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https://mathoverflow.net/questions/249968 | 1 | I was looking through some old notes of mine and stumbled upon a question I had wanted to ask a while back but never got around to it. Here it is now:
Consider a convex set $X\subseteq \mathbb{R}^n$ with non-empty interior and define a topology $\tau$ on $X$ as follows:
1. The neighbourhoods of interior points are ... | https://mathoverflow.net/users/3041 | Name for a certain topology on boundary points of convex sets | Non-tangential convergence. This is a common theme in e.g. boundary values of harmonic functions, Hardy-Littlewood maximal functions, etc.
| 4 | https://mathoverflow.net/users/13650 | 249970 | 113,550 |
https://mathoverflow.net/questions/249934 | 1 | Suppose the following linear recurrence sequence
$$
C\_n:=C\_{n-2}+C\_{n-4}+C\_{n-6}\, .
$$
With the initial values
$$
C\_0=0 \, , \, C\_1=1 \, , \, C\_2=0 \, , \, C\_3=0 \, , \, C\_4=1 \, , \, C\_5=1 \, , \, C\_6=1 \, .
$$
It can be proved that another form of the $C\_n$ sequence is as follows
\begin{equation}
C\_n... | https://mathoverflow.net/users/64181 | The closed-form expression for $C_n$ sequence | The roots of your denominator $x^6 - x^4 - x^2 - 1$ are $\pm \sqrt{r\_i}$ where $r\_1, \ldots, r\_3$ are the roots of $z^3 - z^2 - z - 1$, namely
$$ \eqalign{r\_1 &= \dfrac{1}{3} + \dfrac{1}{3} (19 + 3 \sqrt{33})^{1/3} + \dfrac{4}{3} (19+3 \sqrt{33})^{-1/3}\cr
r\_2, r\_3 &= \dfrac{1}{3} - \dfrac{1}{6} (19 + 3 \sqrt{33... | 3 | https://mathoverflow.net/users/13650 | 249974 | 113,551 |
https://mathoverflow.net/questions/249522 | 3 | I am trying to understand the proof of [Theorem 3.9, p.9](https://arxiv.org/pdf/math/0610266v1.pdf).
---
We consider NLS $$i\partial\_t u + \Delta + |u|^{4/(d-2)}u=0, u(x,0)=u\_0 \in H^1(\mathbb R^d)$$
where $u:\mathbb R^{d+1} \to \mathbb C, u\_0:\mathbb R^d \to \mathbb C$
Assume that $\int\_{\mathbb R^d} |\nab... | https://mathoverflow.net/users/96950 | energy trapping for NLS | Let me quote Lemma 3.4 from the paper, with some paraphrasing to make extremely clear certain points. The function $W$ here is some function $\mathbb{R}^N\to \mathbb{R}$ solving a certain nonlinear elliptic equation (see start of Section 3 of the paper). Its precise form does not matter for our discussion, other than t... | 4 | https://mathoverflow.net/users/3948 | 249980 | 113,556 |
https://mathoverflow.net/questions/249985 | 12 | I heard the following fact a while back from Joel Hamkins, who told me at the time that he learned it from Hugh Woodin:
>
> **Observation**: Let $M$ be a countable transitive model of $\mathsf{ZFC}$. Then there are forcing extensions $M[c]$ and $M[d]$, each by adding a Cohen real, which are non-amalgable. By this, ... | https://mathoverflow.net/users/64676 | A Cohen real which always amalgamates with others | Let $c$ be Cohen over $M$. Construct $d$ as follows. Let $X$ be an infinite set of integers all of whose infinite subsets compute the height of $M$. Inductively construct $d$ such that $d$ lies in every open dense set coded in $M$ and the set of positions at which $c, d$ disagree is an infinite subset of $X$.
| 11 | https://mathoverflow.net/users/2689 | 249986 | 113,558 |
https://mathoverflow.net/questions/249665 | 6 | Every $A \in \text{GL}\_n(\mathbb{R})$ has a unique [Polar decomposition](https://en.wikipedia.org/wiki/Polar_decomposition):
$A=O\_AP\_A$, $O \in \operatorname{O}\_n, P \in \operatorname{Psym}\_n$.
In particular the orthogonal factor is given by $$O\_A=A(\sqrt{A^TA})^{-1}.$$ **Question:**
Let $A,B \in \operatorna... | https://mathoverflow.net/users/46290 | Bounding the non-multiplicativity of isometric projection | If the constant is allowed to depend upon the dimension, the estimate is simple. Let $A=O\_AP\_A,B=O\_BP\_B$, then $AB=O\_AO\_B(O\_B^\*P\_AO\_B)P\_B$ and we are left with showing that if the product of two positive definite self-adjoint operators $X=O\_B^\*P\_AO\_B$ and $Y=P\_B$ is $\delta$-close to a unitary operator ... | 5 | https://mathoverflow.net/users/1131 | 249988 | 113,560 |
https://mathoverflow.net/questions/249982 | 10 | **Conjecture:** If I have an elliptic curve with j-invariant 0 of the form $y^2 = x^3 + b$ over some prime-order field $\mathbb{F}\_p$ (where $p$ is not 2 or 3), and the group of rational curvepoints has prime order $q$ (which is not 2 or 3), then there is some curve of the form $y^2 = x^3 + b'$ over $\mathbb{F}\_q$ wi... | https://mathoverflow.net/users/34823 | Elliptic-curve related equivalence between fields of different characteristic? | If $\# E(\mathbb{F}\_p) = q$ and $j=0$, then the endomorphism ring is an order in the field of third roots of unity so $(p+1-q)^2 - 4p = -3u^2$ for some integer $u$. Now note that $(p+1-q)^2 - 4p$ is symmetric in $p$ and $q$. Hence, if there is an elliptic curve at all over $\mathbb{F}\_q$ with $p$ points, then it auto... | 12 | https://mathoverflow.net/users/2290 | 249991 | 113,562 |
https://mathoverflow.net/questions/249743 | 2 | Let $f:[0,1]\to\mathbb{R}$ be a bounded (Lebesgue) measurable function.
Consider the function $$w(p)=\int\_0^1|f|^p\,d\mu$$.
Is $w(p)$ differentiable at any $0<p<\infty$? I.e. does $w'(p)$ exist for **all** (not just almost all) $0<p<\infty$?
I hope this is not too easy a question. I have asked on Math.SE (few da... | https://mathoverflow.net/users/83274 | Differentiate an integral (Lebesgue integral) | Yes, we need the boundedness of $|f|^p \ln|f|$ when both $x$ and $p$ vary, but important note is that $p$ may vary on a given segment $[p\_1,p\_2]$, $1<p\_1<p\_2$ (this would imply that the derivative $d/dp(\int)$ exists at all points $p\in (p\_1,p\_2)$, and since the segment is arbitrary, it exists for all $p>1$.) Now... | 2 | https://mathoverflow.net/users/4312 | 250005 | 113,566 |
https://mathoverflow.net/questions/248157 | 2 | Let $B$ be a topological space.
Call a subset $A\subset B$ *ultrafilter-like* iff $A$ is dense in $B$ and
each decomposition $A=A\_1\cup A\_2$,
into the union of two open subsets extends to a decomposition of
$B=B\_1\cup B\_2$, $A\_1\subset B\_1$ and $A\_2\subset B\_2$ into the union of two open subsets $B\_1$ and $B... | https://mathoverflow.net/users/97621 | a characterisation of proper maps via ultrafilters | Let $C$ be a connected Tychonoff space and $a,b\in \beta C\setminus C$ be two distinct points. Let $X=\beta C$, $Y=X/\{a,b\}$ be the quotient space and $f:X\to Y$ be the quotient map. It is clear that $f$ is perfect and hence proper.
Let $B=Y$, $A=B\setminus \{q(a)\}$, $h:B\to Y$ be the identity map, $g=q^{-1}|A:A\t... | 1 | https://mathoverflow.net/users/61536 | 250030 | 113,576 |
https://mathoverflow.net/questions/250046 | 2 | Consider the action of $\mathbb{Z}\_3\subset SL\_2(k)$ on $\mathbb{A}^2$, we have the quotient $Y$ as in the title. According to the classification of Du Val singularity, we know that the crepant resolution $X$ of $Y$ has two exceptional curves with self-intersection -2, which intersects at a points.
On the other ha... | https://mathoverflow.net/users/48616 | Crepant resolution of $Y=k[x,y,z]/(xz-y^3)$ | It is easy to see that the canonical class of $X'$ is trivial (combine the formula for the canonical class of the blowup of $A^3$ and the adjunction formula). Hence $X'$ is crepant. But a crepant resolution of a surface is unique, so $X = X'$.
By the way, if you look at the exceptional divisor of $X'$, you will note ... | 9 | https://mathoverflow.net/users/4428 | 250047 | 113,581 |
https://mathoverflow.net/questions/250004 | 3 | Let $S\_n$ be the symmetric group on $\{1, \ldots, n\}$. Let
\begin{align}
T=\sum\_{g\in S\_n} g.
\end{align}
Are there some references about the factorization of $T$?
In the case of $n=3$, we have
\begin{align}
& T=1 + (12) + (23) + (12)(23) + (23)(12) + (12)(23)(12) \\
& = 1 + (12) + (23) + (12)(23) + (23)(12) +... | https://mathoverflow.net/users/11877 | Factorization in the group algebra of symmetric groups | A famous factorization is $(1+X\_1) (1+X\_2) \cdots (1+X\_n)$ where $X\_1=0$, $X\_k=
(1,k)+(2,k)+\cdots +(k-1,k)$ for $2\leq k\leq n$. $X\_k$ is called a
*Jucys-Murphy element*, though this factorization is due to Alfred
Young in 1902. Jucys gave the $q$-analogue
$$ (q+X\_1)(q+X\_2)\cdots(q+X\_n) =\sum\_{\pi\in S\_n} ... | 11 | https://mathoverflow.net/users/2807 | 250054 | 113,584 |
https://mathoverflow.net/questions/249814 | 6 | Given a compact Kähler manifold $M$, let $D$ be an effective divisor on $M$.
1. Is $M\setminus D$ pseudoconvex? That is, can we find a smooth plurisubharmonic function that exhausts $M\setminus D$ ?
2. Can we find a complete Kähler metric on $M\setminus D$ ?
If $1$ and $2$ are not true, can we find any obstruction... | https://mathoverflow.net/users/nan | Plurisubharmonic function and complete Kähler metric on certain Kähler manifold | Question 1: Plurisubharmonic functions extend across codimension 2 subvarieties . Let X be the complex projective plane blown up at one point and D be the exceptional divisor then any plurisubharmonic function on the complement of D in X extends to X and is therefore a constant.
| 5 | https://mathoverflow.net/users/4696 | 250064 | 113,589 |
https://mathoverflow.net/questions/250068 | 18 | My question is: Is it possible to write any scheme as a (1-categorical) colimit of a diagram of affines? If no, what are some examples?
I ask this question because I have read that one can write any derived scheme as a colimit (in the $\infty$-categorical sense) over a diagram of affine derived schemes. So I am curio... | https://mathoverflow.net/users/38075 | Is it always possible to write a scheme as a colimit of affine schemes? | Yes, this is just a basic fact in category theory, if interpreted correctly. For $C$ any category, and $F$ any preheaf on $C,$ $F$ is the colimit in presheaves of the diagram $C/F \to C \stackrel{y}{\hookrightarrow} Psh(C),$ which sends a morphism $f:y(C) \to F,$ to $y(C),$ where $y$ is the Yoneda embedding. This follo... | 24 | https://mathoverflow.net/users/4528 | 250076 | 113,595 |
https://mathoverflow.net/questions/250060 | 1 | I'm reading Atiyah-Bott's paper "The Yang-Mills equations over Riemann surfaces" and have a couple of questions on page 547. They define a connection $A$ as a $G$-invariant splitting of the exact sequence
$ 0 \rightarrow T\_FP \rightarrow TP \rightarrow \pi^{-1}TM \rightarrow 0 $.
