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https://mathoverflow.net/questions/231939 | 2 | Let $f:X\rightarrow Y$ be a morphism with connected fibers between projective varieties (not necessarily flat). Let $D\subset Y$ be an irreducible divisor.
Let us look at the cycle $f^{-1}(D)\subset X$.
What can we say about the dimension of the irreducible components of $f^{-1}(D)$?
| https://mathoverflow.net/users/nan | Inverse image of a divisor | I don't know what you mean by "the cycle $f^{-1}(D)\subset Y$" (and not just because $f^{-1}(D)\subset X$), but for your question you don't need it to be a "cycle", so let's just assume you want the set.
The answer depends on $f:X\to Y$. For instance, if $X=X\_1\cup X\_2$ where $X\_i$ are irreducible and closed in $... | 3 | https://mathoverflow.net/users/10076 | 231971 | 107,826 |
https://mathoverflow.net/questions/231964 | 12 | For $n\ge m\ge 2$, define $$I(n,m):= \int \_0^\infty\dfrac{\text{arcsinh}^nx}{x^m}dx$$ Computer algebra systems say that the indefinite integral can be expressed in terms of polylog functions (of rapidly increasing complexity), but I see no way of doing the limit $\to\infty$. On the other hand, I have found numerically... | https://mathoverflow.net/users/29783 | How to prove that $\int _0^\infty\frac{\text{arcsinh}^nx}{x^m}dx$ is a rational combination of zeta values? | Let me first establish the case $m=2$. The general case is established in the "Added" section below.
By making the substitution $x=\sinh t$ we get
\begin{align\*}
I(n,2)&=\int\_0^\infty t^n\frac{\cosh t}{\sinh^2 t}\,dt
=2\int\_0^\infty t^n\frac{e^t+e^{-t}}{(e^t-e^{-t})^2}dt\\
&=2\int\_0^\infty t^n\left(\sum\_{r=0}^\i... | 16 | https://mathoverflow.net/users/11919 | 231974 | 107,827 |
https://mathoverflow.net/questions/231975 | 3 | Let $(X, \mathcal O\_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O\_X$-modules:
$$
\cdots \overset{f\_5}\longrightarrow I\_5\overset{f\_4}\longrightarrow I\_4 \overset{f\_3}\longrightarrow I\_3 \overset{f\_2}\longrightarrow I\_2\overset{f\_1}\longrightarrow I\_1 \over... | https://mathoverflow.net/users/88073 | Homology in the sections of an infinite exact sequence of injective sheaves of $\mathcal O_X$-modules? | Partial answer: if $X$ has underlying topological space of finite dimension $k$, then what you want is true, because $H^i(X, F)=0$ for any $O\_X$-module $F$ and $i>k$. Let $F=\ker(f\_{k+n})$, so that we have an injective resolution
$$ 0\to F\to I\_{n+k+1} \to I\_{n+k} \to \ldots \to I\_{n+1}\to I\_n \to I\_{n-1} \to\ld... | 3 | https://mathoverflow.net/users/3847 | 231976 | 107,828 |
https://mathoverflow.net/questions/231553 | 0 | Let $A$ be a noetherian integral domain (may be regular). Let $\pi:X \to \mathrm{Spec}(A)$ be a flat morphism. Suppose that each fiber of $\pi$ are quasi-projective. Let $\mathcal{F}$ be a coherent sheaf on $X$, flat over $\mathrm{Spec}(A)$. What can we say about the Euler characteristic of $\mathcal{F} \otimes \mathca... | https://mathoverflow.net/users/45397 | Euler characteristic on flat families of quasi-projective schemes | I am not aware of such results in full generality, but I know that working without the properness assumtpion was in part the main motivation for Grothendieck to write SGA 2.
Let me focus on a related question (but not exactly the same). Let $\pi : X \rightarrow Y$ be a flat morphism of finite type with $Y$ a smooth ... | 6 | https://mathoverflow.net/users/37214 | 231977 | 107,829 |
https://mathoverflow.net/questions/231689 | 27 | For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The sequence of moments determines the distribution of the trace.
For the adjoint representations of $G\_2,F\_4$, and $E\_7$,... | https://mathoverflow.net/users/18060 | Why do the adjoint representations of three exceptional groups have the same first eight moments? | Yes, look for "Deligne's exceptional series". There are no theorems, but several beautiful conjectures.
The basic idea is that there should be a symmetric pivotal category generated by a trivalent vertex, with just a few local relations, depending on a parameter. At special values of the parameter, the category becom... | 14 | https://mathoverflow.net/users/3 | 231979 | 107,831 |
https://mathoverflow.net/questions/231244 | 12 | For naturals $n\ge m$, define
$$I(n,m):=\int\_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$
with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $.
Is it possible to find closed form expressions of $I(n,m)$? I mean closed form in a broad sense, i.e. involving any other "kn... | https://mathoverflow.net/users/29783 | Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$? | Letting $y=\text{arcsinh}\, x$ and integrating by parts, we have
$$I(n,m)=-\frac{2^{m-1}a^n}{m-1}+\frac n{m-1}\,[J(n-1,m-1;a)-J(n-1,m-1;0)]
$$
if $n\ge m\ge2$, where $a:=\text{arcsinh}\,\frac12$ and
$$J(p,q;y):=\int\frac{y^p}{\sinh^qy}\,dy.$$
Formula 1.4.24.1 in Prudnikov--Brychkov--Marichev (PBM, Vol. 1, ISBN 5-9221... | 3 | https://mathoverflow.net/users/36721 | 231981 | 107,832 |
https://mathoverflow.net/questions/230878 | 7 | Is there a topological group which is Hausdorff, first countable,
locally connected and has finite topological dimension, yet fails
to be locally compact?
| https://mathoverflow.net/users/85994 | A non locally compact group of finite topological dimension? | The Gleason-Montgomery theorem on the local compactness of locally path-connected finite-dimensional topological groups was generalized by Banakh and Zdomskyy (<http://topology.auburn.edu/tp/reprints/v36/tp36027.pdf>) who proved that a topological group $G$ is locally compact if it is compactly finite-dimensional and l... | 4 | https://mathoverflow.net/users/61536 | 232011 | 107,839 |
https://mathoverflow.net/questions/231958 | 4 | Let $A,B,C$ be positive integers, and let $Z$ be a positive parameter. Let $M$ be a positive integer, and consider the set of points
$$\displaystyle \{(x,y,z) \in \mathbb{Z}^3 : Ax^2 + By^2 + Cz^2 \leq MZ, Ax^2 + By^2 + Cz^2 \equiv 0 \pmod{M}\}.$$
Let $N(M;Z)$ denote the cardinality of the set above. How does $N(M... | https://mathoverflow.net/users/10898 | Counting lattice points inside an ellipsoid subject to congruence conditions | *Bill Duke*, [**On ternary quadratic forms**](http://dx.doi.org/10.1016/j.jnt.2004.06.013), *J. Number Theory* **110** (2005), no. 1, 37--43, [(preprint)](http://www.math.ucla.edu/~wdduke/preprints/ternary.pdf) gives a uniform estimate for the number of representations of a number by a ternary positive definite quadrat... | 3 | https://mathoverflow.net/users/11142 | 232016 | 107,841 |
https://mathoverflow.net/questions/231990 | 6 | The hyperoctahedral group $H\_n$ can be seen as the centralizer of the permutation $(12)(34)\cdots (2n-1\,2n)$ in $S\_{2n}$. It has $2^nn!$ elements.
The quantities $$ \omega\_\lambda(\pi)=\frac{1}{2^nn!}\sum\_{h\in H\_n}\chi\_{2\lambda}(h\pi)$$ are called the zonal spherical functions of the Gelfand pair $(S\_{2n},H... | https://mathoverflow.net/users/78061 | A sum over characters of $S_{2n}$ and zonal spherical functions of $(S_{2n},H_n)$ | The essential thing here is that the characters $\chi\_{2\lambda}$ are exactly the irreducible constituents of the induced character $(1\_{H\_n})^{S\_n}$. The result generalizes to an arbitrary subgroup $H\leq G$ of a finite group $G$ as follows: For $x$, $y\in G$, we have $\DeclareMathOperator{\Irr}{Irr}$
$$ \frac{ |H... | 8 | https://mathoverflow.net/users/10266 | 232031 | 107,848 |
https://mathoverflow.net/questions/168582 | 11 | So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it might be worth posting here, since if it's true, I think it'd be interesting.
Suppose I've got an inclusion of (at lea... | https://mathoverflow.net/users/11546 | Thom Spectra and Hopf-Galois Extensions of Ring Spectra | So this can definitely be done. It took me a while to figure out all the details, but in the end it's not so conceptually complex.
The basic idea is that if you've got a fibration $F\overset{i}\to E\overset{p}\to B$ of connected $\mathbb{E}\_n$-spaces and a map $f:E\to BGL\_1(\mathbb{S})$ then you want to take the l... | 1 | https://mathoverflow.net/users/11546 | 232037 | 107,851 |
https://mathoverflow.net/questions/227960 | 6 | If $M$ is a Riemannian manifold that is not compact, is it true that the Sobolev spaces on $M$, $W^{k,p}(M)$, still be separable (for $p < \infty$)?
| https://mathoverflow.net/users/85045 | Are Sobolev spaces on non-compact manifolds separable? | Yes they are.
