for each $i$ we have that $n\_i $ is a non-negative integer, and $A\_i$ is a set, does there exist a set $A$ such that $|A\cap A\_i|=n\_i$?
Example 1 : Given as an input the list : $(3... | https://mathoverflow.net/users/95470 | a question about complexity of Boolean problem | The problem is visibly in NP, as we can check in polynomial time whether a given set $A$ is a correct witness.
In fact, the problem is NP-complete, and one can show this by reduction from the NP-complete problem [1-in-3-SAT](https://en.wikipedia.org/wiki/Boolean_satisfiability_problem#Exactly-1_3-satisfiability): giv... | 2 | https://mathoverflow.net/users/12705 | 254645 | 115,193 |
https://mathoverflow.net/questions/254669 | 24 | What is some current research going on in the foundations of mathematics about?
Are the foundations of mathematics still a research area, or is everything solved? When I think about foundations I'm thinking of reducing everything to ZFC set theory + first order logic. Is this still contemporary?
| https://mathoverflow.net/users/101121 | What is some current research going on in foundations about? | It is quite difficult to answer this question comprehensively. It's a bit like asking "so what's been going on in analysis lately?" It is probably best if logicians who work in various areas each answer what is going on in their area. I will speak about logic in computer science. I am very curious to see what logicians... | 38 | https://mathoverflow.net/users/1176 | 254673 | 115,201 |
https://mathoverflow.net/questions/254563 | 4 | In symplectic geometry, Darboux's theorem says that locally, any symplectic manifold of dimension $2n$ looks like symplectic Euclidean space (that is, there is some set of coordinates $(x\_i, y\_i)$ such that the symplectic form looks like $\sum dx\_i \wedge dy\_i$. Equivalently (I think), there are no local invariants... | https://mathoverflow.net/users/44191 | Kahler version of Darboux's Theorem | Let $(M,g,J)$ be a Kahler manifold. This means that $J$ is a (integrable) complex structure, so that $(M,J)$ is a complex manifold, $g$ is a smooth metric such that $J$ is $g$-orthogonal, i.e.
$g(JX,JY) = g(X,Y)$
for any smooth vector fields $X$, $Y$ on $M$ and, moreover, the Kahler form is closed.
Ok, now let us ass... | 8 | https://mathoverflow.net/users/81645 | 254674 | 115,202 |
https://mathoverflow.net/questions/254302 | 6 | The Maurey-Pisier theorem states that if $p\_X$ is the supremum of those $p$ such that the Banach space $X$ has Rademacher type $p$, then $\ell\_{p\_X}$ is finitely representable in $X$.
For $1\leq p<\infty$, let us say the Schauder basis $(e\_i)\_{i=1}^\infty$ has \emph{block type} $p$ if there exists a constant $C... | https://mathoverflow.net/users/100897 | Block version of Maurey Pisier theorem | I believe that the answer is NO. I do not know a counterexample, however, in positive direction one can do the following.
If the basis is unconditional you can define type/cotype on disjointly supported vectors. Then the corresponding result is true. This is written [here](http://www.math.unt.edu/~bunyamin/pdf/envelo... | 2 | https://mathoverflow.net/users/3675 | 254675 | 115,203 |
https://mathoverflow.net/questions/254666 | 2 | I want to see the precise statement and a proof for a theorem of Stoilow on "inner" functions (I do not know what this exactly means, I suppose it is an open map with other natural properties). A vague statement is this:
An inner function is a continuous deformation of a holomorphic function.
I am looking for a si... | https://mathoverflow.net/users/83948 | Stoilow Theorem | First of all, it seems that every author has his own favourite definition of "inner mapping". The best thing would probably be to adopt [the one used by Martin Jurchescu](https://books.google.com/books?id=8v7sCgAAQBAJ&pg=PA420): a continuous, open, zero-dimensional mapping.
Stoilow's theorem says, then, that if $\phi... | 2 | https://mathoverflow.net/users/54780 | 254679 | 115,204 |
https://mathoverflow.net/questions/254664 | 2 | I will use a local coordinate formalism here, since this is related to research in general relativity, and my supervisor only tolerates local coordinate formalisms. Plus the research papers I base my research on also use local coordinates.
If $M$ is an $n$ dimensional $C^\infty$ manifold, with local coordinates $(x^\... | https://mathoverflow.net/users/85500 | Induced connection on null hypersurfaces | The simple answer is:
**You can't project to a null hypersurface**
This is tied to the fact that the "Lorentzian normal" vector field is in fact a tangent vector field.
