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https://mathoverflow.net/questions/257308 | 6 | Let $\Sigma^\infty: Top\_\* \to Spectra$ be a functor sending a pointed topological space $X$ to its suspension spectrum, that is $(\Sigma^\infty X)\_n=\Sigma^nX$ with isomorphisms $\Sigma(\Sigma^\infty X)\_n \to (\Sigma^\infty X)\_{n+1}$. Is it true that this map make $Top\_\*$ a full subcategory of $Spectra$, that is... | https://mathoverflow.net/users/43639 | Do topological spaces form a full subcategory of spectra? | As noted in the comments, this is most definitely false in homotopy categories: the set $[S^1, S^0]$ of homotopy classes of based maps is trivial, but the set $[\Sigma^\infty S^1, \Sigma^\infty S^0]$ of maps in the homotopy category of spectra is $\Bbb Z/2$.
The statement about the suspension spectrum functor being f... | 13 | https://mathoverflow.net/users/360 | 257320 | 116,213 |
https://mathoverflow.net/questions/257310 | 25 | Hope that the following soft question is still appropriate on MathOverflow. I was wondering if there is any communal protocol or etiquette with regard to the resubmission of a research paper after it has been superseded by another as yet unpublished paper.
Here's the situation in detail. Suppose that a paper of yours... | https://mathoverflow.net/users/nan | Would you resubmit a research paper after it has been superseded by another as yet unpublished paper? | This is a tricky situation, though just one spin off of the current absurd state of academic publishing, and particularly the preposterously slow and uneven peer review process in math. My personal feeling is that you should try about as hard to get paper A published as you would any other paper (though strategically, ... | 32 | https://mathoverflow.net/users/66 | 257323 | 116,215 |
https://mathoverflow.net/questions/255736 | 4 | The answer to this question might be known, but I don't know of any reference and will appreciate any references.
Let $n>0$. By Snaith splitting there is a stable splitting of $\Omega^kS^{n+k}$ into wedge of spaces $D\_{k,r}S^n=F(\mathbb{R}^k,r)\ltimes\_{\Sigma\_r}(S^n)^{\wedge r}$ as
$$\Sigma^\infty\Omega^kS^{n+k}... | https://mathoverflow.net/users/51223 | Stable summands of $\Omega^kS^{n+k}$ | The full story is eluded to in the comments of Arone and Rognes. First of all, $D\_{k,r}\Sigma^n X$ is always an $n$-fold suspension. More exciting is that, for all pairs $(k,r)$, there is a natural number $d = d(k,r)$, such that
$D\_{k,r} \Sigma^dX = \Sigma^{rd}D\_{k,r}X$.
So what is $d$? It is the order of the cano... | 7 | https://mathoverflow.net/users/102519 | 257328 | 116,216 |
https://mathoverflow.net/questions/257315 | 6 | I need a reference for the following result (which I can prove myself, but my proof is rather ugly and I would prefer to just cite the statement instead or re-proving it):
Let $M$ be a closed smooth manifold and let $S^1$ act smoothly on $M$. Then the set of fixed points of the action, $M^{S^1}$, is again a (not nece... | https://mathoverflow.net/users/14233 | Smooth circle action, $\chi(M^{S^1}) = \chi(M)$ | See [this wonderful blog post by Pawlowski.](http://chromotopy.org/blog/torus-actions-maximal-tori-2)
| 4 | https://mathoverflow.net/users/11142 | 257329 | 116,217 |
https://mathoverflow.net/questions/256887 | 3 | I have an abelian category $A$ that is AB4, AB3\* and has an injective cogenerator. Do these conditions "help" in checking whether a given family $a\_i$ of (compact) objects of $A$ is generating in it? So, where can I find any "non-trivial" conditions that ensure that $a\_i$ generate $A$? I would prefer not to assume t... | https://mathoverflow.net/users/2191 | Does the existence of an injective cogenerator "help" in finding generators of an abelian category? | A few thoughts a bit too long for a comment. I should preface this by saying that I'm not that familiar with the sort of generator conditions you're using, so I can only really talk about analogous cases using stronger generating conditions. I'm really hoping that someone who actually knows these things might respond t... | 2 | https://mathoverflow.net/users/2362 | 257331 | 116,219 |
https://mathoverflow.net/questions/257027 | 3 | **Question:** Consider a distribution $D$, and $n$ i.i.d. random variables $X\_i$, all distributed according to $D$. Let $p^D\_2:=\Pr[X\_1=X\_2]$. What is a lower bound for $p^D\_n:=\Pr[\exists i\neq j. X\_i=X\_j]$ (as a function of $p^D\_2$)?
**Conjecture:** $p^D\_n \geq 1-\bigl(1-p^D\_2\bigr)^{n\choose 2}$. [**EDIT... | https://mathoverflow.net/users/101775 | Birthday inequality for non-uniform distributions for fixed collision probability (random allocation, collision probability) | I reformulate slightly, please check.
You are considering a sequence $X\_1,X\_2,\ldots$ of (discrete) i.i.d
random variables
and want an upper bound for the probability $\mathbb{P}(R>n)$
in terms of $\sqrt{\beta}$, where $\beta:= {1 \over \mathbb{P}(X\_1=X\_2)}$,
and $R:=\inf\{ n\geq 2\,:\,X\_n\in\{X\_1,\ldots,X\_... | 3 | https://mathoverflow.net/users/48831 | 257340 | 116,222 |
https://mathoverflow.net/questions/257344 | 9 | Let $\mu$ be the Lebesgue measure, and $+$ be addition modulo $1$ in the interval $[0,1)$.
**Question1:** Is there a closed set $C\subset [0,1)$ of positive measure such that for any countable set $D\subset [0,1)$, we have $\mu(C+D)<1$?
**Question2:** Is there a closed set $C\subset [0,1)$ of positive measure such... | https://mathoverflow.net/users/91504 | Countable shifts of closed positive sets | The answer to question 2 (and therefore question 1 as well) is no. This follows from the [Lebesgue Density Theorem](https://en.wikipedia.org/wiki/Lebesgue's_density_theorem), which says, roughly, that a measurable set that is neither null nor co-null cannot be too evenly spread out, but it must be "lumpy" in places, li... | 9 | https://mathoverflow.net/users/70618 | 257345 | 116,224 |
https://mathoverflow.net/questions/257316 | 3 | I have a dumb question.
Given a Hopf algebra $H$, take an invertible element $J\in H\otimes H$ and define $\Delta^J=J^{-1} \Delta J$. This becomes a new coproduct when $J$ satisfies a certain condition. Such $J$ are called twists, and let's denote the new Hopf algbera obtained this way by $H^J$. Two twists are called... | https://mathoverflow.net/users/5420 | Twists of commutative Hopf algebras | This wa really dumb. As Julian says in the comment above, $\Delta^J$ is equal to $\Delta$, so there was nothing to be done.
I posted the question around the midnight, just before going to bed, and I'm writing this on my way to work. This shows we shouldn't think too hard before going to sleep, we are prone to do dum... | 0 | https://mathoverflow.net/users/5420 | 257360 | 116,233 |
https://mathoverflow.net/questions/257354 | 7 | Let $Sec\_r(V)$ be the $r$-secant variety of a Veronse variety $V\subset\mathbb{P}^N$, that is
$$Sec\_r(V) = \bigcup\_{p\_1,...,p\_r\in V}\left\langle p\_1,...,p\_r\right\rangle\subset\mathbb{P}^N$$
where $V$ is the image of $\mathbb{P}^n$ via the embedding induced by $\mathcal{O}\_{\mathbb{P}^n}(d)$.
Is it true tha... | https://mathoverflow.net/users/nan | Degree of equations of secant varieties of Veronese varieties | The answer by JM Landsberg in the link from aginensky's comment precisely says that the $r+1$ lower bound on the degree of generators is true.
See top of page 2 in the article ["Prolongations and computational algebra"](https://arxiv.org/abs/math/0611696) by Sidman and Sullivant where they point to the article ["On the... | 5 | https://mathoverflow.net/users/7410 | 257361 | 116,234 |
https://mathoverflow.net/questions/257365 | 11 | I'm reading Ravenel's book Nilpotence and periodicity in stable homotopy theory. In section 1.1, it says the set of homotopy classes of maps of maps between compact manifolds or between algebraic varieties over real or complex numbers is countable. I know that the set of homotopy classes between triangulable spaces is ... | https://mathoverflow.net/users/102515 | The set of homotopy classes of maps between compact manifolds is countable | This may be somewhat heavy-handed but it follows from work of Kirby-Siebenmann that any compact manifold is homotopy equivalent to a finite CW-complex (implying what you want because any finite CW-complex is homotopy equivalent to its regular neighborhood in some Euclidean space, which in turn can be triangulated as a ... | 11 | https://mathoverflow.net/users/1573 | 257368 | 116,239 |
https://mathoverflow.net/questions/257242 | 4 | While reading about accessible categories in [Locally Presentable and Accessible Categories](https://books.google.de/books?id=iXh6rOd7of0C&pg=PR12&lpg=PR12&dq=locally%20presentable%20and%20accessible%20categories&source=bl&ots=fDlJtqD6Fg&sig=GdynAilkiO-WB1fNdDxbKvMOrSA&hl=en&sa=X&ved=0ahUKEwjXneDinfTQAhWK6xQKHdh8CYcQ6A... | https://mathoverflow.net/users/99988 | Intuition behind $\lambda$-pure subobjects | Think of the category of structures in some signature $\Sigma$. The functor represented by a finitely presentable object $F$, say, corresponds to some term in the language generated by $\Sigma$. For example, suppose that $\Sigma$ is the signature for fields, and $F = \mathbb{Q}[x]/(f(x))$ for some irreducible polynomia... | 2 | https://mathoverflow.net/users/2362 | 257381 | 116,242 |
https://mathoverflow.net/questions/254758 | 1 | Let $\pi:X \rightarrow Y$ be a double cover between compact manifolds $X$, $Y$ and $\theta$ be the deck transformation. Let $H^2(X, \mathbb Z)^\theta$ be a group of $\theta^\*$-invariant elements in $H^2(X, \mathbb Z)$.
My Question is:
Is $H^2(X, \mathbb Z)^\theta$ a subset of $\pi^\* (H^2(Y, \mathbb Z) )$?
You c... | https://mathoverflow.net/users/38823 | Cohomologies of double covers | Using spectral sequences is a bit heavy here. Given a map $s\colon\Delta\_p\to Y$, there are precisely two different lifts to $X$, and we can add them to get a singular chain in $C\_p(X)$. This construction extends linearly to give a chain map $\tau\colon C\_\*(Y)\to C\_\*(X)$ (called the transfer), with $\pi\_\*\circ\... | 7 | https://mathoverflow.net/users/10366 | 257395 | 116,245 |
https://mathoverflow.net/questions/257385 | 3 | I'm currently writing my master thesis about the j-invariant and his q-expansion.