**Step 1** There exists measurable sections $e\_1, e\_2, \dotsc, e\_m$, where $m = \dim M$, of $TM$ (measurable functions mapping a point $x$ to a vector of its tangent plane $T\_xM$) such that for each $x \in M$, $e\_1 (x), e\_2 (x), \dotsc, e\_m (x)$ forms an *orthonormal basis* of the tangent space $... | 6 | https://mathoverflow.net/users/42047 | 232038 | 107,852 |
https://mathoverflow.net/questions/231996 | 5 | What this boils down to is: if $\mathcal{K}$ is cofibrantly-generated model category which permits the small object argument and $\mathcal{D}$ is a small category, then when does $\mathcal{K}^\mathcal{D}$ permit the small object argument? This might be trivial depending on what exactly it means to "permit the small obj... | https://mathoverflow.net/users/2362 | When does the projective model structure on functors exist? | I found it in Hirschhorn's book. It's in 11.6 Diagrams in a cofibrantly generated model category. The answer is always, provided K
is cofibrantly generated (Theorem 11.6.1). For him (as well as for Hovey), the small object argument is (2), actually replacing cof with cell (Definitions 10.5.15 and 10.5.12).
| 5 | https://mathoverflow.net/users/12166 | 232042 | 107,854 |
https://mathoverflow.net/questions/232047 | 3 | How big is the weak-\* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures?
I mean, we are in the symbolic space $\{1,2,\ldots,d\}^{\mathbb{N}}$ and I would like to know, how big the (weak-\*) closure of the set $C$ is, where
$$C = \left\{\sum\_{i=1}^... | https://mathoverflow.net/users/66009 | Convex combinations of Bernoulli Measures | The set of Bernoulli measures $B$ is closed, and the closure of $C$ is precisely the set of measures of the form $\int m\,d\mathbb{P}$ where $\mathbb{P}$ is a Borel probability measure on the set of Bernoulli measures.
Let me unpack that statement a little. Let $\mathcal{M}\_\sigma$ denote the set of shift-invariant ... | 3 | https://mathoverflow.net/users/1840 | 232050 | 107,856 |
https://mathoverflow.net/questions/229788 | 11 | **Question:** Given an arbitrary number of real matrices of the form $ A\_i=
\biggl(\begin{matrix}
C\_i+E\_i & B\_i \\
B\_i^T & D\_i-F\_i
\end{matrix} \biggr)
$, where $B\_i$ is an arbitrary $n\times n$ real matrix, $C\_i$ and $D\_i$ are $n\times n$ real anti-symmetric matrices, $E\_i$ and $F\_i$ are $n\times n$ real s... | https://mathoverflow.net/users/71225 | How to prove this determinant is positive-II? | Let $q(x,y) = x^H J y$ for $x,y \in \mathbb{C}^{2n}$ where $J = diag(I\_n,-I\_n)$ and let $S = \{A \in M\_{2n}(\mathbb{R}) : q(Ax, Ax) \ge q(x,x) $ $\forall x \in \mathbb{C}^{2n}\}$.
Obviously $S$ is a semi group .
Furthermore the $e^{t A\_i}$ are in $S$ since $$\frac{d}{dt} q(e^{t A\_i} x,e^{t A\_i} x) = 2 (e^{t A\_i}... | 5 | https://mathoverflow.net/users/17261 | 232055 | 107,858 |
https://mathoverflow.net/questions/232049 | 1 | Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\phi:M\rightarrow M$$ be an isometry with $\phi^2=Id\_M$. Consider the fixed point set $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ and
suppose $F\subset M$ is a closed $1$-dimensional submanifold. Let $u:M\rightarrow \mathbb{R}$ be an eigenfunction ... | https://mathoverflow.net/users/88111 | Zero set of eigenfunction along a sub manifold | This appears to be a result of Dong. See Theorem 3.4.
*Rui-Tao Dong*, [**Nodal sets of eigenfunctions on Riemann surfaces**](http://projecteuclid.org/euclid.jdg/1214448750), *J. Differential Geom.* **36** (1992), no. 2, 493--506.
| 1 | https://mathoverflow.net/users/2627 | 232068 | 107,866 |
https://mathoverflow.net/questions/231951 | 2 | Given absolutely continuous random variables $(X, Y)$ with joint distribution $P\_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N\_\mathsf{G}$ where $N\_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$. The conditional variance of $Y$ given $Z$ is defined as
$$\mathsf{var}(Y|Z)=\mathbb{E}[(Y-\mathbb{E}[Y|Z])^2|Z].$$
Le... | https://mathoverflow.net/users/41666 | An Inequality Regarding the Squared Conditional Variance | The inequality does not hold in general.
Indeed, without loss of generality $\gamma=1$; otherwise, replace $\sqrt{\gamma} Y$ by $Y$, so that $Z=Y+N$, where $N:=N\_\mathsf{G}$.
Let us write $E$ and $V$ for $\mathbb{E}$ and $\mathsf{var}$, respectively.
First here, the heuristics. Suppose that, in an appropriate sen... | 1 | https://mathoverflow.net/users/36721 | 232073 | 107,868 |
https://mathoverflow.net/questions/231863 | 5 | Let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := \epsilon \phi(u/\epsilon).$$Do we necessarily have that$$\int\_U Du^\epsilon \cdot Du\,dx = \int\_U \phi'(u/\epsilon)|Du|^2\,dx \to 0?$$
| https://mathoverflow.net/users/nan | If $u \in H^1(U)$, then $Du = 0$ almost everywhere on the set $\{u = 0\}$, auxiliary result | We can assume $U$ is bounded.
On one hand, $\|u^{\epsilon}\|\_{L^2}\rightarrow0$ as $\epsilon\rightarrow0$. And $\|Du^{\epsilon}\|\_{L^2}$ is uniformly bounded. Thus we infer $u^\epsilon$ converges weakly to $0$ in $H^1(U)$, which implies that
$$\int\_UDu^\epsilon\cdot Dudx=(u^\epsilon,u)\_{H^1}-\int\_Uu^\epsilon udx... | 0 | https://mathoverflow.net/users/88134 | 232082 | 107,870 |
https://mathoverflow.net/questions/56301 | 11 | [Infinitary logic](https://en.wikipedia.org/wiki/Infinitary_logic) considers languages being infinite by infinite conjunctions and disjunctions.
I wonder why it not considers languages being infinite by relations and functions of infinite arity.
Relations of finite arity $n$ over a base set $A$ can be seen as unary... | https://mathoverflow.net/users/2672 | Predicates of infinite arity | I have several thoughts about this question.
First, to my way of thinking, there is little difference between
an infinite-ary relation $R(a\_0,a\_1,\dots)$ on a set $X$ and a
unary relation on a suitable power of that set, such as $X^\omega$
or $X^\alpha$. For example, an $\omega$-ary relation on $\{0,1\}$
is essenti... | 8 | https://mathoverflow.net/users/1946 | 232089 | 107,872 |
https://mathoverflow.net/questions/232105 | 1 | $c\_0(C[0,1])$ is the $c\_0$-direct sum of countably many $C[0,1]$.How to prove
$C[0,1]$ is Banach-space isomorphic to $c\_0(C[0,1])$.
Here,Banach-space isomorphism means a bounded invertible operator from $C[0,1]$ onto $c\_0(C[0,1])$.
| https://mathoverflow.net/users/38854 | $C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$ | There is a useful simple
**Lemma.** If $X\sim X\oplus X$, $Y\sim Y\oplus Y$, and each of $X,Y$ is isomorphic to a complemented subspace of another, then $X\sim Y$.
**Proof.** We have $X\sim Y\oplus A$, then $X\sim (Y\oplus Y)\oplus A=Y\oplus(Y\oplus A)=Y\oplus X$, analogously $Y\sim X\oplus Y$.
Now let $X=C([0,1]... | 3 | https://mathoverflow.net/users/4312 | 232108 | 107,879 |
https://mathoverflow.net/questions/232087 | 67 | In August 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known from the beginning that it would take experts months to understand his work enough to be able to verify the proof. Are ... | https://mathoverflow.net/users/36586 | Have there been any updates on Mochizuki's proposed proof of the abc conjecture? | In January, Vesselin Dimitrov [posted to the arXiv](http://arxiv.org/abs/1601.03572) a preprint showing that Mochizuki's work, if correct, would be effective. While this doesn't validate Mochizuki's work it does do a few things:
1. It shows that people are understanding more of the proof.
2. It gives another avenue t... | 40 | https://mathoverflow.net/users/3199 | 232118 | 107,882 |
https://mathoverflow.net/questions/232111 | 7 | I have asked this [question](https://math.stackexchange.com/questions/1668628/every-self-adjoint-trace-class-operator-on-l2-has-integral-kernel) on MSE but did not receive an answer. I thought I could try it here.