I'm not sure how the group $G$ acts o... | https://mathoverflow.net/users/98558 | Definition of Connection as G-invariant splitting of a sequence which is a pulled back sequence of bundles | The bundle $\pi^{-1}TM$ is the set of tuples $(p,m,v)$ so that $m \in M$, $p \in \pi^{-1}\{m\}$, and $v \in T\_m M$. The $G$-action is $g(p,m,v)=(gp,m,v)$, only acting on $p$. So you are correct. They are correct (of course) that $E(P)\_m$ should consist of the $G$-equivariant sections of $TP$ along the fiber $P\_m = \... | 2 | https://mathoverflow.net/users/13268 | 250082 | 113,597 |
https://mathoverflow.net/questions/250072 | 5 | I noticed that there is a class of TQFT's that exists for every dimension $n\geq1$. It's probably well-known because it's quite simple, but I'm looking for a standard name or a better way to think about it.
Let $n\text{-}\mathrm{Cob}$ denote the symmetric monoidal category of (unoriented) closed $(n-1)$-manifolds $M,... | https://mathoverflow.net/users/2811 | Tell me something about these "component tensor" TQFT's | The theory you describe is Dijkgraaf-Witten theory with target space a discrete set with $r$ elements. In general, if $X$ is a $\pi$-finite space (i.e., a space with finite homotopy groups, all but finitely many of which are non-trivial), then Dijkgraaf-Witten theory with target space $X$ is the functor $n\text{-}\math... | 5 | https://mathoverflow.net/users/51164 | 250097 | 113,604 |
https://mathoverflow.net/questions/250032 | 12 | Let $S\_g$ be a closed oriented smooth surface of genus $g>1$, and let us consider $\text{Diff}\_0(S\_g)$, the identity component of the diffeomorphism group of orientation preserving diffeomorphisms of $S\_g$.
Is this a torsion-free group?
Sorry if this question is too elementary for experts in low-dimensional t... | https://mathoverflow.net/users/14233 | Is $\mathrm{Diff}_0(S_g)$ torsion-free? | Here is a proof that $Homeo\_0(S)$ is torsion-free for every compact hyperbolic surface $S$. With more analytic assumptions on homeomorphisms one can get the same conclusion for noncompact hyperbolic surfaces.
1. Every element $f\in Homeo\_0(S)$ has nonzero Lefschetz number (here we use hyperbolicity of $S$) and, th... | 11 | https://mathoverflow.net/users/21684 | 250105 | 113,607 |
https://mathoverflow.net/questions/250042 | 1 | Let $E=\mathbb{F}\_p(\!(u)\!)$, the Laurent series field over $\mathbb{F}\_p$. Let $K/E$ be a finite normal separable extension. Consider the field $L=K(x \mid x^p-x-a=0 \text{ for some } a \in K)$. Let $\hat{L}$ be the $u$-adic completion of $L$. We write $G\_E=\mathrm{Gal}(E^s/E)$ for the absolute Galois group of $E$... | https://mathoverflow.net/users/33573 | Another fix field of a certain galois group action | The right way to do this would be to give a short, efficient, abstract argument. I, however, will give a fairly messy example to show that $u^{1/p}\in\hat L$.
I’ll use $K=E=\Bbb F\_p((u))$, and set $a\_n=u^{1-pn}$ for $n>0$, and set $f\_n(x)=x^p-x-a\_n$, an irreducible and separable polynomial over $E$. Its roots are... | 4 | https://mathoverflow.net/users/11417 | 250116 | 113,611 |
https://mathoverflow.net/questions/249693 | 2 | For a complex manifold $X$ there is the Hodge filtration on cohomology, induced by the filtration on the complex of holomorphic forms given by:
$$ F^r\Omega\_X^p:=\begin{cases}\{0\}\qquad\text{if }r>p\\\Omega^p\_X\qquad \text{if } r\leq p.
\end{cases} $$
If $X$ is compact Kähler, Hodge Theory tells us that this fil... | https://mathoverflow.net/users/76748 | Are induced morphisms on cohomology strict with respect to the hodge filtration in the non Kähler case? | The following is a counterexample of what you ask for non-compact Kähler manifolds.
Let $X$ be a smooth projective variety and $D \subset X$ a very ample divisor. Since $U:= X- D$ is affine, the filtration $F^\bullet H^k(U,\mathbf{C})$ induced by the filtered complex in your question is trivial in the sense that $F^p... | 2 | https://mathoverflow.net/users/14037 | 250123 | 113,614 |
https://mathoverflow.net/questions/250113 | 6 | I've asked this on MathSE without success:
<https://math.stackexchange.com/questions/1929559/is-being-an-ndr-a-local-property>
A pair of topological spaces $(X,A)$ is an NDR (neighborhood deformation retract) pair if there are continuous maps $u:X\to I$ and $h:X\times I\to X$ satisfying,
(1) $u^{-1}(0)=A$,
(2) For ... | https://mathoverflow.net/users/15891 | Is being an NDR a local property? | There is a theorem of Dold to this effect:
Dold, Albrecht
Die Homotopieerweiterungseigenschaft (=HEP) ist eine lokale Eigenschaft. (German)
Invent. Math. 6 1968 185–189.
| 7 | https://mathoverflow.net/users/3634 | 250127 | 113,616 |
https://mathoverflow.net/questions/250147 | 0 | I am looking for some references to learn about viscosity solutions. Any suggestions would be greatly appreciated.
| https://mathoverflow.net/users/98607 | references for theory of viscosity solutions | My suggestion is "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations" by Bardi, Martino, and Capuzzo-Dolcetta, Italo.
| 2 | https://mathoverflow.net/users/98603 | 250150 | 113,624 |
https://mathoverflow.net/questions/250163 | 1 | Let $x, y$ and $z$ be positive integers with $x<y$. It appears that the integer
$$(y^2z^3-x)(y^2z^3+3x)$$
is never a perfect square. Why? A proof? I'm not sure if it is easy.
| https://mathoverflow.net/users/66131 | Why is this not a perfect square? | Rewrite your expression as $(y^2z^3+x)^2-4x^2$. This is clearly less than $(y^2z^3+x)^2$. On the other hand, it is larger than $(y^2z^3+x-2)^2=(y^2z^3+x)^2-4(y^2z^3+x)+4$, since $4(y^2z^3+x)-4\geq 4y^2+4x-4\geq 4y^2>4x^2$. So if this expression is a square, it must be $(y^2z^3+x-1)^2=(y^2z^3+x)^2-2(y^2z^3+x)+1$, so $4x... | 10 | https://mathoverflow.net/users/30186 | 250168 | 113,630 |
https://mathoverflow.net/questions/250172 | 19 | While on some work on symmetric functions, we encountered
$$f(a,b)=\frac{a^2+b^2}{1+ab}$$
is an integer iff it is a perfect square, where $(a,b)\in\mathbb{N}^2$. Well, it turns out that this is a well-known (well-documented) problem.
So, I propose a slight alteration to the question:
>
> When is $f(a,b)$ a perfec... | https://mathoverflow.net/users/66131 | When is $f(a,b)=\frac{a^2+b^2}{1+ab}$ a perfect square rational number? | **Edit:** in the original formulation, it wasn't clear that $a,b$ were supposed to be positive integers. This answer solves the question for $a,b$ *rational* instead.
---
The function $f$ takes every square value, once rational $a,b$ are allowed. Let $t$ be any rational number, and take the equation $(a^2+b^2)/(1+a... | 12 | https://mathoverflow.net/users/17907 | 250176 | 113,633 |
https://mathoverflow.net/questions/236661 | 2 | Recall a topological space $X$ is *path connected* if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$.
Say that $X$ is *continuously path connected* if there is a continuous function $f\colon [0,1] \times X \times X \to X$ such that $f(0,x,y) = x$ and $f(1,x,y)... | https://mathoverflow.net/users/12978 | Is every path connected space continuously path connected | As pointed out by Anton Petrunin, the condition is stated is equivalent to the space being [contractible](https://en.wikipedia.org/wiki/Contractible_space). Following Omar Antolín-Camarena, this can be seen since $(t,x) \mapsto f(t,x,y\_0)$ constitutes a homotopy between the identity on $X$ and the constant function $y... | 6 | https://mathoverflow.net/users/15002 | 250184 | 113,634 |
https://mathoverflow.net/questions/250181 | 6 | Consider the following block matrix
$$
X = \begin{bmatrix} A & C \\ C^\top & B\end{bmatrix},
$$
where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, and $C\in\mathbb{R}^{n\times m}$.
**To Prove (or disprove)**: If $X$ is positive definite, i.e. $X>0$, then the following trace inequality holds
$$
\left[\m... | https://mathoverflow.net/users/62673 | On a trace condition for positive definite $2\times 2$ block matrices | For any unitarily invariant norm it can be shown that
\begin{equation\*}
\|X\| = \left\Vert
\begin{bmatrix}
A & C\\
C^\* & B
\end{bmatrix}
\right\Vert \le \|A\| + \|B\|.
\end{equation\*}
Thus, using the squared Frobenius norm on both sides and cancelling, we obtain
\begin{equation\*}
\|C\|\_F^2 \le \|A\|\_F\|B\... | 6 | https://mathoverflow.net/users/8430 | 250186 | 113,635 |
https://mathoverflow.net/questions/249539 | 14 | This is a followup to my question [here](https://mathoverflow.net/questions/249523/crux-of-dworks-proof-of-rationality-of-the-zeta-function).
Here is a note of Michael Larsen where he gives a very simple proof of a slightly weaker result than Dwork's rationality of the zeta function.
<http://mlarsen.math.indiana.ed... | https://mathoverflow.net/users/nan | Weaker version of Dwork's rationality of zeta function, what is needed to beef up into a complete proof? | Not quite an answer, but a heuristics from the point of view of Weil philosophy about why the rationality mod $p$ is much easier.
