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https://mathoverflow.net/questions/244197 | 1 | For a set of under-determined linear equations, I was wondering if there is any closed form for all non-negative solutions? Is there a way to analytically characterize the feasibility set of such equations?
More formally, for given $M\in \mathbb{R}^{n\times m}\_{+}$ and $y\in \mathbb{R}^{m\times 1}$, find all solutio... | https://mathoverflow.net/users/95000 | nonnegative solution of nonhomogeneous under-determined linear system of equations | All such vectors $x\geq 0$ form a polyhedron in $\mathbb{R}^n$, which may be empty. The latter can be certified using Farkas Lemma, as David points out in the comment. On the other hand, a nonempty polyhedron is a Minkowski sum of a subspace, a polyhedral cone, and a polytope.
(see Wikipedia for terms I used).
Your ... | 1 | https://mathoverflow.net/users/11100 | 244201 | 111,823 |
https://mathoverflow.net/questions/241932 | 5 | Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy of $\mathrm{SL}(2,\mathbb{C})$ such that
$\bullet$ its action preserves a decomposition $\mathbb{C}^n = \mathbb{C}^2 \... | https://mathoverflow.net/users/25511 | Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$ | The only groups $G$ of the type you are looking for are $SL(n)$, $SO(n)$ or $Sp(n)$.
This is proved in a paper by Beukers and Heckman (see Proposition (6.4) in that paper). See the math review
<http://www.ams.org/mathscinet-getitem?mr=974906>
for a reference.
Some explanation: Beukers and Heckman talk about ... | 4 | https://mathoverflow.net/users/23291 | 244206 | 111,824 |
https://mathoverflow.net/questions/241565 | 13 | I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL\_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}^3$ is preserved by all elements of $G$, and if every element of $G$ has all eigenvalues equal in modulus, then there ... | https://mathoverflow.net/users/1840 | Groups of matrices in which all elements have all eigenvalues equal in modulus | The specific question for $SL(3,\mathbb{R})$ has already been answered by Misha and Uri Bader. I just want to make some additional remarks which say that this holds in greater generality.
Lemma: Suppose $\Gamma \subset GL(n,\mathbb{C})$ is a subgroup such that all the eigenvalues of all the elements of $G$ have abso... | 4 | https://mathoverflow.net/users/23291 | 244208 | 111,825 |
https://mathoverflow.net/questions/244214 | 12 | One major approach to the theory of forcing is to assume that ZFC has a countable *transitive* model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, uses the fact that $M$ is *countable* to prove that there exists a generic set $G \in V$, then defines $M[G]$ as an a... | https://mathoverflow.net/users/56878 | Why do we need a transitive model in forcing arguments? | Yes, one can undertake forcing without the transitivity assumption,
and even the countability of the model is not important.
One of the standard ways to do this is with the Boolean-valued
model quotient construction, which has been described in many places. Basically, given a forcing notion $B$,
a complete Boolean al... | 20 | https://mathoverflow.net/users/1946 | 244218 | 111,832 |
https://mathoverflow.net/questions/244204 | 7 | How can I evaluate
$$ \int\_{-1}^1 P\_n(x)P\_l(x)x^k dx $$
when $k$ is even?
Or what might be a source where I could find integrals like this?
| https://mathoverflow.net/users/94200 | Legendre Polynomial Integral | The integral $$\int\limits\_0^1 x^k P\_m(x)P\_n(x)dx$$ is evaluated in terms of the hypergeometric function $\_3F\_2$ in <http://link.springer.com/article/10.1007/BF01650571> (Some integrals containing products of legendre polynomials, by L. Carlitz). The evaluation is based on
$$P\_m(x)P\_n(x)=\sum\limits\_{r=0}^{\mat... | 7 | https://mathoverflow.net/users/32389 | 244219 | 111,833 |
https://mathoverflow.net/questions/244215 | 4 | Let
$$
\tilde{\mathbb C}={\mathbb C}\smallsetminus (-\infty,0]
$$
the complex plane without the negative real axis. Let $V$ denote the set of all holomorphic functions $f:\tilde{\mathbb C}\to{ \mathbb C}$ such that
$$
f^+(0)=\lim\_{\lambda\searrow 0}f(\lambda)
$$
exists.
If $K\subset \{\mathrm{Re}(z)>0\}$ is a compac... | https://mathoverflow.net/users/nan | Integral representation of a limit | No. The existence of such a representation would imply that the linear functional $\Phi:V\to \mathbb C$, $f\mapsto f^+(0)$ is a continuous with respect to the topology of uniform convergence on compact subsets of $K$. By Hahn-Banach one could thus extend $\Phi$ to $H(\tilde{\mathbb C})$. This is certainly not true: The... | 4 | https://mathoverflow.net/users/21051 | 244220 | 111,834 |
https://mathoverflow.net/questions/244173 | 2 | Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f\_{1},f\_{2},f\_{3} \in R$ be nonzero elements of $R$.
(a) Consider the varieties $M:=V(f\_{1},f\_{2})$ and $N:=V(f\_{2},f\_{3})$ in $\mathbb{C}^{3}$ and suppose that $M$ and $N$ are curves with at least two irreducible components and exactl... | https://mathoverflow.net/users/81588 | Intersection of two curves is not Cohen Macaulay | The curve $Q$ can indeed be Cohen-Macaulay, and you can use the Hilbert-Burch-Schaps theorem to construct examples by considering the three $2\times 2$ minors of a $3\times 2$ matrix of elements of $R$. One simple example is $(f\_1,f\_2,f\_3) = (x-y,x^2-yz,x-z)$. The complete intersection ideal $\langle x-y,x^2-yz \ran... | 4 | https://mathoverflow.net/users/13265 | 244225 | 111,838 |
https://mathoverflow.net/questions/244222 | 4 | It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
The picture is less clear if $K$ is infinite-dimensional over $\mathbb{Q}$.
I believe the best result is that of Kobayashi who proved (modulo the usual conjectu... | https://mathoverflow.net/users/70751 | Mordell-Weil rank of an elliptic curve over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$? | I think the answers and comments in [Argument for unboundedness of integral points of elliptic curves over number fields](https://mathoverflow.net/questions/164629/argument-for-unboundedness-of-integral-points-of-elliptic-curves-over-number-fie) show the rank is unbounded in *finite* extensions of the rationals and in ... | 2 | https://mathoverflow.net/users/12481 | 244229 | 111,841 |
https://mathoverflow.net/questions/244180 | 5 | I would like to consider the quaternionic projective space $\mathbb{PH}^{n-1}\subset\mathbb{G}\_2(\mathbb{C}^{2n})$ as a subvariety of the Grassmannian of complex 2-planes.
For a real vector $e\in\mathbb{R}^{4n}$ the inclusion is just given by
$$
\mathbb{H}\cdot e
=
\mathbb{C}\cdot e
\oplus
\mathbb{C}\cdot Je.
$$
One ... | https://mathoverflow.net/users/94991 | Quaternionic projective space in complex Grassmannian | Here is how I understand David Treumann indications.
* We take a flag $F$ of $\mathbb{C}^{2n}$ such that $JF$ is the opposite flag.
* The complex codimension of $\mathbb{HP}^{n-1}$ in $\mathbb{G}\_2(\mathbb{C}^{2n})$ being $2(n-1)$ (half of the dimension of the Grassmannian), we consider middle dimensional Schubert v... | 3 | https://mathoverflow.net/users/94991 | 244237 | 111,843 |
https://mathoverflow.net/questions/244254 | 10 | This question came up in our algebraic topology class and our Professor didn't know the answer. I also couldn't find an answer so far.
>
> What is the cardinality of the set of subgroups of $F\_2$?
>
>
>
Here $F\_2 = \mathbb Z \* \mathbb Z$ denotes the free group on two generators. The cardinality of the set o... | https://mathoverflow.net/users/76299 | The set of subgroups of $F_2$ | It is clear that there are $2^{\aleph\_0}$ subgroups of the free group $F\_\infty$ on countably many generators (because each subset of a free generating set generates a different subgroup). In addition, it is well known that $F\_2$ contains a subgroup that is not finitely generated. Since all subgroups of a free group... | 21 | https://mathoverflow.net/users/68305 | 244258 | 111,846 |
https://mathoverflow.net/questions/244261 | 1 | I would like to convert this problem into a Linear Programming Problem :
$\min |x|+|y|+|z|$
subject to $x+y \leq 1$
$2x+z=3$.
The solution to this problem is given [chapter](https://www.calvin.edu/~pribeiro/courses/Power%20Systems%20Interim/Sakis-Book/PSA_Appendix_B.pdf) and [here](http://orinanobworld.blogsp... | https://mathoverflow.net/users/45857 | convert absolute form into linear programming problem | Let's call the following model model $A$:
$\min |x|+|y|+|z|$
subject to $x+y \leq 1$
$2x+z=3$.
and the following model model $B$:
$\min x\_1+x\_2+|y|+|z|$
subject to $x\_1-x\_2+y \leq 1$
$2x\_1-2x\_2+z=3$.
$x\_1,x\_2 \geq 0$
I claim that the two models are equivalent.
To see this, suppose the opt... | 3 | https://mathoverflow.net/users/95037 | 244265 | 111,848 |
https://mathoverflow.net/questions/244268 | 6 | Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?
| https://mathoverflow.net/users/95040 | Numbers divisible only by primes of the form 4k+1 | Yes, $A(X) = cX/\sqrt{\log(X)} + O(X/\log^{3/2}(X))$ for a positive real number $c$ which I think is 1 (edit: This remark on the constant was just a vague recollection which is wrongly remembered as it turns out. See the comments below.). This result holds in much much more generality, e.g. for primes in congruence cla... | 10 | https://mathoverflow.net/users/3384 | 244270 | 111,850 |
https://mathoverflow.net/questions/244184 | 82 | I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course).
I hope this is not too broad.
| https://mathoverflow.net/users/24478 | Examples of algorithms requiring deep mathematics to prove correctness | * **Group Isomorphism of simple groups.** There is a trivial polynomial time algorithm for testing if two (finite) simple groups $G$ and $H$, specified by their multiplication tables, are isomorphic: guess at most two generators $g\_1, g\_2$ from $G$, then guess two elements of $H$ that they map into, and check if the ... | 53 | https://mathoverflow.net/users/35733 | 244274 | 111,852 |
https://mathoverflow.net/questions/125688 | 25 | Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and we know that this $F$ must now satisfy a linear ordinary differential equation
$$P\_{k+1}(T)F^{(k+1)} + P\_{k}(T)F^{(k)} ... | https://mathoverflow.net/users/2024 | When does a modular form satisfy a differential equation with rational coefficients? | Let $K$ be the algebraic closure of the differential field $\mathbb{C}(T)$.