Let $T$ be a self-adjoint trace-class operator on $L^2(\mathbb{R})$. Is is true that it can be represent... | https://mathoverflow.net/users/47482 | Every self-adjoint trace class operator on $L^2$ has integral kernel | Yes, this is correct. Actually, "self-adjoint trace-class" is more than you need; any Hilbert-Schmidt operator can be represented as an integral operator. The Hilbert-Schmidt operators from $L^2(X)$ to $L^2(Y)$ are precisely the integral operators with kernel in $L^2(X\times Y)$ (at least for $\sigma$-finite $X$ and $Y... | 5 | https://mathoverflow.net/users/23141 | 232119 | 107,883 |
https://mathoverflow.net/questions/229743 | 2 | A set of positive integers $d\_1, \dots, d\_n$ describe a n-dimensional tetrahedron $T$ with the vertices
$$ (0,\dots,0), (1/d\_1,0,\dots,0), (0, 1/d\_2,\dots,0), \dots, (0,\dots,1/d\_n).$$
Let $L\_T(t)$ be the Ehrhart (quasi-)polynomial of $T$, i.e. the number of integer lattice points in $tT$.
I understand that typ... | https://mathoverflow.net/users/51478 | Approximating Ehrhart Polynomial of Rational n-Tetrahedron | Actually, $L\_T(t)$ is *not* a polynomial (it's an honest quasipolynomial except for very special cases of $d\_1, \dots, d\_n$) and already the second "coefficient" is nontrivial, as is the last "coefficient". (The leading coefficient is a constant--the volume of T.) I'm not sure what exactly you're after in terms of a... | 2 | https://mathoverflow.net/users/3193 | 232130 | 107,886 |
https://mathoverflow.net/questions/232124 | 3 | Let us assume $X$ is a smooth, projective and unirational variety of dimension $n$ over $\mathbb{C}$.
Given a conic bundle $\pi: Y\rightarrow X$ such that $\omega\_{\pi}^{-1}$ is relatively very ample with respect to $\pi$, here $\omega\_{\pi}=\omega\_Y\otimes \pi^{\*}\omega\_X^{-1}$. Assume all three possible types ... | https://mathoverflow.net/users/70593 | What is known about the cohomology of the relative tangent bundle on a conic bundle? | Such conic bundle is given by a rank three vector bundle, say $E$ on $X$, and a line subbundle $L \subset Sym^2E$ (just take $E$ to be the pushforward of $\omega^{-1}\_\pi$, and $L$ corresponds to the equation of the conic bundle). Then the pushforward of $T\_\pi$ to $X$ is isomorphic to $L \otimes E \otimes \det E^\ve... | 4 | https://mathoverflow.net/users/4428 | 232144 | 107,892 |
https://mathoverflow.net/questions/232141 | 6 | In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid.
Could you give me a reference for this fact or help me to sketch a proof?
Thank you in advance.
| https://mathoverflow.net/users/69558 | Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything) | I think that the original reference is "Zariski's Main Theorem für affinoide Kurven" by K.-H. Fieseler (Mat. Ann. 251, 1980). He proves that a finite union of affinoid domains of a one-dimensional affinoid space is affinoid, but this is probably not enough to answer your question.
More generally, though, J. Fresnel ... | 6 | https://mathoverflow.net/users/4069 | 232146 | 107,893 |
https://mathoverflow.net/questions/231074 | 2 | In a commutative field $K$, the Zariski dimension of an algebraic subset of $K^n$ over $K$ does not vary if one enlarges $K$ if I understood well. In particular, for two Zariski-closed vector spaces $V(K)\subset W(K)\subset K^n$, if one considers an extension field $L/K$, one has $$[V(K):W(K)]=[V(L):W(L)].$$
Let $D$ ... | https://mathoverflow.net/users/18583 | For a division ring $D$, does $[D:C_D(a)]_{right}$ vary when $D$ is enlarged? | There is a result of Brauer (1932) stating that :
> **Theorem (Brauer).** Let $K$ be a division ring with centre $k$ and $A$ a $k$-algebra. Then the centralizer $A'$ of $A$ in $K$ is again a $k$-algebra and the bicentralizer $A''$ contains $A$. Moreover $$[K:A']\_{left}=[A:k],$$ whenever either side is finie, and wh... | 2 | https://mathoverflow.net/users/18583 | 232153 | 107,897 |
https://mathoverflow.net/questions/232142 | 3 | Suppose $$f(x)=\sum\_{k=0}^\infty a\_k\frac{(i x)^k}{k!}$$ where $$a\_k=k!\int\_0^1 p\_k(y\_{k-1})\int\_0^{y\_{k-1}}p\_{k-1}(y\_{k-2})\cdots \int\_0^{y\_1}p\_1(y\_0) dy\_0\cdots dy\_{k-2}\;dy\_{k-1}$$ for functions $p\_i>0$ and $p\_i=p\_j$ if $i=j \mod 2$. Is $f$ is bounded on $\mathbb{R}$?
This is a generalization o... | https://mathoverflow.net/users/18261 | bounded analytic function as a power series | If $f(x)$ is bounded, the Laplace transform
$${\mathscr L}f(s) = \int\_0^\infty f(x) e^{-sx}\; dx$$
is analytic in the open right half plane, and the same goes for the Laplace transform of $\widetilde{f}(x) = f(-x)$.
On the other hand, $|a\_k| \le C^k$ implies that $\sum\_{k=0}^\infty a\_k i^k s^{-k-1}$ converges ab... | 4 | https://mathoverflow.net/users/13650 | 232158 | 107,899 |
https://mathoverflow.net/questions/231543 | 11 | Let $V\_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi\_{2q + 1} V\_{n-q}(\mathbb{C}^n) \cong \mathbb{Z}.$$Given a complex $n$-bundle $\omega$ over a CW-complex $B$ with typical fiber... | https://mathoverflow.net/users/nan | Primary obstruction to the existence of a cross-section of $V_{n - q}(\omega)$ is a cohomology class in $H^{2q+2}(B, \pi_{2q+1} V_{n - q}(F))$? | The fiber $V\_{n - q}(F)$ is $2q$-connected, so it is not hard to construct a equivalence over the $(2q + 1)$-skeleton. We clearly can take sections over each vertex in the $0$-skeleton in the same connected component, then we can connect them on the 1-skeleton via paths because it is connected, then we can fill in wit... | 8 | https://mathoverflow.net/users/nan | 232166 | 107,904 |
https://mathoverflow.net/questions/232081 | 5 | The following problem is listed here: <http://www-personal.umich.edu/~erman/Papers/Questions2.pdf> and attributed to Vistoli:
Let $\mathcal A\_g$ denote the moduli stack of principally polarized abelian varieties over a field $k$, let $A\_g$ denote the associated coarse moduli space, and $K(A\_g)$ its function field.... | https://mathoverflow.net/users/35353 | Essential dimension and the moduli space of abelian varieties | The two notions are related using Theorems 4.1 and 6.1 of the paper of Brosnan, Reichstein and Vistoli:
Theorem 6.1 reduces the computation of the essential dimension of the stack to that of the generic gerbe $\mathcal{X}\_g$. Theorem 4.1 says that the essential dimension of $\mathcal{X}\_g$ is $\mathrm{cd}(\mathcal... | 4 | https://mathoverflow.net/users/519 | 232172 | 107,905 |
https://mathoverflow.net/questions/232187 | 1 | Let $\kappa$ be an infinite cardinal and suppose $$n, d: \kappa \to \big((\kappa+1)\setminus \{0\}\big) = \{1, \ldots, \kappa\}$$ are arbitrary functions.
Is there $E \subseteq \big\{\{x,y\}: x\neq y \in \kappa\big\}$ such that the graph $G=(\kappa,E)$ has the following property?
>
>
> >
> > For all $k\in \kapp... | https://mathoverflow.net/users/8628 | Infinite graph with degrees given | You might find the following paper useful: [Degree sequences of infinite graphs, by Andreas Blass and Frank Harary](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/135582/jlms0010.pdf).
>
> The degree sequences of finite graphs, finite connected graphs, finite trees and finite forests have all
> been chara... | 6 | https://mathoverflow.net/users/25485 | 232190 | 107,910 |
https://mathoverflow.net/questions/232152 | 6 | This is probably a well-known issue, but I could not find a clear discussion in the literature, and I think others could find it useful.
Consider a real-analytic function germ $f:(\mathbb R^2,0) \rightarrow \mathbb R$, it is represented by a convergent power series $\sum\_{i,j}a\_{ij}x^iy^j \in \mathbb R\{x,y\}$. Sup... | https://mathoverflow.net/users/48737 | Complexifying a real-analytic singularity | As David Speyer has already suggested, we have $Q\_f^\mathbb{C}\simeq Q\_f\otimes\mathbb{C}$, so that answers the first question (and the second question).