---
In more details:
Start with a tangent vector field $T^\nu$ to the image of $\Sigma$ in your Lorentzian manifold. If you lower the in... | 4 | https://mathoverflow.net/users/3948 | 254680 | 115,205 |
https://mathoverflow.net/questions/254683 | 0 | Not really research level but here goes anyway: Suppose I have a topological space $X$ with a closed subset $K$ and a continuous map $f : X \times [0,1) \to X$ such that:
0) $f(x,0) = x$.
1) For all $k \in K, 0 \leq t < 1$, $f(k,t) \in K$.
2) For all $x \in X$, and all sequences $\{t\_i\} \to 1$, there exists a s... | https://mathoverflow.net/users/40460 | Technical but elementary homotopy question | No. Let $X$ be the topologist's sine curve, i.e. $X \subset \mathbb{R}^2$ is $\{0\} \times [-1,1] \cup \{(x,\mathrm{sin}(1/x)\mid x \in (0,1)\}$, and set $K = \{0\} \times [-1,1]$. Let $f\_t$ fix $K$ for all times, while $f\_t(x,\mathrm{sin}(1/x) = (x\_t,\mathrm{sin}(1/x\_t))$ where $x\_t = \min(x,1-t)$. Then the homot... | 6 | https://mathoverflow.net/users/2362 | 254689 | 115,208 |
https://mathoverflow.net/questions/254641 | 3 | It is fairly easy to see from the formalism of the L group that the L group of a quasisplit unitary group will be a nontrivial semidirect product of $GL\_n(\mathbb{C})$ (for the appropriate value of $n$), and a group of order two. When $n$ is odd this determines the isomorphism class uniquely. When $n$ is even, it does... | https://mathoverflow.net/users/21252 | Why is the L group of an even unitary group what it is? | There are three separate issues here: exactly which unitary or special unitary groups you're thinking about (in effect: exactly which non-degenerate hermitian spaces), how to define $L$-groups in general, and how that definition works out in the quasi-split case.
In general, if $G$ is a pinned split connected reducti... | 5 | https://mathoverflow.net/users/81332 | 254692 | 115,210 |
https://mathoverflow.net/questions/254715 | 0 | Let $H=(V,E)$ be a hypergraph, let $n\in\mathbb{N}$. A *vertex coloring* is a map $c: V\to \{1,\ldots, n\}$ such that for $v\neq w \in V$ we have $c(v)\neq c(w)$ whenever there is $e\in E$ such that $v, w\in e$. We call the least $m\in\mathbb{N}$ such that there is a coloring map from $V$ to $\{1,\ldots,m\}$ the *chrom... | https://mathoverflow.net/users/8628 | Maximizing the chromatic number of regular hypergraphs | The answer is $n^{3/2}(1+o(1))$.
For the upper bound, observe that there are at most $n\cdot{n\choose 2}$ pairs of vertices lying in one edge. Thus, if $|V|>n^{3/2}(1+o(1))$, then there are two vertices not belonging to an edge, and we may identify them harmlessly (they will be of the same color). Repeating the proce... | 3 | https://mathoverflow.net/users/17581 | 254716 | 115,217 |
https://mathoverflow.net/questions/254717 | 8 | For any real square matrix $A$ there is an invertible matrix $P$ such that $A^t = P^{-1}AP$. I have two binary ($0,1$) matrices $A$ and $B$. When does there exist a $P$ such that $A^t = P^{-1}AP$ and $B^t = P^{-1}BP$ hold simultaneously? I am particularly looking for some easy conditions on these matrices $A$ and $B$.
... | https://mathoverflow.net/users/36977 | When are two binary matrices simultaneously equivalent to their transpose? | Clearly, a necessary condition is that for every word $w$ in two letters, one has
$${\rm Tr}\,w(A^t,B^t)={\rm Tr}\,w(A,B).$$
Equivalently,
$${\rm Tr}\,\hat w(A,B)={\rm Tr}\,w(A,B),$$
where $\hat w$ is the reverse word. Namely, if $w=x^\ell y^mx^n\cdots$, then $\hat w=\cdots x^ny^mx^\ell$.