Let $\partial$ denote differentiation w.r.t. $T$. Now $\mathbb{C}(T)[\partial] \subseteq K[\partial]$ are rings of differential operators. Your function $F$ is a solution of $L(F)=0$ where $L \in K[\partial]$ is the differential operator $L ... | 5 | https://mathoverflow.net/users/51091 | 244279 | 111,854 |
https://mathoverflow.net/questions/244294 | 3 | It is a well-known exercise that $C\_n = \chi\_{(n,n)}(1)=\chi\_{(n,n)}^{1^n}$ where $C\_n$ is the $n$th Catalan number and $\chi\_{(n,n)}^{1^n}$ is the character of the irrep $(n,n)$ on conjugacy class $1^n = (1,1,\cdots,1)$.
A related identity that I stumbled across for fixed-point-free involutions is that $\chi\_{... | https://mathoverflow.net/users/32968 | A Combinatorial Identity Involving Characters of $S_n$ (Reference Request?) | This is a special case of Corollary 2.2 in S. Fomin and N. Lulov, [On the number of rim-hook tableaux](http://www.math.lsa.umich.edu/~fomin/Papers/lulov.ps), taking $r=2$ and $\lambda = (n,n)$. It's probably easier to use the formula in the proof than the corollary itself.
| 7 | https://mathoverflow.net/users/7709 | 244298 | 111,856 |
https://mathoverflow.net/questions/244305 | 4 | I am familiar with the nLab web page that nicely lays out the axioms needed to define strict 2-categories using whiskering as opposed to horizontal composition of 2-cells. However, I am old fashioned and, when writing an article, would prefer to use a journal reference over a web URL reference.
Does anyone know of a ... | https://mathoverflow.net/users/15980 | Whiskering approach to strict 2-categories: literature reference needed | The formal laws for defining 2-categories along these lines were first spelled out by Godement:
* Roger Godement, *Topologie algébrique et theorie des faisceaux*, Hermann, Paris, 1958.
See the "five rules of functorial calculus" given in Appendix 1. Indeed, the horizontal composition of 2-cells is sometimes called... | 7 | https://mathoverflow.net/users/2926 | 244306 | 111,858 |
https://mathoverflow.net/questions/244230 | 3 | It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
Much less is known if $K$ is infinite-dimensional over $\mathbb{Q}$.
The rank of $E$ over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$ is infinite. It... | https://mathoverflow.net/users/70751 | Is there an infinite family of primes $q_{1},q_{2},...$ so that the rank of $E(\mathbb{Q}(\sqrt{-q_{i}}))$ equals that of $E(\mathbb{Q})$? | Pasten has answered the question: Murty (MS1106677, Corollary to Theorem 2) has shown that the quadratic twist of $E$ by a prime $q$ has rank zero for infinitely many primes $q$, if GRH holds.
| 3 | https://mathoverflow.net/users/70751 | 244310 | 111,859 |
https://mathoverflow.net/questions/244300 | 3 | I'm learning Arakelov theory on arithmetic surfaces and I have the following general question.
Let $K$ be a number field and consider its ring of integers $O\_K$. Moreover let $S:=\operatorname{Spec} O\_K$ and consider a regular, projective arithmetic surface $\pi: X\to S$. For every embedding $\sigma:K\to\mathbb C$ ... | https://mathoverflow.net/users/47136 | Arakelov divisors and the meaning of real coefficients | Changing the coefficient of a divisor by some $\epsilon > 0$ can have a very significant effect, for example it can move you out of the nef cone. Indeed, even in non-Arakelov algebraic geometry it is often very useful to allow real coefficients on divisors. As a specific example, let $A$ be a simple abelian variety wit... | 4 | https://mathoverflow.net/users/11926 | 244314 | 111,860 |
https://mathoverflow.net/questions/244088 | 14 | Is it consistent (from suitable large cardinals) that there is a ccc poset which forces PFA?
This seems quite implausible to me. If we could force PFA via ccc forcing, the ground model would have to be quite close to a model of PFA (having the correct continuum, no squares, SCH holding etc.). However, the ground mode... | https://mathoverflow.net/users/1058 | Forcing PFA with ccc forcing | I think a negative answer can be derived from the following observations.
One. A nontrivial c.c.c. forcing adds a subset of $\omega\_{1}$ (consider the least cardinal $\kappa$ for which it adds a subset of $\kappa$, and the tree of possible initial segments for this subset; the splitnodes in the tree give rise to a ... | 13 | https://mathoverflow.net/users/31807 | 244320 | 111,862 |
https://mathoverflow.net/questions/244313 | 5 | I have solve this following
>
> Question: Complex numbers ${x\_i},{y\_i}$ satisfy $\left| {{x\_i}} \right| = \left| {{y\_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits\_{i=1}^n{{x\_i}}$, $y=\frac{1}{n}\sum\limits\_{i=1}^n{{y\_i}}$ and $z\_i=x{y\_i}+y{x\_i}-{x\_i}{y\_i}$. Prove that $$\sum\li... | https://mathoverflow.net/users/38620 | maybe this conjecture also hold to this complex inequality | The following seems to be a counterexample:
Let $n=2$, $x\_1 = 1 = - y\_1$, $x\_2= -1 = - y\_2$, $z\_1=z\_2 =1$.
Then $x=y=0$, $z=1$. So $w\_i = -2 x\_i y\_i z\_i = 2$ for $i=1,2$.
Finally, $w\_1 + w\_2 = 4$.
| 5 | https://mathoverflow.net/users/85570 | 244324 | 111,865 |
https://mathoverflow.net/questions/244042 | 1 | According to the answer of znt to the previous version, I revise the question as follows:
Is there a real $(n-1)\times n$ matrix $A$
such that $A$ is not a full rank matrix and satisfy $a\_{ii}<0$ and $a\_{ij}>0$ for $i \neq j$ and $\sum\_{i} a\_{ij}<0$ for every $j\leq n-1$.
A stronger version:
>
> Is there a... | https://mathoverflow.net/users/36688 | Is this a full rank matrix? | No, to your stronger question. If an $n \times n$ matrix satisfies the condition you specified, it would be a [strictly diagonally dominant matrix](https://en.wikipedia.org/wiki/Diagonally_dominant_matrix#cite_note-3) which is non-singular.
| 3 | https://mathoverflow.net/users/56140 | 244340 | 111,869 |
https://mathoverflow.net/questions/244342 | 3 | Consider the following family of polynomials over $\mathbb{Q}$:
$$f\_n = x^n - x^{n-1} - \dots - 1$$
Notice that these polynomials satisfy the recurrence
$$ f\_{n+1} = x f\_n - 1 $$
I would like to show that every polynomial in this family is irreducible over $\mathbb{Q}$. My strategy for showing this is to sho... | https://mathoverflow.net/users/95083 | Irreducibility of family of polynomials | This family of polynomials is studied in [The Galois group of $x^n - x^{n-1} - \cdots - x - 1$](http://www.sciencedirect.com/science/article/pii/S0022404903002457) by Paulo Martin. It is shown that each $f\_n$ is irreducible in Corollary 2.2.
| 7 | https://mathoverflow.net/users/51668 | 244344 | 111,870 |
https://mathoverflow.net/questions/244297 | 8 | The Kelley-Morse set theory can be thought as the "full-secondorderification of $\sf ZFC$", where we switch from sets to classes and allow the comprehension schema to include quantifiers on class variables.
As in the usual case of class-set theory, sets are exactly those classes which are elements of other classes. S... | https://mathoverflow.net/users/7206 | Uncountable models of Kelley-Morse set theory with only a countable number of sets | The first relevant theorem is the following classical result:
**Theorem A.** (Mostowski, Keisler) *If $M$ is a countable model of Kelley-Morse + Choice Scheme, then there is an elementary extension $M^{\*}$ of $M$ such that $\mathrm{Set}^{M}= \mathrm{Set}^{M^{\*}}$ and $M^{\*}$ has cardinality $\aleph\_1$.*
In the... | 11 | https://mathoverflow.net/users/9269 | 244349 | 111,871 |
https://mathoverflow.net/questions/244350 | 3 | [Hurwitz's theorem](https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)) states that a real possibly-non-associative algebra (meaning just a real vector space $V$ equipped with a map $m: V \otimes V \to V$) along with a *positive definite* quadratic form $|\cdot|^2 : V \to \mathbb R$ which is multi... | https://mathoverflow.net/users/78 | Normed algebras of indefinite signature? | I believe the answer to my question is that these are called [composition algebras](https://en.wikipedia.org/wiki/Composition_algebra) and are essentially fully understood. I was hoping the answer would be more interesting.
| 1 | https://mathoverflow.net/users/78 | 244353 | 111,872 |
https://mathoverflow.net/questions/244318 | 4 | Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points
$$
F(X,n):=\{(x\_{1}, \dots, x\_{n})\in X^{n}\: : \: x\_{i}\neq x\_{j}\text{ for each }i,j \}
$$
In <https://www.jstor.org/stable/2946581?seq=1#page_scan_tab_contents> there is an explicit rational dg algeb... | https://mathoverflow.net/users/41970 | cohomology of configuration space of punctured variety | If you read Totaro's paper "Configuration spaces of algebraic varieties" he derives the same cdga calculating the cohomology of $F(X,n)$ as Kriz, but in a different way, via the Leray spectral sequence for $F(X,n) \to X^n$. First he calculates what the first nontrivial page of the spectral sequence looks like for an ar... | 4 | https://mathoverflow.net/users/1310 | 244360 | 111,874 |
https://mathoverflow.net/questions/244364 | 4 | If you factor $x^n-1\in\mathbb{Q}[x]$, then for $n\leq 104$ the coefficients of the factors are in $\{-1, 0, 1\}$. (This is not true for $n=105$, however). Let $U$ be the set of positive integers $n$ such that all the coefficients of the irreducible factors of $x^n-1$ over $\mathbb{Q}$ are either $-1$, $0$, or $1$.
F... | https://mathoverflow.net/users/8628 | Coefficients of factors of $x^n-1\in\mathbb{Q}[x]$ | The density is zero. Let $A(n)$ denote the maximal size of coefficients of the $n$-th cyclotomic polynomial. Erdos conjectured that the density of $n$ for which $A(n) \ge C$ is $1$ for any constant $C$. [Maier](http://link.springer.com/chapter/10.1007%2F978-1-4612-3464-7_22) established this in a strong form, showing t... | 17 | https://mathoverflow.net/users/38624 | 244367 | 111,877 |
https://mathoverflow.net/questions/244368 | 2 | Is it provable, in ZF (without Choice), that every filter can be extended to one of cardinality continuum?
The extended filter is not requested to be an ultrafilter.
| https://mathoverflow.net/users/2415 | Without Choice: Are there filters of cardinality continuum? | If we talk about filters on $\omega$, then the answer is an easy yes.