For the third question, consider $f = (x^2+y^2)^2$. The singularity at $(x,y)=0$ is clearly isolated, but it is not algebraically isolated, since $\mu\_\mathbb{R}... | 7 | https://mathoverflow.net/users/13972 | 232202 | 107,913 |
https://mathoverflow.net/questions/232197 | 3 | Let
* ${A\_j} \in {\mathbb{C}^{n \times n}},0<{w\_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.
* ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}\_m}{\lambda ^m} + .....{A\_1}\lambda + {A\_0}$ is a matrix polynomial.
* ${\rm{Q(}}\lambda {\rm{) = }}{{\... | https://mathoverflow.net/users/78479 | A question on determinant of a matrix polynomial | The statement is false if $P^\*$ is taken to mean the element by element complex conjugate of $P(\lambda)$. A counterexample: let $m=1$, $\omega\_0 = \omega\_1 = 1$, and
$$A\_0 = \left( \begin{array}{cc} 0&i\\2i&0 \end{array} \right) \\
A\_1 = \left( \begin{array}{cc} 1&i\\0&2 \end{array} \right)
$$
Then coefficients i... | 3 | https://mathoverflow.net/users/82067 | 232214 | 107,920 |
https://mathoverflow.net/questions/232215 | 6 | Let $M$ be a smooth closed simply-connected $4$-manifold with $w\_1 = w\_2 = 0$. Can $TM$ be trivialized in the complement of a point?
| https://mathoverflow.net/users/nan | Tangent bundle of smooth closed simply-connected $4$-manifold $w_1 = w_2 = 0$ can be trivialized in complement of point? | $M$ is smooth, so there is a cellular decomposition with a single 4-cell. Thus we want $TM$ trivializable over the 3-skeleton, so we look to analyze sections of the frame bundle which has Lie group fiber $SO(4)$ (here $M$ is orientable because $w\_1=0$). Obstruction theory says we can extend over the 2-skeleton, becaus... | 4 | https://mathoverflow.net/users/12310 | 232219 | 107,922 |
https://mathoverflow.net/questions/232220 | 2 | See my previous question [here](https://mathoverflow.net/questions/232215/tangent-bundle-of-smooth-closed-simply-connected-4-manifold-w-1-w-2-0-ca).
>
> Let $M$ be a smooth closed simply-connected $4$-manifold with $w\_1 = w\_2 = 0$. Can $TM$ be trivialized in the complement of a point?
>
>
>
This was answered... | https://mathoverflow.net/users/nan | Smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$ without a point admits symplectic structure? | By an application of Gromov's $h$-principle, an open manifold $W$ admits a symplectic structure precisely if $TW$ admits an almost complex structure. In the case you are considering it does, as $TW$ is trivial.
| 2 | https://mathoverflow.net/users/318 | 232223 | 107,924 |
https://mathoverflow.net/questions/232224 | 1 | A $(v,k,t)$ *covering design* is a collection of $k$-subsets of $V=\{1,\ldots,v\}$ chosen so that any $t$-subset of $V$ is contained in (or "covered by") at least one $k$-set in the collection. Existence is trivial, so instead one focuses on obtaining "small" coverings.
The best lower bound on the size of a covering ... | https://mathoverflow.net/users/30734 | Covering designs where $v$ is linear in $k$ | Partition all $v$ elements onto groups of size $[k/t]$ and for any $t$ groups choose a $k$-set containing them all. This is of constant (but large) size.
| 0 | https://mathoverflow.net/users/4312 | 232226 | 107,926 |
https://mathoverflow.net/questions/232236 | 4 | Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. I know the following.
* There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$.
* Any two $1$-forms $\alpha$, $\alpha'$ with this property satisfy $\alpha = f\alp... | https://mathoverflow.net/users/nan | Closed $3$-manifold, $2$-dimensional subbundle of this manifold, is this form exact or not? | In fact $\omega\wedge d\omega$ is closed its cohomology class is a well-known invariant of foliation named the Godbillon-Vey invariant. Thus $\omega\wedge d\omega-\omega'\wedge d\omega'$ is exact.
See for example this paper. <http://homepages.math.uic.edu/~hurder/papers/54manuscript.pdf>
See also p.3 of this paper
... | 9 | https://mathoverflow.net/users/80891 | 232239 | 107,928 |
https://mathoverflow.net/questions/232238 | 5 | Suppose you are given a 3-ball $B$ in $\mathbb{R}^3$ that is bounded by a PL sphere, a triangulation $T$ of $B$ by Euclidean tetrahedra. Is that triangulation necessarily shellable?
I know that if $T$ can be lifted to a convex hypersurface in $\mathbb{R}^4$, then it is shellable; this applies if $T$ comes from a Dela... | https://mathoverflow.net/users/5010 | Is every triangulation of a Euclidean ball by convex tetrahedra shellable? | I haven't had time to check through the details of the construction, but the example B\_3\_9\_18 found in the proof of Theorem 2 [here](http://www.eg-models.de/models/Simplicial_Manifolds/2003.05.004/_preview.html) by Frank Lutz appears to be embeddable in 3-space.
Frank specializes in creating wickedly tiny polytope... | 8 | https://mathoverflow.net/users/18263 | 232241 | 107,930 |
https://mathoverflow.net/questions/232157 | 7 | Let $K\subset \Bbb{C}$ be a compact subset of the complex plane, and let $C(K)$ be the space of all complex continuous functions on $K$.
We say that $f\in C(K)$ is a **generator** of $C(K)$ when the set $\{p(f) \mid p\ \style{font-family:inherit;}{\text{is a polynomial}}\}$ is dense in $C(K)$.
If $K$ is any set con... | https://mathoverflow.net/users/38854 | On what kind of condition of a compact set $K$ in the plane, $C(K)$ has a generator? | I believe Yemon's conjecture is correct: $C(K)$ has a generator in the sense of the question if and only if $K$ has empty interior and $\mathbb{C}\setminus K$ is connected. As he points out, the reverse direction follows from Lavrentiev's theorem. For the forward direction, if $K$ has nonempty interior then Mr. Li has ... | 3 | https://mathoverflow.net/users/23141 | 232242 | 107,931 |
https://mathoverflow.net/questions/232235 | 4 | Over an algebraically closed field $k$, which smooth hypersurfaces $X \subset \mathbb{P}^n$ are abelian varieties?
If $n=2$, then the smooth hypersurfaces of degree 3 (i.e. elliptic curves) are abelian varieties. Furthermore, the degree $d$ of the hypersurface $X$ must be $n+1$ (so that the canonical bundle is trivi... | https://mathoverflow.net/users/88213 | Which hypersurfaces in $\mathbb{P}^n$ are abelian varieties? | Really this is mostly just consolidating what has been (implicitly) said in the comments and cleaning it up a bit (e.g. using the Chow ring instead of singular cohomology), but might as well make it an answer...
Let $A$ be an abelian variety of dimension $d$. We prove that $A$ cannot embed in $\mathbb{P}^{2d-1}$ and ... | 22 | https://mathoverflow.net/users/51424 | 232245 | 107,932 |
https://mathoverflow.net/questions/232253 | 2 | Suppose we have a Lie algebra with structure constants
$$\mathrm{d}e^i=\sum\_{j<k}a\_{ijk}e^j\wedge e^k$$
for some coefficients $a\_{ijk}$.
In this setting, **how may be checked (perhaps computationally?) that our algebra is nilpotent?** I wonder whether there is a somewhat nice algorithm involving the coefficien... | https://mathoverflow.net/users/62367 | Nilpotency of Lie Algebra from Structure Constants | By Engel's theorem, the Lie algebra is nilpotent just when, for any fixed $k$, the matrix $A\_k=(a\_{ijk})$ is nilpotent.
Edit: By Engel's theorem, the Lie algebra is nilpotent just when every linear combination of the matrices $A\_k=(a\_{ijk})$ is nilpotent.
| 2 | https://mathoverflow.net/users/13268 | 232259 | 107,935 |
https://mathoverflow.net/questions/232056 | 4 | Let $[n]=\{1,...,n\}$ and $[\hat n]=\{\hat 1,...,\hat n\}$. Realize the hyperoctahedral group $H\_n$ as the centralizer of the permutation $(1\hat 1)\cdots (n \hat n)$. It has $2^n n!$ elements.
Let $\pi$ be the special permutation $\pi=(12\cdots n)(\hat 1)\cdots(\hat n)$, i.e. the elements of $[n]$ are arranged in a... | https://mathoverflow.net/users/78061 | Hyperoctahedral group acting on a special permutation | It sounds to me like you are trying to compute zonal spherical functions by projecting the corresponding irreducible characters, that is,
$$ \omega^\lambda((1,3,\cdots,2n-1)) = \frac{1}{|H\_n|}\sum\_{h \in H\_n} \chi\_{2\lambda}((1,3,\cdots,2n-1)h),$$
where $H\_n \leq S\_{2n}$ is the hyperoctahedral group of order $2^n... | 2 | https://mathoverflow.net/users/32968 | 232261 | 107,936 |
https://mathoverflow.net/questions/232232 | 12 | Suppose $H\_1,\ldots,H\_{2n}$ are open hemispheres which cover $S^{n-1}$ with the property that removing any one of them leaves $S^{n-1}$ uncovered. Is it necessarily the case that the hemispheres can be grouped into antipodal pairs? That is, possibly after reordering we have $H\_i=-H\_{i+1}$ for $i=1,3,5,\ldots,2n-1$?... | https://mathoverflow.net/users/34448 | Covering the unit sphere by open hemispheres | It looks so. We start with inductive proof of
**Steinitz Theorem.** Let $A$ be a finite set of rays starting from the origin in $\mathbb{R}^d$. Assume that positive span of these rays is the whole $\mathbb{R}^d$. Then there exists a subset of at most $2d$ rays from $A$ with the same property.