Weil reduced the rationality of zeta-function to the existence of a good cohomology theory. The crucial property of a cohomology which satisifies the Weil axioms is that it has zero-charac... | 3 | https://mathoverflow.net/users/39304 | 250191 | 113,637 |
https://mathoverflow.net/questions/250079 | 1 | Given a deterministic function $h\in L^2([0,T]; \mathbb{R})$, we can define the associated exponential martingale
\begin{align}
M\_t = \exp\left[\int\_{0}^{t} h\_s \,dB\_s - \frac{1}{2}\int\_{0}^{t} h\_s^2\,ds\right], \quad t\in [0,T]
\end{align}
with the "kernel" $h$. Here $B$ is a standard Brownian motion. By Ito's ... | https://mathoverflow.net/users/91196 | Approximate an exponential martingale through its kernel | The OP seems to be an $\epsilon$ or so away from answering the question. Let me try to plug the gap, by introducing the approximating SDEs that $M\_t^n$ satisfies:
$$
d M\_t^n = h\_t^n M\_t^n d B\_t \;, \qquad M\_0^n = 1 \;, \tag{a}
$$ which we will compare to
$$
d M\_t = h\_t M\_t d B\_t \;, \qquad M\_0 = 1 \;. \tag{... | 3 | https://mathoverflow.net/users/64449 | 250204 | 113,639 |
https://mathoverflow.net/questions/249998 | 1 | We consider the Fourier multiplier operator $T\_0$ defined by the explicit expression
$$(T\_0f)(x)=\int\_{\mathbb{R}^n}{e^{ix\cdot \xi}m(\xi)\hat{f}(\xi)d\xi}, \ f\in S(\mathbb{R}^n),$$ where $S(\mathbb{R}^n)$ is the Schwartz function space. Here we assume that the multiplier $m(\xi)\in L^\infty(\mathbb{R}^n)$ satisfie... | https://mathoverflow.net/users/98145 | Does the bounded extension of the Fourier multiplier operator agrees with its original explicit definition? | We construct a sequence $f\_k\in S(\mathbb{R}^n)$ s.t. $||f\_k-f||\_p\to 0$ and $$T\_0f\_k-\int\_{\mathbb{R}^n}e^{ix\cdot\xi}m(\xi)\hat{f}(\xi)d\xi\to 0,\ a.e..$$ By the bounded extension of the multiplier operator, there is a function $g\in L^p$ s.t. $||T\_0f\_k- g||\_p\to 0$. Then there is a subsequence $T\_0f\_{k\_j... | 0 | https://mathoverflow.net/users/98145 | 250213 | 113,640 |
https://mathoverflow.net/questions/250206 | 2 | I am cross-posting this wuestion from [mathSE](https://math.stackexchange.com/q/1931718/60713). I hope it is of an adequate level for MO.
Let $K$ be a Kan complex, let $f,g:K\to K$ be morphisms of simplicial sets. Consider a simplex $\sigma\in K\_n$, and let $\alpha$ be a vertex of $\sigma$. Assume I have a $1$-simpl... | https://mathoverflow.net/users/44134 | Constructing a homotopy from some starting data | The answer to your question (1) is yes. What you're doing is showing that this inclusion
$$(\{0,1\} \times \Delta^n) \cup (\Delta^1 \times \{\alpha\}) \to \Delta^1 \times \Delta^n,$$
as an iterated pushout of horn-fillers, is an anodyne extension -- a special type of acyclic cofibration. The property of being an acycli... | 2 | https://mathoverflow.net/users/360 | 250220 | 113,643 |
https://mathoverflow.net/questions/250217 | 3 | **In the real projective plane, I am given three points and two lines. I want to find out how many conic sections there are that are incident to each of the three points and tangent to each of the two lines.** What is the easiest description for how many such conic sections exist, as a function of the points and lines?... | https://mathoverflow.net/users/5340 | When do two lines and three points determine exactly two conics? Exactly four? | In general there will be four solutions, possibly complex. In [*Conic by three points and two tangent lines*](https://math.stackexchange.com/q/1122145/35416) I've asked for ways to compute these, and based on a comment there found a way which is in my opinion pretty intuitive to understand since it relates to a 3d setu... | 5 | https://mathoverflow.net/users/25563 | 250228 | 113,645 |
https://mathoverflow.net/questions/250214 | 0 | I have a question about Sobolev spaces.
In the following, we assume $d \ge 2$.
Let $D$ be a domain of $\mathbb{R}^d$. That is, $D$ is a connected open subset of $\mathbb{R}^d$. **Note that $D$ is not necessary bounded.** $H^{1}(D)$ denotes first order $L^2$-Sobolev space on $D$ with Neumann boundary condition.
**I ... | https://mathoverflow.net/users/68463 | Continuous Sobolev embedding | The abstract condition to have the Sobolev embeddings on domains in their known form as on Euclidean space is the existence of an *extension operator* for $D$, that is, a continuous linear mapping $E \colon W^{k,p}(D) \to W^{k,p}(\mathbb{R}^n)$ which serves as a right inverse for the *restriction operator* $R \colon W^... | 3 | https://mathoverflow.net/users/85906 | 250230 | 113,646 |
https://mathoverflow.net/questions/249458 | 6 | The following is a toy version of something I've been fiddling with and I thought it might be more efficient to post it as a question here.
Just to fix definitions: for me, a complex-valued function $f$ on a group $G$ is said to be *positive-semi-definite* if, for every choice of points $x\_1, \dots, x\_n\in G$, the ... | https://mathoverflow.net/users/763 | Restricting a continuous positive-semidefinite function to a finite subset | The answer for *countably infinite* has been given by fedja and Uri Bader in the comments and is yes: put $f$ to $0$ outside of the subgroup generated by $E$, and leave $f$ unchanged on this subgroup.
For *finite*, the answer is no. For an example, take $f=1$ and $E=\{-1,0,1\}$. Since the only PSD function of the for... | 3 | https://mathoverflow.net/users/10265 | 250232 | 113,647 |
https://mathoverflow.net/questions/250207 | 1 | Consider the family of proper complex genus two curves with affine equation $y^2 = x(x-1)(x-a)(x-b)(x-c)$, defined over an open subset $U$ of $\mathbb{C}^3$. Here, $U$ consists of all triples $(a,b,c)$ such that the corresponding curve is smooth.
Fix a point $x \in U$, and let $C$ denote the corresponding curve. The... | https://mathoverflow.net/users/98640 | Simple closed curves on genus 2 surfaces | The action is not transitive on simple closed curves. For instance, some of the simple closed curves lift to the following unbranched, $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ cover, yet others do not: the projective smooth model of the affine curve $\text{Zero}(y^2-x(x-1)(x-a)(x-b)(x-c), 4xz^2-(z^2+1)^2) ... | 1 | https://mathoverflow.net/users/13265 | 250240 | 113,648 |
https://mathoverflow.net/questions/250221 | 5 | Let $X$ be a projective variety over $\mathbb{C}$. Is there a way to define some number $\tilde{\chi}(X)\in \mathbb{Z}$ satisfying both of the following two properties?
>
> $\boldsymbol{(1)} \;$ When $X$ is smooth, $\tilde{\chi}(X)=\chi(X)$, where $\chi(X)$ is the usual Euler number of $X$ (as a topological space).... | https://mathoverflow.net/users/44651 | Topological Euler number of a singular variety | The following example shows that the answer to abx's down-to-earth question
>
> Is the Euler number of the general fiber independent of the smoothing?
>
>
>
is in general *no*.
Take $X \subset \mathbb{P}^5$, the cone over a rational normal curve $C\_4 \subset \mathbb{P}^4$. It is well-known that $X$ admits ... | 14 | https://mathoverflow.net/users/7460 | 250248 | 113,650 |
https://mathoverflow.net/questions/249773 | 1 | Exchange pattern, see Section 2 in "cluster algebras I: foundations" by Fomin and Zelevinsky or [How to understand exchange pattern?](https://mathoverflow.net/questions/249230/how-to-understand-exchange-pattern)
Given an example $\cdots \overset{2}{-} t\_1 \overset{1}{-} t\_2 \overset{2}{-} t\_3 \overset{1}{-} t\_4 \... | https://mathoverflow.net/users/89288 | A question about exchange pattern | Using Fomin-Zelevinsky's notation, we can start with $(M\_1(t\_1),M\_1(t\_2))=(x\_2,1)$, $(M\_2(t\_2),M\_2(t\_3))=(x\_1,1)$. This can be considered initial data, considering $t\_2$ as the initial cluster: I don't think the axioms can be used to deduce the values of the second pair from the first. However, note that thi... | 3 | https://mathoverflow.net/users/98678 | 250258 | 113,654 |
https://mathoverflow.net/questions/250177 | 11 | There is the following result on page 160 of Thurston's book "The Geometry and Topology of Three-Manifolds", as follows.
>
> The volume of an ideal simplex in $\mathbb{H}^3$ with dihedral angles $\alpha$, $\beta$, $\gamma$ is equal to$$\Lambda(\alpha) + \Lambda(\beta) + \Lambda(\gamma),$$where $\Lambda: \mathbb{R} ... | https://mathoverflow.net/users/nan | Volume of an ideal simplex in $\mathbb{H}^3$, idea/intuition behind result | A longer but more intuitive proof of this formula is by "integrating" the Schlaefli identity for ideal tetrahedra.
One starts from a euclidean triangle with angles $\alpha$, $\beta$, $\gamma$ and side lengths $a$, $b$, $c$. Put $x=\log a$, $y=\log b$, $z=\log c$ and consider the function $$f(x,y,z) = \alpha x + \beta... | 12 | https://mathoverflow.net/users/98590 | 250259 | 113,655 |
https://mathoverflow.net/questions/250260 | 2 | Consider the following properties of a compact connected Lie group $G$:
(a) $G$ is semi-simple,
(b) $G$ has a finite fundamental group.
The well known Weyl's theorem states that (a) implies (b).
There are results in the structure theory of compact groups that lead me to believe that the converse implication is ... | https://mathoverflow.net/users/50457 | Converse to the Weyl's theorem | This is in (e.g.) Bourbaki, [*Lie groups and Lie algebras*, Chap. 9, §1, no. 4, Cor. 4](https://books.google.com/books?id=m_bKwNLBZk4C&pg=SL267-PA6).
| 5 | https://mathoverflow.net/users/19276 | 250261 | 113,656 |
https://mathoverflow.net/questions/249823 | 1 | Let A be an algebra. We denote by by A-proj the full subcategory of A-mod consisting of projective modules. An A-module T is called a tilting module if $proj.dim(\_{A}T)=n < \infty$, $Ext\_{A} ^{j} (T,T) =0$ for all $j > 0$, and there is an exact sequence $ 0 \rightarrow \_A A \rightarrow X\_0 \rightarrow X\_1 \rightar... | https://mathoverflow.net/users/83554 | The projective modules of an algebra and the tilting module? | A more general argument is that if you delete T from the minimal projetive resolution, you get a tilting complex. (see for example the book by Happel)
But by definition, for a tilting complex M, add(M) generates the bounded homotopy category of finitely generated projective modules, and this is here only possible if th... | 1 | https://mathoverflow.net/users/61949 | 250265 | 113,657 |
https://mathoverflow.net/questions/250268 | 8 | Originally posted [here](https://math.stackexchange.com/questions/1931924/construction-of-irreps-of-path-algebra-of-cyclic-quiver-classification-of-all-f) on Mathematics Stack Exchange.
Let $Q$ be a quiver with vertex set $\{1, 2, \ldots, n\}$ such that $Q$ has a single edge $i \to i + 1$, for every $i = 1, 2, \ldots... | https://mathoverflow.net/users/98682 | Construction of irreps of path algebra of cyclic quiver, classification of all finite-dimensional irreps | Every irreducible finite-dimensional representation is either $1$-dimensional (and there are exactly $n$ of them, corresponding to the vertices) or $n$-dimensional and they can be indexed by non-zero, complex numbers.