If $\cal F$ is a filter on $\omega$ such that there is at least one $A\in\cal F$ which is co-infinite, then for every $B\subseteq\omega\setminus A$ we have that $A\cup B\in\cal F$.
But every filter can be extended (perhaps trivially) to include a... | 1 | https://mathoverflow.net/users/7206 | 244373 | 111,881 |
https://mathoverflow.net/questions/244366 | 0 | Let $A$ be a normal local domain with field of fractions $K$. Let $L$ be a finite separable extension of $K$ (if relevant, I'm happy to assume all possible ramification is tame), and let $B$ be the normalization of $A$ in $L$.
Must $B$ have finite projective dimension over $A$?
Disclaimer: I know very little about ... | https://mathoverflow.net/users/88840 | Does the integral closure of a normal local ring in a finite extension of its fraction field have finite projective dimension? | The result of Kantorovitz that you link proves that in many cases $B$ cannot have finite projective dimension as an $A$-module. For instance, let $B$ be $k[x,y]$ and let $A$ be the subring $k[x^2,xy,y^2] \subset k[x,y]$. This is etale in codimension $1$, yet purity does not hold.
In fact, you can write down a free re... | 3 | https://mathoverflow.net/users/13265 | 244376 | 111,882 |
https://mathoverflow.net/questions/244380 | 4 | The introduction [here](https://arxiv.org/pdf/math/0211021v1.pdf) states 'A formal perturbation argument
of Funk later indicated that, modulo isometries and rescalings, the general Zoll
metric on $\mathbb{S}^2$ depends on one odd function $f:\mathbb{S}^2\rightarrow\mathbb{R}$'. This implies that every Zoll metric on $\... | https://mathoverflow.net/users/95090 | Does every Zoll metric on $\mathbb{S}^2$ arise from a perturbation of the round metric? | The last I checked, it was *unknown* whether the set of Zoll metrics on the $2$-sphere was connected. What Guillemin (V. Guillemin, *The Radon transform on Zoll surfaces*, Advances in Mathematics **22** (1976), 85–119.) proved (roughly speaking) is that the set of Zoll metrics conformal to the round metric that are suf... | 7 | https://mathoverflow.net/users/13972 | 244382 | 111,883 |
https://mathoverflow.net/questions/244247 | 5 | Is the following property true for every stable holomorphic bundle of rank 2 with trivial determinant on a compact Riemann surface:
The space of trace-free Higgs fields, whose determinant have only simple zeros, is open and dense in the space of all trace-free Higgs fields.
I would like to see a proof, a reference or... | https://mathoverflow.net/users/4572 | Higgs fields whose determinant have only simple zeros | The statement is easily checked to be true for very stable bundles, so when you look for a counter example you will need to look for a wobbly (i.e. stable but not very stable) bundle.
Suppose that $E$ is a rank two bundle with $E$ stable and $\det E = \mathcal{O}$. We are trying to decide if it can happen that for a... | 7 | https://mathoverflow.net/users/439 | 244385 | 111,886 |
https://mathoverflow.net/questions/244390 | 6 | Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums.
A triangulated category $\mathcal{T}$ is called compactly generated if there is a set $\mathcal{S}$ of objects of $\mathcal{T}$ wh... | https://mathoverflow.net/users/24965 | Is there a compact generated triangulated category which does not have a compact generator? | Examples can be found amongst the derived categories of algebraic stacks, see [Hall--Rydh: Algebraic groups and compact generation of their derived categories of representations](http://arxiv.org/abs/1405.1890v3), in particular theorem A. The easiest example should be $\mathrm{B}\mathbb{G}\_{\mathrm{m}}$.
| 12 | https://mathoverflow.net/users/6263 | 244392 | 111,888 |
https://mathoverflow.net/questions/244393 | 0 | I have a function which lives in $f(x,t)∈L^2(0,T;H^{1/2})∩L^\infty(0,T;L^2)$
for a certain time interval. I also know that $\partial\_{t} \ f(x,t)∈L^2(0,T;H^{−1})$. Can I assure that the function lives in $f(x,t)∈C(0,T;L^2)$, i.e., is continuous in time with values in $L^2$?
| https://mathoverflow.net/users/43026 | Space time Lesbesgue spaces | Because $f\in H^1(0,T;H^{-1})$, hence $f\in C(0,T;H^{-1})$, you may infer that $t\mapsto f(t)$ is continuous into $L^2$ equipped with its weak topology. To prove that it is continous into $L^2$ equipped with its strong (normed) topology, you need that $t\mapsto\|f(t)\|$ be continous ; this is not guaranted by your assu... | 1 | https://mathoverflow.net/users/8799 | 244394 | 111,889 |
https://mathoverflow.net/questions/244405 | 2 | How can one maximize the following function:
$ f(V) = || V \otimes V - U\_1 \otimes U\_2 ||$
where $U\_1, U\_2 \in SU(n)$ are given and we seek to maximize over $V \in SU(n)$.
Both the maximum value of $f$ and the $V$ which maximizes it are interesting.
| https://mathoverflow.net/users/41654 | Maximize inner product of a tensor of unitary matrices | $\newcommand{\diag}{\operatorname{diag}}%
$The maximum distance achieved is 2, except possibly when $n=2$. By left multiplying by the unitary $U\_1^{-1} \otimes U\_1^{-1}$, we are reduced to $U\_1 = I$ and $U\_2 = U$, a unitary.
Rewrite $V\otimes V - I \otimes U = (V \otimes VU^{-1} - I)(I \otimes U)$. Since the lat... | 6 | https://mathoverflow.net/users/42278 | 244412 | 111,894 |
https://mathoverflow.net/questions/244387 | 4 | Let $X$ be a complex projective manifold. Let $B$ be the closed subscheme of $H^1(X,T\_X)$ defined by $\mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal defining the origin. In other words, $B$ is a fat point with tangent space equal to $H^1(X,T\_X)$.
There should be a universal infinitesimal deformation $\mathcal{X... | https://mathoverflow.net/users/48866 | Surjectivity of the Kodaira-Spencer map | **Edit.** I corrected the second short exact sequence.
Here is a "global" description of the sheaf of algebras $\mathcal{O}\_{\mathcal{X}}$. For the affine space $\mathbb{A}$ associated to the vector space $\text{Ext}^1\_{\mathcal{O}\_X}(\Omega\_{X/k},\mathcal{O}\_X)$, on the product $\mathbb{A}\times\_{\text{Spec}(k... | 2 | https://mathoverflow.net/users/13265 | 244423 | 111,898 |
https://mathoverflow.net/questions/244420 | 0 | For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2\_{L^2\_{\alpha}}= \int\_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x \rangle = (1+|x|^2)^{\frac{1}{2}}$
Let $ S = \{x\in \mathbb{R}^3: a<x\_1<b \}$. Note the following easy Poincaré i... | https://mathoverflow.net/users/50438 | Global Poincaré type estimate | $$\int \partial\_1 (\langle x\rangle^{\delta-1} x\_1 u^2) \mathrm{d}x = 0 $$
So
$$ \int [ (\delta - 1) \langle {x}\rangle^{\delta - 3} |x\_1|^2 + \langle{x}\rangle^{\delta - 1}] u^2 ~\mathrm{d}x = -\int 2 \langle x\rangle^{\delta - 1} x\_1 u \partial\_1 u ~\mathrm{d}x $$
so by C-S
$$ | \mathrm{LHS} | \leq 2 \left( \lan... | 3 | https://mathoverflow.net/users/3948 | 244424 | 111,899 |
https://mathoverflow.net/questions/244432 | 4 | Let $G$ be *simply connected, simple* algebraic group over $\mathbb{C}$.
Let $H\subset G$ be a *self-normalizing* spherical subgroup of $G$,
not necessarily connected or reductive.
Here "self-normalizing" means that $\mathcal{N}\_G(H)=H$.
I am interested in a certain quotient $H^{\mathrm{mt}}$ of $H$.
Namely, let $H^... | https://mathoverflow.net/users/4149 | A quotient group of a self-normalizing spherical subgroup | Q1: Yes. More generally, for any spherical subgroup $J\subseteq G$, the quotient $N\_G(J)/J$ is multiplicative (see the Brion-Pauer paper). Apply this to $J=H^0$. Then $\pi\_0(H)\subseteq N\_G(H^0)/H^0$ shows that $\pi\_0(H)$ is multiplicative.
Q2: No. Take $H=N\_G(T)$ where $G=SL(2)$ and $T$ is a maximal torus. Then... | 5 | https://mathoverflow.net/users/89948 | 244433 | 111,903 |
https://mathoverflow.net/questions/244244 | 6 | On p. 127 of Kashiwara-Schapira's paper "Deformation Quantization Modules", there is the following situation: $X$ is a smooth complex (quasi?)projective variety and $\delta\colon X\to X\times X$ is its diagonal embedding. The goal is to prove the Hochschild-Kostant-Rosenberg theorem in this context, but I'm having trou... | https://mathoverflow.net/users/39713 | A mysterious quasi-isomorphism in Kashiwara-Schapira's proof of HKR | First, $P\_k$ is not a direct sum. It is, in fact, an extension (non-trivial!) corresponding to the Atiyah class. Of course, $\delta^\*P\_k$ is not isomorphic to $\Omega^k$, but there is a canonical map $\delta^\*P\_k \to \Omega^k$ (induced by the projection $P\_k \to \delta\_\*\Omega^k$ and the adjunction for $\delta$... | 4 | https://mathoverflow.net/users/4428 | 244441 | 111,907 |
https://mathoverflow.net/questions/243686 | 9 | Let $G$ be a finitely generated group and $S$ a symmetric generating set. Define density (lower density, say) with respect to the sequence of balls $S^n$.
>
> Is it true that a subgroup of $G$ has positive density iff it has finite index?
>
>
>
Low-hanging fruit:
1. This is certainly true if $G$ has subexpon... | https://mathoverflow.net/users/20598 | Finite-index iff positive density? | It's not true. Let $\def\Z{\mathbf{Z}} G = F\_2 \times \Z$, let $H = F\_2$, and let $S$ be the generating set consisting of the identity, $x,x^{-1},y,y^{-1}\in F\_2$ and $1,-1\in\Z$. Then for all $n\geq 0$ and $k\in\Z$ we have
$$ |S^n\cap(F\_2 + k)| = \begin{cases} 2\cdot 3^{n-|k|} - 1 & \text{if}~ n\geq |k|, \\0 & \te... | 3 | https://mathoverflow.net/users/20598 | 244444 | 111,909 |
https://mathoverflow.net/questions/244445 | 0 | How many subsets $I$ of $S:=\{1,\cdots,n\}$ exist such that
$\sum\_{i \in I} x\_i \neq \sum\_{j \in S-I} x\_j$
for all $0 < x\_1 < \cdots < x\_n$?