**Proof.** Induction i... | 8 | https://mathoverflow.net/users/4312 | 232262 | 107,937 |
https://mathoverflow.net/questions/232257 | 29 | * Kronheimer and Mrowka showed that the Khovanov homology detects the unknot.
* Bar-Natan showed a program to compute the Khovanov homology fast: there was no rigorous complexity analysis of the algorithm, but it is estimated by Bar-Natan that the algorithm runs in time proportional to the square root of the number of ... | https://mathoverflow.net/users/88227 | What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka) | EDIT: Marc Lackenby has just announced a quasi-polynomial time algorithm. That is, given an $n$—crossing diagram, the algorithm takes $n^{O(\log(n))}$ time to either find a spanning disk (proving the knot is trivial) or a hierarchy (proving the knot is non-trivial).
PREVIOUS: A quick skim of the paper you linked to f... | 38 | https://mathoverflow.net/users/1650 | 232267 | 107,939 |
https://mathoverflow.net/questions/232260 | 17 | How can we show the following equation
$$\sum\_{n\text{ odd}}\frac1{n\sinh(n\pi)}=\frac{\mathrm{ln}2}8\;?$$
I found it in a physics book(David J. Griffiths,'Introduction to electrodynamics',in Chapter 3,problem 3.48), which provided no proof.
| https://mathoverflow.net/users/75153 | Evaluating an infinite sum related to $\sinh$ | Here is a proof, surely not the simplest. Using
$$\frac1{\sinh(n\pi)}=\frac{2e^{-\pi n}}{1-e^{-2\pi n}}=2\sum\_{m\text{ odd}}e^{-mn\pi}$$
and
$$2\sum\_{n\text{ odd}}\frac{x^n}{n}=-2\ln(1-x)+\ln(1-x^2)=\ln\frac{1+x}{1-x},\qquad |x|<1,$$
we get that
$$\sum\_{n\text{ odd}}\frac1{n\sinh(n\pi)}=\sum\_{m\text{ odd}}\ln\f... | 23 | https://mathoverflow.net/users/11919 | 232279 | 107,944 |
https://mathoverflow.net/questions/232278 | 9 | I have a really basic question about cluster algebras and cluster varieties. According to the definition of Fomin-Zelevinsky a cluster algebra is generated by a bunch of polynomial rings inside the ring of Laurent polynomials. It means that the corresponding variety is covered by a bunch of affine spaces (all which hav... | https://mathoverflow.net/users/3891 | Cluster algebras and cluster varieties | A bunch of points:
$\def\Spec{\mathrm{Spec}\ }$
• Let $A$ be a cluster algebra over a field $k$, let $(x\_1, \ldots, x\_n)$ be a cluster and let $L$ be the Laurent polynomial ring $k[x\_1^{\pm}, \ldots, x\_n^{\pm}]$. I imaging your intended question is whether the map $\Spec L \to \Spec A$ is an open immersion. (You ... | 13 | https://mathoverflow.net/users/297 | 232287 | 107,947 |
https://mathoverflow.net/questions/232320 | 6 | Given a commutative ring $k$ and for $i = 1,2$ a homomorphism of $k$-modules $X\_i \overset {f\_i} \longrightarrow Y\_i$ with $X\_i$ flat over $k$.
Is the following conclusion true for general $k$? If $f\_1$ and $f\_2$ are injective, so is their tensor product $f\_1 \otimes\_k f\_2: X\_1 \otimes\_k X\_2 \longrightarr... | https://mathoverflow.net/users/88277 | Tensor product of monomorphisms is a monomorphism? | Let $F$ be a field, and $k=F[x,y]/(x^2,xy,y^2)$. Since $k$ is a finite-dimensional $F$-algebra, flat=projective, and for $k$-modules $M,N$, there is a natural isomorphism $\operatorname{Hom}\_k(M,N^\ast)\cong (M\otimes\_kN)^\ast$, where $L^\ast=\operatorname{Hom}\_F(L,F)$ denotes $F$-dual.
So we have a counterexampl... | 8 | https://mathoverflow.net/users/22989 | 232343 | 107,957 |
https://mathoverflow.net/questions/232349 | 2 | Let $f:(0, \infty) \longrightarrow (0, \infty)$ be a monotonously increasing function (in fact, a step function) and let $P$ be a polynomial of degree $N$. Suppose I know that for some $k$, the limit
$$A:= \lim\_{\varepsilon\rightarrow 0} \int\_0^\infty \frac{e^{-\varepsilon s}}{s^k}(f(s) - P(s))\, \mathrm{d}s$$
exists... | https://mathoverflow.net/users/16702 | Regularized integral and asymptotic expansion | No. The limit can exist due to cancellations, without $|f-P|$ being as small pointwise as you hoped for.
Let's take $k=0$, $P(s)=s$, and $f(s)=n+1/2$ on $n<s<n+1$. Then
$$
\int\_n^{n+1}e^{-\epsilon s} (f(s)-P(s))\, ds =-e^{-\epsilon(n+1/2)}\int\_{-1/2}^{1/2} te^{-\epsilon t}\, dt = \frac{\epsilon}{12}e^{-\epsilon(n+1... | 2 | https://mathoverflow.net/users/48839 | 232353 | 107,960 |
https://mathoverflow.net/questions/232359 | 1 | If I have a diagonalizable matrix $A = V\Lambda V^{-1}$, is there a way to show that for any similar $B$ such that $B = T\Lambda T^{-1}$, the Euclidean condition number $\kappa\_2(B) \geq \kappa\_2(\Lambda)$? Or is this statement false?
(and if it's a well-known result, is there a good linear algebra reference that m... | https://mathoverflow.net/users/1305 | Similarity transform of a diagonalizable matrix that minimizes the Euclidean condition number | It is a consequence of the fact that the largest singular value is greater than the largest eigenvalue (in modulus): $\lvert\lambda\_\max(B)\rvert \leq \sigma\_\max(B)$, and $\lvert\lambda\_\max(B^{-1})\rvert \leq \sigma\_\max(B^{-1})$, so $\kappa\_2(\Lambda) = \lvert\lambda\_\max(B)\rvert \lvert\lambda\_\max(B^{-1})\r... | 2 | https://mathoverflow.net/users/1898 | 232363 | 107,963 |
https://mathoverflow.net/questions/232345 | 0 | I've gone through many texts in algebraic geometry, specifically, Schubert calculus. They all claim that the Schubert classes $[\Omega\_{\lambda}]$ form a basis for the cohomology ring of the complex Grassmannian of $k$-dimensional subspaces of $n$-dimensional complex space, i.e. $H^{\*}(Gr(k, n))$, but without any pro... | https://mathoverflow.net/users/86315 | How do Schubert classes form a basis for $H^{*}(Gr(k, n))$? | What Fulton is suggesting is (an easy special case of) the fact that cellular and usual cohomology are equal. Consider the long exact sequence attached to the pair $(Y\_p,Y\_{p-1})$. We have maps
$$ \cdots \to H^\*(Y\_{p}, Y\_{p-1}; \mathbb Z)\to H^\*(Y\_p; \mathbb Z) \to H^\*(Y\_{p-1}; \mathbb Z) \to \cdots.$$
The k... | 6 | https://mathoverflow.net/users/66 | 232369 | 107,964 |
https://mathoverflow.net/questions/232358 | 3 | I've isolated a property of rings (integral domains, associative, unitary, non necessarily commutative) that is useful to me :
$$xR\cap yR\neq\{0\}\quad\text{ whenever $x$ and $y$ are non zero.}$$
> **Question.** Does this property has a name, or falls into a known and labelled class of rings ?
Commutative integ... | https://mathoverflow.net/users/18583 | On rings $R$ such that $xR\cap yR$ is non zero whenever $x$ and $y$ are non zero | These rings are called [*right uniform rings.*](https://en.wikipedia.org/wiki/Uniform_module)
Generally, a right ideal $I$ is called (right) *uniform* if all nonzero right subideals $J,K\subseteq I$ have nonzero intersection: $J\cap K \neq 0$.
Note: Your condition is equivalent to the condition that any two nonzer... | 9 | https://mathoverflow.net/users/17734 | 232376 | 107,967 |
https://mathoverflow.net/questions/232372 | 8 | See [here](https://galoisrepresentations.wordpress.com/2012/12/27/classic-papers-in-number-theory/) for a comment of Matt Emerton.