Proof: Let $V$ be an irreducible representation, $V=\bigoplus\_{i\in\mathbb{Z}/n} V\_i$ its standard... | 9 | https://mathoverflow.net/users/3041 | 250271 | 113,659 |
https://mathoverflow.net/questions/250273 | 3 | With $p\_i$ being the $i$-th prime, I'm wondering whether there is a tighter bound than $\alpha = 4$ in the relation
$$
\prod\_{i=1}^n p\_i < \alpha^{p\_n}
$$
$\alpha = 4$, which is tight enough to be used to prove Bertrand's postulate (Chebyshev's theorem) that there is always a prime in the interval $[k,2k]$, is easi... | https://mathoverflow.net/users/82067 | Is there a tighter bound than $\alpha=4$ in $ \prod_{i=1}^n p_i < \alpha^{p_n} $? | It follows from the Prime Number Theorem that the $p\_n$-th root of the product tends to $e=2.7182\dots$. In practice one takes the logarithm of the product and divides by $p\_n$, which then tends to $1$. This means that any $\alpha>e$ produces an upper bound for sufficiently large $n$, while any $\alpha<e$ produces a ... | 10 | https://mathoverflow.net/users/11919 | 250276 | 113,660 |
https://mathoverflow.net/questions/250266 | 1 | What is the expected value for the minimum distance between $n$ points placed randomly, assuming a uniform distribution, within a cube of volume $V$?
| https://mathoverflow.net/users/98681 | Minimum distance between $n$ points in a cube | This question (or at least approximations to the distribution that hold beyond lowest order in the number of points) has practical importance. The "DIEHARD" suite of tests for pseudo-random generators has as one of its tests the generation of many cases of the minimum distance of $N$ points in a 2 or 3 dimensional box,... | 6 | https://mathoverflow.net/users/82067 | 250279 | 113,662 |
https://mathoverflow.net/questions/250275 | 6 | Let $H^d:\mathcal{P}(\mathbf{R}^n) \to \mathbf{R}\cup \{\infty\}$ be the $d$-dimensional Hausdorff outer measure on $\mathbf{R}^n$, for some $0<d<n$ with $n$ integer, which is constructed in the following way:
For each $\delta>0$ and $X \subseteq \mathbf{R}^n$, define
$$
H\_\delta^d(X)=\inf\left\{\sum\_{n=1}^\infty ... | https://mathoverflow.net/users/32898 | Subsets $X$ such that their Hausdorff outer measure is not finite | Yes, if $X$ is an analytic set. See:
* Besicovitch, *On existence of subsets of finite measure of sets of infinite measure*, 1952
* R. O. Davies, *Subsets of finite measure in analytic sets*, 1951.
For discussion you may consult e.g.
<http://arxiv.org/abs/1408.1999>
| 5 | https://mathoverflow.net/users/4600 | 250280 | 113,663 |
https://mathoverflow.net/questions/249770 | 13 | 1. Trivial example. First, suppose $X$ is finite. Then we have a finite set $S := X(\overline{\mathbb{F}}\_q)$ with an action of $\text{Fr}\_q$. How can one explain why the rationality of the zeta function should be true in this case (it is still nonobvious from the definition of $Z(X, t)$)? Let $\mathbb{Q}[S]$ be the ... | https://mathoverflow.net/users/nan | Rationality of zeta function and Grothendieck-Lefschetz fixed point formula, cohomology can be computed as the de Rham cohomology | The following is sections 3-4 of van der Put's [The cohomology of Monsky and Washnitzer](https://eudml.org/doc/94862) with all $p$-adic issues and analytic subtleties (necessary to make the argument correct, of course!) removed.
Let $X$ be a smooth manifold of dimension $n$ and let
$F: X \to X$ be a degree $q^n$ fin... | 8 | https://mathoverflow.net/users/297 | 250289 | 113,665 |
https://mathoverflow.net/questions/250284 | 19 | It is well-known that one can get the Lebesgue measure on [0, 1] by tossing a fair coin infinitely (countably) many times and mapping each sequence to a real number written out in binary.
I was trying to explore what happens if you follow the same procedure with a biased coin. I managed to prove that if the induced m... | https://mathoverflow.net/users/94232 | Measure induced on [0, 1] by infinite tosses of biased coin | For $\omega \in [0,1]$, let $X\_i(\omega)$ be the $i$th binary digit of $\omega$. (If $\omega$ is a dyadic rational and thus has two binary expansions, let's say we choose the expansion that ends with all 0s; it makes no difference). Under Lebesgue measure, the $X\_i$ are iid Bernoulli $1/2$ random variables, so by the... | 20 | https://mathoverflow.net/users/4832 | 250291 | 113,666 |
https://mathoverflow.net/questions/250263 | 18 | There is the following analogy:
$$\begin{array}{cc} \text{frames} & - & \text{commutative rings} \\ | && | \\\text{locales} & - & \text{affines schemes}\end{array}$$
Here, $\bigvee$ is analagous to $\sum$ and $\bigwedge$ is analogous to $\prod$. The category of locales is defined as the dual of the category of fram... | https://mathoverflow.net/users/98306 | Locales as geometric objects | First, if you haven't already you should have a look at this introductory paper by P.T. Johnstone [The Art of pointless thinking](http://www.heldermann.de/R&E/RAE18/ctw06.pdf) which gives a lot of insight on how locale theory works.
Here are some observations which I hope will answer your questions:
1) As I said in... | 19 | https://mathoverflow.net/users/22131 | 250307 | 113,670 |
https://mathoverflow.net/questions/250312 | 14 | How to solve a Diophantine equation like $$3^n-1=2x^2$$. One can easily see that the parity of $n$ and $x$ will be same and equation further can be seen taking if $$n\equiv0\pmod3\quad \text{then }x \equiv0\pmod{13} $$ but I don't know what to do further.
| https://mathoverflow.net/users/97687 | Diophantine equation $3^n-1=2x^2$ | This problem happens to have appeared on the Polish Mathematical Olympiad camp in 2015. Here is the official solution of the problem: (I use $m$ in place of $x$ because this is how the problem was stated there)
Suppose first $n$ is even, say $n=2k$. Then the equation is equivalent to $(3^k+1)(3^k-1)=2m^2$. Clearly $\... | 23 | https://mathoverflow.net/users/30186 | 250327 | 113,679 |
https://mathoverflow.net/questions/250303 | 13 | From work of Pontryagin and Whitney, as I understand it, the homotopy 4-type of $BSO(3)$ is $K(\mathbb{Z}/2,2) \times\_{K(\mathbb{Z}/4,4)} K(\mathbb{Z},4)$, where the pullback is along the maps $\mathfrak{P}\_2\colon K(\mathbb{Z}/2,2) \to K(\mathbb{Z}/4,4)$ (Pontryagin square) and the obvious $K(\mathbb{Z},4) \to K(\ma... | https://mathoverflow.net/users/4177 | The fifth k-invariant of BSO(3) | Recall that $SU(2)\cong Spin(3)$ and $PU(2)\cong SU(2)/\mathbb{Z}\_2\cong SO(3)$. Originally Woodward calculated the lower stages of a Postnikov decomposition of $PU(n)$ in “The Classification of Principal PU(n)-bundles Over a 4-complex.” If I recall, his cacluations did not apply fully to the case $n=2$ and they were ... | 11 | https://mathoverflow.net/users/54788 | 250329 | 113,681 |
https://mathoverflow.net/questions/249944 | 2 | Let $\mathsf{MK^-}$ be the theory "$\mathsf{MK}-\text{Foundation}-\text{Limitation of size}-\text{Union}+\text{Subsets}$", where $\mathsf{MK}$ is Morse-Kelley set theory with axioms mentioned in:
<https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory>
Of course axioms of pairing and union and one direction ... | https://mathoverflow.net/users/95347 | Proper classes subnumerous to $V$ in a model of a Morse-Kelley related theory | I believe the following gives a positive answer to your first question, modulo large cardinals:
Let $\kappa$ be measurable, and consider a [Prikry-generic](http://blog.assafrinot.com/?p=2156) extension of the universe $V[G]$. Look at $\mathcal{M}=(V\_{\kappa+1})^{V[G]}$. We can view this as a model of $MK^-$ by takin... | 3 | https://mathoverflow.net/users/8133 | 250337 | 113,684 |
https://mathoverflow.net/questions/250336 | 5 | I have been stuck for a few days in a seemingly harmless question.
Given a simply connected open set $\Sigma\subset\mathbb{R}^2$, with smooth boundary $\partial\Sigma$, I am interested in estimating
$$
\int\_{\Sigma}d(x,\partial\Sigma)\;dx,
$$
where $d(x,\partial\Sigma)=\inf\_{y\in\partial\Sigma}|x-y|$.
I would lov... | https://mathoverflow.net/users/39062 | Integral of the distance function to the boundary of a planar set | The correct constant is $1/2$. Let $L(s)$ be the perimeter of the set of points whose distance to the boundary is $\ge s$. For simply-connected domains $L(s)$ is non-increasing, the integral of the distance to the boundary is $\int\_0^\infty sL(s)\,ds$ and the area is $\int\_0^\infty L(s)\,ds$. Now we want to minimize ... | 5 | https://mathoverflow.net/users/1131 | 250359 | 113,692 |
https://mathoverflow.net/questions/250308 | 6 | For motivation and related questions, see below.
**Rough sketch of the question.**
View $\bigsqcup\_{p \text{ prime}} (\mathbb{Z}/p\mathbb{Z})$ as a ‘subset’ of the unit circle, via $a\pmod{p} \mapsto e^{2\pi i \cdot a/p}$. (This is not injective for $0 \pmod{p}$, but I wish to ignore that for the moment.)
Let $f \... | https://mathoverflow.net/users/98708 | On the distribution of roots modulo primes of an integral polynomial | In my paper, [Polynomial congruences and density](http://www.maa.org/sites/default/files/Myerson10-200716952.pdf), Mathematics Magazine 80 (2007) 299-302, I cite the result of Christopher Hooley, On the distribution of roots of polynomial congruences, Mathematika 11 (1964) 39-49, MR 29 #1173, to the effect that if $f$ ... | 9 | https://mathoverflow.net/users/3684 | 250362 | 113,694 |
https://mathoverflow.net/questions/250370 | 3 | I think I've encountered a question about 4-manifolds which maybe easy but I'm not familiar with. Can anyone give me an example of a simply connected 4-manifold $M$ (with boundary, of course) with $H\_2(M)\cong \mathbb{Z}\_k$? It must exist, I thought?
Thanks.
| https://mathoverflow.net/users/70446 | Simply connected 4-manifolds with boundary | That can't happen if $k>1$. First, notice that $H\_2(M, \partial M) \cong H^2(M) \cong 0$ by Poincare duality and universal coefficients. Then the exact sequence on homology for $(M, \partial M)$ forces $H\_1(\partial M) = 0$, and then $H\_2(\partial M) = 0$, and then you have a contradiction unless $k=1$. (I'm assumin... | 9 | https://mathoverflow.net/users/65952 | 250372 | 113,698 |
https://mathoverflow.net/questions/250287 | 7 | Consider a generic nontrivial $d$-cocycle $\omega\_d^G \in H^d(G,U(1))$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. In otherwords, here the $d$-cocycle $\omega\_d^G$ is a complex $U(1)=\mathbb{R}/\mathbb{Z}$ function with the norm $|\omega\_d^G|=1$ but with a $U(1)$ complex pha... | https://mathoverflow.net/users/27004 | $G$ cocycle split to a coboundary in $J$, via a group extension | In case d=1, the answer is always negative: 1-cocycles are homomorphisms, 1-coboundaries are always trivial, and inflation is injective.
If you do not restrict yourself to the case where $N$ is abelian, the answer is positive: take a short exact sequence $1\to R\to F\to G\to 1$ where $F$ is free. For the abelian case ... | 6 | https://mathoverflow.net/users/41644 | 250397 | 113,703 |
https://mathoverflow.net/questions/249780 | 7 | Let $X\_0$ be a trace-one positive definite matrix, i.e. $X\_0>0$, $\mathrm{tr}(X\_0)=1$. Let $A>0$ and consider the following iteration
$$
X\_{k+1} = X\_k^{1/2}AX\_k^{1/2},\quad k\geq 0,\quad (\star)
$$
where $X\_k^{1/2}$ denotes the (principal) square root of $X\_k$.