Let $b\_n$ be this number. Then we are interested in the number $a\_n := 2^n - b\_n$.
Is there any reference which considers this situation, where maybe I can read t... | https://mathoverflow.net/users/nan | How many subsets I of $\{1,\cdots,n\}$ exist? | I claim that $b\_n=2\binom{n}{[n/2]}$ for $n\geqslant 1$. If $\sum\_{i\in I} x\_i\ne \sum\_{i\notin I} x\_i$, then either the sign is always '$>$' or always '$<$' (since the set of strictly increasing sequences is convex and therefore connected.) So, we need to prove that there are exactly $\binom{n}{[n/2]}$ subsets $I... | 9 | https://mathoverflow.net/users/4312 | 244455 | 111,911 |
https://mathoverflow.net/questions/244451 | 1 | Bessaga-Pelczynski Selection Principle states that if $(x\_{n})\_{n}$ is a basis for a Banach space $X$, then every normalized weakly null sequence $(y\_{n})\_{n}$ in $X$ admits a subsequence that is equivalent to a block basic sequence with respect to $(x\_{n})\_{n}$. I am wondering whether the sequence $(y\_{n})\_{n}... | https://mathoverflow.net/users/41619 | Bessaga-Pelczynski Selection Principle | If $(x\_n)\_{n=1}^\infty$ is normalized then yes, that is true. It is even true if we replace the normalization criterion with $\lim\|y\_n\|=1$. The proof is in Albiac-Kalton, although it is not formally stated there. For a reference, see Theorem 1.1 [here](http://www.eweb.unex.es/eweb/extracta/Vol-29-1-2/29J1_2Wallis.... | 1 | https://mathoverflow.net/users/73784 | 244460 | 111,912 |
https://mathoverflow.net/questions/244379 | 6 | Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure that the corepresentations of $G \otimes H$ are direct sums of corepresentations of the form $V \otimes W$, where $V$ is a ... | https://mathoverflow.net/users/81477 | Corepresentations of Tensor Products of Hopf Algebras | Let $A$ be a coalgebra over a field $\mathbb K$ and $\mathcal A = \mathrm{Comod}^A$ its category of comodules. A well-known result of Sweedler writes $A = \mathrm{colim}\_i A\_i$ where $A\_i$ are finite-dimensional subcoalgebras of $A$. Let $A\_i^\*$ denote the linear dual to $A\_i$ and $\mathcal A\_i = \mathrm{Mod}\_{... | 6 | https://mathoverflow.net/users/78 | 244461 | 111,913 |
https://mathoverflow.net/questions/244473 | 2 | Let $f(x)$ be a non-linear polynomial over $\mathbb{Z}$.
Consider the following sum $\pi\_f(x):= \#\{y: y< x \text{ and } f(y) \text{ is prime} \}$.
Can we get an upper bound for $\pi\_f(x)$?
| https://mathoverflow.net/users/31356 | An upper bound for the number of prime numbers in non-linear progressions | Yes, you can get sharp upper bound, matching (up to a multiplicative constant) the asymptotic predicted by the Bateman-Horn conjecture (<https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture>). This is a standard (and almost the first) application of sieve methods. I recommend the books by Halberstam and Richer... | 8 | https://mathoverflow.net/users/95145 | 244474 | 111,916 |
https://mathoverflow.net/questions/244000 | 5 | Let $n$ be postive integer number, and $x\_{i}\ge 0$, such
$$x\_{i}x\_{j}\le 4^{-|i-j|},1\le i,j\le n$$
then I have prove
$$x\_{1}+x\_{2}+\cdots+x\_{n}<\dfrac{5}{3}$$
**Edit Add Proof**:since $x^2\_{i}\le 1,0\le x\_{i}\le 1$,Let $S\_{j}=\sum\_{i=1}^{j}x\_{i},S=\sum\_{i=1}^{n}x\_{i}$,then we have
$$0=S\_{0}\le S\_{1}\... | https://mathoverflow.net/users/38620 | Find the best constant to this bounded inequality | These are some thoughts concerning the **Question**. We have that
\begin{align\*}
&S\_n(3)=\sum\_{1\leq i\leq n}x\_i^3< \sum\_{i\geq 1}4^{-i}=1/3,\\
&S\_n(1,2)=\sum\_{1\leq i<j\leq n}x\_i x\_j^2< \sum\_{1\leq i<j}4^{-(2j-i)}=1/45,\\
&S\_n(2,1)=\sum\_{1\leq i<j\leq n}x\_i^2 x\_j< \sum\_{1\leq i<j}4^{-j}=1/9,\\
&S\_n(1,1... | 4 | https://mathoverflow.net/users/94262 | 244485 | 111,921 |
https://mathoverflow.net/questions/244404 | 7 | I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds.
The motivation comes from gauge theories, commonly used in theoretical physics. The simplest example is a vector field $A\_i$ defined on a E... | https://mathoverflow.net/users/95116 | Geometric Construct for Integrating Symmetric Tensors? | Here is one way to construct all "local" gauge-invariant quantities out of a symmetric tensor $A\_{ij}$. It would be up to you to decide how it meshes with the intuition you gained from your investigations on the lattice.
First, define $R[A]\_{ij:kl} = \partial\_i\partial\_k A\_{jl} - \partial\_j\partial\_k A\_{il} -... | 6 | https://mathoverflow.net/users/2622 | 244495 | 111,923 |
https://mathoverflow.net/questions/244430 | 5 | (This question is cross-posted on [math.stackexchange](https://math.stackexchange.com/q/1860105/44643))
I'm playing with p-adic valuations, and find that the odd harmonic sums, $\tilde{H}\_k=\sum\_{i=1}^{k}\frac{1}{2i-1}$, has 2-adic valuation $||k^2||\_2=2||k||\_2$.
E.g.) $\tilde{H}\_4=\frac{176}{85}$ has $||\til... | https://mathoverflow.net/users/69521 | 2-adic valuation of odd harmonic sums | It is sufficient to prove this result for the sum where $i$ ranges from $m 2^n + 1$ to $(m+1)2^n$. That is because $\tilde{H}\_k$ is a sum of an odd number of terms of this form with $n= ||k||\_2$, so if they all have $2$-adic valuation $2||k||\_2$ then $\tilde{H}\_k$ does as well.
To that end, consider what happens ... | 8 | https://mathoverflow.net/users/18060 | 244501 | 111,925 |
https://mathoverflow.net/questions/244463 | 5 | Let $R$ be an integral $\bar{k}$-algebra of finite type. Let $V(I) \subseteq \mathbb{A}\_R^n$ be a reduced (closed) subgroup scheme such that $V(I)\backslash \{0\} \neq \emptyset$ and the $\mathbb{G}\_m^R$-action on $\mathbb{A}\_R^n \backslash \{0\}$ restricts to a free action on $V(I)\backslash \{0\}$. Is it true that... | https://mathoverflow.net/users/33573 | Subgroup schemes of $\mathbb{A}^n$ | I am posting my comments above as an answer. If one does not impose a condition on fibers, then there are counterexamples such as when $R=\overline{k}[t]$ and the ideal $I$ in $R[x,y]=\Gamma(\mathbb{A}^2\_R,\mathcal{O})$ equals $\langle x^p-ty^p\rangle$.
On the other hand, if the fiber over every $\overline{k}$-poin... | 2 | https://mathoverflow.net/users/13265 | 244508 | 111,928 |
https://mathoverflow.net/questions/244454 | 3 | Let $K$ be an algebraically closed valued field which is $C$-minimal, as defined, for example, in [this article](http://www.sciencedirect.com/science/article/pii/0168007294900647). Examples include pure algebraically closed valued fields, as well as Lipschitz and Robinson's expansions by rigid subanalytic sets (Lipshit... | https://mathoverflow.net/users/2234 | the structure on the value group sort of a C-minimal field | I think the answer is no. Consider an algebraically closed valued field in the three sorted language. Using a relative quantifier elimination argument, any o-minimal expansion of the value group preserves $C$-minimality of the structure (meaning, definable subsets in one variable of the field sort are finite boolean co... | 3 | https://mathoverflow.net/users/8145 | 244509 | 111,929 |
https://mathoverflow.net/questions/244486 | 2 | Following problem though not a research problem
if $x,y,z,w$ are postive integers,and such
$$xyzw=504(x^2+y^2+z^2+w^2)$$
such example $(x,y,z,w)=(21,63,84,84)$ hold,
Now My problem there exist distinct postive integer solution? or find this equation all solution?
| https://mathoverflow.net/users/38620 | Find a distinct postive integer solution to this $xyzw=504(x^2+y^2+z^2+w^2)$ diophantine equation | The basic idea for Markoff-Hurwitz type equations
$$ x\_1^2+\cdots+x\_n^2=Ax\_1x\_2\cdots x\_n $$
with $A\in\mathbb Z$ is that if you have a solution $(x\_1,\ldots,x\_n)$ in integers, then by fixing $n-1$ of the variables to be the given values, you get a monic quadratic equation for the last variable, so since there i... | 13 | https://mathoverflow.net/users/11926 | 244516 | 111,930 |
https://mathoverflow.net/questions/244514 | 7 | In [this question](https://mathoverflow.net/questions/4590/when-are-dual-modules-free) it was claimed that if a module $M$ over a noetherian domain $R$ satisfies $\rm{Ext}^i(M,R)=0$ for $i=1,2$, then $M$ is reflexive. Is this true? Does someone know a reference or a proof for this? If it is not correct, is there anothe... | https://mathoverflow.net/users/36563 | Criterion for being reflexive via Ext | **Edit.** Hailong Dao points out a **serious error** in what I originally wrote. I have edited the statement below. Unfortunately, the corrected condition on the Ext modules is now rather complicated: a dimension condition on the support of every Ext module, not just vanishing of the final two Ext modules.
I am just ... | 8 | https://mathoverflow.net/users/13265 | 244517 | 111,931 |
https://mathoverflow.net/questions/244483 | 2 | **GHC for triangulated motives:** The Hodge conjecture holds and an object $\rm M \in Dmg$ is effective if and only if its Hodge realization is effective.
>
> I would like to know some references on GHC for triangulated motives (other than Huber's).