>
> There are also various seminar reports of Serre, e.g. his report on mod p modular forms, but also his report from the late 60s on the possibility of Galois reps. attached to modular... | https://mathoverflow.net/users/88308 | Reference request: seminar report of Serre from late 60s on possibility of Galois representations attached to modular forms? | Googling reveals [these lectures slides of Ken Ribet](https://math.berkeley.edu/%7Eribet/red_lodge2.pdf), which say the following.
>
> After Serre's article on elliptic curves was written in the early 1970s, his techniques were generalized and extended in different directions. In particular, Serre and Swinnerton-Dy... | 5 | https://mathoverflow.net/users/nan | 232377 | 107,968 |
https://mathoverflow.net/questions/232209 | 7 | By my understanding, Gödel's first incompleteness theorem says that any theory with sufficient1 interpretability strength is *essentially incomplete*, that is, any consistent recursively enumerable extension of the theory must be incomplete.
Meanwhile, the second theorem says that any theory with sufficient2 interpre... | https://mathoverflow.net/users/4336 | Relationship between first and second incompleteness theorems | It is possible to construct first-order theories which prove their own consistency (and which are incomplete). Here is one example.
Consider a first-order theory with equality with a constant symbol 0 (zero), a unary predicate symbol N (the natural numbers), and a binary predicate symbol σ (the sequential relationshi... | 5 | https://mathoverflow.net/users/20716 | 232388 | 107,972 |
https://mathoverflow.net/questions/232374 | 4 | I'm considering a situation where I have the linear restriction map of Fréchet spaces
$$
C^\infty(C\_1) \to C^\infty(C\_2)
$$
where $C\_2 \hookrightarrow C\_1$ are a pair of compact, connected subsets of $\mathbb{R}^n$ homeomorphic to closed balls, and interiors diffeomorphic to open balls. I believe I can assume that... | https://mathoverflow.net/users/4177 | Linear extension operators for smooth functions: from compact sets to compact sets | In general, there are several candidates for the definition of $C^\infty(K)$:
One is the space $\lbrace f|\_K: f\in C^\infty(\mathbb R^n)\rbrace$ of all restrictions (endowed with the quotient topology), another is the intersection $\bigcap\limits\_{k\in\mathbb N\_0} \lbrace f|\_K: f\in C^k(\mathbb R^n)\rbrace$ (which ... | 7 | https://mathoverflow.net/users/21051 | 232390 | 107,973 |
https://mathoverflow.net/questions/2369 | 21 | Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G\_1(A)\},$$ where $G\_1(A)$ is the connected component of the group of invertibles $G(A)$ that contains the identity.
>
> Is it true that $e(ab)\cu... | https://mathoverflow.net/users/1162 | In a Banach algebra, do ab and ba have almost the same exponential spectrum? | Just to update this: a negative solution was recently given by Klaja and Ransford. See [arXiv 1510.08109.](http://arxiv.org/abs/1510.08109)
| 5 | https://mathoverflow.net/users/763 | 232396 | 107,977 |
https://mathoverflow.net/questions/232356 | 10 | Let $C$ be a smooth projective curve. Is it true that
$$\textrm{Aut}(C\times C)\cong S\_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$
and in case, what would be a reference for this? Thanks.
| https://mathoverflow.net/users/nan | Automorphisms of cartesian products of curves | This is a particular case of a more general rigidity result, whose proof (similar to the one given in abx's answer) can be found in Lemma 3.8 of
F. Catanese, [*Fibred surfaces, varieties isogenous to a product and related moduli spaces*](http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.1catanese... | 9 | https://mathoverflow.net/users/7460 | 232398 | 107,979 |
https://mathoverflow.net/questions/231886 | 9 | Let $(X\_{1},\ldots,X\_{k},\ldots)$ be a martingale difference sequence, i.e.
$$
E[X\_{k}|\mathcal{F}\_{k-1}] = 0
$$
where $\mathcal{F}\_{k-1}$ is the $\sigma$-algebra filtration at $k-1$.
Let $\sigma\_{k}^2 = E[X\_{k}^2|\mathcal{F}\_{k-1}]$. Here note that $\sigma\_{k}^2$ is a random variable measurable w.r.t. $\m... | https://mathoverflow.net/users/88033 | Berry-Esseen bound for martingale sequence with varying and dependent variances | In the setting you describe, there's generally no CLT. Let me describe a counterexample showing that $\sum\_{i=1}^{k-1} X\_i / \sqrt{\sum\_{i=1}^{k-1} \sigma\_i}$ doesn't tend to normal. I'm pretty sure the same holds for the discounted averages as well.
The example is a random walk which is lazy when positive. More ... | 5 | https://mathoverflow.net/users/1061 | 232401 | 107,980 |
https://mathoverflow.net/questions/232400 | 11 | I would like to study Nori motives and I am a complete outsider of the subject. I do, however, have background on Chow motives, Voevodsky motives $\mathrm{DM}$ and his stable homotopy category $\mathrm{SH}$.
My question is:
**Do you know an introductory reference and the main papers about Nori motives?**
Thank yo... | https://mathoverflow.net/users/12204 | Reference for Nori motives | Probably the best introduction has already been mentioned by Donu Arapura, and it is available online:
* Marc Levine, [Mixed Motives](https://www.uni-due.de/~bm0032/publ/MixMotKHB.pdf) (2005)
Section 1 ("Essentials of Nori Motives") of this paper might also be useful:
* Annette Huber & Stefan Müller-Stach, [On th... | 11 | https://mathoverflow.net/users/43108 | 232423 | 107,988 |
https://mathoverflow.net/questions/232418 | 2 | The complex spin groups $Spin^C(n)$ appear in the fibration
$Spin(n)\rightarrow Spin^C(n)\rightarrow\ S^1$
which must split since $BSpin(n)$ is 3-connected to give a homotopy equivalence
$Spin^C(n)\simeq Spin(n)\times S^1$
This is, however, only an equivalence of spaces, and not of topological groups.
In low... | https://mathoverflow.net/users/54788 | 'Accidental' isomorphisms for $Spin^C(n)$ | We have $Spin^C(n) = (Spin(n)\times S^1)/(\mathbb{Z}/2)$, where $\mathbb{Z}/2$ is generated by $(-1,-1)$ and the first $-1$ is the nontrivial element of the kernel of $Spin(n)\to SO(n)$. Since the accidental isomorphisms come from the spin representation of $Spin(n)$, this element corresponds to $-id$ in $SU(2),Sp(2),S... | 6 | https://mathoverflow.net/users/35687 | 232425 | 107,989 |
https://mathoverflow.net/questions/232432 | 4 | I am reading Bridgeland's paper on stability condition, where he defined a slicing of a triangulated category. See
<http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf>
After the definition he remarked that the decomposition in axiom (c) is unique and I am having trouble showing this.
I tri... | https://mathoverflow.net/users/48616 | Uniqueness of decomposition for a slicing of triangulated category | The first step is the following. Assume $E'$ and $E''$ are two objects with filtrations as in axiom (d). Then if $\phi\_n(E') > \phi\_1(E'')$ then $Hom(E',E'') = 0$. This follows from (c) by induction.
The second step. Assume $E$ and $F$ are two objects, $\phi \in \mathbb{R}$, and $E' \to E \to E''$, $F' \to F \to F'... | 3 | https://mathoverflow.net/users/4428 | 232436 | 107,992 |
https://mathoverflow.net/questions/232424 | 1 | A finite group $G$ satisfies property $P\_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. Let for $x\in G$, $x^G$ be a conjugacy classe of $G$. Also Let $G$ be a solvable group satisfying property $P\_5$, $M=... | https://mathoverflow.net/users/54820 | Finite groups $G$ satisfying property $P_n$ | There are no finite groups $G$ such that all your assumptions are met (so in a sense, the answer is "Yes").
Let $A$ be the normal $3$-complement of $G$, so that $M/A$ is a $3$-group. I claim that $|M/A| = 3$. Otherwise, the number of conjugacy classes of $M/A$ is at least $9$. Since $|C\_G(x)|=6$, we have $|C\_M(x)|... | 4 | https://mathoverflow.net/users/10266 | 232442 | 107,995 |
https://mathoverflow.net/questions/232406 | 2 | Given four independent, identically distributed Gaussian random variables with zero mean and unit variance $x\_1$, $x\_2$, $y\_1$, $y\_2$, consider
\begin{equation}
u \equiv \max(x\_1+C\, y\_1, x\_2+C \, y\_2) - \max(x\_1-C \, y\_1, x\_2-C \, y\_2),
\end{equation}
where $C$ is a real number.