>
> **My question:** Is it true that if *there... | https://mathoverflow.net/users/62673 | Trace of a nonlinear matrix equation (cont'd) | Actually, you have completely solved it yourself, just didn't dare to acknowledge it. In my notation, you have $(X\circ X^T)v(A)=(Y\circ Y^T)v(I)$ when $Y^2=XAX$. Similarly, $(Z\circ Z^T)v(I)=(Y\circ Y^T)v(A)$ when $Z^2=YAY$. Taking the trace, we must have
$$
1=\langle(X\circ X^T)v(I),v(I)\rangle=\langle(Y\circ Y^T)v(... | 5 | https://mathoverflow.net/users/1131 | 250399 | 113,704 |
https://mathoverflow.net/questions/250393 | 0 | The exchange relation in a cluster algebra is
\begin{align}
x\_k' = \frac{1}{x\_k} (\prod\_{j \to k}x\_j + \prod\_{k \to j} x\_j).
\end{align}
Do we have some tropical version of this relation? Are there some references? Thank you very much.
| https://mathoverflow.net/users/11877 | Tropical version of exchange relations in cluster algebras | This maybe too naive and not helpful to your problem: pretend $x\_i$ are positive real number, and let $x\_i = e^{-t a\_i}$ for $a\_i \in \mathbb{R}$. One can consider the limit $t \to \infty$ and taking $log$ on both sides to get
$$ a\_k' = \min(\sum\_{j \to k} a\_j, \sum\_{k \to l} a\_l) - a\_k $$
Of course, this is ... | 3 | https://mathoverflow.net/users/64506 | 250420 | 113,711 |
https://mathoverflow.net/questions/151973 | 13 | Recall the well-known proof that a unique factorization domain is a GCD domain:
>
> Let $x, y \in R \setminus \{ 0 \}$. Factor $x$ and $y$ into pairwise non-associated irreducible elements: $$\begin{align\*}x &= p\_1^{e\_1} \cdots p\_n^{e\_n}, \\ y &= p\_1^{f\_1} \cdots p\_n^{f\_n}.\end{align\*} $$ Then one can che... | https://mathoverflow.net/users/31233 | Constructively correct notion of unique factorization domain | It seems that Lombardi and Quitté in their book "[Commutative Algebra: Constructive Methods](http://hlombardi.free.fr/CACM.pdf)", Ch. XI, §3, define UFDs as GCD-domains such that every regular element is a product of irreducibles. In classical logic, this coincides with the usual definition. (Ingo sketched the proof in... | 7 | https://mathoverflow.net/users/98306 | 250439 | 113,716 |
https://mathoverflow.net/questions/233664 | 7 | This question is in reference to Gaitsgory's preprint *[Contractibility of the space of rational maps](http://arxiv.org/abs/1108.1741)*. On p. 5 of the preprint, Gaitsgory defines a prestack $\mathscr{Y}$ (say over affine $\mathbb{C}$-schemes) to be *homologically contractible* if the functor $\text{Vect}\longrightarro... | https://mathoverflow.net/users/64153 | Homological contractibility of a prestack | This is proven in some detail in section 3 [Gaitsgory's writeup](http://www.math.harvard.edu/~gaitsgde/GL/Tamagawa.pdf) of his the Atiyah-Bott formula. He starts with the fully faithfulness definition, then proves the equivalance with homological statement at the very end of the section.
| 2 | https://mathoverflow.net/users/24706 | 250447 | 113,718 |
https://mathoverflow.net/questions/249833 | 6 | The Setup
---------
Suppose I have a stochastic process $f(Z\_t)$ where $Z\_t$ solve the $d$-dimensional SDE
$$
dZ\_t = \mu(t,Z\_t)dt + \sigma(t,Z\_t)dW\_t
$$
and $f$ is a smooth function.
---
My Question
-----------
Is there a notion of *time-derivative* "$d\_t$" of the process $f(Z\_t)$ which satisfies:
... | https://mathoverflow.net/users/36886 | Does there exist a stochastic time derivative? | The Malliavin derivative satisfies the requisite chain rule. However, if $Z\_t$ is no longer a function of the Wiener process, then the Malliavin derivative of $f(Z\_t)$ is identically zero.
To see this, we briefly recall the nice link between the Girsanov theorem and Malliavin calculus. Let $\epsilon>0$ be a pertur... | 3 | https://mathoverflow.net/users/64449 | 250448 | 113,719 |
https://mathoverflow.net/questions/250430 | 3 | Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety.
By Losev's theorem, the spherical $G$-variety $X$ is uniquely determined by its spherical datum, see below.
In particular, the group ${\rm Aut}\_G(X)=\mathcal{N}\_G(H)/H$ is uniquely determined by the spherical datum.
... | https://mathoverflow.net/users/4149 | The group of $G$-automorphisms of a spherical variety from the spherical datum? | This question has been answered by Losev in Theorem 2 of his paper "Uniqueness property for spherical homogeneous spaces".
The answer is roughly as follows: The space $\overline X=G/\mathcal N\_G(H)$ is also spherical with weight lattice $\overline M$ and spherical roots $\overline\Sigma$. It is well known that $\mat... | 5 | https://mathoverflow.net/users/89948 | 250458 | 113,723 |
https://mathoverflow.net/questions/250467 | 3 | The classic theory of correspondences between smooth algebraic curves can be found in André Weil's *Foundations of algebraic geometry*. However, this reference works in a pre-modern algebraic geometry way. My question is:
>
> Do you know about a modern reference for the theory of correspondences for smooth complete... | https://mathoverflow.net/users/12204 | Modern reference for the theory of correspondences for curves | Griffiths and Harris, *[Principles of Algebraic Geometry](http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471050598.html)*.
See in particular Chapter 2 (*Riemann Surfaces and Algebraic Curves*), Section 5 (*Correspondences*).
| 3 | https://mathoverflow.net/users/7460 | 250469 | 113,727 |
https://mathoverflow.net/questions/250470 | 6 | Let $X$ and $Y$ be metric spaces. The $(\varepsilon,\delta)$-definition of continuity of single-valued maps can be rephrased as:
>
> Let $f$ be a single-valued map from $X$ to $Y$. $f$ is continuous at $x\_0 \in X$ if for every neighborhood $N\_Y$ of $f(x\_0)\in Y$, there exists a neighborhood $N\_X$ of $x\_0$ such... | https://mathoverflow.net/users/98759 | Upper semicontinuity of set-valued maps with open values | I think there is not much more to say than that most interesting results about upper semicontinuous set-valued maps involve closed-valued or even compact-valued maps. Indeed some authors choose to define upper semicontinutiy only for such maps. But there are good reasons not to. It helps to rephrase continuity notions ... | 5 | https://mathoverflow.net/users/35357 | 250471 | 113,728 |
https://mathoverflow.net/questions/250479 | 1 | Let $X\neq \emptyset$ be a set. We say that $U\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a *proper covering* if
* $\bigcup U = X$, and
* for $a\neq b\in U$ we have $a\not\subseteq b$.
Let $\text{Cov}(X)$ denote the collection of all (proper) coverings of $X$. For $A, B\in \text{Cov}(X)$ we set $A\leq B$ if $A... | https://mathoverflow.net/users/8628 | Does the lattice of coverings embed in the lattice of partitions? | The answer is no, because in general there can be more proper coverings than partitions, which will prevent any injective mapping from coverings to partitions.
For example, if $X$ is countably infinite, then there are only
continuum many partitions, but I claim that there are
$2^{\frak{c}}$ many proper coverings. So ... | 3 | https://mathoverflow.net/users/1946 | 250482 | 113,731 |
https://mathoverflow.net/questions/250452 | 8 | Suppose $X$ is a variety of dimension $n$ over $k$ and there exists a dominant rational map $\mathbb{A}^N\dashrightarrow X$, where $N$ can be larger than $n$. Is it true that there is a dominant rational map $\mathbb{A}^n\dashrightarrow X$?
If $k$ is characteristic 0, this is true because we can apply generic smooth... | https://mathoverflow.net/users/16356 | Dominant map from affine space implies unirationality | It seems to me that the following arguement doses not depend on characteristic. Consider a general fiber of $\varphi:\mathbb{A}^N\dashrightarrow X$. Take its closure in $\mathbb{P}^N$. It is a subvariety of dimension $N-n$. Then a general $\mathbb{P}^n$ intersects it in a finite subscheme, so restriction of $f$ to this... | 6 | https://mathoverflow.net/users/39304 | 250483 | 113,732 |
https://mathoverflow.net/questions/250442 | 2 | When are the $l$-local $p$-adic Galois representations of Siegel modular forms semistable? By this I mean $\rho\_{f}: G\_{\mathbb{Q}}\to \operatorname{GSpin}\_{2n+1}(\overline{\mathbb{Q}}\_p)$ restricted to the decomposition group at $l$. Is this controlled by the level? I am primarily interested in this when $l=p$ and... | https://mathoverflow.net/users/47195 | Semistability of local Siegel Galois rep: | As I commented above, the question needs some adjustment for $n \ge 3$, since the Galois representation doesn't go into $\operatorname{GSp}\_{2n}$ but rather into its $L$-group, which is $\operatorname{GSpin}\_{2n + 1}$; this is isomorphic to $\operatorname{GSp}\_{2n}$ if $n = 1$ or $n = 2$, but totally different if $n... | 2 | https://mathoverflow.net/users/2481 | 250486 | 113,733 |
https://mathoverflow.net/questions/250040 | 17 | I am interested in the following innocent looking statement:
>
> Let $A \leftarrow R \rightarrow B$ be two homomorphisms of commutative rings. Assume that their kernels consist of nilpotent elements. Then, the kernel of $R \to A \otimes\_R B$ consists of nilpotent elements, too.
>
>
>
Geometrically, this means... | https://mathoverflow.net/users/98306 | Constructive proof that a kernel consists of nilpotent elements | This answer provides a scheme how to construct a constructive proof, though I'm still working to actually explicitly extract the constructive proof, so please don't accept the answer just yet. (Update: See below.) We'll prove the following statement:
>
> Let $R$ be a reduced ring. Let $A$ be a finitely generated $R... | 11 | https://mathoverflow.net/users/31233 | 250490 | 113,735 |
https://mathoverflow.net/questions/250489 | 3 | Let $A$ be a nonnegatively graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A\_{>0}M$ is a graded $A$-submodule of $M$.
(Here, given a nonnegatively graded algebra $A$, we've defined $A\_{>0} := \oplus\_{i > 0} A\_i$.)