>
>
>
[Slice filtration on motives and the hodge conjecture, ... | https://mathoverflow.net/users/83957 | Reference - Generalized Hodge conjecture for triangulated motives | You may have a look at Proposition 4.3.1 in our preprint <https://arxiv.org/abs/1411.6354> (see page 47; $l^{c-1}$ denotes the localization of all effective motives by $c$-effective ones, whereas the only objects of "infinitely large weights" are zero ones). It easily yields that conjectures A and B in part II (that ar... | 3 | https://mathoverflow.net/users/2191 | 244522 | 111,933 |
https://mathoverflow.net/questions/244362 | 1 | For a $[n,k,n-k+1]\_q$ Reed Solomon code is there a polynomial time algorithm to find at least one minimum weight $(n-k+1)$ codeword? I searched in literature and I could not find one and hence I am suspecting there is a decision version of this problem which might be $\mathsf{NP}$-complete.
| https://mathoverflow.net/users/10035 | Finding minimum weight codeword of MDS RS code | The solution holds for any MDS code. I'm assuming the code is given by its generator matrix $G$. In that case, simply convert $G$ into its reduced row echelon form $G'$. This will be a matrix of the form $G'=[I | A]$, where $I$ is the $k\times k$ identity matrix, and $A$ is $k\times (n-k)$. (Note here: since the code i... | 2 | https://mathoverflow.net/users/9044 | 244543 | 111,938 |
https://mathoverflow.net/questions/244541 | -2 | I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ is generator of $G$.
Am I interpreting this action correctly. I am thinking of it as following, think $P^1$ as extende... | https://mathoverflow.net/users/92070 | Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ | Not sure whether this is what you are after, but one way to give a geometric meaning is the following.
You can think of $P^1{\mathbb C}$ as the "boundary at infinity" of the 3-dimensional hyperbolic space. To every "ideal tetrahedron" (i.e., a tetrahedron in hyperbolic 3-space with vertices at infinity) you can assoc... | 1 | https://mathoverflow.net/users/39082 | 244544 | 111,939 |
https://mathoverflow.net/questions/244519 | 22 | This question was asked on MathStackexchange [here](https://math.stackexchange.com/questions/1850242/x-1-2-x-n-1-x-nx-n-1-over2-what-can-we-say-bout-x-n-text), but there was no answer, so I am asking it here.
Let$$x\_1 = 2, \quad x\_{n + 1} = {{x\_n(x\_n + 1)}\over2}.$$What can we say about the behavior of $x\_n \tex... | https://mathoverflow.net/users/nan | $x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$? | As remarked by Joe Silverman, a natural way to look at this question is by phrasing it in terms of the map $f(x):=\frac{1}{2}x(x+1)$ on the $2$-adic integers $\mathbb{Z}\_2$. We are then asking about the behaviour of the orbit $f^n(2)$ with respect to the partition of $\mathbb{Z}\_2$ into two clopen sets $U\_1:=2\mathb... | 23 | https://mathoverflow.net/users/1840 | 244551 | 111,942 |
https://mathoverflow.net/questions/244558 | 12 | I have tried searching the literature for a result like the following, but have not found anything.
For a positive integer $m$, is it known that
$$\{ \sin (k \pi / m): 1 \leq k \leq m/2, (k,m)=1 \}$$
is linearly independent over the rationals?
References or a proof would be greatly appreciated.
| https://mathoverflow.net/users/95204 | linear independence of $\sin(k \pi / m)$ | Note: Fedor and Vladimir have already answered the question, but this is a partial answer in the other direction, under a stronger hypothesis. (This answer, which I had earlier deleted, has been edited in response to some helpful comments.)
If $m$ is odd and square-free, then the claim of the OP holds.
Let $S$ be ... | 15 | https://mathoverflow.net/users/2926 | 244563 | 111,945 |
https://mathoverflow.net/questions/244539 | 1 | Let $G$ be a finite abelian group and $\widehat{G}$ the character group.
Let $S \subset \widehat{G}$ be a Galois-stable subset i.e. if $\chi \in S$, then the Galois conjugates $\chi^{\sigma} \in S$ for any $\sigma \in Gal(\overline{\mathbb{Q}}/\mathbb{Q})$. Let $H\_{S} \subset \widehat{G}$ be the subgroup generated by ... | https://mathoverflow.net/users/95196 | Characters and Galois stability | Suppose that $G\_i=C\_2^i$, and let $S\_i$ be a minimal generating set of $\widehat{G\_i}$. Since all characters are rational, $S\_i$ is stable, but we have $|H\_{S\_i}|=|G\_i|=2^i$, $|S\_i|=i\asymp\log|G\_i|$. So the bound $\log|G\_i|$ is in fact optimal.
| 0 | https://mathoverflow.net/users/37555 | 244564 | 111,946 |
https://mathoverflow.net/questions/244548 | 1 | Is there a closed form expression for
$$ \sum\_{k=n}^\infty\frac{k!^2}{(k+x)(k-n)!(k+n+1)!} $$
where $0<x<1$ ?
(For $n=0$, I know that
$$\sum\_{k=0}^\infty\frac{1}{(k+x)(k+1)}=\frac{\psi(x)+\gamma}{x-1}$$
where $\psi(x)$ is the digamma function and $\gamma$ is the Euler-Mascheroni constant).
| https://mathoverflow.net/users/94200 | Factorial Series | [Wolfram Programming Lab](http://www.wolfram.com/programming-lab/) gives the slightly "simpler"
$$
\sum\_{k=n}^{\infty} \frac{(k!)^2}{(k+x)(k-n)!(k+n+1)!} \\ = \frac{(n!)^2}{(n+x)(2 n+1)!}\ {}\_3F\_2(n+1,n+1,n+x; 2n+2,n+x+1;1),
$$
with a generalized hypergeometric function instead of Meijer's G-function.
**Edit:** In... | 1 | https://mathoverflow.net/users/37436 | 244565 | 111,947 |
https://mathoverflow.net/questions/244562 | 10 | I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition.
>
> **Definition.** For a normal projective surface $X$ with quotient singularities, a $\mathbb Q$-Gorenstein smoothing is a one-parameter flat family of projective surfaces $\psi \colon \ma... | https://mathoverflow.net/users/20282 | Some examples of $\mathbb Q$-Gorenstein smoothing | **Question 1**. The answer is *yes*, and the classical example is as follows. Is it possible to find a one-parameter family $\psi \colon \mathcal{X} \to \Delta$ such that $X\_0$ is isomorphic to the cone over a rational normal curve $C\_4 \subset \mathbb{P}^4$, whereas $X\_t (t \neq 0)$ is isomorphic to a smooth ration... | 8 | https://mathoverflow.net/users/7460 | 244571 | 111,951 |
https://mathoverflow.net/questions/244459 | 8 | I am studying the congruent number problem
and I heard that there is a paper by Kazuma Morita
which claims to solve this problem from my colleague.
I saw the paper on his homepage but it is very short
and I cannot belive it is true because it is too short.
While I try to find his mistakes, I don't have the knowledge ... | https://mathoverflow.net/users/95139 | congruent number problem | Objection to nfdc23. Sorry if it is wrong!
His $p$-adic avatar carries the uniformizer to Frob
and one gets the equality of L-functions but
this is related just by $\chi=\iota\circ \sigma\circ Art$ which
is not a continuous character in general because of a field isomorphism $\iota:\bar{Q}\_{p}\simeq C$. He looke... | -1 | https://mathoverflow.net/users/85711 | 244580 | 111,957 |
https://mathoverflow.net/questions/3965 | 22 | Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space
and $r$ is the distance from $\Sigma$ to the center of the ball.
Is it true that
$$\mathop{\rm area} \Sigma\ge \pi\cdot(1-r^2).$$
Comments:
---------
* The problem is solved in all dimensions and codimension, see ["Area bounds... | https://mathoverflow.net/users/1441 | Minimal surface in a ball | This has just been [solved](http://arxiv.org/abs/1607.04631) (in full generality) by Brendle and Hung using the first variation formula together with a clever (if mysterious) choice of vector field.
| 7 | https://mathoverflow.net/users/26801 | 244582 | 111,958 |
https://mathoverflow.net/questions/244545 | 2 | Assume we have a Levy process $(X\_t)\_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X\_1\right]=1$. Let $\tilde{X}\_t:=X\_t-t\cdot E\left[X\_1\right]$. Define $\bar{X}\_t(u):=\tilde{X}\_{tu}$.
Can we say, that for fixed $u\in[0,1]$
$$
\frac{\bar{X}\_{t}(u)}{\sqrt... | https://mathoverflow.net/users/82744 | Version of Donsker-Invariance-Principle | Let $Y\_t(u):=\frac{\bar{X}\_{t}(u)}{\sqrt{t}}$ and $W(u):=W\_u$. The convergence (as $t\to\infty$) in distribution (in the Skorokhod space $D[0,1]$) of $Y\_t$ to $W$ can be proved quite similarly to the way it was done e.g. in the proof of Theorem 16.14 in
*Foundations of Modern Probability* by Kallenberg.
Alterna... | 2 | https://mathoverflow.net/users/36721 | 244591 | 111,961 |
https://mathoverflow.net/questions/244600 | 2 | The Lebesgue density theorem in $\mathbb{R}^n$ may be stated as follows. For a Lebesgue-measurable $A\subseteq\mathbb{R}$ and $r>0, x\in\mathbb{R}^n$, define
$$ \chi\_{A,r}(x)=\frac{\mu(A\cap B\_r(x))}{\mu(B\_r(x))},$$
where $\mu(\cdot)$ is the Lebesgue measure and $B\_r(x)$ is the closed ball about $x$ with radius $r$... | https://mathoverflow.net/users/12518 | A uniform Lebesgue density theorem | For the first version of your claim, consider $n=1$, $A = [0,1]$. For any $r \in (0,1)$, $\chi\_{A,r}(x)$ is near $1/2$ in a neighbourhood of $0$, so
$$\text{ess-sup}\_{x \in B\_R(0)} |\chi\_{A,r}(x) - \chi\_A(x)| \ge 1/2$$
| 4 | https://mathoverflow.net/users/13650 | 244603 | 111,964 |
https://mathoverflow.net/questions/219829 | 9 | On p. 13 of "Vertex Algebras for Beginners", 2nd edition, Kac writes:
"Under certain assumptions and with certain additional data one may reconstruct the whole QFT from these chiral algebras, but we shall not discuss this problem here".
However, he does not give any references to support this claim (at least not ne... | https://mathoverflow.net/users/69505 | The proof that a vertex algebra can lead to a Wightman QFT | I found Nikolov's paper "Vertex Algebras in Higher Dimensions and Globally Conformal Invariant Quantum Field Theory" [arXiv:hep-th/0307235](https://arxiv.org/abs/hep-th/0307235). The abstract:
>
> We propose an extension of the definition of vertex algebras in arbitrary space-time dimensions together with their bas... | 2 | https://mathoverflow.net/users/69505 | 244616 | 111,969 |
https://mathoverflow.net/questions/244409 | 8 | Let $k$ be a field which is complete with respect to a non-trivial non-archimedean rank-1 valuation, and let $X$ be scheme which is locally of finite type over $k$. In section of 3.5 of [Berkovich's book](https://books.google.com/books/about/Spectral_Theory_and_Analytic_Geometry_Ov.html?id=6x5PIl2-DkkC), he defines the... | https://mathoverflow.net/users/47692 | Good analytic spaces over a field into locally ringed spaces is fully faithful | First, I would like to say that I do not understand why you need the full faithfullness of the analytification functor. It seems to me that the main point is to prove that giving a morphism from an analytic space $X$ to the affine analytic space of dimension $n$ is equivalent to giving $n$ global section of $X$.