Do you know how to c... | https://mathoverflow.net/users/45356 | Difference between maxima of random variables | Using $\max(a,b) = \dfrac{a+b}{2} + \left| \dfrac{a-b}{2}\right|$, write $u = w\_1 + |w\_2| - |w\_3|$ where
$$ \eqalign{ w\_1 &= C (y\_1 + y\_2) \cr
w\_2 &= \dfrac{1}{2} (x\_2 - x\_1 + C (y\_2 - y\_1))\cr
w\_3 &= \dfrac{1}{2} (x\_2 - x\_1 - C (y\_2 - y\_1))\cr}$$
are jointly normal with mean $0$ and covariance matri... | 6 | https://mathoverflow.net/users/13650 | 232446 | 107,997 |
https://mathoverflow.net/questions/230037 | 7 |
> **Definition.** According to Shelah, a field $K$ **does not have the independence property** (*i.e.* is **NIP**) if for every first order formula $\varphi(x, \bar y)$ in the language of fields $(+,\times,0,1)$, the Vapnik–Chervonenkis dimension of the family of subsets$\{\varphi(K, \bar k) : \bar k \in K^n\}$ is fin... | https://mathoverflow.net/users/18583 | Examples of NIP fields of characteristic $p$ | Understanding concretely which fields (in the "pure" language of rings, $\mathcal{L} = \{0, 1, +, \cdot\}$) are NIP is a topic of current interest in model theory. The 2015 paper "Dp-minimal valued fields" by Jahnke, Simon, and Walsberg begins:
"Very little is known about NIP fields. It is widely believed that an NIP... | 4 | https://mathoverflow.net/users/93 | 232448 | 107,998 |
https://mathoverflow.net/questions/231926 | 13 | There are very many papers in the area of (possibly non-uniformly) hyperbolic dynamical systems whose aim is to prove the Central Limit Theorem. In a dynamical context, this means that one:
* has a metric space $\Omega$, a map (or flow, but let's stick to maps) $T:\Omega\to\Omega$ and a (Borel) invariant probability ... | https://mathoverflow.net/users/4961 | Applications of the Central Limit Theorem in dynamical systems | These limit theorems can be useful when studying systems preserving an infinite invariant measure.
For example, Jean Pierre Conze has used it in his paper "Sur un critere de recurrence en dimension 2 pour les marches stationnaires" to get a criteria for recurrence for $\mathbb{Z}^2$ skew product extensions. See also... | 11 | https://mathoverflow.net/users/78465 | 232449 | 107,999 |
https://mathoverflow.net/questions/232227 | 3 | Fix some $n \geq 1$ and some prime $p$. I'm looking for finite $p$-groups $G$ and finite-dimensional complex representations $V$ of $G$ with the following two properties:
1. The abelianization of $G$ has rank $n$; for instance, it could be $(\mathbb{Z}/p)^n$.
2. For all nonidentity $g \in G$ and all nonzero $v \in V$... | https://mathoverflow.net/users/88205 | Representations of p-groups where 1 is never an eigenvalue | (This question has been answered in comments by Geoff Robinson:)
You have already found (almost) all examples there are, namely cyclic $p$-groups and generalized quaternion groups in case $p=2$. The point is that every abelian subgroup of such a group $G$ is cyclic: Suppose $A\leq G$ is abelian. Let $\lambda \in \op... | 3 | https://mathoverflow.net/users/10266 | 232450 | 108,000 |
https://mathoverflow.net/questions/232451 | 1 | It has been shown, by elementary methods, that every positive integer can be expressed as the sum of $4$ squares. This type of result has been proven for many different powers $p$, for example, when $p=5$,
$$
\forall n \in \Bbb{N}: \exists \{x\_i\} x\_i \in \Bbb{N}^{37}, 1 \leq i \leq 37 : n = \sum x\_i^5
$$
(and the ... | https://mathoverflow.net/users/82067 | Is it proved that for every integer $p>0$ there exists an integer $k>0$ such that every integer $n>0$ can be expressed as $j_1^p+\dots+j_k^p$? | You ask if Waring's conjecture has been proved. The answer is yes (Hilbert 1909), and you can read about its history [here](https://en.wikipedia.org/wiki/Waring's_problem).
Your $k(p)$ is usually denoted by $g(p)$, and its minimal value is almost precisely known. It is harder to estimate the related quantity $G(p)$ ... | 10 | https://mathoverflow.net/users/11919 | 232453 | 108,001 |
https://mathoverflow.net/questions/232301 | 1 | Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Assume further that $H^1(\mathcal{O}\_X)=0$. Is $H^0(R^1j\_\*\mathcal{O}\_{X \backslash S})=0$?
| https://mathoverflow.net/users/58203 | Global sections of higher direct image sheaf | Since $S$ is finite, $X$ being projective is a red herring, so is anything about the (global) cohomology of $\mathscr O\_X$, and
(to some extent) it being a surface, or regular.
If $X$ is $S\_2$, and $\dim X\geq 2$, then the sheaf version of Exercise III.2.3 in [Hartshorne] (or the exercise itself noting that by $S$... | 3 | https://mathoverflow.net/users/10076 | 232460 | 108,004 |
https://mathoverflow.net/questions/232379 | 3 | Let $\vec{\mu}\_1, \vec{\mu}\_2,\ldots, \vec{\mu}\_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|\_1 = 1,\mu\_j \geq 0\}.$$
(Here $\mu\_j$ means the $j$-th entry of $\vec{\mu} \in R^d$.) We define $\Delta^{k-1}$ similarly.
O... | https://mathoverflow.net/users/82358 | Moment matching on the standard simplex | It is a standard result that the matrices of the form $\mu^{\otimes 2}$ for nonzero $\mu$ are the extreme rays of the positive semidefinite cone. That is to say, your condition on the second moments implies that $\mu^{\otimes 2}$ is a scalar multiple of $\mu\_i^{\otimes 2}$ for some $i$. The normalization then gives $\... | 3 | https://mathoverflow.net/users/5963 | 232473 | 108,007 |
https://mathoverflow.net/questions/232470 | 3 | Here I try to seek if restricting the structure of permutations would still keep GI property.
Given a $2n$ vertex undirected graph whose vertices are partitioned arbitrarily in pairs to say WLOG $(1,2)$, $(3,4)$, $\dots$, $(2n-1,2n)$. Call these vertices pairs as super vertices.
Call two such graphs $2n$ vertex lab... | https://mathoverflow.net/users/nan | A possible GI isomorphic problem | It is GI-complete. Take two arbitrary connected graphs with degrees at least 2 (obviously a GI-complete class). For each vertex $v$, add a new vertex $v'$ and join it only to $v$. The pairs $\{v,v'\}$ have the property you describe.
| 3 | https://mathoverflow.net/users/9025 | 232486 | 108,011 |
https://mathoverflow.net/questions/232459 | 10 | Suppose you have an abelian category $\bf A$, and $A\to B\to C$, $A'\to B'\to C'$ two exact sequences, in a diagram
$$
\begin{array}{cccccccc}
0 &\to & A &\to& B &\to& C &\to & 0\\
&&\downarrow && \downarrow && \downarrow \\
0 &\to & A' &\to& B' &\to& C' &\to & 0
\end{array}
$$
(i.e., suppose you have a morphism of exa... | https://mathoverflow.net/users/7952 | A general version of the 5 lemma | One condition on the factorization system that should work is that "factorizations preserve exact sequences", i.e. if you have a map between exact sequences and you $(E,M)$-factor it componentwise, then the intermediate objects also form an exact sequence. If this is true, then you can factor your given map of exact se... | 3 | https://mathoverflow.net/users/49 | 232487 | 108,012 |
https://mathoverflow.net/questions/232468 | 2 | I asked the following question on Stackexchange and got no reply so I am reposting it here. Let $G$ be a finite group. A $G$-module C is a **class module** if, for all subgroups $H \subset G$:
1) $H^1(H,C)=0$
2) $H^2(H,C)$ is cyclic of order $\#H$
Remark: If $G$ is cyclic then $\mathbb{Z}$ is a class module.
**... | https://mathoverflow.net/users/47195 | Existence of class modules for finite groups | The answer is yes. Let $G$ be a finite group. You can realise $G$ as a Galois group $G(L/K)$ of a finite Galois extension $L/K$ of number fields. Let $C\_L$ be the idele class group of $L$. $G$ acts on $L$ and on $C\_L$.
By p.196 of Tate's article on "global class fields" (in the book Algebraic Number Theory" by Cas... | 4 | https://mathoverflow.net/users/23291 | 232489 | 108,013 |
https://mathoverflow.net/questions/232085 | 0 | Consider the scalar elliptic equation of divergence form
$$div((1+a)\nabla\pi)=div F\ \ in\ \ R^3,$$
where $a$ is a Schwartz function with $1+a\geq c=const>0$, $F=(F\_1,F\_2,F\_3)$ is a vector-valued Schwartz function.