Now i have the result that the growth of the coefficients is asymptotically
$$c(n) \sim \frac{e^{4\pi \sqrt{n}}}{\sqrt{2}n^{\frac{3}{4}}}.$$
Is there any special reason to investigate this growth? Is it possible to use this result to ... | https://mathoverflow.net/users/102557 | Asymptotic Formula of the coefficients of the q-expansion of the J-Invariant | There are lots of reasons why it's interesting to know the growth rate of modular functions. For the $j$-invariant, the coefficients are closely related to the dimensions of the irreducible representations of the monster group (the largest sporadic finite simple group), so the size of those coefficients tells you somet... | 2 | https://mathoverflow.net/users/11926 | 257398 | 116,247 |
https://mathoverflow.net/questions/257392 | 7 | A polynomial
$$f(x) = x^n + a\_{n-1} x^{n-1} + \cdots + a\_1 x + a\_0$$
where $a\_{n-1}, a\_0$ are known. How to determine whether polynomial $f$ has integer roots? Can we give a bound of $\max\{a\_i \mid i=1,\dots,n-2\}$ and $\min\{a\_i \mid i=1,\dots,n-2\}$ decide by $n,a\_{n-1}$, and $a\_0$ such that the polyn... | https://mathoverflow.net/users/102562 | How to determine whether a polynomial has integer roots? | I'm not sure if I understand just what you're asking, but perhaps it is the following:
>
> Suppose we know $a\_0$ and $a\_{n-1}$, and we know that $f(x)$ has only integer roots. Can we use this information to meaningfully bound the coefficients?
>
>
>
If that's what you're asking, then the answer is no. Consid... | 7 | https://mathoverflow.net/users/22512 | 257405 | 116,251 |
https://mathoverflow.net/questions/257391 | 2 | I asked this question on MSE a few weeks ago but got no response.
The question arose out of some research in group theory on random group presentations. The actual question is more complicated, but the following captures the essence of it. It is similar to the birthday coincidence problem, but involves a condition on... | https://mathoverflow.net/users/35840 | A condition on triples of elements in a random collection of ordered pairs | In the comments to your question, @MattF. is correct, and @IlyaBogdanov is also spot on (in probabilistic combinatorics, this approximation comes as a routine knee-jerk reaction). This problem is extremely well-known. I'll summarize very quickly, but you can find more online by searching "probability random graph has a... | 1 | https://mathoverflow.net/users/22512 | 257409 | 116,253 |
https://mathoverflow.net/questions/255574 | 17 | In quantum integrability and related topics a lot of not-so imaginative terminology is used. One may hear people talk about "*Q*-operators", "*R*-matrices", "*S*-matrices", "*T*-operators", as well as "*L*-operators". In fact, this has further led to even less poetic names such as "*RTT*-relations" and "*RLL*-relations... | https://mathoverflow.net/users/45956 | Where does the name "R-matrix" come from? | **(See update at the bottom)**
I could not resist: in the meanwhile I have searched the literature further. I think the following is an interesting addition to my best guess from the OP, which is why I have chosen to share it, even if it isn't the definitive answer to my question.
As I (almost correctly) mentioned,... | 9 | https://mathoverflow.net/users/45956 | 257410 | 116,254 |
https://mathoverflow.net/questions/257408 | 5 | Siegel's theorem states the following:
Let $C$ be a smooth projective curve over a number field $K$. Let $\tilde C\subset C$ be an open affine subvariety, and $i:\tilde C\hookrightarrow \mathbb{A}^m\_K$ be a closed immersion. Then if $i(\tilde C)$ lies over infinitely many $\mathbb{A}^m\_{\mathcal{O}\_K}(\mathcal{O}\... | https://mathoverflow.net/users/98901 | Understanding Siegel's Theorem on integral points | Every finite set is a Siegel set, in your sense for curves of positive genus. Proof: Let $S$ be the set in question, take an affine open subset $U$ of your curve containing $S$ and embed this open set $U$ in affine space. Change variables by clearing denominators such that the points of $S$ are integral with respect to... | 3 | https://mathoverflow.net/users/2290 | 257417 | 116,255 |
https://mathoverflow.net/questions/257259 | 5 | Let $G$ be a finite group with finite-dimensional irreducible representations $\rho\_i:G\to\mathrm{GL}\_{n\_i}(k)$ over a field $k$ indexed by $i=1,...,m$. These compose with the canonical map $\mathrm{GL}\_{n\_i}(k)\to\mathrm{GL}(k)$ to give $\rho\_i:G\to\mathrm{GL}(k)$ for $i=1,\cdots,m$. Taking classifying spaces an... | https://mathoverflow.net/users/102390 | Surjectivity of representations in algebraic K-theory | I don't actually know places in the literature where induced maps from representations have been studied (except from the Brauer lift used in Quillen's computation of K-theory of finite fields). The following are some remarks and observations which partially answer the question.
As a first remark, I assume that the ... | 2 | https://mathoverflow.net/users/50846 | 257419 | 116,256 |
https://mathoverflow.net/questions/257411 | 3 | I could not decide if I should post this question in MO or Mathstackexchange, so feel free to downvote it if you think it does not belong here. I will delete my post and post it in MathSE in that case.
What I am wondering about is the following: Given a projective map $\pi:X\to B$, where $B$ is integral and $\mathscr... | https://mathoverflow.net/users/48522 | Fibers of pushforward of a bundle when the fiber dimension is not constant | This is an explanation and a correction of Mohan's comment. Let us use the base change isomorphism (of derived functors)
$$
Lj^\*R\pi\_{2\*}\mathcal{L} \cong R\Gamma(C,Li^\*\mathcal{L}),
$$
where $j$ is the embedding of the point $[K\_C]$ and $i$ is the embedding of the fiber $C$ over this point. Since $\mathcal{L}$ is... | 1 | https://mathoverflow.net/users/4428 | 257423 | 116,258 |
https://mathoverflow.net/questions/257425 | 1 | Do you have any information if the completely monotonic function of $n$-variables function has been defined yet?
| https://mathoverflow.net/users/102576 | Do you have any information if the completely monotonic function of n-variables has been defined yet? | You may like to look into [Ressel paper here](http://link.springer.com/article/10.1007/s11117-013-0244-6). If you don't have access then check out his [presentation slides](http://websites.math.leidenuniv.nl/positivity2013/presentations/ressel.pdf).
| 2 | https://mathoverflow.net/users/66131 | 257427 | 116,259 |
https://mathoverflow.net/questions/255508 | 6 | The question is motivated by Eckmann-Hilton duality and certain flaws of the homotopy category of CW-complexes. Unfortunately, I do not know the formalism of model categories, so excuse me if it is a basic fact concerning them.
First consider a circle $S^1$ with a fixed point. All spaces are supposed to be connected ... | https://mathoverflow.net/users/43639 | Is there an analogue of CW-complexes built from $K(\mathbb Z, n)$ instead of $S^n$? | Repeating my comment above as answer:
You may find Peter May's article "The Dual Whitehead Theorems" interesting in this context. It's at math.uchicago.edu/~may/PAPERS/47.pdf
| 2 | https://mathoverflow.net/users/4042 | 257429 | 116,260 |
https://mathoverflow.net/questions/257416 | 25 | From time to time, I run into the **finite** product $\prod\_{j=1}^n(1+q^j)$. And, the more it happens, the more fascinated I've become. So, herein, I wish to get help in collecting such results. To give some perspective into what I look for, check out the below examples. First, some nomenclature: $(q)\_k=(1-q)(1-q^2)\... | https://mathoverflow.net/users/66131 | what else is in $\prod_{j=1}^n(1+q^j)$? | Up to scaling, $\prod\_{j=1}^n(1+q^j)$ is the character of the principal specialization of
the spinor representation of $\mathfrak{so}(2n+1)$. This was first
explicitly stated by J. W. B. Hughes, Lie algebraic proofs of some theorems on partitions, in *Number Theory and
Algebra*, Academic Press, 1977, pp. 135--155.
| 22 | https://mathoverflow.net/users/2807 | 257433 | 116,262 |
https://mathoverflow.net/questions/256673 | 2 | We say two permutations $\pi\_1$ and $\pi\_2$ in the symmetric group $\mathfrak{S}\_n$ are **$k$-equivalent**, denoted
$\pi\_1 \sim\_k \pi\_2$, if one can be
determined from the other after a finite number of switching of neighborly terms that differ by exactly $k$.
The above concept is motivated by [this paper of R.... | https://mathoverflow.net/users/66131 | "flavored" equivalence classes of permutations | This answer will come in two parts, first we will transform the problem into a form that is ammenable to the methods of section 2 of [Stanley's paper](http://www-math.mit.edu/~rstan/papers/multfree.pdf), then we will show that the above summation is equivalent to that answer.
If we renumber the elements of the permut... | 2 | https://mathoverflow.net/users/4422 | 257436 | 116,263 |
https://mathoverflow.net/questions/257442 | 1 | Let $\ c\in \mathbb R.\ $ Let
$$ D\_n(c)\ := \frac {4^n}{\binom {2\cdot n}n\cdot\sqrt{4\cdot n + c}} $$
Then, more or less by the Wallis product theorem, we have this well-known convergence:
$$ \lim\_{n\rightarrow\infty}\ D\_n(c)\ \ =\ \ \frac {\sqrt{\pi}}2 $$
This holds regardless of the choice of the constant... | https://mathoverflow.net/users/8385 | Approximating the central binomial coefficient | We have $$\left(\frac{D\_{n+1}(c)}{D\_n(c)}\right)^2=\frac{4(4n+c)(n+1)^2}{(2n+1)^2(4n+c+4)}=1+\frac{4n(c-1)+3c-4}{(2n+1)^2(4n+c+4)},$$
the convergence is the fastest if this is most close to 1, i.e. for $c=1$. In this case we get $$\frac{D\_{n+1}(1)}{D\_n(1)}=1+O(n^{-3}),\\ \frac{2D\_n(1)}{\sqrt{\pi}}=\prod\_{k=n}^\in... | 7 | https://mathoverflow.net/users/4312 | 257443 | 116,265 |
https://mathoverflow.net/questions/257439 | 1 | The term "intersecting permutations" is used for a family of permutations $A \subset S\_n$ such that for all $\pi,\sigma \in A$, $\pi(i)=\sigma(i)$ for some $i \in [n]$.
Is there a term for a family of permutations with the "opposite" property, that is for all $\pi,\sigma \in A$, $\pi(i)\neq \sigma(i)$ for all $i \i... | https://mathoverflow.net/users/102584 | Common name for totally non-intersecting permutations | These are mutual derangements. Equivalently, the rows of a latin rectangle. Probably other names have appeared.
| 7 | https://mathoverflow.net/users/9025 | 257446 | 116,267 |
https://mathoverflow.net/questions/257401 | 5 | The $\textsf{AD}\_{\mathbb R}$-hypothesis is the statement that there is a $\lambda$ which is both a limit of Woodins and a limit of ${<}\lambda$-strongs. Are there any results relating the consistency of this statement to the consistency of a proper class of Woodins?