Seco... | 5 | https://mathoverflow.net/users/4069 | 244628 | 111,975 |
https://mathoverflow.net/questions/241790 | 6 | I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature.
Let us fix a non-archimedean complete valuation field $k$ (the valuation is not necessarily non-trivial)
The first definition is as in [TEMKIN](http://arxiv.org/pdf/1010.223... | https://mathoverflow.net/users/80490 | Relations between two definitions of non-archimedean analytic spaces | Let me give more a few more details than in nfdc23's comment. The most general definition of a analytic space is the one that you find in Berkovich's IHES paper. One requirement is that for every point $x$ and every open subset $U$ containing $x$, there exist finitely many affinoid domains $V\_1,\dotsc,V\_n$ of $U$ tha... | 5 | https://mathoverflow.net/users/4069 | 244630 | 111,976 |
https://mathoverflow.net/questions/244641 | 3 | Given a smooth, compact complex surface with ample canonical bundle satisfying $c\_1^2=3c\_2$, is it true that every Kahler class is a multiple of $c\_1$? This seems to be the case for fake projective planes, because $b\_2=1$.
| https://mathoverflow.net/users/3709 | Kähler classes for surfaces of general type with $c_1^2=3c_2$ | The answer is **no**.
A counterexample is provided by the so-called *[Cartwright-Steger surface](https://arxiv.org/pdf/1412.4137v2.pdf)* $X$, namely a surface of general type with $p\_g(X)=q(X)=1$ and $K\_X^2=9$.
Such a surface has Picard rank $3$, and moreover $\mathrm{NS}(X)=H^{1, 1}(X)$, in other words all Hodge... | 6 | https://mathoverflow.net/users/7460 | 244646 | 111,980 |
https://mathoverflow.net/questions/244642 | 2 | I am trying to understand the proof in Sec. A2 of [Gretton *et al.*](http://www.jmlr.org/papers/volume13/gretton12a/gretton12a.pdf). To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question.
Given a Reproducing Kernel Hilbert Space $\mathcal{F}$, we know ... | https://mathoverflow.net/users/95238 | How to compute bounding coefficients for McDiarmid's inequality? | The paper you refer to says (in line 1 of Sect. 2.2) that $\mathcal{F}$ is the **unit ball** of a reproducing kernel Hilbert space (not the entire RKHS). So, $|f(a)|=|\langle f, \phi(a)\rangle|\le\|f\| \|\phi(a)\|\le1\times\sqrt{\langle\phi(a),\phi(a)\rangle}=\sqrt{k(a,a)}\le\sqrt K$ for all $f\in\mathcal{F}$ and all p... | 2 | https://mathoverflow.net/users/36721 | 244647 | 111,981 |
https://mathoverflow.net/questions/244652 | 6 | Continuing in my attempts to understand bits and pieces of Borceux and Janelidze's *Galois Theories*, I've just realized that I don't have any geometric intuition for the most convenient characterization of Galois descent for [covering morphisms](https://mathoverflow.net/questions/239994/categorification-of-covering-mo... | https://mathoverflow.net/users/69037 | Geometric intuition for the condition of Galois descent | Let $G \to E \to B$ be a principal bundle. It's classified by a map $f : B \to BG$ in the sense that $E$ is the homotopy fiber of this map. This means that $E$ has a certain universal property: namely, it is the universal map to $B$ such that the composition with $f$ is equipped with a nullhomotopy. This says precisely... | 6 | https://mathoverflow.net/users/290 | 244654 | 111,983 |
https://mathoverflow.net/questions/244625 | 5 | I am trying to understand [this short paper](http://math.caltech.edu/SimonPapers/75.pdf) and I am getting stuck right at the end.
Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$).
We are considering the operator
$$A=-\dfrac{d^2}{dx^2}+V$$
on $L^2([0,1],dx)$ where $A$ has periodic boundary conditi... | https://mathoverflow.net/users/20838 | Gap-opening perturbations of the periodic Schrödinger operator | Here's a slightly different (direct) method that gives similar conclusions: Since we have a double periodic eigenvalue at $\widehat{E}$ (let me in fact assume that $\widehat{E}=0$, for convenience), the transfer matrix
$$
T\_0(x) = \begin{pmatrix} u\_1(x) & u\_2(x) \\ u'\_1(x) & u'\_2(x) \end{pmatrix}
$$
satisfies $T\_... | 2 | https://mathoverflow.net/users/48839 | 244663 | 111,984 |
https://mathoverflow.net/questions/244586 | 10 | Suppose we have Grothendieck abelian categories $\mathcal{A}, \mathcal{B}$. Suppose also we have given an exact functor of triangulated categories
$$
F \colon D(\mathcal{A}) \to D(\mathcal{B})
$$
where $D(\mathcal{A})$ and $D\mathcal({B})$ denote appropriate derived categories of complexes (possibly bounded below or bo... | https://mathoverflow.net/users/917 | When is a functor a right derived functor? | Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories with enough injective objects. Let me use the notation $D^+(\mathcal{A})$ and $D^+(\mathcal{B})$ to denote the stable $\infty$-categories whose homotopy categories are the (cohomologically bounded below) derived categories of $\mathcal{A}$ and $\mathcal{B}$, res... | 16 | https://mathoverflow.net/users/7721 | 244670 | 111,987 |
https://mathoverflow.net/questions/244469 | 5 | Let $M$ be a smooth manifold, $g\_M$ a Riemannian metric, and consider for $x\in M$ the *volume growth function*, $gr\_x$ that maps $r>0$ to the volume $vol\_{g\_M}(B(x,r))$. My interest is to see whether or not any differentiable manifold can be endowed with a metric structure such that the family of growth functions ... | https://mathoverflow.net/users/44172 | Does every smooth manifold admit a metric with bounded geometry and uniform growth? | The answer is "yes".
Read page 96 in ["Volume and bounded cohomology"](http://www.numdam.org/item?id=PMIHES_1982__56__5_0) by Gromov or "Manifolds with quadratic curvature decay and slow volume growth" by Lott and Shen.
| 4 | https://mathoverflow.net/users/1441 | 244674 | 111,989 |
https://mathoverflow.net/questions/244471 | 15 | It is well-known that a C\*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is *isometrically* isomorphic to $X^\*$. Does the same hold for isomorphic preduals?
>
> Let $A$ be a C\*-algebra such that $A$ is (not n... | https://mathoverflow.net/users/24916 | Is a C*-algebra with an isomorphic predual a von Neumann algebra? | Via my colleague Garth Dales, some observations which answer your question in the negative, even in the abelian case:$\newcommand{\N}{{\mathbb N}}$
>
> We know that $K$ is hyper-Stonean iff $C(K)$ is isometrically dual. So you are asking for locally compact spaces $K$ such that $C\_0(K)$ is isomorphically dual, but... | 10 | https://mathoverflow.net/users/763 | 244675 | 111,990 |
https://mathoverflow.net/questions/244532 | 2 | Minimize the following function:
$ f(V) = || V \otimes V - U\_1 \otimes U\_2 ||$
where $U\_1, U\_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product.
The minimum value of $f$ and the $V$ which minimizes it are both important.
Cross posted on mse.
| https://mathoverflow.net/users/41654 | Minimize matrix distance to tensor product | I was hoping someone else would do this, because, although it's not difficult, it's complicated. Here is a qualitative version. Unless $U$ is itself relatively close to $I$ (this will be formulated properly), the minimum will be close to $2$. For example, if $U$ has eigenvalues the $n$ $n$th roots of unity, then the mi... | 1 | https://mathoverflow.net/users/42278 | 244679 | 111,991 |
https://mathoverflow.net/questions/244665 | 2 | As stated in this question,
[Lebesgue differentiation theorem holds on locally doubling space?](https://mathoverflow.net/questions/218457/lebesgue-differentiation-theorem-holds-on-locally-doubling-space)
and proved here,
<http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf>
the Lebesgue differentiation theorem (LDT) hol... | https://mathoverflow.net/users/12518 | Doubling metrics, doubling measures, Lebesgue density | A version of Lebesgue differentiation for non-doubling measures on doubling metric space appears in section 3 of:
T. Hytonen, A FRAMEWORK FOR NON-HOMOGENEOUS ANALYSIS ON METRIC SPACES, AND THE RBMO SPACE OF TOLSA
<http://www.raco.cat/index.php/PublicacionsMatematiques/article/download/191394/387513>
I leave it to y... | 3 | https://mathoverflow.net/users/95266 | 244688 | 111,993 |
https://mathoverflow.net/questions/244684 | 4 | Let $\mathcal{M}$ be a model category presenting an $\infty$-category $\mathcal{C}$. I believe that every left Bousfield localization $\widetilde{\mathcal{M}}$ of $\mathcal{M}$ corresponds to a reflective $\infty$-subcategory $\widetilde{\mathcal{C}}$ in $\mathcal{C}$ -- perhaps some hypotheses are needed for this.
W... | https://mathoverflow.net/users/2362 | When does every $\infty$-localization correspond to a Bousfield localization? | You definitely need $M$ to be combinatorial for these types of statements. I believe Lurie has shown that every accessible localization of a presentable infinity category can be expressed as a left Bousfield localization. See chapter 5 section 5 of HTT. He uses *strongly reflective* to mean it comes from an *accessible... | 5 | https://mathoverflow.net/users/11540 | 244690 | 111,995 |
https://mathoverflow.net/questions/244687 | 5 | Recall that a group $G$ is called **residually finite** if for any nontrivial element $g\in G$ there exists a finite group $H$ and a homomorphism $f$ from $G$ to $H$ such that $f(g)\neq1$.
My question is
>
> for which kind of finitely presented group $G=\{x\_1, \cdots, x\_n|r\_1, \cdots, r\_m\}$, there exists a f... | https://mathoverflow.net/users/27111 | Mapping a group to a finite group s.t. the image of each generator is nontrivial | I'm a bit unclear on what you're asking, but I'll assume that
you mean that given $G$, is there a presentation $G=\{x\_1, \ldots, x\_n|r\_1, \ldots, r\_m\}$ for which the property holds (note that your
question only concerns the generators $x\_1,\ldots x\_n$, so the
relators $r\_j$ are irrelevant)? Otherwise, if
the ... | 13 | https://mathoverflow.net/users/1345 | 244691 | 111,996 |
https://mathoverflow.net/questions/157049 | 10 | For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of p-groups of order $p^4$.