Unless the word $\hat w$ be co... | 6 | https://mathoverflow.net/users/8799 | 254723 | 115,218 |
https://mathoverflow.net/questions/254678 | 1 | Assume we have the projective plane $\mathbb{A}^2=Spec(\mathbb{C}[r,s])$. Now take the projective plane over this affine plane $\mathbb{P}^2\_{\mathbb{A}^2}$ with homogenous coordinates $[u:v:w]$.
Define a threefold $Y$ by the vanishing of $u^2+rv^2+sw^2$ in $\mathbb{P}^2\_{\mathbb{A}^2}$, i.e. $Y=V(u^2+rv^2+sw^2)$.
... | https://mathoverflow.net/users/70593 | How to test if these two threefolds are birationally equivalent? | These $\mathbb{C}$-varieties are both rational threefolds, so they are birationally equivalent. First, in $Y$ consider the open subset $$D\_+(w) = \text{Spec} \ \mathbb{C}[u/w,v/w,r,s]/\langle (u/w)^2 + r(v/w)^2 + s \rangle.$$ The linear projection of $D\_+(w)$ to $\mathbb{A}^3$ by $(u/w,v/w,r,s)\mapsto (u/w,v/w,r)$ is... | 3 | https://mathoverflow.net/users/13265 | 254727 | 115,220 |
https://mathoverflow.net/questions/254711 | 4 | Hardy's uncertainty principle states that a real function $f$ and its Fourier transform $\widehat{f}$ may not both decay faster at infinity than the standard Gaussian $e^{-\pi t^2}$, unless $f = 0$. In a sense, $e^{- \pi t^2}$ is the closest a non-zero function can come to having both $f$ and $\widehat{f}$ almost compa... | https://mathoverflow.net/users/26522 | Is there an uncertainty principle for Fourier pairs everywhere dominated by $t^{-A}$? | Using compactness arguments one can show that such an $A$ exists, although this argument does not make it easy to extract an effective value for $A$ (but presumably one of the real variable proofs of the Hardy uncertainty principle, such as the one in [this blog post of mine](https://terrytao.wordpress.com/2009/02/18/h... | 5 | https://mathoverflow.net/users/766 | 254729 | 115,222 |
https://mathoverflow.net/questions/254660 | 6 | Let $f: X\to Y$ be a submersion between projective manifolds. Then $Rf^i\_\*(\mathbb{C}\_X)$ are local systems on $X$, for all $i$.
My question is whether the converse is true. More precisely, let $f: X\to Y$ be a holomorphic map between projective manifolds. Suppose that for all $i$, $Rf^i\_\*(\mathbb{C}\_X)$ are g... | https://mathoverflow.net/users/78808 | When is a proper map between smooth varieties a submersion? | I am just posting my comment above as an answer. There are counterexamples that depend on the fact that the divisible coefficient sheaf $\mathbb{C}\_X$ (for the analytic topology, presumably) does not detect certain torsion phenomena. Begin with a smooth, projective, geometrically integral curve $E$ of genus $1$ and a ... | 6 | https://mathoverflow.net/users/13265 | 254731 | 115,223 |
https://mathoverflow.net/questions/254739 | 5 | I'm trying to prove the following statement:
>
> Let $F\dashv U\colon {\cal C}\leftrightarrows {\cal D}$ be an adjunction, and $G \colon {\cal C}^\text{op}\times{\cal D}\to \cal E$ a functor; then there is an isomorphism
> $$\tag{$\star$}
> \int\_{C\in\cal C} G(C,FC) \cong \int\_{D\in\cal D} G(UD,D)
> $$
>
>
>
... | https://mathoverflow.net/users/7952 | A criterion for $F,U$ to be adjoint | Yes, this is true if we assume that the isomorphisms are natural in $G$. We may even restrict to $\mathcal{E}=\mathsf{Set}$. Here is a sketch of the proof.
Consider $G=\mathrm{Hom}\_{\mathcal{D}}(F(-),-) : \mathcal{C}^{op} \times \mathcal{D} \to \mathsf{Set}$. Then
$$\int\_x G(x,Fx) = \int\_x \mathrm{Hom}\_{\mathcal{... | 7 | https://mathoverflow.net/users/98306 | 254744 | 115,225 |
https://mathoverflow.net/questions/254718 | 4 | The Baire's simple limit theorem states that if the functions $f\_n : \mathbb{R} \to \mathbb{R}$ are continuous and converge everywhere to a function $f$ then $f$ has a dense set of continuity points. We say that $f$ is of Baire class one when $f$ is a limit everywhere of continuous functions.