In [this paper](http://www.math.uci.edu/~mzeman/R... | https://mathoverflow.net/users/38602 | Proper class of Woodins and $\textsf{AD}_{\mathbb R}$-hypothesis | The existence of a proper class of Woodin cardinals and a proper class of strong cardinals is strictly stronger in consistency strength over ZFC than the existence of a cardinal $\lambda$ that is a limit of Woodin cardinals and a limit of $<\lambda$-strong cardinals. The former implies the consistency of the latter and... | 5 | https://mathoverflow.net/users/1946 | 257463 | 116,272 |
https://mathoverflow.net/questions/257470 | 6 | I have searched this in the literature but could not find any reference, so I would like to post it here. Hope this is at the research level.
Assume that
$$
A=\begin{vmatrix}
a\_1 & b\_1 \\
c\_1 & a\_2 & b\_2 \\
& c\_2 & \ddots & \ddots \\
& & \ddots & \ddots & b\_{n-1} \\
& & & c\_{n-1} & a\_n
\end{vmatrix}
$$
where ... | https://mathoverflow.net/users/83815 | Are Diagonally dominant Tridiagonal matrices diagonalizable? | ### Counterexample:
$$
\begin{bmatrix}
-1 & 1 & 0 & 0\\
0 & -1 & 1 & 0\\
0 & 0 & -2 & 2\\
0 & 0 & 2 & -2
\end{bmatrix}
$$
is defective: its eigenvalues are $-1,-1, 0, -4$ (it is block triangular, so its eigenvalues are those of the $2\times 2$ blocks on the diagonal), but $A+I$ has rank 3.
### Strategy to construct... | 13 | https://mathoverflow.net/users/1898 | 257472 | 116,277 |
https://mathoverflow.net/questions/257190 | 7 | In the paper by Verity and Riehl "Fibrations and Yoneda lemma in an ${\infty}$-cosmos" (<https://arxiv.org/abs/1506.05500>), they prove a Yoneda lemma that holds in any ${\infty}$-cosmos (see Corollary 6.2.13). Therefore, in particular, it holds in $\textbf{qCat}$.
If we unravel their definition of cartesian fibratio... | https://mathoverflow.net/users/57280 | Equivalent definitions of Cartesian Fibrations between Quasi-Categories | Riehl and Verity prove that their definition agrees with Lurie's in Corollary 4.1.24 (cf. Remark 2.4.1.4 in HTT).
| 3 | https://mathoverflow.net/users/1100 | 257474 | 116,279 |
https://mathoverflow.net/questions/257418 | 6 | I would like to construct hyperelliptic curves whose Jacobians are isogenous to the square of a supersingular elliptic curve over $\mathbb{F}\_{p^2}$
My question is motivated by the following example.
Let $H/\mathbb{F}\_{5^2}$ be the hyperelliptic curve given by $y^2 = x^6 + 1$ and $E/\mathbb{F}\_{5^2}$ an elliptic... | https://mathoverflow.net/users/91023 | Jacobians of genus 2 curves isogenous to a square of a supersingular elliptic curve over $\mathbb{F}_{p^2}$ | Such curves are constructed in my paper "Familles de courbes et de variétés abéliennes sur $\mathbb{P}^1$, II", Astérisque vol. 86 (1981).
| 6 | https://mathoverflow.net/users/7666 | 257476 | 116,280 |
https://mathoverflow.net/questions/256763 | 5 | Edit: In case that there is no solution for the original question, I modify to enrich the question.
We like to ask a possible specific inflation a $H^3(Q, \mathbb{R} /\mathbb{Z})$ cocycle with a finite group $Q$ into a coboundary in the following two cases in quaternion group or dihedral group:
1. Inflate the 3-coc... | https://mathoverflow.net/users/44768 | Inflate a finite-group cocycle into coboundary in non-Abelian groups | You can determine the possible inflation of $d$-cocycle in $G$ by lifting to a larger group $G$, from the Lyndon-Hochschild-Serre spectral sequence.
For $H/K= G$, with $BG$ path connected and $\pi\_1(BG)$
acting trivially on $H^\*(K, U(1))$, there is a spectral sequence $\{E^{p,q}\_n, d\_n\}$
with $E^{p,q}\_2 = H^p(G, ... | 1 | https://mathoverflow.net/users/27004 | 257484 | 116,283 |
https://mathoverflow.net/questions/257349 | 10 | I am reading a paper and I am trying to understand an equality which is given without proof:
$$\sum\_{k=1}^s\binom{2s-k}{s}\frac{k}{2s-k}v^k(v-1)^{s-k}=v\sum\_{k=0}^{s-1}\binom{2s}{k}\frac{s-k}{s}(v-1)^{k} $$
Here, $s>0$, $k$ and $v$ are positive integers.
The equality in question appears in Lemma 2.1 of
<http://web.wi... | https://mathoverflow.net/users/102529 | Equality with binomials | Here is a non-automated proof. We divide by $v$, and then we expand $v^{k-1}$ on the left hand side as
$$v^{k-1}=(1+v-1)^{k-1}=\sum\_{j=0}^{k-1}\binom{k-1}{j}(v-1)^j.$$
Then, looking at the coefficients of $v-1$ on the two sides, we are left with proving the identity
$$ \sum\_{\substack{0\leq k\leq s\\0\leq j\leq k-1\\... | 4 | https://mathoverflow.net/users/11919 | 257498 | 116,285 |
https://mathoverflow.net/questions/257502 | 10 | The [prime number theorem](https://en.wikipedia.org/wiki/Prime_number_theorem) says on average we can find $\frac n{\log n}$ primes of magnitude $n$.
[Erdos-Kac law](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem) state a typical number of magnitude $n$ has $\log\log n$ primes.
Somehow the fact $e^{\l... | https://mathoverflow.net/users/nan | Does the Prime Number Theorem have anything to do with Erdos-Kac law or vice versa? | Yes. The number of prime factors of a number is distributed roughly like a Poisson process of expectation $\log \log n$, so the probability of exactly one prime factor is roughly $e^{- \log \log n} = 1/\log n$.
Remember that natural numbers of size roughly $n$ correspond, in the number field / function field dictiona... | 18 | https://mathoverflow.net/users/18060 | 257504 | 116,287 |
https://mathoverflow.net/questions/257510 | 1 | Let $X$ be an $n \times n$ invertible square matrix over some field $\mathbb{F}$, and let $Y = XX^T$ be the product of the matrix with its transpose.
When $\mathbb{F} = \mathbb{R}$, $Y$ is positive-definite, so it is easy to see that for any subset $S \subseteq \{ 1, 2, \dots, n \}$, the matrix $Y\_{S,S}$ obtained by... | https://mathoverflow.net/users/39521 | 'Positive-definite' matrices over finite fields | No, already in the case when $S$ is a set of one element. In this case we ask that the diagonal elements of $X X^T$ are nonzero. These are the sums of squares of row vectors in $X$.
The row vector may be any nonzero vector, so your desired statement implies that, for any $x\_1,\dots,x\_n$ not all zero, we have $\sum... | 5 | https://mathoverflow.net/users/18060 | 257512 | 116,290 |
https://mathoverflow.net/questions/257493 | 5 | $\newcommand{\i}{\iota}$
The general notion that I am trying to disprove is that if we are given a fibration $X \to Y$ with fiber $F$ such that the delooping $BF$ exists, that there is a map $Y \to BF$ such that $F \to X \to Y \to BF$ is a fibration sequence.
This question came up when I was trying to see what went w... | https://mathoverflow.net/users/68932 | The inability to continue a fibration sequence even when a delooping exists | I haven't thought about the example that you describe, but here is a different kind of example.
Consider a topological group $F$, and let $M$ denote the monoid of weak equivalences $F\to F$, so there is an evident inclusion $F\to M$. Let $X$ be any based space, and let $i\colon X\to CX$ be the inclusion of the base ... | 6 | https://mathoverflow.net/users/10366 | 257514 | 116,292 |
https://mathoverflow.net/questions/257496 | 2 | I was wondering if anyone knew of any results regarding the limiting distribution of singular values for sparse random real-valued matrices?
Specifically, let $X$ be an $N\times M$ matrix with real-valued entries, with $X\_{ij} = a\_{ij}K\_{ij}$, where the nonzero entries $K\_{ij}\sim N(0, \sigma^2)$ i.i.d, and $a\_{... | https://mathoverflow.net/users/36751 | Singular values of sparse random real-valued matrix | A general theoretical framework has been developed in [Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices](https://arxiv.org/abs/0803.1553) (2008), and an alternative approach is in [Spectral Density of Sparse Sample Covariance Matrices](https://arxiv.org/abs/cond-mat/0612584) (2006). Here is a... | 2 | https://mathoverflow.net/users/11260 | 257518 | 116,295 |
https://mathoverflow.net/questions/257495 | 35 | So I've been reading about derived categories recently (mostly via Hartshorne's *Residues and Duality* and some online notes), and while talking with some other people, I've realized that I'm finding it difficult to describe what a "triangle" is (as well as some other confusions, to be described below).
Let $\mathcal... | https://mathoverflow.net/users/15242 | What is a triangle? | To answer your first precise questions:
* Yes, every distinguished triangle in $D(A)$ comes from a short exact sequence. For every distinguished triangle $X \to Y \to \mathrm{Cone}(f) \stackrel{+1}\to $ there is a short exact sequence
$$ 0 \to Y \to \mathrm{Cone}(f) \to X[-1] \to 0$$
of complexes in $A$, and our dis... | 38 | https://mathoverflow.net/users/1310 | 257519 | 116,296 |
https://mathoverflow.net/questions/187757 | 15 | It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference or two to help me get some things straightened out.
One can construct a category $\mathcal{M}$ of 0-dimensional motives over $\mathbb{Q}$. If I'm not mistaken, one possible constr... | https://mathoverflow.net/users/5263 | Galois group for 0-dimensional motives | Motives of $0$-dimensional varieties are usually called Artin motives. The different fiber functors (essentially) all give rise to automorphism group isomorphic to the absolute Galois. There is one difference though: the cohomology theories have different coefficients ($\mathbb{Q}$ for de Rham cohomology, $\mathbb{Q}\_... | 10 | https://mathoverflow.net/users/50846 | 257520 | 116,297 |
https://mathoverflow.net/questions/256515 | 4 | Let $p \in H^4(\mathbb{CP}^2)$ and $\ell \in H^2(\mathbb{CP}^2)$ be the cohomology classes Poincaré dual to a point and a line respectively.
**Question.** What is the Gromov-Witten invariant $\langle p, p, \ell\rangle\_{0, 1}$ counting degree $1$, genus $0$ curves in $\mathbb{CP}^2$?
| https://mathoverflow.net/users/102036 | Gromov-Witten invariant $\langle p, p, \ell\rangle_{0, 1}$ counting degree $1$, genus $0$ curves in $\mathbb{CP}^2$? | As Dan says, you can use the divisor equation:
$$\langle e\_{\alpha\_1}, \ldots, e\_{\alpha\_n}, \ell \rangle\_{g,d} = d\ \langle e\_{\alpha\_1}, \ldots, e\_{\alpha\_n} \rangle\_{g,d} $$
where the $e\_{\alpha\_i}$ are any cohomology classes of $\mathbb{CP}^2$, to reduce your invariant $\langle p,p,\ell \rangle\_{0,1}$ ... | 3 | https://mathoverflow.net/users/27069 | 257544 | 116,304 |
https://mathoverflow.net/questions/257535 | 7 | An SFT (shift of finite type) is a set of maps to some finite alphabet that is defined by a finite number of disallowed finite words.