My question is as follows. How do I see that $M$ is finitely generated as an $A$-... | https://mathoverflow.net/users/98682 | $M$ is finitely generated as $A$-module iff $M/A_{>0}M$ is finitely generated as $A$ module? | This is essentially Nakayama's lemma (not literally, but the same proof). More generally, the result is that a graded module map $\phi\colon N\to M$ is surjective if and only if the induced map $N\to M/A\_{>0}M$ is surjective. "Only if" is obvious; consider the "if" direction. The induced map being surjective is the sa... | 8 | https://mathoverflow.net/users/66 | 250493 | 113,737 |
https://mathoverflow.net/questions/250485 | 4 | In the case the modular curve $X\_0(N)$ has genus one, is there a reference for the explicit map between such curve and the corresponding elliptic curve in the lmfdb database? (just having the explicit map to the j-line would be enough).
| https://mathoverflow.net/users/4685 | Isomorphism between genus 1 modular curves and elliptic curves | The maps $j : X\_{0}(N) \to \mathbb{P}^{1}$ are given in Magma's ''Small modular curves'' database. In each case, they construct functions on the modular curves $X\_{0}(N)$ out of eta products, modular forms attached to elliptic curves, theta series of binary quadratic forms, or weight $2$ Eisenstein series. It appears... | 7 | https://mathoverflow.net/users/48142 | 250498 | 113,739 |
https://mathoverflow.net/questions/250495 | 2 | This is related to my question [here](https://mathoverflow.net/questions/250489/m-is-finitely-generated-as-a-module-iff-m-a-0m-is-finitely-generated-as). My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A\_0$ is finitely generated as ... | https://mathoverflow.net/users/98682 | Nonnegatively graded algebra $A$ finitely generated as $k$-algebra iff $A_0$ finitely generated, $A_{>0}$ finitely generated as $A$-module? | Seems like the "greedy algorithm" works...
Suppose $A\_0$ is finitely generated as a $k$-algebra, say by $a\_1, \ldots, a\_m$ and $A\_{>0}$ is finitely generated as an $A$-module, say by $b\_1, \ldots, b\_n$. I claim that $\{a\_1, \ldots, a\_m, b\_1, \ldots, b\_n\}$ generate $A$ as a $k$-algebra.
Indeed, given any... | 5 | https://mathoverflow.net/users/84144 | 250501 | 113,740 |
https://mathoverflow.net/questions/250507 | -1 | Can the derivative $f^\prime$ of a smooth function $f\in C^\infty[0,1]$ change sign infinitely many times (or $f$ have infinitely many isolated critical points)? If yes, how about an analytic function over a closed finite interval or more generally—what do I need to assume about $f$ in order to exclude this possibility... | https://mathoverflow.net/users/98834 | Derivative of smooth function change sign infinitely on [0,1]? | Yes, for example $f(x)=e^{-1/x}\sin(1/x)$. No for analytic functions -- such function has all derivatives equal to zero at accumulating point of its zeroes.
| 2 | https://mathoverflow.net/users/39304 | 250509 | 113,741 |
https://mathoverflow.net/questions/246877 | 5 | Let $K$ be a $p$-adic field and $\Gamma$ be Schottky group of $g$ generators and $L \subset \mathbb{P}^1\_K$ be the limit set of $\Gamma$. Let $\mathbb{P}^1\_K - L= \Omega$ and we know that there exists a covering of $\Omega$ such that the reduction of $\Omega$ is a tree of projective lines whose dual graph is isomorph... | https://mathoverflow.net/users/46460 | Schottky groups, Mumford curves and $p$-adic uniformization | $T(\Gamma)$ is constructed by gluing semi-stable skeletons and hence the inverse image of any vertice $v$ is the closed unit disc punctured by finitely many maximal open rational discs and the number of these maximal open discs corresponds to the index of $v$ and it is equal to $g+1$ since $T(\Gamma)$ is the universel ... | 1 | https://mathoverflow.net/users/46460 | 250510 | 113,742 |
https://mathoverflow.net/questions/250545 | 4 | Following [the case of groups](https://math.stackexchange.com/questions/541590/free-object-isomorphisms), I asked in [this MSE question](https://math.stackexchange.com/questions/1934096/quick-proof-that-free-objects-on-sets-of-different-cardinality-are-not-isomorphi) for a quick proof that given a free-forgetful adjunc... | https://mathoverflow.net/users/69037 | Free algebras on sets of different cardinality – for what theories are they non-isomorphic? | It is not true for "usual theories", at least if you accept that left $R$-modules for noncommutative $R$ is a usual theory. Take $R=\mathrm{End(}V)$ for some infinite-dimensional vector space $V$, then $R \cong R^2$ as left $R$-modules. Rings for which $\forall n,m : \mathbb{N}. R^n \cong R^m \Rightarrow n=m$ holds (th... | 5 | https://mathoverflow.net/users/98306 | 250549 | 113,760 |
https://mathoverflow.net/questions/250550 | 0 | Let $A$ be a k-algebra, where k is a fixed field. Let $S$ be a simple, non-injective $A$-module such that $Ext^{i}\_{A}(S,S)=0$ for $1 \leq i \leq n$. Let $P(S)$ be the projective cover of $S$, and let $Q$ be the direct sum of all non-isomorphic indecomposable projective $A$-module which are not isomorphic to $P(S)$. $... | https://mathoverflow.net/users/83554 | How to get $I_i \in add(\nu_A(Q))$ for $1 \leq i \leq n$ by $Ext^{i}_A(S,S)=0$? | You just need the following fact for any simple module $S$ and any other module $M$: $Ext^{i}(S,M)=0$ iff in the minimal injective resolution $(I\_i)$ of $M$, $S$ is not a submodule of $I\_i$. You can find this in the first volume of Bensons book "Representations and Cohomology", somehwere in the beginning (have not th... | 0 | https://mathoverflow.net/users/61949 | 250554 | 113,761 |
https://mathoverflow.net/questions/228137 | 6 | In Görtz and Wedhorn's *Algebraic Geometry I*, there's the following proposition:
**Proposition 3.4.** Let $(X,\mathcal O\_X)$ be a locally ringed space. If $Y$ is an affine scheme then the natural map below is an isomorphism $$\mathsf{Hom}(X,Y)\overset{\cong}{\longrightarrow} \mathsf{Hom}(R,\Gamma(X,\mathcal O\_X))$... | https://mathoverflow.net/users/69037 | Morphisms of locally ringed spaces into affine schemes | Here is a sketch: Let $\alpha : R \to \Gamma(X,\mathcal{O}\_X)$ be a ring homomorphism. We want to define a morphism $f:X \to \mathrm{Spec}(R)$ which is $\alpha$ on global sections. Let $x \in X$, and consider the composition of $\alpha$ with $\Gamma(X,\mathcal{O}\_X) \to \mathcal{O}\_{X,x}$, $s \mapsto s\_x$. Pull bac... | 6 | https://mathoverflow.net/users/98306 | 250556 | 113,762 |
https://mathoverflow.net/questions/250444 | 5 | This is cross-posted in [MSE](https://math.stackexchange.com/q/1935700/9464).
I have seen two different kinds of definitions of the notation $C^k(\overline{\Omega})$ — by "extension" of functions on $\Omega$ or by "restriction" of functions on $\mathbb{R}^n$. I'm not sure about how different these two kinds of defini... | https://mathoverflow.net/users/nan | Two different kinds of definitions of $C^k(\overline{\Omega})$ — extension and restriction | The standard reference for the extension problem is [Whitney's extension theorem](http://www.ams.org/journals/tran/1934-036-01/S0002-9947-1934-1501735-3/S0002-9947-1934-1501735-3.pdf), which, has also many more modern presentations (a pretty good one is found in Stein's *Singular Integrals and Differentiability Propert... | 6 | https://mathoverflow.net/users/3948 | 250559 | 113,763 |
https://mathoverflow.net/questions/218124 | 10 | This is a crosspost of [this MSE question](https://math.stackexchange.com/questions/1351300/how-do-the-direct-and-inverse-image-sheaf-functors-interact-with-homotopy).
The direct image sheaf functor $f\_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by ... | https://mathoverflow.net/users/69037 | How do the direct and inverse image sheaf functors interact with homotopy? | By naturality, it suffices to consider the projection map $p : X \times I \to X$. If $X$ and $I$ are locally conctractible Hausdorff spaces and $\mathcal{F}$ is a sheaf of abelian groups on $X$, then the Vietoris-Begle theorem (Bredon, *Sheaf theory*, II.13) states that the unit $\mathcal{F} \to p\_\* p^\* \mathcal{F}$... | 2 | https://mathoverflow.net/users/98306 | 250560 | 113,764 |
https://mathoverflow.net/questions/250563 | 6 | For an abelian variety $A$, the $p$-adic Selmer group is defined to be the subset of $H^1(G\_k,H\_1^{et}(A;\mathbb{Q}\_{p}))$ whose restriction to $G\_{k\_v}$ is in the image of $A(k\_v)$ for all places $v$ of $k$.
[Minhyong Kim's paper](http://www.ucl.ac.uk/~ucahmki/alb.pdf) states on its third page (p.91) that his ... | https://mathoverflow.net/users/1355 | Selmer Group versus Selmer Variety | You should read the Bloch--Kato paper in the Grothendieck Festschrift. This was, I believe, the first paper to consider Selmer groups of Galois representations defined by local conditions coming from p-adic Hodge theory (e.g. crystalline at p).
The definition you quote from Kim is exactly the Bloch--Kato definition ... | 9 | https://mathoverflow.net/users/2481 | 250564 | 113,766 |
https://mathoverflow.net/questions/250517 | 5 | [I found in the literature](https://arxiv.org/abs/1501.00741) that, in characteristic 0, Kodaira vanishing holds for log-canonical pairs. On the other hand, the usual statement for Kawamata-Viehweg vanishing talks about a klt pair $(X,\Delta)$.
**Question:** Are there (good) examples of failure of Kawamata-Viehweg va... | https://mathoverflow.net/users/89459 | Log canonical counterexample to Kawamata-Viehweg vanishing | In Prop. 3.13 of <http://arxiv.org/pdf/1212.5105.pdf> there is an example of a generically finite map $\lambda : T\to A$ where $A$ is an abelian variety and $T$ is a Gorenstein variety with log canonical singularites, and $R^1\lambda \_\* \omega \_T\ne 0$ is supported at a closed point of $A$ (everything is defined ove... | 3 | https://mathoverflow.net/users/19369 | 250567 | 113,768 |
https://mathoverflow.net/questions/250565 | 4 | In Goodwillie's "Calculus I", speaking of a commutative diagram of spaces
$$\begin{array}{c} Y & \rightarrow & Y\_1 \\ \downarrow & & \downarrow & \\ Y\_2 & \rightarrow & Y\_{12} \end{array}$$
there is the following statement
>
> 'Cartesian' implies that (for every basepoint in $Y$) the relative homotopy groups... | https://mathoverflow.net/users/39004 | Homotopy pullbacks/relative homotopy groups vs homotopy pushouts/relative homology groups | Relative homotopy groups are the homotopy groups of the homotopy fiber. A homotopy pullback square induces an equivalence of the homotopy fibers of two of its parellel maps by the cancellation property of homotopy pullbacks.
Similarly, relative homology groups are the homology groups of the homotopy cofiber. The stat... | 8 | https://mathoverflow.net/users/12547 | 250569 | 113,769 |
https://mathoverflow.net/questions/250340 | 1 | Suppose I have a functional
$$
E=\int F(y\_{1,1},..y\_{1,n},y\_{2,1}\ldots,y\_{n,n})d\boldsymbol{x}\,,
$$
where $\boldsymbol{y}:\mathbb{R}^{n}\to\mathbb{R}^{n},\,\boldsymbol{y}(\boldsymbol{x})=\left(y\_{1}(x\_{1},...x\_{n}),...,y\_{n}(x\_{1},...x\_{n})\right)$,
and $y\_{1,1},...,y\_{n,n}$ are partial derivatives, i.e... | https://mathoverflow.net/users/97799 | Change of functional derivative under rigid coordinate transformation | (a comment) Isn't it possible to use the same formula (from the post scriptum) but replacing $u(v)$ as it follows from the definition:$u(v) = U^T y(V v)$
and compute derivatives of $u$ with respect to $v$ as for function decomposition.