I duly searched the web for a while and all the group theory manuals I could lay my hands on b... | https://mathoverflow.net/users/26380 | Classification of automorphism groups of groups of order $p^4$ | There is a description of the automorphism groups of groups of order $p^4$ in the Thesis by [B. Girnat: "Klassifikation der Gruppen bis zur Ordnung $p^5$." Staatsexamensarbeit, TU Braunschweig, 2003](https://arxiv.org/abs/1806.07462), whose advisor was B. Eick.
The description is given in terms of the action on a set o... | 4 | https://mathoverflow.net/users/26380 | 244693 | 111,997 |
https://mathoverflow.net/questions/244699 | 2 | Let $F: \mathcal C \to \mathcal D$ be a functor with $\mathcal D$ cocomplete, and let $\mathscr P \mathcal C$ be the free cocompletion of $\mathcal C$ (i.e., the category of small presheaves on $\mathcal C$), so that there is a (unique, up to isomorphism) cocontinuous extension $\hat F$ of $F$ along the Yoneda embeddin... | https://mathoverflow.net/users/95276 | Yoneda extension of a faithful functor is faithful | As for the first question, it's false. This is obvious in the case of posets $C, D$ (with $D$ a sup-lattice): any poset map $C \to D$ is faithful when considered as a functor, but the induced functor $Set^{C^{op}} \to D$ can't be faithful (for most presheaves $F, G$, there will be more than one natural transformation $... | 5 | https://mathoverflow.net/users/2926 | 244700 | 111,998 |
https://mathoverflow.net/questions/244704 | 0 | Build a random tournament $T=(V,E)$ on $V=\{1,\ldots, n\}$ in the following fashion: for $i < j\in \{1,\ldots, n\}$ let the probability be $0.5$ whether $(i,j)\in E$ or $(j,i)\in E$ (in a tournament, exactly one of these is the case).
Let $E(n)$ be the expected value of the longest directed circuit in $T$.
Is there... | https://mathoverflow.net/users/8628 | Length of longest directed circuit in random tournament | A random tournament is strongly connected with probability tending to 1 exponentially fast, and all strongly connected tournaments have hamiltonian cycles.
| 6 | https://mathoverflow.net/users/9025 | 244706 | 112,000 |
https://mathoverflow.net/questions/244702 | 6 | Denote $A=\{0\}, B=\{0,1\}$. Then any subset of $\Omega:=\{A,B\}^{\mathbb N}$ is a continuum provided the number of $B$'s is infinite. We treat these as binary expansions of numbers in $[0,1]$.
For instance, $(AB)^\omega$ is the set $\left\{\sum\_{n=1}^\infty a\_n4^{-n}\mid a\_n\in\{0,1\}\right\}$, which is a Cantor ... | https://mathoverflow.net/users/8131 | Random Cantor sets on the unit interval | Like the classical Cantor set, your set has a natural representation $C=\bigcap C\_n$, where $C\_1\supseteq C\_2\supseteq C\_3\supseteq\ldots$, and each $C\_n$ is a disjoint union of $2^{n-X\_n}$ intervals of length $2^{-n}$, with $X\_n$ denoting the number of $A$'s in your sequence up to position $n$.
So, covering b... | 4 | https://mathoverflow.net/users/48839 | 244715 | 112,005 |
https://mathoverflow.net/questions/244717 | 8 | Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois group"...that is...one has the universal covering $p\_G:EG\rightarrow BG$ for which $$Gal(p\_G):=Deck(p\_G)\simeq\pi\_1(B... | https://mathoverflow.net/users/nan | Inverse galois problem and étale homotopy | The inverse Galois problem is a statement about the étale fundamental group of a specific scheme. So constructing a space whose fundamental group is $G$ does not tell you whether $G$ is a quotient of the fundamental group of $\operatorname{Spec} \mathbb Q$.
An analogous problem in topology would be.
Q: Does every f... | 13 | https://mathoverflow.net/users/18060 | 244723 | 112,008 |
https://mathoverflow.net/questions/244698 | 3 | Let $(X\_n,\mathcal{X}\_n)$, $n=1,2,\ldots$ be measurable spaces. Define $Y\_n = \prod\_{k=1}^n X\_k$ and let $\mathcal{Y}\_n$ be the corresponding product $\sigma$-algebra. Similarly let $Y=\prod\_{k=1}^\infty X\_k$ and $\mathcal{Y}$ the corresponding $\sigma$-algebra.
Let $\mu\_n$ be probability measures on $(Y\_n,... | https://mathoverflow.net/users/8457 | Necessary and sufficient conditions for Kolmogorov's Extension Theorem | Gnedenko and Kolgomorov defined perfect measures for this purpose. Perfectness is a sufficient condition: If all the marginal measures (projections to $X\_n$) are perfect then the resulting finitely additive measure on the product is countably additive. In this form perfectness is not necessary. But it is necessary in ... | 3 | https://mathoverflow.net/users/95282 | 244725 | 112,010 |
https://mathoverflow.net/questions/244617 | 10 | Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\pi$ be an irreducible unitary representation of $G$ on $H$.
Now consider the restriction $\pi|\_N$ of $\pi$ to $N$. Is... | https://mathoverflow.net/users/55895 | Restriction of irreducible unitary representation to normal subgroup of finite index | I like this question!
Restricted to the finite index subgroup $N$, the representation $\pi$ splits into a direct sum of irreducible representations.
I could not see an easy proof of this, but the proof goes along the following lines.
Suppose $I$ is a totally ordered indexing set and for each $i\in I$, $W\_i$ i... | 8 | https://mathoverflow.net/users/23291 | 244742 | 112,014 |
https://mathoverflow.net/questions/244748 | 1 | In [his Senior Thesis](https://www.math.washington.edu/Undergrad/Handbook/Senior%20Theses/SamuelCoskey_June2003.pdf), Samuel Coskey answered the question of which axioms of $ZFC$ hold at each stage of the cumulative hierarchy. Here is the list of his results:
>
> Axioms that always hold: Extensionality, Foundation,... | https://mathoverflow.net/users/20597 | Partial Universes and the Axioms of $ZF$ Set Theory Without Choice | If you drop choice, then of course choice is no longer in the "always holds" category, but there's nothing to determine at what level of the cumulative hierarchy it first fails. The first failure of choice could occur just a few levels past $\omega$, or it could occur far beyond the first inaccessible cardinal. (The va... | 11 | https://mathoverflow.net/users/6794 | 244755 | 112,015 |
https://mathoverflow.net/questions/244753 | 4 | How could people classify all rank $2$ complex vector bundles over $S^2\times S^2$ up to isomorphism?
Could you give a rank 2 complex vector bundle which cannot be split as a sum of two line bundles?
| https://mathoverflow.net/users/95296 | Rank 2 complex vector bundles over $S^2\times S^2$ | This is a special case of [The space of homotopy classes of maps of products of spheres](https://mathoverflow.net/questions/234367/the-space-of-homotopy-classes-of-maps-of-products-of-spheres/234615#23461).
Classifying rank 2 complex vector bundles on $S^2\times S^2$ is the same as calculating the set of pointed homo... | 5 | https://mathoverflow.net/users/8103 | 244760 | 112,018 |
https://mathoverflow.net/questions/244739 | 1 | Let $\Omega:=(0,1)^n$ and define $ACL\_i(\Omega)$ as the set of all Borel functions $u:\Omega\to\mathbb{R}$ such that
$$ t\mapsto u(x\_1,\dots,x\_{i-1},t,x\_{i+1},\dots,x\_n) $$
is $AC$ for a.e. $(x\_1,\dots,x\_{i-1},x\_{i+1},\dots,x\_n)$. Let $ACL(\Omega):=\cap\_i ACL\_i(\Omega)$ and denote by $ACL^p(\Omega)$ the set ... | https://mathoverflow.net/users/36952 | Every $W^{1,p}$ has a representative in ACL | This is addressed in section 4.9.2 of the Evans and Gariepy's book on measure theory. The key is to show that the representative in $ACL\_i$ is the precise representative of the function (defined in the book), so it does not depend on $i$.
| 2 | https://mathoverflow.net/users/35800 | 244766 | 112,020 |
https://mathoverflow.net/questions/244720 | 1 | Let $X$ be a smooth projective variety with an action of linear algebraic group $G$. Theorem 5.6.1 in Criss/Ginzburg (*Representation Theory and Complex Geometry*) lists a bunch of equivalent conditions for when this space satisfies a "Kunneth formula:"
(a) for any $Y$ with an action of $G$, the exterior tensor map $... | https://mathoverflow.net/users/6059 | Counterexamples to Kunneth formula in algebraic K-theory | **Edit.** User hic points out that with the first action I specified, there are infinitely many orbits. So please change the group to $\textbf{PGL}\_2\times \textbf{SL}\_2$ with its left-right action.
The examples in my comment are only quasi-projective, not projective. However, there are also projective examples. Be... | 1 | https://mathoverflow.net/users/13265 | 244788 | 112,026 |
https://mathoverflow.net/questions/244721 | 1 | Let $x\_1,\dots,x\_n$ be i.i.d. $N(0,I\_{p\times p})$, with $n>p$. Let $\hat S=\frac1n\sum\_{i=1}^n x\_i x\_i^T$ be the sample covariance.
Assume the asymptotic setting where $\frac pn\to \alpha<1$.
Is there a result about the concentration of $\mathbb{E}\left[\operatorname{tr}\left(\hat{S}^{-1}\right)\right]$ in t... | https://mathoverflow.net/users/13542 | Trace of the inverse sample covariance as the number of samples and dimension scale to infinity | Writing $\hat{S}= \frac1n XX^T$ where $X$ has columns $x\_1,\dots,x\_n$, we can express
$$
\text{tr}(\hat{S}^{-1}) = \sum\_{i=1}^p \lambda\_i(\hat{S})^{-1} = n\sum\_{i=1}^p \sigma\_i(X)^{-2}
$$
where $\sigma\_1(X), \dots, \sigma\_p(X)$ are the nonzero singular values of $X$. Then we can use the inverse second moment i... | 1 | https://mathoverflow.net/users/82504 | 244793 | 112,029 |
https://mathoverflow.net/questions/244749 | 5 | On $S^4$, we know that rank 2 complex vector bundles are classified by $\pi\_3(U(2))=\mathbb Z$. Any element $g\in\pi\_3(U(2))=\mathbb Z$ determines a complex vector bundle $E$ over $S^4$.