By simple I mean has a small alphabet and a small number of disallowed words.
>
> **QUESTION.** What are some of the simplest SFTs on $\mathbb{Z}^2$ that have no periodic points?
>... | https://mathoverflow.net/users/23661 | What is the simplest SFT on $\mathbb{Z}^2$ that has no periodic points? | Wang tiles are unit squares with edges marked with colors, and the problem of whether a given set of Wang tiles can tile the plane such that edges of adjacent squares match has been studied exhaustively (see <https://en.wikipedia.org/wiki/Wang_tile>). In particular, there is a set of 11 Wang tiles using four colors whi... | 11 | https://mathoverflow.net/users/8112 | 257546 | 116,306 |
https://mathoverflow.net/questions/257571 | 8 | I was thinking about a possible *hierarchy* for the top three Grothendieck's works: EGA,SGA,FGA. But I haven't read all these works, and so I'm asking if there is actually such a hierarchy.
Here the word hierarchy means a possible "reading path" of all these three works, which would be the first to be read in order ... | https://mathoverflow.net/users/82222 | Hierarchy of Grothendieck's SGA, EGA, FGA | In a few words: EGA is previous to everything, though one can use SGA 1 to complement some aspects of EGA IV. FGA goes "in between". There is a complicated tree for SGA. SGA 3 is independent of the rest while SGA 4, SGA 5 and SGA 7 are a full saga. SGA 6 is more or less independent but you need Verdier's thesis (or its... | 19 | https://mathoverflow.net/users/6348 | 257572 | 116,311 |
https://mathoverflow.net/questions/257553 | 0 | 1-Do we have some relation between the projective dimension of $k[X]$ as $K[x\_{1},..x\_{n}]$-module and the Krull dimension of the affine variety $X$ of $A^{n}$.
2- If we have to affine subvarieties $X$,$Y$ of $A^{n}$. Do we have some relation between
$pdk[X]$ and $pdk[Y]$?
Thanks!
| https://mathoverflow.net/users/92487 | Homological dimension in algebraic geometry | 1) Assuming for simplicity that $X$ is integral, what you can say is $\ \mathrm{pd}(K[X])\geq n-\dim(X)$. After localizing at the (prime) ideal of $X$, this is an easy case of the Auslander-Buchsbaum theorem. And, of course, $\ \mathrm{pd}(K[X])\leq n$.
2) I just don't understand the question. If $X$ and $Y$ are arbi... | 3 | https://mathoverflow.net/users/40297 | 257576 | 116,312 |
https://mathoverflow.net/questions/257293 | 1 | I find I frequently have to refer to the set of ordinal notations below some given notation. For instance given a notation $\alpha$ I often need to refer to the set $\lbrace \beta \mid \beta <^{\mathcal{O}}\_s \alpha\rbrace$. Higher Recursion Theory simply writes such sets as $W\_{g(\alpha), s}$ (where $g$ is the appro... | https://mathoverflow.net/users/23648 | Using Ordinal Notations in Computability Theory Is There A Standard Notation For The Notations Below $\alpha$ | In my thesis (<https://www.lacl.fr/~benoit.monin/ressources/misc/Thesis_report_benoit_monin_v1.9.pdf>), I use $\mathcal{O}\_{<\alpha}$, $\mathcal{O}\_{\leq\alpha}$, and $\mathcal{O}\_{=\alpha}$, for the set of notations of ordinals respectively strictly smaller than $\alpha$, smaller than or equal to $\alpha$, and equa... | 1 | https://mathoverflow.net/users/14490 | 257582 | 116,314 |
https://mathoverflow.net/questions/257584 | -1 | If I have a known matrix A, is there a method to find a matrix B that share all the eigen vectors of Matrix A?
| https://mathoverflow.net/users/101548 | Finding a matrix with shared eigen vectors with a given matrix | Any analytic function of $A$ (including such diverse ones as $p(A),$ where $p$ is a polynomial, and $\exp(A)$) will work.
| 1 | https://mathoverflow.net/users/11142 | 257586 | 116,315 |
https://mathoverflow.net/questions/257585 | 6 | From Rademacher's book (Topics in Analytic Number Theory) I'm using the functional equation of $\vartheta\_2(0|\tau) = 2\sum\_{m=0}^\infty q^{\left(m+\frac12\right)^2} = \vartheta\_2(\tau)$ and the fact that it vanishes at the cusps $\tau = \frac{a}{b}$, $a$ odd, $b$ even.
The order of vanishing at infinity is just $... | https://mathoverflow.net/users/102659 | Finding order of vanishing for Jacobi Theta function | The convention when working with modular forms $f(\tau)$ of weight $k$ is for the order of vanishing at a cusp $a/c$ to mean the order of vanishing of $(c\tau+d)^{-k} f\left(\frac{a\tau+b}{c\tau+d}\right)$ *as a function of $q = e^{2 \pi i \tau}$*. One also speaks of the value of a modular form at a cusp as the constan... | 6 | https://mathoverflow.net/users/48142 | 257590 | 116,317 |
https://mathoverflow.net/questions/257313 | 37 | Consider $27$ (pairwise distinct!) lines in $\mathbb{P}^3$ whose intersection graph is that expected¹ of the $27$ lines on a smooth cubic surface. **Question:** Is there a simple necessary and sufficient condition for these $27$ lines to indeed lie on a smooth cubic surface?
For a long time I thought this was always ... | https://mathoverflow.net/users/17064 | When do 27 lines lie on a cubic surface? | It turns out that condition (T) is, indeed, sufficient for the $27$ lines (distinct and intersecting as expected) to lie on a cubic surface.
To see this, consider the lines $a\_1,a\_2,a\_3,a\_4,a\_5$ and $b\_6$, where the labeling is as in note (2) of the question: $a\_1$ through $a\_5$ are pairwise skew, and $b\_6$ ... | 16 | https://mathoverflow.net/users/17064 | 257593 | 116,320 |
https://mathoverflow.net/questions/257589 | 2 | It is well known, that the characterizing property of Clothoids is, that their curvature is proportional to length; that is also the reason, why they are used as design elements e.g. in road design.
>
> **Question:**
>
> Has the analogue to clothoids, where the slope and not the curvature is proportional to len... | https://mathoverflow.net/users/31310 | "Slope Analogue" of Clothoids | Note that $y''(x)=\sqrt{1+y'(x)^2}$ so $u=y'$ satisfies $u'(x)=\sqrt{1+u^2}$. This is separable, so we get
$$
\frac{du}{\sqrt{1+u^2}}=dx,\quad \sinh^{-1}(u)=x + C.
$$
| 3 | https://mathoverflow.net/users/4600 | 257595 | 116,322 |
https://mathoverflow.net/questions/256752 | 2 | Let $p>1$ be a real number. It is known that if $(X\_n)\_{n\geq 0}$ is a martingale bounded in $L^p$ (i.e. $\sup\{\mathbb{E}(|X\_n|^p), n\geq 0\} < +\infty$ ), then $(X\_n)\_{n\geq 0}$ converges a.s. and in $L^p$ to $X\in L^p$. This results is still true if $(X\_n)\_{n\geq 0}$ is a positive submartingale.
However, do w... | https://mathoverflow.net/users/nan | Submartingales bounded in $L^p$, $p>1$ | The almost sure convergence is true for any submartingale with an integrable upper majorant. This implies the $L^1$ convergence through uniform integrability.
However, in general, there is no convergence in $L^p$. I'll give a counterexample for $p=2$, but it can be modified for any $p>1$.
Let $X\_n = - e^{W\_n - ... | 1 | https://mathoverflow.net/users/8146 | 257602 | 116,325 |
https://mathoverflow.net/questions/257390 | 4 | I am dealing with the basics of Radin forcing but there are some formal details which I was not able to keep with. For example, given $u$ a measure sequence with length an uncountable regular cardinal $\lambda<\kappa(u)$ it is known that forcing with the Radin forcing $\mathbb{P}\_u$ changes the cofinality of $\kappa(u... | https://mathoverflow.net/users/64649 | Changing cofinalities using Radin Forcing | By induction on $2 \leq \delta=$length of (u)$< \kappa,$ one can show that $otp(C)= \omega^{\delta-1}$, if $\delta < \omega$ and $otp(C)= \omega^{\delta}$ if $\omega \leq \delta < \kappa$ where $C$ is the Radin club, in particular if $\delta$ is regular uncountable, then $otp(C)=\delta$. Further one can manage to not a... | 4 | https://mathoverflow.net/users/11115 | 257604 | 116,326 |
https://mathoverflow.net/questions/257563 | 1 | Let $Z\_N = \displaystyle{\sum\_{k+j\leq N}} \frac{N!N^{k+j}}{N^{N+1}}\frac{u^kv^j}{k!j!}\binom{N-j}{N-j-k}$ where $u$ and $v$ are two unknowns.
My question is: Is there a closed-form for $Z\_N$ or is $Z\_N$ the multivariate taylor series expansion of some function $f(u,v)$ such that $f(u,v)$ has a closed form? Than... | https://mathoverflow.net/users/61149 | A closed form of a summation or the taylor series expansion of some function with a closed form? | Let's focus on $P\_N:=\sum\_{k+j\leq N}\frac{N^{k+j}}{k!j!}\binom{N-j}ku^kv^j$. To get $Z\_N$, just multiply out by $\frac{N!}{N^{N+1}}$.
The notation $[z^m]F(z)$ means the coefficient $a\_m$ of $z^m$ in the series expansion $F(z)=\sum\_na\_nz^n$.
Start by changing variables $m=k+j$ so that $j=m-k$ and hence
\begi... | 3 | https://mathoverflow.net/users/66131 | 257611 | 116,329 |
https://mathoverflow.net/questions/257617 | 10 | I would define Stiefel-Whitney classes as the pullbacks of generators of $H^\*(BO, \mathbb{Z}/2)$ under a classifying map, and I gather this is a pretty common definition.
However, the book "Characteristic classes" by Milnor-Stasheff contains a different definition, as ``eigenvalues'' for the Steenrod squares acting... | https://mathoverflow.net/users/84144 | Who discovered this definition of Stiefel-Whitney classes? | For the relation $Sq(U) = \Phi(w)$, where $Sq$ is the total Steenrod squaring operation, $U$ is the Thom class, $\Phi$ is the Thom isomorphism and $w$ is the total Stiefel-Whitney class, I would cite Rene Thom's 1951 thesis, published as "Espaces fibres en spheres et carres de Steenrod" in Ann. Sci. Ecole Norm. Sup. (3... | 17 | https://mathoverflow.net/users/9684 | 257618 | 116,330 |
https://mathoverflow.net/questions/257614 | 4 | *I asked this on [math.stackexchange](https://math.stackexchange.com/questions/2052956/cardinal-arithmetic-in-inner-models) but did not receive an answer, so I'm asking here.*
Assume large cardinals. Can we have $\omega\_2^{L(\mathbb{R})}=\omega\_2$?