Then you'll obtain a formula which may take some nice form in tensor notations. Or n... | -1 | https://mathoverflow.net/users/97620 | 250576 | 113,773 |
https://mathoverflow.net/questions/250572 | 10 | In this article by Katz and Tate [here](http://www.ams.org/notices/199903/mem-dwork.pdf), there's a nice account of Dwork's argument for showing the rationality of the zeta function part of the Weil conjectures. Here is an excerpt.
>
> To recapitulate, we now know that the zeta function as power series has integer ... | https://mathoverflow.net/users/nan | Dwork's proof of rationality of zeta function, crux of his generalization of a result of Borel along the way | Why don't you read [Dwork's paper](https://www.evernote.com/shard/s24/sh/4d614105-26d3-487f-92b9-6f0b199fc06f/cb0b02f3ddfa9cea015df533e6623724) - it is quite clear (see p. 643)?
| 6 | https://mathoverflow.net/users/11142 | 250579 | 113,775 |
https://mathoverflow.net/questions/250521 | 0 | We have two Bernoulli distributions with success probabilities $\mu\_1$ and $\mu\_2$. We sample $n$ times from distribution 1 and the sequence we get is $X\_1, \ldots, X\_n$. Let
$
\hat{kl}\_s = \sum\_{t=1}^s \ln\frac{\mu\_1 X\_t + (1-\mu\_1)(1-X\_t)}{\mu\_2 X\_t + (1-\mu\_2)(1 - X\_t)},
$
represent the empirical ... | https://mathoverflow.net/users/10071 | How to derive this change of measure identity in multi-armed bandits? | Shishir, this is quite elementary: write the probability $P\_2(\omega)$ of any individual bit sequence $\omega$ as $P\_1(\omega) f(\omega)$ where by definition, $f(\omega)=\exp(-\hat{kl}\_n)$. Finally, sum over $\omega \in A$.
| 3 | https://mathoverflow.net/users/7691 | 250581 | 113,777 |
https://mathoverflow.net/questions/250573 | 0 | The "local" Hardy-Littlewood maximal function is given by $$(M\_\phi f)(x)= \sup\_{0<\epsilon<1}|\phi\_\epsilon \ast f|(x),$$ which is similar to the classical Hardy-Littlewood maximal function : $$(Mf)(x)= \sup\_{\epsilon>0}|(\chi\_B)\_\epsilon \ast f|(x),$$
where $\phi\in S(\mathbb{R}^n)$ is some nonnegative, radial,... | https://mathoverflow.net/users/98145 | How much do we know about this "local" Hardy-Littlewood maximal function? | The failure of the maximal function to be a bounded map in $L^1$ is not just an issue of "failure at infinity".
Consider the case of $n = 1$. Let $f$ be a non-negative function that is compactly supported, and smooth away from the origin. Near the origin suppose that
$$ f(x) = \frac{1}{|x| (\ln |x|)^2} $$
This func... | 1 | https://mathoverflow.net/users/3948 | 250583 | 113,778 |
https://mathoverflow.net/questions/250588 | 2 | Algebraic $K\_0$ group for an algebra $A$ may be defined in terms of stable isomorphism classes of idempotents in $M\_n(A)$ or equivalently in terms of isomorphism clasess of finitely generated projective modules. If $f:A \to B$ is a morphism of algebras and $e=(e\_{ij})\_{i,j}$ is an idempotent in $M\_n(A)$ then we de... | https://mathoverflow.net/users/24078 | Morphisms in K-theory: comparison of two pictures | It seems that by $M\_n(A)$ you denote two different objects: actual $M\_n(A)$ -- which is the same as $\text{Hom}(A^n, A^n)$ -- and $A^n$ itself.
One can show that any finitely generated projective $A$-module $P$ can be written in the form $P = eA^n$ for some idempotent $e = (e\_{i,j}) \in \text{Hom}(A^n,A^n)$. And, ... | 4 | https://mathoverflow.net/users/95546 | 250591 | 113,781 |
https://mathoverflow.net/questions/250597 | 3 | Let me motivate my question with this example.
The volume integral of a ball $\int\_{B(0,R)} dx$ can be written as an integral over the surface of balls, i.e.
$$\int\_{B(0,R)} dx = \int\_0^R \int\_{\partial B(0,r)}dS dr.$$
This shows, that the derivative w.r.t. $R$ is just the surface-integral
$$\frac{d}{dR} \i... | https://mathoverflow.net/users/98868 | Differentiate a growing volume | You need the coarea formula:
$$
\int\_E g(x)|\nabla u(x)|dx=
\int\_{-\infty}^{+\infty}
\left(
\int\_{\{x\in E\ :\ u(x)=t\}}
g(x)\ dH^{n-1}\_ x
\right) dt
$$
where $E$ is an open subset of $R^n$, $u:E\to R$ is a Lipschitz function, $g\in L^1$ and $dH^{n-1}$ is the Hausdorff measure (surface measure). If you pick $g(x)= ... | 6 | https://mathoverflow.net/users/7294 | 250608 | 113,786 |
https://mathoverflow.net/questions/250434 | 1 | nLab has an article on [internal logic](https://ncatlab.org/nlab/show/internal+logic). Homotopy type theory also discusses its application in logic. Any one knows references to the correspondence study of the internal logic induced from higher category theories and higher topo theories ?
| https://mathoverflow.net/users/43138 | internal logic from higher categories | In addition to the [reference](http://arxiv.org/abs/1507.02648) stated in the comment. nLab does provide an entry for [internal logic from $(\infty,1)-$topos.](https://ncatlab.org/nlab/show/internal+logic+of+an+(infinity,1)-topos)
| 1 | https://mathoverflow.net/users/43138 | 250610 | 113,787 |
https://mathoverflow.net/questions/250599 | 4 | Let $G$ be a compact, connected Lie group and $S$ a torus in $G$ **not** assumed maximal. Then conjugation in $G$ induces a faithful representation of $N = N\_G(S)/Z\_G(S)$ in the Lie algebra $\mathfrak s$ of $S$.
In all the instances I know where the image of $N \to \mathrm{GL}(\mathfrak s)$ is a reflection group, ... | https://mathoverflow.net/users/5792 | If $N_G(S)/Z_G(S)$ is a reflection group, is it a Weyl group? | The answer is negative, even if $S$ is a circle, i.e., $1$-dimensional. To see that let $G$ be simple such that its Weyl group contains $-1$, i.e., $G$ is not of type $A\_n$, $D\_{2n+1}$, or $E\_6$. Then $N\_G(S)/Z\_G(S)=\{\pm1\}$ is a reflection group for *any* $1$-dimensional subgroup. $S\subseteq T$. There are infin... | 7 | https://mathoverflow.net/users/89948 | 250615 | 113,790 |
https://mathoverflow.net/questions/250609 | 9 | Ira Gessel "dubbed" the name *super Catalan* to
$$S(m,n)=\frac{(2m)!(2n)!}{m!n!(m+n)!}$$
and offers a combinatorial proof in [his paper](http://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf)
**Note.** The numbers $\frac12S(m,n)$ are also integers.
I would like to extend the discussion by asking for a... | https://mathoverflow.net/users/66131 | Extending the discussion on "super Catalan" numbers | I have a somewhat more algebraic proof than Jan-Christoph Schlage-Puchta's carry-based proof. I believe the idea goes back to Landau.
Let's start with your super super Catalan numbers type 1. A rational number $a$ is integer iff $v\_p(a)\ge 0$ for all primes $p$. Hence, to prove integrality of those numbers, it suffi... | 7 | https://mathoverflow.net/users/31469 | 250622 | 113,792 |
https://mathoverflow.net/questions/250548 | 4 | Let $G$ be a finite group (non $p$-group) with the following properties:
a: For every prime divisor $p$ of $\vert G\vert$, there exists only one minimal subgroup of order $p$.
b: For every pair of distinct prime divisors of $\vert G\vert$ say $p$ and $q$, If $P\_{1}, P\_{2},..., P\_{m}$ are all of the $p$-subgroups... | https://mathoverflow.net/users/97247 | Finite groups with unique subgroup of order $p$, for all prime divisors of $G$ | I believe that it is possible to prove by elementary methods that a finite group $G$ which has a unique subgroup of order $p$ for each prime divisor $p$ of its order is solvable.
Since the derived $[G,G]$ inherits the property, (though its order may have fewer prime divisors), we may suppose that $G= [G,G]$, in which ... | 3 | https://mathoverflow.net/users/14450 | 250625 | 113,794 |
https://mathoverflow.net/questions/249688 | 4 | Let $G$ be a finite group with the following property:
For every nontrivial proper subgroups $H$ and $K$ for which $H\cap K=1$, if the number of their minimal subgroups are $m$ and $n$ respectively, then the number of minimal subgroups of the subgroup generated by $H$ and $K$,$\langle H,K\rangle$, is $m+n$.
What a... | https://mathoverflow.net/users/97247 | Minimal Subgroups | We show that $G$ satisfies the given properties iff $G$ has a unique subgroup of order $p$ for every prime divider $p$ of $|G|$ and that every two Sylow subgroups of distinct orders permute.
Necessity: If $A$ and $B$ are distinct cyclic subgroups of prime order $p$, then $[A,B]\neq1$ otherwise $\langle A,B\rangle\con... | 2 | https://mathoverflow.net/users/40723 | 250631 | 113,796 |
https://mathoverflow.net/questions/250641 | 3 | $\omega \subseteq \mathbb{R}^+$ is called a Sidon sequence,
if all the sums $a + a' \ (a, a' \in \omega, a \leq a')$ are distinct,
and it is an asymototic basis of order $2$,
if any positive integer $n$ sufficiently large can be expressed as a sum of $2$ elements of $\omega$.
According to the article I am reading, ap... | https://mathoverflow.net/users/84272 | No Sidon sequence which is an asymptotic basis of order $2$ | Sidon set $A$ has at most $\sqrt{n}(1+o(1))$ elements not exceeding $n$ (\*). So, $A+A$ contains at most $|A|(|A|+1)/2=n(1/2+o(1))$ elements not exceeding $n$, unlike an asymptotic basis of order 2.
(\*) may be proved as follows: fix $M$, denote your elements $x\_1<x\_2<\dots<x\_m\leqslant n$ and consider all differ... | 8 | https://mathoverflow.net/users/4312 | 250644 | 113,800 |
https://mathoverflow.net/questions/250640 | 2 | Let $E$ be an elliptic curve over a field $k$ of characteristic $p = 2$ or $3$, and suppose $j(E) = 0\equiv 1728\mod p$.
In this case, $Aut(E)$ has size either 12 or 24.
Let $\omega$ be an invariant differential for $E$. How does $Aut(E)$ act on $\omega$? (Ie, it should multiply it by some nonzero element of $k$. W... | https://mathoverflow.net/users/88840 | Automorphisms of elliptic curves in characteristic 2 and 3 | There are exact formulas for the automorphisms, for example in *The Arithmetic of Elliptic Curves* Appendix A, proof of Proposition 1.2. From those you can write down explicitly the action of $\text{Aut}(E)$ on $\omega=dx/(2y+a\_1x+a\_3)$. Here's the characteristic 3 case. $E$ has a Weierstrass equation of the form
$$... | 8 | https://mathoverflow.net/users/11926 | 250649 | 113,803 |
https://mathoverflow.net/questions/250639 | 5 | Let $k$ be a fixed field. A skein relation defined on $k$ is a local relation on $k$-linear combination of oriented planar link diagrams saying that one can trade an overcrossing for an undercrossing by paying a price in the form of an error term which contains no crossings, and vice versa. More formally, it is a linea... | https://mathoverflow.net/users/8320 | Knot invariants from skein relations | Yes, this condition is sufficient.