**Question**: Can we say that the corresponding second Chern class $c\_2(E)$ equals $g$, i.e. $\left<c\_2(E),[S^4]\right>=\deg(g... | https://mathoverflow.net/users/95296 | Rank 2 complex vector bundles over $S^4$ | Choose a homotopy equivalence $U(2)\simeq \Omega BU(2)$ to use as an identification and let $g:S^3\rightarrow U(2)\simeq \Omega BU(2)$ represent a homotopy class in $\pi\_3U(2)$. Then the $U(2)$-bundle $E\rightarrow S^4$ corresponding to $g$ is classified by the map $\tilde{g}:S^4\rightarrow BU(2)$ which is adjoint to ... | 1 | https://mathoverflow.net/users/54788 | 244797 | 112,030 |
https://mathoverflow.net/questions/244789 | 5 | Is the following true (and if yes, where the best proof is written?)?
For any $c>0$ for large enough positive integers $N$ we have $\sum\_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$?
Hm, if true, it should be hard: it allows to get signs of certain Gauss type sums, for example.
| https://mathoverflow.net/users/4312 | uniform one-sided van der Corput inequality | The answer is negative.
As you said if you take $\frac{t}{2\pi}$ to be a rational number, say $\frac{p}{q}$ then it converges to the Gaus-sums:
$$
\lim\_{N \to \infty} \frac{1}{N} \sum\_{k=0}^{N-1} \cos(k^{2} t) = \Re \frac{1}{q} \sum\_{k=0}^{q-1} \exp\left( 2 \pi i \cdot \frac{p}{q} k^{2}\right)
$$
Now if you take ... | 5 | https://mathoverflow.net/users/50901 | 244799 | 112,032 |
https://mathoverflow.net/questions/244805 | 4 | While reading the statement of Roth's theorem I started asking myself **what are examples of sets of positive upper density**? It's not hard to come up with a few:
* Flip a coin with probability $\mathbb{P}(H) = p < 1$ and take all the heads
* Consider $\{ n : \{ n \theta\} > 0.001$ where $\theta$ is an irrational nu... | https://mathoverflow.net/users/1358 | Examples of Sets with Positive Upper Density | The *Birkhoff ergodic theorem* produces many integer sets of positive upper density. Given a probability space $(X, \mu)$, a transformation $T : X \rightarrow X$ preserving $\mu$, and some measurable set $A \subset X$, we have the convergence for almost all $x \in X$,
$$
{1\over N} Card\{k< N \mid T^k(x)\in A\} \righta... | 4 | https://mathoverflow.net/users/6129 | 244806 | 112,035 |
https://mathoverflow.net/questions/244801 | 4 | One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the additive group as does the right-sided ideal generated by $Q$.
I know this is a fact, but I don't know a simple abstrac... | https://mathoverflow.net/users/3902 | Left- and right-sided principal ideals of quaternions have same index? | Here's something reasonably quick: if $R$ is a domain and $\mathcal{O}$ is an $R$-order in an $F$-algebra $B$ with $F=\mathrm{Frac}(R)$, then the $R$-index $[\mathcal{O}:\mathcal{O}\alpha]\_R$ is (either by definition, or after some argument) equal to the $R$-ideal generated by $\det\_{B/F}(\cdot \alpha)$, the determin... | 5 | https://mathoverflow.net/users/4433 | 244813 | 112,037 |
https://mathoverflow.net/questions/244808 | 19 | If $f\in\mathbf{C}[[q]]$ is non-constant, and algebraic over $\mathbf{C}[q]$ (in the sense that it is a root of a polynomial with coefficients in in $\mathbf{C}[q]$) then can $f$ be the $q$-expansion of a modular form (for some congruence subgroup of $SL(2,\mathbf{Z})$)?
I ask for the following reason. There are geom... | https://mathoverflow.net/users/1384 | Can something finite over $\mathbb{C}(q)$ be a modular form? | A non-constant modular form has a natural boundary on the real line. A power series that's algebraic in $q = e^{2\pi i \tau}$ can't.
| 22 | https://mathoverflow.net/users/14830 | 244818 | 112,039 |
https://mathoverflow.net/questions/244769 | 4 | Let us consider permutations $\pi$ on $\{1,\dots,n\}$ as sequences $\pi(1),\pi(2),\dots,\pi(n)$. For a permutation $\pi$ let $R(\pi)$ be the ternary relation with $(x,y,z)\in R(\pi)$ whenever element $y$ appears in between $x$ and $z$ (i.e., x is left of y and z is right of y, or, x is right of y and z is left of y). S... | https://mathoverflow.net/users/1330 | Betweenness in permutations | Without loss of generality, we may assume that $\pi\_1=12\dots n$.
By the Erdős–Szekeres lemma, $\pi\_2$ has a monotone subsequence of length at least $\sqrt{n}$. In other words, there is a subset $S\_2 \subseteq \{1,2,\dots,n\}$ of size at least $\sqrt{n}$ where $R(\pi\_1)$ and $R(\pi\_2)$ coincide. By induction, fo... | 6 | https://mathoverflow.net/users/24076 | 244822 | 112,041 |
https://mathoverflow.net/questions/244834 | 2 | Let $\mathcal{M}\_{g,n}$ be the moduli stack over $\mathbb{Q}$ of smooth curves of genus $g$ with $n$ marked points. I've seen in many sources an exact sequence:
$$1\rightarrow\pi\_1((\mathcal{M}\_{g,n})\_{\overline{\mathbb{Q}}})\rightarrow\pi\_1(\mathcal{M}\_{g,n})\rightarrow\text{Gal}(\overline{\mathbb{Q}}/\mathbb{... | https://mathoverflow.net/users/88840 | reference request for homotopy exact sequence of moduli stacks of curves | V. Zoonekynd, The fundamental group of an algebraic stack, arxiv.org math.AG/0111071
B. Noohi (2004). FUNDAMENTAL GROUPS OF ALGEBRAIC STACKS. Journal of the Institute of Mathematics of Jussieu, 3, pp 69-103
| 4 | https://mathoverflow.net/users/2290 | 244836 | 112,046 |
https://mathoverflow.net/questions/244833 | 6 | Recently I am reading Wall's paper "*On the Orthogonal Groups of Unimodular Quadratic Forms II*". In this paper, I encountered with the map $E^1\_\omega$, which now I am interested in.
Let $X$ be an unimodular integral lattice, and $H$ denotes the unimodular lattice which has a basis $x$, $y$ satisfying $x\cdot x=y\c... | https://mathoverflow.net/users/85988 | Intuition behind the definition of the Siegel-Eichler transformation | This is explained very clearly in Wall's paper, Diffeomorphisms of 4-manifolds, J. London Math. Soc. (1964) 131-140. The idea is that if you do surgery on a circle $C$ in a simply-connected $4$-manifold $X$, you get $X \# T\_r$ where $T$ is a 2-sphere bundle over $S^2$ with fiber $S^2$, and intersection form generated ... | 3 | https://mathoverflow.net/users/3460 | 244838 | 112,047 |
https://mathoverflow.net/questions/244837 | 17 | It seems like Gödel didn't use the letter $L$ for his model before his book "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory", which is probably the first place it got used.
Do anyone of you know why he used the letter $L$? It does seem like a bit ad ho... | https://mathoverflow.net/users/38602 | Why did Gödel name his constructible universe $L$? | I heard from [Kai Hauser](http://page.math.tu-berlin.de/~hauser/) that the letter $L$ comes from "**law**", and it is because the model is constructed using some laws.
| 14 | https://mathoverflow.net/users/11115 | 244840 | 112,048 |
https://mathoverflow.net/questions/244767 | 7 | Is there any classification of the rank 2 complex vector bundles over $\mathbb CP^2$ up to diffeomorphism?
| https://mathoverflow.net/users/95296 | Rank 2 vector bundles over $\mathbb CP^2$ | The question was essentially answered by Sasha's comment, but I will give a couple of additional/alternative references/remarks.
As a first remark, the continuous and smooth classifications coincide, this is discussed e.g. in the answers to this MO-question: [From Topological to Smooth and Holomorphic Vector Bundles]... | 11 | https://mathoverflow.net/users/50846 | 244847 | 112,050 |
https://mathoverflow.net/questions/244812 | 8 | Suppose $G$ is a semi-simple group of adjoint type over an algebraic closed field, and $X$ its wonderful compactification a la De Concini and Procesi. Let $P=MU$ be a parabolic subgroup in $G$, and let $Y$ be the wonderful compactification of $M$.
Question. Can you describe the closure of $P$ in $X$? In particular, ... | https://mathoverflow.net/users/10458 | Wonderful compactification | I am just writing my comment above as an answer, just in the **special case** that the parabolic is a Borel subgroup $B$. In this case, the reductive part $M$ is a maximal torus $T$ in $G$. Denote by $W\subset N\_G(T)$ a Weyl group. There is a conjugation action of $G$ on itself, and there is an induced conjugation act... | 6 | https://mathoverflow.net/users/13265 | 244864 | 112,056 |
https://mathoverflow.net/questions/244352 | 0 | Let $R$ be a reduced finite type $\bar{k}$-algebra, a projective morphism $\pi \colon V \rightarrow \mathrm{Spec}(R)$ and ideals $I, J \subseteq R$. Assume there is a split $s\_{IJ} \colon \mathrm{Spec}(R/IJ) \rightarrow V \times\_{\mathrm{Spec}(R)} \mathrm{Spec}(R/IJ)$ of $\pi$ and that the sections $s\_{J} \colon \ma... | https://mathoverflow.net/users/33573 | Lifting sections to completion of closed subschemes | No. Take $R=k[x, y]$ and $I = (x)$ and $J = (y)$. Then let $V$ be the glueing of two copies of $\text{Spec}(R)$ at $0 = (0, 0)$. Let $s\_{IJ}$ be the morphism which sends $V(I)$ into the first copy and $V(J)$ into the second. Then $\hat s\_{IJ}$ does not exist but $\hat s\_I$ and $\hat s\_J$ do (and therefore also $\ha... | 1 | https://mathoverflow.net/users/95365 | 244881 | 112,060 |
https://mathoverflow.net/questions/244877 | 8 | Using, e.g., properties of iterated finite differences it is easy to show that for any pair of integers $n$ and $m$ with $n>\!>m$ one has the identity
$$
\sum\_{k=0}^m(-1)^{k-m} {n-k\choose m}{m\choose k}=(-1)^{m}.
$$
Motivated by [Colored sl(N) link homology via matrix factorizations](http://arxiv.org/abs/1110.2076v2... | https://mathoverflow.net/users/8320 | q-analog of a combinatorial identity involving binomial coefficients | You can see this as an instance of the [q-Vandermonde identity](https://en.wikipedia.org/wiki/Q-Vandermonde_identity). The q-binomial theorem tells us that the coefficient of $t^k$ in $\prod\_{i=0}^{m-1}(1+q^{-2i}t)$ is $q^{-k(m-1)}{m \brack k}\_q$ and the coefficient of $t^{n-m-k}$ in $\prod\_{i=0}^{m}\frac{1}{1+q^{-2... | 8 | https://mathoverflow.net/users/2384 | 244887 | 112,062 |
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