Note that $\omega\_1=\omega\_1^{L(\mathbb{R})}$ always: clearly $... | https://mathoverflow.net/users/8133 | Cardinal arithmetic in $L(\mathbb{R})$ | Should be possible, at least for $\omega\_2$. Under large cardinals, $\omega\_2^{L(\mathbb{R})}$ is the supremum of the lengths of the boldface $\Delta^1\_2$-prewellorderings of the reals (i.e., $\delta^1\_2$), which is the same whether it's computed in V or in $L(\mathbb{R})$. Woodin showed that, for example, if the n... | 8 | https://mathoverflow.net/users/102670 | 257620 | 116,332 |
https://mathoverflow.net/questions/257623 | 1 | Is there any closed form, asymptotics, and/or approximations for the following integral:
$$f(c) := \frac{1}{\sqrt{2\sigma^2\pi}}\int e^{-x^2/2\sigma^2}\frac{x^2}{1-cx^2}dx,$$ where $\sigma^2$ is real valued and $c$ is complex? I am integrating over $\mathbb{R}$.
Even in the case where $c$ is real would be interesti... | https://mathoverflow.net/users/36751 | Evaluation of Gaussian density integral | Obviously you need $\sigma^2 > 0$ (and I'll assume $\sigma > 0$), and $c$ should not be a positive real to avoid having singularities at $x = \pm \sqrt{1/c}$. According to Maple, the result is
$$ -{\frac {\sqrt {\sigma}}{c}}- \frac{\sqrt{-\pi c}}{ c^2 \sqrt{2\sigma}}
\exp\left(-\frac{1}{2c\sigma^2}\right) + \frac{\sq... | 1 | https://mathoverflow.net/users/13650 | 257629 | 116,334 |
https://mathoverflow.net/questions/257636 | 6 | Let $T$ be a sheaf topos and $I$ a small category. Then the functor category $[I,T]$ is also a sheaf topos. Now let $E$ be an elementary topos (cartesian closed category with finite limits + subobject classifier).
Are there additional conditions that we can impose on $E$ to guarantee that $[I,E]$ is an elementary to... | https://mathoverflow.net/users/84563 | Diagrams in an Elementary Topos | If $E$ is small complete, then $[I, E]$ is an elementary topos, by the following argument.
If $I\_0$ is the discrete category of objects of $I$, then $[I\_0, E]$ is just an $I\_0$-indexed product of elementary toposes, and such is always an elementary topos.
Then $[I, E]$ is the category of coalgebras for a left ... | 7 | https://mathoverflow.net/users/2926 | 257639 | 116,340 |
https://mathoverflow.net/questions/257608 | 3 | I have two related questions about numerical methods for root solving:
1) $f: R \to R$ is continuous and piece-wise smooth, with $f(a)f(b) < 0$.
$f$ has very high number of knot-points and computing analytic expression for $f'$ is not possible.
To find a root of $f$ in $[a,b]$, I can try the following:
* Use... | https://mathoverflow.net/users/102664 | root solving without analytic derivative | Obviously, within the realm of piecewise-smooth functions one can find examples where any derivative-based approach fails. I believe you're looking for the term "**derivative free optimization**".
* [Here is the relevant page on Wikipedia](https://en.wikipedia.org/wiki/Derivative-free_optimization), which lists abou... | 2 | https://mathoverflow.net/users/20507 | 257645 | 116,344 |
https://mathoverflow.net/questions/257613 | 3 | *This is a spinoff of [this earlier question of mine](https://mathoverflow.net/questions/256998/potentially-club-filters-on-omega-2).*
**Short version:**
>
> What measures in $L(\mathbb{R})$ can be gotten from "potentially club" filters, under appropriate hypotheses?
>
>
>
---
**Long version:** *EDIT: M... | https://mathoverflow.net/users/8133 | Getting measures (especially on $\omega_2$) from potential clubs | The claim you are trying to prove is false: $L(\mathbb R)$ has no measurable cardinals greater than $\Theta$. Work in $L(\mathbb R).$ We will use Woodin's theorem that $\text{HOD} = L[\mathbb P]$ for $\mathbb P\subseteq \Theta$ a partial order encoded as a subset of $\Theta$ and that $L(\mathbb R)$ is an inner model of... | 8 | https://mathoverflow.net/users/102684 | 257646 | 116,345 |
https://mathoverflow.net/questions/257653 | 2 | Does anybody have a good reference on the Grassmanian and its universal property?
I am reading this paper on Quot schemes: <https://arxiv.org/abs/math/0504590>
Where the Grassmanian is constructed, but its representability and its universal quotient are "exercises". In particular exercise (2) in chapter 1.
| https://mathoverflow.net/users/81957 | Universal property of the Grassmanian | I suggest you:
1. Eisenbud, Harris - *The Geometry of Schemes*, (2000) Springer Verlag, paragraph III.2.7;
2. Eisenbud, Harris - *3264 & All That, Intersection Theory in Algebraic Geometry*, chapters 3 and 4 ([click](http://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf));
3. Görtz, Wedhorn - *Algebraic... | 3 | https://mathoverflow.net/users/57030 | 257657 | 116,349 |
https://mathoverflow.net/questions/257630 | 5 | I am surprised that I didn't find a reference for the following question.
**Q:** Is there any characterization of the finite subgroups of $GL\_n( \mathbb{F}\_p [T\_1, \dots, T\_n])$? Can we do so more generally for $GL\_n(\mathbb{Z}/ p^n \mathbb{Z} \ [T\_1, \dots, T\_n] )$.
**Edit:** I had previously state an incor... | https://mathoverflow.net/users/53100 | Finite subgroups of GL_n of polynomial rings over finite fields | Well, polynomial rings are very complicated. I assume the question is more generally about $GL\_n(\mathbb{F}\_q[T\_1,\dots,T\_m])$.
The case of one variable can be deduced from the following paper:
* C. Soulé: Chevalley groups over polynomial rings. In: Homological group theory (Proc. Sympos. Durham) London Math. S... | 11 | https://mathoverflow.net/users/50846 | 257658 | 116,350 |
https://mathoverflow.net/questions/257670 | 4 | Given an undirected unweighted graph $G(V, E)$, there is an efficient algorithm to find the shortest paths between every pair of nodes. I am interested in the reverse problem, we want to reconstruct the original graph given the shortest distances between every pair of nodes.
**Reconstruction from shortest paths**
*... | https://mathoverflow.net/users/8784 | Reconstructing a graph from shortest paths information | This problem is solvable in polynomial-time. Given a $V \times V$ distance matrix $A$, let $G$ be the graph with vertex set $V$, where $uw \in E(G)$ if and only if $A\_{uw}=1$. Note that $G$ is the only possible graph that has shortest distance matrix $A$. Now just compute all the shortest distances between all pairs o... | 3 | https://mathoverflow.net/users/2233 | 257673 | 116,357 |
https://mathoverflow.net/questions/257668 | 7 | I am trying to simplify an expression and find a closed form for
$$\sum\_{m=0}^l \binom{s-m}{s-l} \binom{s-1+m}{s-1}x^m$$
How could I get rid of this summation?
| https://mathoverflow.net/users/102529 | Closed expression for hypergeometric sum | You may argue as GH from MO from [your other post](https://mathoverflow.net/questions/257349/equality-with-binomials).
* the coefficient of $y^m$ in $(1-xy)^{-s}$ equals $\binom{s+m-1}{s-1}x^m$;
* the coefficient of $y^{\ell-m}$ in $(1-y)^{\ell-s-1}$ equals $\binom{s-m}{s-\ell}$.
Therefore, the sum on your LHS equa... | 9 | https://mathoverflow.net/users/66131 | 257674 | 116,358 |
https://mathoverflow.net/questions/257567 | 8 | Assuming [Bishop's](http://www.springer.com/gp/book/9783642649059) constructive mathematics, is it true that any real-valued square matrix with **distinct** roots of the characteristic polynomial can be diagonalized? By distinct, I mean **apart**: $x \neq y \triangleq \exists q \in \mathbb{Q}.|x - y| > q$. (There may b... | https://mathoverflow.net/users/42302 | Matrix diagonalization and eigenvector computation constructively | The following works constructively over an arbitrary local ring $R$ (constructively, $\mathbb{R}$ is a local ring).
Assume that you matrix $M$ is canceled by a polynomial $Q$, of degree $m$ (with leading coefficent $1$), that $Q$ can be factored into
$$Q= \prod^m\_{i=1}(X-q\_i)$$
and that for each $i \neq j$, $(q... | 3 | https://mathoverflow.net/users/22131 | 257676 | 116,359 |
https://mathoverflow.net/questions/257671 | 1 | In *Set Theory* Jech defines a cone to be a subset of the Baire Space $\mathcal{N}$ of the form
$$\operatorname{cone}(x\_0)= \{x : x\_0 \in L[x]\}$$
where $x\_0 \in \mathcal{N}$. Jech then defines the equivalence relation $\equiv$ with
$$x \equiv y \iff (x \in L[y] \land y \in L[x]).$$
In a similar fashion, in *Set T... | https://mathoverflow.net/users/102300 | Defining cones and Turing cones | Simon Thomas already mentioned that a winning strategy can be recursively coded into a real. I thought it might be a good idea to write down one such coding explicitly:
Let $A \subseteq \mathcal{N}$ and let $\Sigma$ be a winning strategy for $G\_{A}$ (say for player $II$, the other case is virtually the same). Then
$... | 4 | https://mathoverflow.net/users/57114 | 257681 | 116,360 |
https://mathoverflow.net/questions/257659 | 5 | According to Google Scholar original Gallager's article [Low-density parity-check codes](http://ieeexplore.ieee.org/document/1057683/) is cited more than 10000 times. It looks scary for non-experts.
I suspect that the number of algorithms for constructing good sparse matrices for [LDPC-codes](https://en.wikipedia.org... | https://mathoverflow.net/users/5712 | LDPC codes construction | [Ryan and Lin's *Channel Codes*](https://books.google.com/books?id=0gwqxBU_t-QC) has several chapters (computer-based, finite geometries, finite fields, combinatorial designs) devoted to various constructions of LDPC codes, as well as a chapter devoted to nonbinary codes that also details constructions for that case.
... | 3 | https://mathoverflow.net/users/1847 | 257682 | 116,361 |
https://mathoverflow.net/questions/257678 | 6 | As is well known, the following theory is equiconsistent with $PA$:
>
> $ZFC$ with the axiom of infinity replaced by its negation.