The $b=0$ case is trivial. For the $b\ne 0$ case, embed $k$ in a larger field $K$ where $-a$ has a square root; then your knot invariant can be derived from the HOMFLY-PT knot invariant $P\_H(\ell,m)$ (which lives in $K[\ell,\ell^{-1},m]$), which satisfies the skein relation
$$
\e... | 6 | https://mathoverflow.net/users/3075 | 250652 | 113,804 |
https://mathoverflow.net/questions/250013 | 6 | Consider an interval exchange transformation that is, a bijective piecewise continuous map $[0,1] \rightarrow [0,1] $ whose restriction to its continuity intervals are translations. Assume that it is minimal and non uniquely ergodic.
Do the ergodic probability measures necessarily give the same weight to each of its... | https://mathoverflow.net/users/25511 | Non uniquely ergodic interval exchange transformations | The space of invariant probability measures of a minimal interval exchange transformation $T$ on $d$ intervals $I\_1, \dots, I\_d$ is parametrized by a subsimplex $\mathcal{M}$ of the standard simplex $\Delta=\{(x\_1,\dots,x\_d)\in\mathbb{R}^d\_+: \sum x\_i=1\}$ in the sense that:
1) a point $(x\_1,\dots, x\_d)\in\m... | 8 | https://mathoverflow.net/users/1568 | 250664 | 113,807 |
https://mathoverflow.net/questions/250665 | 2 | Let $k$ be an algebraically closed field of characteristic zero. We consider $f:\mathfrak{X}\to \mathbb{A}^1$ an smooth projective morphism of pure relative dimension $n$, where $\mathfrak{X}$ is an smooth algebraic variety.
Suppose that $\mathcal{F}$ is a torsion-free coherent sheaf on $\mathfrak{X}$ which is flat o... | https://mathoverflow.net/users/31724 | Flat family of torsion-free coherent sheaves over $\mathbb{A}^1$ | Let $i:Spec(k) \to A^1$ be the embedding if the point zero, $j:X\_0 \to X$ the embedding of its fiber, and $p:X\_0 \to Spec(k)$ the restriction of $f$. Then we have a Cartesian diagram. A flat base change implies that
$$
Li^\*\circ Rf\_\* \cong Rp\_\* \circ Lj^\*,
$$
an equality of derived functors. Since $i$ and $j$ a... | 4 | https://mathoverflow.net/users/4428 | 250667 | 113,808 |
https://mathoverflow.net/questions/250679 | 11 | Let $q = e^{2\pi i\,z}$.
>
> **I. 24th power**
>
>
>
The Ramanujan tau function $\tau(n)$ is given by the expansion of the Dedekind eta function $\eta(z)$'s $\text{24th}$ power. Then
$$\begin{aligned}\eta(z)^{24} &= \sum\_{n=1}^\infty\tau(n)q^n\\&=q - 24q^2 + 252q^3 - 1472q^4 + 4830q^5 - 6048q^6 - 16744q^7 +... | https://mathoverflow.net/users/12905 | Ramanujan's tau function, $691$ congruence, and $\eta(z)^{12}$ | Yes, this is true, as a consequence of an identity in a space of
modular forms of weight $6$.
The form $\eta(2z)^{12}$ is in this space; and $\sigma\_5(n)$ for $n$ odd
are the coefficients of the weight-$6$ form
$$
\frac1{1008} \Bigl(E\_6(z+\frac12) - E\_6(z)\Bigr) =
q + 244 q^3 + 3126 q^5 + 16808 q^7 + 59293 q^9 + \... | 23 | https://mathoverflow.net/users/14830 | 250690 | 113,816 |
https://mathoverflow.net/questions/250590 | 2 | I'm basically wondering how to make "curved" the first column of the diagram
$\require{AMScd}$
\begin{CD}
P\_1 @>\textrm{inclusion} >> G\\
@V \omega\_0 V P\_1\cap P\_2 V @V\omega V P\_2 V\\
P\_1/(P\_1\cap P\_2) @= G/P\_2
\end{CD}
where $M:=G/P\_2$ is a compact homogeneous space, such that the smaller group $P\_1\sub... | https://mathoverflow.net/users/22606 | What is the space parametrising the curved sub-Cartan geometries of a flat Cartan geometry? | I don't think that this is a real answer to your question, but there is a general concept of extension functors for Cartan geometries. Such a functor extending Cartan geometries of type $(G,P)$ to Cartan geometries of type $(K,L)$ is determined by a homomorphism $i:P\to L$ and a linear map $\alpha:\mathfrak g\to\mathfr... | 3 | https://mathoverflow.net/users/64141 | 250692 | 113,817 |
https://mathoverflow.net/questions/250593 | 1 | Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points.
Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $km$ at $s$ general points.
The expected number of conditions imposed by ... | https://mathoverflow.net/users/nan | Number of conditions imposed by fat points to a linear system | In general your polynomial $P(k)$ has degree $n$ as soon as there is a positive dimensional subscheme in the base locus of the linear system. Consider for instance the case $n=3$, $s = 2$, $m=2$, $d = 3$. Then the line through the two base points is in contained in the base locus of the linear system with multiplicity ... | 0 | https://mathoverflow.net/users/14514 | 250697 | 113,818 |
https://mathoverflow.net/questions/221258 | 4 | By Cauchy identity, $${}\_1\phi\_0(a;—;q,z)=\sum\_{n\geq0}\frac{(a;q)\_n}{(q;q)\_n}z^n=\frac{(az;q)\_{\infty}}{(z;q)\_\infty},\quad|z|<1,|q|<1,$$
we can obtain the $q-$analogue of $(1-z)^{-a}(1-z)^{-b}=(1-z)^{-a-b},$ $${}\_1\phi\_0(a;—;q,z){}\_1\phi\_0(b;—;q,az)={}\_1\phi\_0(ab;—;q,z),$$
which is $$\sum\_{n\geq0}\frac{... | https://mathoverflow.net/users/81507 | How to prove that $\sum_{i=0}^n\frac{(a;q)_i}{(q;q)_i}\frac{(b;q)_{n-i}}{(q;q)_{n-i}}a^{n-i}=\frac{(ab;q)_n}{(q;q)_n}$? | Here's a proof that doesn't involve Cauchy's identity. I will begin by stating two elementary lemmas.
**Lemma 1:** Let $f \in \mathbb{C}[x,y,z]$ be a polynomial. If $f$ vanishes on the set
$$\{ (a,a^i,a^j) : a \in \mathbb{C} , i,j \in \mathbb{N}\_{>0} \},$$
then $f$ is identically 0.
**Lemma 2:** Let $\binom{n}{k}\... | 3 | https://mathoverflow.net/users/31469 | 250700 | 113,819 |
https://mathoverflow.net/questions/250698 | 3 | I'm looking for a reference for the following statement:
Let $G$ be a reductive algebraic group acting on a projective variety $X$. Let $x, y\in X$ be such that $y$ lies in the closure of $Gx$.
Then there exists a $1$-dimensional algebraic subtorus $T \subset G$ and a point $x'\in Gx$ so that $y$ lies in the closure ... | https://mathoverflow.net/users/98919 | Each point of an orbit closure is a limit of one-parameter subgroup | The statement is not correct. Observe that $T$ would be contained in the stabilzer of $y$ but is may very well happen that $G\_y$ is unipotent. What you have in mind might be the Hilbert-Mumford criterion, though. In that case $X$ should be affine and the closure of $Tx$ intersects only the closure of the orbit $Gy$ bu... | 2 | https://mathoverflow.net/users/89948 | 250704 | 113,821 |
https://mathoverflow.net/questions/250703 | 5 | In the this [paper](http://arxiv.org/pdf/math/9811185.pdf) by Friedlander and Iwaniec, it is said that they are "able to avoid much of the high-powered technology frequently used in modern analytic number theory such as the bounds of Weil and Deligne." What bounds are being referred to here?
| https://mathoverflow.net/users/40983 | Reference to "bounds of Weil and Deligne" | The first of these bounds was explicitly stated by Weil in the short paper
* André Weil, "On some exponential sums" (1948)
It depends on the Riemann hypothesis for curves over finite fields, which he had proved back in the early 40s, and shows that
$$|S(m,n;p)|\leq 2 \sqrt{p}$$
where $S(m,n;p)$ is the classic K... | 4 | https://mathoverflow.net/users/43108 | 250708 | 113,822 |
https://mathoverflow.net/questions/250682 | 5 | Let $A$ be a finite dimensional algebra and $M$ a right $A$-module. Then $M$ is a left $B=End\_A(M)$-module.
$M$ is said to have the double centralizer property in case the canonical map given by right multiplication $F: A \rightarrow End\_{B^{op}}(M)^{op}$ is an isomorphism (of algebras). That $F$ is injective means t... | https://mathoverflow.net/users/61949 | Double centralizer properties | An example for the second question: Let $A$ be the $4$-dimensional algebra $$A=\begin{pmatrix} \Bbbk & 0\\ \Bbbk & \Bbbk \end{pmatrix} \times \Bbbk$$ and $$f=(\begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix},0)$$ Then $B={\rm{End}}\_A(fA) \cong \Bbbk$ and $${\rm{End}}\_{B^{\rm{op}}}(fA)^{\rm{op}}={\rm{End}}\_{\Bbbk}(\Bbbk^2... | 7 | https://mathoverflow.net/users/18756 | 250710 | 113,824 |
https://mathoverflow.net/questions/250707 | 2 | Consider the linear matrix differential equation
$\def\diag{\mathrm{diag}}$
\begin{align}
U(0) &= I\\
\frac{\mathrm{d}U}{\mathrm{d}t}(t) &= U(t) \phantom{.} Q(t) & & \quad(1)
\end{align}
where $Q(t),U(t)$ are $n\times n$ real valued matrices and $Q(t)$ is a transition rate matrix, which means that the off diagonal ... | https://mathoverflow.net/users/nan | Solving a matrix ODE | Let's suppose all the $V^\delta(t)$ have a common initial condition
$V^\delta(0) = U\_0$. If $W^\delta(t) = \dfrac{\partial}{\partial \delta} V^\delta(t)$, then we have
$$ \dfrac{dW^\delta}{dt} = \dfrac{\partial}{\partial \delta} \dfrac{dV^\delta}{dt} = \dfrac{\partial}{\partial \delta} (1+\delta) V^\delta(t) Q(t)
= V... | 4 | https://mathoverflow.net/users/13650 | 250711 | 113,825 |
https://mathoverflow.net/questions/250712 | -1 | Suppose that we have Galois covering $$X\longrightarrow P^{1}\_{\mathbb{Q}}$$ defined over the rationals which is cyclic, in the sense that the Galois group associated to the covering is a cyclic group.
Then $X$ can not be $P^{1}\_\mathbb{Q}$. I read this result somewhere. Is this result true? I tried this over the co... | https://mathoverflow.net/users/92070 | Question related to Galois covering of Projective line over rational numbers | *Your (!)* [MO-question](https://mathoverflow.net/questions/244541/action-of-mathbbz-3-mathbbz-on-p1) (together with the answer/comments) shows that the group of order $3$ is also a Galois group of a Galois cover $\mathbb P^1\_{\mathbb Q}\to\mathbb P^1\_{\mathbb Q}$. Besides this and order $2$, the only other possible ... | 1 | https://mathoverflow.net/users/18739 | 250718 | 113,826 |
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