In the following, "mea... | https://mathoverflow.net/users/94415 | Baire's simple limit theorem "almost everywhere" | No.
Your first hypothesis is true for any Lebesgue measurable function $f$; there is a sequence of continuous functions $f\_n$ such that $f\_n \to f$ almost everywhere. This is standard.
But there do exist Lebesgue measurable (even Borel) functions which are not a.e. equal to any function of Baire class 1.
For in... | 2 | https://mathoverflow.net/users/4832 | 254752 | 115,229 |
https://mathoverflow.net/questions/254506 | 3 | We say that a set of varieties $S$ lives in a **bounded family** if there exists a projective morphism $\mathcal{X} \to T$ between varieties of finite type, such that for any $X \in S$, there exists a closed point $t \in T$, such that its fibre $\mathcal{X}\_t$ is isomorphic to $X$.
It seems that the following two fa... | https://mathoverflow.net/users/29730 | Finiteness of Gorenstein indexes and volumes for varieties in a bounded family | By Noetherian induction, it suffices to show that indicies and volumes are bounded over an open subset of any irreducible component of $T$.
We may assume that $T$ is smooth and there is a dense set $\{t\_i\}\subset T$ such that the corresponding fibers $\mathcal X\_{t\_i}$ are normal. By EGA IV Theorem 12.2.4(iv), aft... | 1 | https://mathoverflow.net/users/19369 | 254757 | 115,231 |
https://mathoverflow.net/questions/254759 | 5 | Let $E$ be an elliptic curve over a field $K$. I am mostly interested in the case $K$ is a number field or a local field but the following question is valid for all $K$.
Let $p$ be any prime power which is comprime to the characteristics of $K$. Galois cohomology gives a cup product map
$$H^{1}(K,E[p]) \times H^{1... | https://mathoverflow.net/users/nan | A cup product in Galois cohomology of Elliptic curve | One can use the exact sequence
$$ 0 \to E(K)/mE(K) \to H^1(K,E[m]) \to H^1(K,E)[m] \to 0 $$
to define a pairing
$$ E(K)/mE(K) \times H^1(K,E)[m] \to H^2(K,\mu\_m) $$
by taking $(Q,\xi)$ to $\phi(Q'\cup \xi')$, where $Q'$ is the image of $Q$
in $H^1(K,E[m])$ and $\xi'$ is any pullback of $\xi$ to $H^1(K,E[m])$.
For $K... | 9 | https://mathoverflow.net/users/11926 | 254761 | 115,232 |
https://mathoverflow.net/questions/254764 | 1 | I wonder if the following graph problems have been studied and have names.
**Problem(s).**
Given two $n$-vertex unlabeled graphs $G\_1$ and $G\_2$, find their maximum/minimum edge intersection. That is find two labeled graphs $H\_1 = ([n], E\_1)$ and $H\_2 = ([n],E\_2)$ such that $H\_1 \simeq G\_1$, $H\_2 \simeq G\_2... | https://mathoverflow.net/users/83519 | Maximum/minimum intersection of two graphs | This is essentially the [Maximum common edge subgraph problem](https://en.wikipedia.org/wiki/Maximum_common_edge_subgraph), which is at least as hard as the subgraph isomorphism problem.
| 2 | https://mathoverflow.net/users/19029 | 254767 | 115,235 |
https://mathoverflow.net/questions/254724 | 8 | For which $n$ has the $S^1$-equivariant rational cohomology of $LBSO(n)$ been computed? Here, $SO(n)$ means the isometry group of the round sphere (preserving the orientation), $B$ stands for *classifying space* and $L$ for *free loop space*. The $S^1$-action on $LBSO(n)$ is, of course, via rotating the loops.
There ... | https://mathoverflow.net/users/14233 | Rational cohomology of $LBSO(n)$ | 1st method) If $X$ is a 1-connected then you can use the fact that you have an isomorphism of graded algebras (J.D.S. Jones "Cyclic homology and equivariant homology" Inventiones Math. 1987):
$$H^n\_{S^1}(LX;\mathbb{Q})\cong HC\_{-n}^{-}(S^\*(X;\mathbb{Q}))$$
where the right term is the negative cyclic homology of ... | 6 | https://mathoverflow.net/users/27816 | 254768 | 115,236 |
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