>
>
>
Since this theory is equiconsistent with $PA$, it would seem reasonable to infer (wouldn't it?) that the consistency of '$ZFC$ with the axiom of infinity replaced by its nega... | https://mathoverflow.net/users/20597 | Is $PRA$ + $TI({\epsilon_0})$ mutually interpretable with some theory in the language of set theory? | Yes, the consistency of "ZFC with the axiom of infinity replaced by its negation" is provable in "PRA + TI($\epsilon\_0$)". Technically one has to also show that "PRA + TI($\epsilon\_0$)" can prove the equiconsistency result (since it already proves the consistency of PA), but these are fairly natural theories so that ... | 6 | https://mathoverflow.net/users/8991 | 257685 | 116,362 |
https://mathoverflow.net/questions/257649 | 5 | What are some general classes of compact complex manifolds whose universal covers are bounded domains? One class I know are the Kodaira fibered surfaces.
| https://mathoverflow.net/users/36038 | Natural classes of compact complex manifolds whose universal covers are bounded domains | I'm not aware of a sufficient condition, although many classes have been recognized as carrying the "bounded domain" covers. Here is [one interesting paper](http://porto.polito.it/2502314/1/N50Porto.pdf) to look into.
| 5 | https://mathoverflow.net/users/66131 | 257692 | 116,367 |
https://mathoverflow.net/questions/257677 | 6 | Given a graph $G$, its line graph, denoted $L(G)$, is the graph whose vertices are the edges of $G$ and where two edges of $G$ are adjacent in $L(G)$ if they are incident to each other, i.e., they share some endpoint. I am interested in the graph, let's call it $L'(G)$, whose vertices are the edges of $G$, two being ad... | https://mathoverflow.net/users/18606 | Has anyone seen this graph construction that is similar to the line graph? | This graph is known as $\Gamma(G)$ (the corresponding construction in which the edges span a triangle is called $\Delta(G)$) or also the Gallai graph of $G$.
See for example:
V.B. Le
*Gallai Graphs and Their Iteration Behavior*
Dissertation Thesis, TU Berlin 1994
V.B. Le
*Gallai graphs and anti-Gallai graphs*
Dis... | 6 | https://mathoverflow.net/users/47118 | 257702 | 116,369 |
https://mathoverflow.net/questions/257701 | 7 | I've been trying to understand why this result in algebraic geometry is true for a long time.
In the language of classical algebraic geometry, this is what I want to prove:
Let $k$ be a universal field, and let $F \subseteq k$ be a subfield. Let $X \subseteq \mathbb{A}^n$ be an affine (not necessarily irreducible) va... | https://mathoverflow.net/users/38145 | If $Y$ is closed in $X$, and $X(F) \cap Y$ is dense in $Y$, then $Y$ is defined over $F$ | In scheme language, the reformulation of your question (which we'll see really does achieve what you want, and unsurprisingly plays just as essential a role in the scheme version of the theory of linear algebraic groups as it does in the older language) is to show that if $X$ is a scheme of finite type over a field $F$... | 9 | https://mathoverflow.net/users/81332 | 257713 | 116,371 |
https://mathoverflow.net/questions/257732 | 3 | Let $f \colon X\to Y$ be a finite map of normal varieties. Then every irreducible component of $X\times\_{Y}X$ dominates $Y$.
The proof goes as follows. We can suppose that $f \colon X\to Y$ is a Galois covering. Then every irreducible component of $X\times\_{Y}X$ is the image of $f\_{\sigma}:X\to X\times\_{Y}X$, giv... | https://mathoverflow.net/users/102713 | Finite map of normal varieties | By definition we have $$X \times\_Y X =\{(x\_1, \,x\_2) \in X \times X \; | \; f(x\_1)=f(x\_2)\}$$ and, since we are assuming that $f \colon X \to Y$ is a Galois morphism (this is not restrictive, because otherwise we can pass to the Galois closure), we can rewrite this as $$X \times\_Y X= \{(x\_1, \, x\_2) \; | \; x\_... | 6 | https://mathoverflow.net/users/7460 | 257737 | 116,378 |
https://mathoverflow.net/questions/257680 | 3 | Suppose that we have a morphism between profinite groups $f: G\_{1}\rightarrow G\_{2}$ such that $f^{\ast}:H\_{cont}^{\ast}(G\_{2},A)\rightarrow H\_{cont}^{\ast}(G\_{1},A) $ is an isomorphism for any finite trivial $G\_{i}$-module $A$. What can we say in general about $f$ ? and in particular when $G\_{i}=\mathrm{Gal}(\... | https://mathoverflow.net/users/102216 | induced isomorphism in continuous cohomology | Here is one recent result in this direction, taken from I. Efrat and J. Minac, Galois groups and cohomological functors, Trans. of the AMS, <http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2016-06724-0/home.html> :
Let $q=p^s$ be a prime power.
Suppose that $G\_i=\mathrm{Gal}(\overline{K\_i}/K\_i)$ for fields... | 5 | https://mathoverflow.net/users/101929 | 257739 | 116,380 |
https://mathoverflow.net/questions/257709 | 13 | Let $G=\operatorname{SL}\_6$ act on $V=\Lambda^3 \mathbb C^6$. I would like to find the ring of invariants $\mathbb C[V]^G$. ~~There is an obvious invariant
$$Sq: V \to \mathbb C, \quad \omega \mapsto \omega^2 \in \Lambda^6 \mathbb C^6 \simeq \mathbb C,$$
with the last isomorphism $\operatorname{SL}\_6$-invariant.~~ **... | https://mathoverflow.net/users/43639 | Ring of invariants of $\operatorname{SL}_6$ acting on $\Lambda^3 \mathbb C^6$ | The principal isotropy group is $H=SL(3)\times SL(3)$: it has the right dimension (namely 16) and occurs as an isotropy group (namely of a general element of $W$). Now it is a general result of Luna-Richardson that the restriction map $\mathbb C[V]^G\to\mathbb C[W]^N$ is an isomorphism where $W=V^H$ and $N=N\_G(H)/H$.
... | 10 | https://mathoverflow.net/users/89948 | 257741 | 116,381 |
https://mathoverflow.net/questions/257700 | 4 | The following problem bears some similarity to the Additive Basis Conjecture [[ALM91](http://www.sciencedirect.com/science/article/pii/009731659190045I),[JLPT92](http://www.sciencedirect.com/science/article/pii/009589569290016Q)] saying (in characteristic $3$) that there is an absolute constant $N$ such that for any po... | https://mathoverflow.net/users/9924 | Ensuring that a sum of squares is non-zero (in a finite field) | For any $k\geqslant 2m+1$ elements $t\_1,\dots,t\_{k}$ in the additive group $\mathbb{F}\_3^m$ there exist coefficients $\varepsilon\_1,\dots,\varepsilon\_{k}\in \{0,1\}$, not all of them equal to 0, such that $\sum \varepsilon\_i t\_i=0$. This is a result of Olson (John E. Olson. A Combinatorial Problem on Finite Abel... | 4 | https://mathoverflow.net/users/4312 | 257742 | 116,382 |
https://mathoverflow.net/questions/257686 | 1 | In the paper
<http://topo.math.auburn.edu/tp/reprints/v05/tp05011.pdf>
the author claimes (Theorem 2, without proof) that for a completely regular Hausdorff space $X$ the following are equivalent:
(1) The space of continuous functions $C(X)$ with the compact open topology is countably tight.
(2) Every open cove... | https://mathoverflow.net/users/58628 | Lindelöf Property for Open Covers for Compact Sets | The following is adressed to the second part of your question.
For brevity, let's call a collection $\mathcal{U}$ of open sets of $X$ as $k$-cover if for any compact subset $K\subset X$ there exists $U\in\mathcal{U}$ such that $K\subset U$. Keep in mind that (clearly) any $k$-cover of $X$ is an open covering for $X$.... | 3 | https://mathoverflow.net/users/41407 | 257753 | 116,384 |
https://mathoverflow.net/questions/257722 | 13 | $\DeclareMathOperator\Spec{Spec}$Let $k \subset L$ be two algebraically closed fields of characteristic $0$. Let $U \subset \mathbb P^n\_k$ be a smooth quasi-projective variety and let $U\_L$ denote the base change of $U$ to $\Spec (L)$. Does anyone have a reference for why the map on étale fundamental groups $\pi\_1(U... | https://mathoverflow.net/users/75970 | Is the map on étale fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism? | $\DeclareMathOperator\Frac{Frac}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Hilb{Hilb}$This is an expansion of my comments above. You do not need resolution of singularities or SGA 4. The key step is "elimination of
ramification" or "Abhyankar's Lemma". This is proved in Append. 1 of Exposé XIII of SGA 1. Here ... | 8 | https://mathoverflow.net/users/13265 | 257762 | 116,390 |
https://mathoverflow.net/questions/257683 | 3 | For any finite crystallographic reflection group $W = \langle s\_1, \ldots , s\_n\rangle$, every hyperplane reflection is of the form $ws\_iw^{-1}$ for some $i$ and some $w \in W$.
A finite crystallographic reflection group $W$ is a Coxeter group with the presentation
\begin{align}\label{Coxeter system}
S=\langle s\... | https://mathoverflow.net/users/89288 | About reflections of reflection groups | 1) No. $W$ consists of elements of determinant 1 and -1. According to your wikpedia, all elements of determinant 1 are "rotations". Elements of determinant -1 are not necessarily reflections because they are not necessarily of order 2. Just think of a 4-cycle $(1,2,3,4)\in S\_4$: his order is 4, not 2. It is a proper r... | 3 | https://mathoverflow.net/users/5301 | 257768 | 116,394 |
https://mathoverflow.net/questions/257770 | 4 | It is known that for $n \not\equiv 0 \mod 4$, the oriented cobordism ring $MSO\_n$ is finite. That is, for oriented n-dimensional manifold $Y$, there exists $m\in \mathbb{N}$, such that $mY$ bounds.
Does it hold for equivariant oriented cobordism with compact Lie group action?
Addition: @Oscar Randal-Williams show... | https://mathoverflow.net/users/49927 | Is equivariant oriented cobordism finite? | No. For a $G$-manifold $M$, taking the signature of the fixed points $M^G$ defines a homomorphism $\phi : \Omega\_n^G \to \mathbb{Z}$, as if $W : M\_0 \leadsto M\_1$ is a cobordism then so is $W^G : M\_0^G \leadsto M\_1^G$.
Now let $G=S^1$, $X=\mathbb{CP}^k$ with the $G$-action having $(k+1)$ fixed points, $Y=\mathbb... | 8 | https://mathoverflow.net/users/318 | 257772 | 116,396 |
https://mathoverflow.net/questions/257773 | 12 | Let $R$ be a regular ring over a field of char 0. Let $X=Spec R$ and $D=\mathcal{D}\_X$
the algebra of differential operators over it.
The overall vague question is what kind of algebraic object is $D$ and what kind of category is the category of its modules? Here are some points whose answers could together be cons... | https://mathoverflow.net/users/22810 | What kind of algebraic object is $\mathcal{D}_X$? (algebra of diifferential operators). What's special about modules over it? | 1. Proposition 1.2.9 of <http://math.columbia.edu/~scautis/dmodules/hottaetal.pdf> explains that if $M$ and $N$ are both left $D$-modules and $M'$ and $N'$ are both right $D$-modules then
(a) $M\otimes\_{R} N$ is naturally a left $D$-module;
(b) $M'\otimes\_{R} N$ is naturally a right $D$-module;
(c) $\mathrm{Hom... | 10 | https://mathoverflow.net/users/345 | 257776 | 116,398 |
https://mathoverflow.net/questions/257748 | 4 |
>
>
> >
> > I want to show that $End\_0 (B\_n(G)) = \cup\phi\_{\sigma,g} \cup C\_{I(B\_n(G))}$, where $\phi\_{\sigma,g} : B\_n(G) \rightarrow B\_n(G) $ is an endomorphism is defined by $(i,a,j)\phi\_{\sigma,g} = (i\sigma , ag , j\sigma)$ and $\sigma \in S\_n$ and $g \in End(G)$, $C\_X$ is the set of all constant ma... | https://mathoverflow.net/users/46769 | Endomorphism of Brandt Semigroup $B_n(G)$, where $G$ is a finite group | Note that $(i,a,j)=(i,1,1)(1,a,1)(1,1,j)$ and so a homomorphism is determined by what it does to the elements $(i,1,1)$ and $(1,1,j)$ and $(1,a,1)$ with $a\in G$. Moreover, since $B\_n(G)$ is an inverse semigroup and $(i,1,1)$ is the inverse of $(1,1,i)$, in fact an endomorphism is determined by what it does to element... | 3 | https://mathoverflow.net/users/15934 | 257779 | 116,399 |
https://mathoverflow.net/questions/257730 | 4 | Let $\mathfrak{S}\_n$ be the permutation group on $[n]$. Denote the cardinality of $\{\pi\in\mathfrak{S}\_n: \pi^2=id\}$, the set of [involutions](http://mathworld.wolfram.com/PermutationInvolution.html), by $I(n)$.
It is well-known that these numbers have the exponential generating function
$$\sum\_{n\geq0}I(n)\frac{x... | https://mathoverflow.net/users/66131 | permutations rescuing chain/product rules? | Here is what I would consider a "cute proof." Write
\begin{eqnarray\*} \sum\_{m\geq 0} D^mf(x) \frac{t^m}{m!} & = &
f(x+t)\\ & = & f(x)e^{t+\frac 12t^2 +tx}. \end{eqnarray\*}
We get your formula by taking the coefficient of $t^m/m!$ in the
product
$$ e^{t+\frac 12t^2}e^{tx}. $$
Note also that directly from the Expon... | 4 | https://mathoverflow.net/users/2807 | 257780 | 116,400 |
https://mathoverflow.net/questions/257672 | 1 | I rapidly recall a general construction exposed in [this](http://www.math.muni.cz/%7Erosicky/papers/TF2.pdf) paper.
>
> If $(\cal E,M)$ is a factorization system on $\mathbf C$ (let's say $\bf C$ has finite co/limits, or it's even co/complete) such that $\cal M$ satisfies the 3-for-2 property, then there is a rule ... | https://mathoverflow.net/users/7952 | Conditions so that a FS induces a Frobenius adjunction | Let $i\dashv r\dashv i:\mathcal{B}\hookrightarrow \mathcal{A}$ be a bireflective subcategory. $r$ is necessarily left exact hence this a special type of essential localization called a 'quintessential localization' in Johnstone's 1996 TAC paper (see <https://ncatlab.org/nlab/show/quality+type> for references).
An *ex... | 3 | https://mathoverflow.net/users/102745 | 257784 | 116,402 |
https://mathoverflow.net/questions/257699 | 8 | I need to clarify some idea I have in my mind about linear and non-linear regressions. Whatever I know about this topic comes from the book of Taylor "Introduction to error analysis": a set of measurements ${x\_i}$ and ${y\_i}$ for $i= 1, 2, \dots N$ are assumed to have a trend according to a specific function $y = f(x... | https://mathoverflow.net/users/102696 | Gauss-Newton vs gradient descent vs Levenberg-Marquadt for least squared method | The Levenberg-Marquardt method is the most effective optimization algorithm, to be preferred over the methods of steepest descent and Gauss-Newton in a wide variety of problems. You might find this [explanation](https://people.duke.edu/~hpgavin/ce281/lm.pdf) by Henri Gavin instructive:
>
> The Levenberg-Marquardt c... | 12 | https://mathoverflow.net/users/11260 | 257787 | 116,404 |
https://mathoverflow.net/questions/257705 | 7 | Suppose $S$ is a non-compact and complete surface (2 dimensional smooth Riemannian manifold) of constant curvature. I am wondering if there exists a group $G$ which acts by isometries and properly discontinuously on $S$ such that $S/G$ becomes compact?! Are there maybe any reference where I can find results related to ... | https://mathoverflow.net/users/99795 | Does any surface of constant curvature admit a cocompact group action? | As Uri Bader says in the comments, covering-space theory implies that this happens if and only if $\pi\_1S$ is a normal subgroup of $\pi\_1\Sigma$, where $\Sigma$ is some compact surface.
The cases of positive and zero curvature are easy, so we may as well assume that $S$ and $\Sigma$ are of constant negative curvatu... | 14 | https://mathoverflow.net/users/1463 | 257793 | 116,406 |
https://mathoverflow.net/questions/257441 | 4 | Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.
Is then $H^1\_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1\_\mathrm{fppf}(X,\mathscr{A}[p])$ finite?
This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems <http://jmilne.org/math/Bo... | https://mathoverflow.net/users/nan | finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme | Let $k$ be a finite field. Let $X$ be a normal proper variety. Let $G$ be a finite flat commutative group scheme over $X$ of order a power of $p$.
Lemma 1. If $T$ is a $G$-torsor over $X$ and $T$ is trivial over the generic point of $X$, then $T$ is trivial.
Proof. Namely, let $X' \subset T$ be the scheme theoretic... | 3 | https://mathoverflow.net/users/102034 | 257801 | 116,408 |
https://mathoverflow.net/questions/257822 | 26 | Just for fun, I began to play with numbers of two distinct ciphers. I noticed that most of the cases if you consider the numbers $AB$ and $BA$ (written in base $10$), these have few common divisors: for example $13$ and $31$ are coprime, $47$ and $74$ are coprime. Obviously this is not always the case, because one can ... | https://mathoverflow.net/users/102730 | A surprising conjecture about twin primes | Suppose $n-1$ and $n+1$ are both primes.
$\gcd(an+b,bn+a)$ divides $an+b - (bn+a) = (a-b)(n-1)$.
There are two cases. If $n-1$ divides $\gcd(an+b,bn+a)$ then $b=n-1-a$ so $an+b= (n-1) (a+1)$ and $bn+a=(n-1)(b+1)$, so $\gcd(an+b,bn+a) = (n-1)\gcd(a+1,b+1)$.
$(a+1)+(b+1)=n+1$. Because $n+1$ is prime, two numbers th... | 29 | https://mathoverflow.net/users/18060 | 257825 | 116,416 |
https://mathoverflow.net/questions/257832 | 5 | **Q:** What exactly is a power admissible model?
**Background:** Admissible models, introduced by Jon Barwise, form the building blocks of inner model theory. They are transitive models $\mathcal M = (M; \in)$ satisfying a suitable fragment of set theory, namely **K**ripke-**P**latek set theory. Sifting through a cou... | https://mathoverflow.net/users/57114 | What is a 'power admissible model'? | Stefan, in the paper "The Strength of Mac Lane Set Theory," Mathias says that the notion is due to Harvey Friedman, and essentially coincides with what you describe, except that in KP$^P$, foundation is restricted to universal formulae. (See page 47 of Mathias' paper
<https://www.dpmms.cam.ac.uk/~ardm/maclane.pdf>)
| 5 | https://mathoverflow.net/users/102670 | 257835 | 116,419 |
https://mathoverflow.net/questions/257797 | 5 | I have listened to lectures that mention Jonquières automorphisms for affine spaces by name. They don't seem to be found in textbooks on algebraic geometry.
I would like to know the exact reference preferably for the original work where it can be found.
I have access to Hanspeter Kraft's Bourbaki seminar talk *Challe... | https://mathoverflow.net/users/22878 | When and where were Jonquières automorphisms defined first? | E. de Jonquières: De la transformation géométrique des figures planes, et d'un mode de génération de certaines courbes à double courbure de tous les ordres. Nouv. Ann. (2) 3, 97--111 (1864). You can find it in Numdam.
| 7 | https://mathoverflow.net/users/3903 | 257837 | 116,421 |
https://mathoverflow.net/questions/257744 | 1 | Let $A$ be a local selfinjective algebra with indecomposable module $M$.
Let $N=A \oplus M$.
When there is an indecomposable module $U$ not in $add(N)$, having finite $add(N)$-resolution for some choice of $A$ and $M$?
This is not possible in general due to the following examples:
* $A=K[x]/(x^n)$ for arbitary ... | https://mathoverflow.net/users/61949 | Finite add(N)-resolution | I think that this is possible if and only if $\operatorname{Ext}^1\_A(M,M)=0$, and so the question reduces to another question [Ext^1 for a local finite dimensional selfinjective algebra](https://mathoverflow.net/questions/249252/ext1-for-a-local-finite-dimensional-selfinjective-algebra) that you have asked (and which ... | 1 | https://mathoverflow.net/users/22989 | 257839 | 116,422 |
https://mathoverflow.net/questions/257833 | 2 | Let $L$ be a finite extension of $\mathbb{Q}\_p$, $G$ be a locally $L-$analytic group, and let $H$ be a compact open subgroup of $G$. Then in corollary $5.3.19$ of [this paper](http://www.math.uchicago.edu/~emerton/pdffiles/analytic.pdf) Emerton shows that the space of locally analytic distributions $\mathcal{D}^{la}(H... | https://mathoverflow.net/users/69289 | Locally analytic distribution algebra is a Frechet-Stein algebra | The basic idea is that any compact open $H$ contains good open subgroup $H'$. (This is essentially just by taking any sufficiently small neighborhood of the origin in the Lie algebra and applying the exponential map. If $\mathfrak{h}$ is a $\mathbb{Z}\_p$-lattice in the Lie algebra of $H$ and $[\mathfrak{h},\mathfrak{h... | 3 | https://mathoverflow.net/users/93798 | 257855 | 116,428 |
https://mathoverflow.net/questions/257852 | 7 | For the sake of this question I want to focus on an unfair coin.
Assuming we have a number of $i.i.d.$ samples $X\_1, ..., X\_n$ and a precision level requirement $\tau$. I search for the optimal estimator $\hat{P\_n}$ minimizing the expression $P(|\hat{P\_n} - {P(X\_1 = 1)} |\geq \tau)$. Also I would like to know the ... | https://mathoverflow.net/users/102783 | Is there a proof that the Law of Large Numbers is the limit (numerical) for estimation of expected values? | And if you want the *exact* minimax risk rate, take a look at the recent preprint by Iosif Pinelis and myself: <https://arxiv.org/abs/1606.08920>
Essentially, the above shows that the optimal estimator is the maximum-likelihood one (i.e., the obvious one obtained by dividing the number of heads by sample size) and th... | 3 | https://mathoverflow.net/users/12518 | 257856 | 116,429 |
https://mathoverflow.net/questions/257863 | 4 | Let $X$ be a variety and $\varphi : F\_1 \to F\_2$ be a morphism of vector bundles over $X$. Then it is easy to check that the locus on $X$ for which $\varphi$ vanishes is a closed subscheme of $X$. Furthermore, the ideal sheaf cutting out this locus can be described locally by trivializing both vector bundles.
Let ... | https://mathoverflow.net/users/45609 | Zero locus of a family of morphisms of vector bundles | I am just writing my comment above as an answer. This follows from Corollaire 7.7.8, EGA III.
Grothendieck, Alexander
Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Seconde partie.
Publications Mathématiques de l'IHÉS, 17 ... | 6 | https://mathoverflow.net/users/13265 | 257868 | 116,431 |
https://mathoverflow.net/questions/257850 | 3 | Let L be a linear differential operator with eigenfuctions $\phi\_{k}$ then for $a\_{n}\sim N(0,1)$ consider
$$h(z):=\sum a\_{n}\phi\_{k}(z).$$
Is there a general theory for such sums? For $L=\Delta$, h is called the GFF and has some interesting properties such as the GFF Markov property and its circular average be... | https://mathoverflow.net/users/99863 | Reference: random sum of eigenfunctions $\sum a_{n}\phi_{k}(z)$, $a_{n}\sim N(0,1)$ | Check out (on arxiv.org, if you like) work by Nazarov and Sodin and Sarnak and Wigman - they study many aspects of this sort of thing.
| 1 | https://mathoverflow.net/users/11142 | 257872 | 116,433 |
https://mathoverflow.net/questions/257871 | 2 | The Hardy-Littlewood maximal operator
$$Mf(x)=\sup\_{x\in B}\frac1{\vert B\vert}\int\_B\vert f(y)\vert dy$$
where the supremum is taken over all balls $B\subset\mathbb{R}^n$ which contain $x$.
It is well-known that $M$ is both strong (for $p>1$) and weak-type (for $p\geq1$) integral operator; see for example [this p... | https://mathoverflow.net/users/66131 | is this weighted-maximal function unbounded? | It is bounded with constant depending only on $n$ and $p$ (apparently we get strong estimates). This is a result of Forzani et al (see [here](http://www.ams.org/journals/proc/2002-130-01/S0002-9939-01-06156-1/S0002-9939-01-06156-1.pdf)) which states the following:
**Theorem (L.Forzani, R. Scotto,
P. Sjogren & W. Urbi... | 2 | https://mathoverflow.net/users/48438 | 257875 | 116,435 |
https://mathoverflow.net/questions/257867 | 1 | I have $N$ Bernoulli random variables that each one of them is negatively dependent to exactly one of the other variables, for example: $Y\_1$ is dependent to $Y\_2$ and $Y\_3$ is dependent to $Y\_4$ and so on.
All $Y\_i$ with odd indexes are independent and identically distributed and all $Y\_i$ with even indexes are ... | https://mathoverflow.net/users/102790 | Central limit theorem for negatively dependent random variables | Assuming $Y\_{2i+1} + Y\_{2i+2}$ is independent of $Y\_{2i+3} + Y\_{2i + 4}$, then the [answer to this question](https://mathoverflow.net/questions/29508/is-there-a-central-limit-theorem-for-bounded-non-identically-distributed-random) answers yours in the affirmative. To prevent non-degeneracy, your condition that "dis... | 0 | https://mathoverflow.net/users/4923 | 257884 | 116,436 |
https://mathoverflow.net/questions/257838 | 2 | A simplicial category is a category enriched over the monoidal category of simplicial sets (morphism sets are now simplicial sets), and the collection of all such categories forms a category itself (modulo set theoretic issues). It is asserted on p. 23 of "Higher topos theory" that this latter category has all small co... | https://mathoverflow.net/users/70964 | Colimits in the category of simplicial categories | To expand on Dmitri Pavlov's answer, the recipe for colimits, as in any monadic category, will be the following.
1. Take the colimit of the underlying (simplicial) graphs.
2. Apply the free functor.
3. Mod out by all relations that existed in the (simplicial) categories you're taking the colimit of.
Colimits in $\m... | 6 | https://mathoverflow.net/users/2362 | 257890 | 116,438 |
https://mathoverflow.net/questions/257798 | 7 | In [his paper "Forcing in admissible sets"](http://download.springer.com/static/pdf/739/art%253A10.1007%252FBF01978554.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2FBF01978554&token2=exp=1482366949~acl=%2Fstatic%2Fpdf%2F739%2Fart%25253A10.1007%25252FBF01978554.pdf%3ForiginUrl%3Dhttp%253A%252F%252Fl... | https://mathoverflow.net/users/8133 | History of forcing over admissible sets | Forcing over admissible sets was first carried out by Jon Barwise in his 1967 Stanford dissertation.
[This recent paper](https://www.dpmms.cam.ac.uk/~ardm/fifofields3.pdf) of Mathias (it appeared in Fundamenta Mathamaticae in 2015) develops forcing over set theories that are weaker than KP. It also has a nice histor... | 7 | https://mathoverflow.net/users/9269 | 257894 | 116,439 |
https://mathoverflow.net/questions/255758 | 14 | Inspired by [this MO question about integrating binomial coefficients](https://mathoverflow.net/questions/254881/binomial-again-and-again/) and the answers, I was wondering whether integrating powers of binomial coefficients also relates to the respective sums. And indeed I have strong numerical evidence that
$$\int\_... | https://mathoverflow.net/users/6415 | Integral of power of binomials equal to sum of power of binomials? | The generalization looks like this
$$
\int\_{-\infty}^{\infty} \binom{n}{\alpha x}^l dx =\sum\_{k=-\infty}^\infty\binom{n}{\alpha k}^l,\quad 0<\alpha\le 2/l,~l\in\mathbb{N}\tag{1}
$$
where $n$ need not be an integer. The general theorem is given for example in the paper [Surprising sinc sums and integrals](https://www.... | 7 | https://mathoverflow.net/users/82588 | 257903 | 116,442 |
https://mathoverflow.net/questions/257885 | 23 | How does one go from an understanding of basic algebraic topology (on the level of Allen Hatcher's Algebraic Topology and J.P. May's A Concise Course in Algebraic Topology) to understanding the [paper of Hill, Hopkins, and Ravenel on the Kervaire invariant problem](https://annals.math.princeton.edu/2016/184-1/p01)?
I... | https://mathoverflow.net/users/85392 | Roadmap to Hill-Hopkins-Ravenel | There is one major topic that is missing from your list: **spectra**. Spectra (and $E\_\infty$-ring spectra) are the basics of modern stable homotopy theory and are not treated, if not very cursorily, in the references you've looked at. Seriously, spectra are the bread and butter of a homotopy theorist nowadays and the... | 22 | https://mathoverflow.net/users/43054 | 257908 | 116,443 |
https://mathoverflow.net/questions/257904 | 0 | i have this numerical calculation problem :
$$\prod \limits\_{i=121443371}^{455052511} 1+\frac{1}{p(i)} \leq 1.06406506887043952285362856325019948 $$
such that $p(i)$ is the $i$-th prime number
i want to check whether the left side of the inequality is truly less than the number on the right side or not ?
i used W... | https://mathoverflow.net/users/95470 | Numerical Calculations | In 1939 Rosser proved that $p\_n>n\log n$. In 1999 Dusart proved
$p\_n>n(\log n+\log\log n-1)$ and also
$$p\_n\ge n\Bigl(\log n+\log\log n-1+\frac{\log\log n-2.25}{\log n}\Bigr),\qquad
n\ge 2.$$
Finally Dusart in 2010 proved
$$p\_n\le n\Bigl(\log n+\log\log n-1+\frac{\log\log n-2}{\log n}\Bigr),\qquad
n\ge 688383.$$
... | 8 | https://mathoverflow.net/users/7402 | 257914 | 116,445 |
https://mathoverflow.net/questions/257909 | 6 | Let $n$ be a given positive integer, and let $f(x)=\displaystyle\sum\_{k=0}^{n}a\_{k}x^k$, where $a\_{i}\in \mathbb{R}$, $0 \le i \le n$. If
$$|f(x)|\le 1,\qquad \text{for } ~|x|\le 1,$$
what is the maximum of the $|a\_{p}|$ for a fixed $p$？
I conjecture that the answer is $|[x^p]T\_{n}(x)|$, where the $T\_{n}(x)$ a... | https://mathoverflow.net/users/38620 | Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$ | Let $T\_{n}(x)=\sum\_{\nu=0}^{n}t\_{n,\nu}x^{\nu}$ denote the Chebyshev polynomial (of the first kind) of degree $n$ and let $x\_{n,\nu}=\cos\nu\pi/n$ for $0\leq\nu\leq n$.
The answer follows from the following two results :
1) Let $f$ be a polynomial (possibly complex), of degree at most $n$, such that
$$|f(x\_{n... | 14 | https://mathoverflow.net/users/89429 | 257918 | 116,447 |
https://mathoverflow.net/questions/257934 | 2 | There is an isomorphism between (rational) correspondences on a curve $C/\mathbb{F}\_p$ orthogonal to the "valence zero" ones (i.e. orthogonal under intersection pairing to $\{\*\}\times C$ and $C\times \{\*\}$) and endomorphisms of the Jacobian $J(C)$, giving a pretty and applicable demonstration of how $J(C)$ "is" th... | https://mathoverflow.net/users/58688 | Geometric (or at least non-cohomological) proof of Lefschetz trace formula for curves | This was proved by Weil using his intersection theory. For a modern exposition, see 11.2 of Milne, J. S. Jacobian varieties. Arithmetic geometry (Storrs, Conn., 1984), 167--212, Springer, New York, 1986. Available on his website.
| 4 | https://mathoverflow.net/users/102818 | 257935 | 116,449 |
https://mathoverflow.net/questions/257834 | 7 | Take the variety $X$ to be $\mathbb{C}\_\infty \times\mathbb{C}\_\infty $ with the points $(0,0)$ and $(\infty,\infty)$ removed. Use coordinates $(z,w)\in\mathbb{C}\times \mathbb{C}$ for one chart of the product of the two Riemann spheres. Then $\xi\in\Omega^{0,1}X$given by
\begin{eqnarray\*}
\xi\,=\,\frac{z\,\mathrm{d... | https://mathoverflow.net/users/29625 | Cohomology of a projective variety with points removed | If I understand correctly, you are trying to compute the Dolbeault cohomology of your $X$. Also, I am assuming that by $\mathbb C\_\infty$ you mean the Riemann sphere, which in algebraic geometry would be the projective line $\mathbb P^1\_{\mathbb C}$. Of course, this is just rephrasing what you are saying, mainly for ... | 3 | https://mathoverflow.net/users/10076 | 257950 | 116,452 |
https://mathoverflow.net/questions/257961 | 4 | I was wondering if for an arbitrary valuation ring $R$ and an element $\pi$ in the maximal ideal of $R$, the rings
1. $R[x\_1,\ldots,x\_n]$;
2. projective limit of $R/(\pi^l)[x\_1,\ldots,x\_n]$, probably also denoted as $R \langle x\_1,\ldots,x\_n \rangle$ in the case when $R$ is a rank 1 valuation ring and;
3. $R/(\... | https://mathoverflow.net/users/24913 | Is polynomial ring of n variables with coefficients in an arbitrary valuation ring coherent? | Concerning question (1):
>
> Polynomial algebras in finitely many indeterminates over valuation domains are coherent.
>
>
>
This is Theorem 7.3.3 in S. Glaz, *Commutative coherent rings,* Springer LNM 1371 (1989).
| 6 | https://mathoverflow.net/users/11025 | 257962 | 116,455 |
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