Now can we give a counterexample to show that for $1<p<\infty, p\neq2$, the solution map $F\mapsto\... | https://mathoverflow.net/users/88134 | $L^p$ estimates for elliptic equation of divergence form | You may also give a look to
G.Di Fazio Lp estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Un. Mat. Ital. A (7) 10 (1996), no. 2, 409–420
where leading coefficients are VMO and boundedness is obtained for any $1<p<+\infty$.
| 1 | https://mathoverflow.net/users/49136 | 232503 | 108,016 |
https://mathoverflow.net/questions/232433 | 3 | I am assessing the probability distribution on a running time of some algorithm that we've developed. I am looking for a family of probability mass functions $f\_n$ with the following recurrence:
$$
f\_{n}(k)=qf\_{n-1}(k)+p\sum\_{i=1}^{k-1}f\_{n-1}(i)f\_{n-1}(k-i),
$$
with the starting term
$$
f\_{0}(k)=\begin{cases}
1... | https://mathoverflow.net/users/37757 | Solving recursion / finding generating function of a probability mass function | I do not think it is possible to get a closed-form of either $f\_n$ or $F\_n$.
For brevity, let's rewrite your recurrence equation for the generating function $a\_n:=F\_n(x)$ as $a\_n = q a\_{n-1} + p a\_{n-1}^2$ with the initial condition $a\_0=x$. Then, change of variables $b\_n = p a\_n + \frac{q}{2}$ will reduce ... | 2 | https://mathoverflow.net/users/87904 | 232511 | 108,018 |
https://mathoverflow.net/questions/232519 | 1 | Let $\varphi\colon (\mathbb C^\*)^m\times\mathbb A^n\to\mathbb A^n$ be an algebraic effective action of a torus on affine space and $X$ be a Zariski closure of an orbit of this action. Suppose we also have an algebraic action $\psi\colon\mathbb C^\*\times X\to X$ of one-dimensional torus on $X$. Does always exist a mor... | https://mathoverflow.net/users/88385 | Action of $\mathbb C^*$ on a closed orbit of a torus | Comment above posted as an answer. No, that is not true. One counterexample is when $X=\mathbb{A}^1 = \text{Spec} \ \mathbb{C}[x]$, $m$ equals $1$, $\phi(\lambda,x)$ equals $\lambda x$, and $\psi(\lambda,x)$ equals $\lambda(x-1)+1$.
| 4 | https://mathoverflow.net/users/13265 | 232520 | 108,022 |
https://mathoverflow.net/questions/232524 | 2 | Let $X\subseteq\mathbb{P}^{N}$ be a quasiprojective variety of dimension $N-1$, and let
$$
\nu:X^{\nu}\rightarrow X
$$
be its normalization. Let us suppose that $X^{\nu}(\neq X)$ is smooth. I wonder if in this case
$$
\dim (\mathrm{Sing(X)})=\dim (X)-1.
$$
I think Serre's Normality Criterion (See Lemma 12.5 of [this](... | https://mathoverflow.net/users/87998 | If $X$ has non-singular normalization $\dim (\mathrm{Sing(X)})=\dim (X)-1$? | **Proposition:** *If $X$ is S2, then the normalization map $\nu : X^{n} \to X$ is an isomorphism outside a set of pure codimension 1 (in either $X$ or $X^{n}$).*
*Proof:* Let $Z \subseteq X$ be the locus where $\nu$ is not an isomorphism and let $W \subseteq X^{n}$ be the (scheme-theoretic) pre-image of $W$. For a co... | 7 | https://mathoverflow.net/users/3521 | 232529 | 108,025 |
https://mathoverflow.net/questions/232516 | 8 | For the sake of concreteness denote by $M\_0(X)$ the linear space of all signed Borel measures $\sigma$ with $\sigma(X)=0$ on some metric space $(X,d)$ and fix some base point $x\_0\in X$. On this space define the norm
$$
\newcommand{\norm}[1]{\|#1\|}
\newcommand{\Lip}{\mathrm{Lip}}
\norm{\sigma}\_0^\* = \sup\{\int\_X ... | https://mathoverflow.net/users/9652 | Completion of spaces of measures w.r.t. weak norms | This is known as the Arens-Eells space $AE(X)$. In the nonlinear Banach space literature it's also called the Lipschitz-free space $\mathcal{F}(X)$. It is not a dual space in general, but rather the predual of the space ${\rm Lip}\_0(X)$ of Lipschitz functions which vanish at $x\_0$.
In some cases it is a dual space.... | 11 | https://mathoverflow.net/users/23141 | 232530 | 108,026 |
https://mathoverflow.net/questions/232131 | 6 | Let $X$ be a (connected) closed $n$-manifold and $G=\pi\_1(X)$ be the fundamental group of $X$. There is a classifying map $f: X \rightarrow K(G, 1)$ which induces an isomorphism on $\pi\_1$. I would like know when the map $f\_\*: H\_n(X, \mathbb{Z}) \rightarrow H\_n(K(G,1), \mathbb{Z})$ is injective, or even when $f\_... | https://mathoverflow.net/users/88164 | comparing homology of a space and homology of the classifying space of its fundamental group | There can not be such a condition.
For any finitely presented group $G$, we can find a closed 4-manifold $N$ with fundamental group $G$. In dimension $n \geq 6$, we can now take $M = N \times S^{n-4}$, a $n$-manifold with fundamental group $G$. Then the classifying map $f\colon M \to BG$ of the universal cover $\tild... | 6 | https://mathoverflow.net/users/14233 | 232541 | 108,032 |
https://mathoverflow.net/questions/230747 | 8 | A set of $d \times d$ real or complex matrices is commonly called *irreducible* if those matrices do not jointly preserve a linear subspace with dimension strictly between zero and $d$. A stronger hypothesis which is useful in multiplicative ergodic theory - for example, in Furstenberg's theorem on random matrix produc... | https://mathoverflow.net/users/1840 | Sets of matrices which are irreducible but not strongly irreducible | The answer to your second question is **no**.
Let $E\_1$, $E\_2$, $E\_3$ be pairwise transverse $2$-dimensional subspaces of $\mathbb{R}^4$.
Consider the following semigroup:
$$
\Sigma := \{M \in \mathrm{Mat}(4,4) ; \; M(E\_i)=E\_i, i=1,2,3\}.
$$
**[Edit]** The following remark will be useful: Any linear transforma... | 2 | https://mathoverflow.net/users/1516 | 232552 | 108,035 |
https://mathoverflow.net/questions/232550 | 11 | I have a question regarding the Mordell Weil theorem a number field $K$.
I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman.
They presented a proof for the case where $E[2] \in E (\mathbb{Q}) $
where $E : Y^2 = X(X^2 + AX + B)$ and mentioned before hand that it i... | https://mathoverflow.net/users/nan | Weak Mordell-Weil over number fields | That's more or less the right way to do it. The proof of your claim (or something similar) will require (1) finiteness of the ideal class group (or at least, the 2-part of the class group) and (2) finite generation of the group of units in $O\_K$, i.e., Dirichlet's unit theorem, although again what's really needed is t... | 11 | https://mathoverflow.net/users/11926 | 232557 | 108,038 |
https://mathoverflow.net/questions/232559 | 1 | Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains.
Then the colimit of the obvious diagram is an integral domain.
Does this statement also hold in the case where we drop the commutativity condition?
| https://mathoverflow.net/users/nan | Filtered Colimit of associative $k$-algebras that are domains | I don't see how commutativity matters.
Suppose $A$ is the filtered colimit of algebras $A\_i$ and $x,y\in A$ with $xy=0$. Then $x$ is represented by $x\_j\in A\_j$ and $y$ by $y\_k\in A\_k$ for some $j$ and $k$. By "filtered", these map to elements $x\_l,y\_l\in A\_l$ for some $l$ such that $x\_ly\_l=0$. Since $A\_l$... | 5 | https://mathoverflow.net/users/22989 | 232586 | 108,044 |
https://mathoverflow.net/questions/230982 | 4 | What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl\_n$ be the general linear Lie algebra. Classify all Lie cobrackets $\delta: gl\_n \to \Lambda^2 gl\_n$. Any help will be greatly appreciated!
Edit: it seems that the case that $g$ is a se... | https://mathoverflow.net/users/11877 | Classifications of Lie bialgebras | **Semisimple case:** Belavin-Drinfeld result does **not** classify all lie bialgebras on a (finite-dim.) semisimple complex Lie algebra, but only so called *quasi-triangular Lie bialgebras* (those having a non skew-symmetric r-matrix which satisfies CYBE). If one moves from the quasi-triangular case to the triangular c... | 3 | https://mathoverflow.net/users/6032 | 232587 | 108,045 |
https://mathoverflow.net/questions/232596 | 3 | Let $O\subset\mathbb{R}^n$ be a open set which is star shaped with respect to the origin. How does one prove that there exists an increasing sequence of star shaped (w.r.t the origin) domains $O\_i$ such that :
* $O\_i$ has a smooth boundary
* $O\_i\subset O$
* $\lambda (O\backslash O\_i)\to 0$ when $i\to\infty$, or... | https://mathoverflow.net/users/8887 | Can one smooth open star shaped domains from the inside by star shaped domains? | This is not really an answer, but an equivalent and hopefully easier reformulation of the question that does not fit in one comment.
You can describe the starshaped domain $O$ by a function $\rho\colon S^{n-1}\to(0,\infty]$ such that $$O\setminus\{0\}=\{\,x\in\mathbb R^n\setminus\{0\}\mid |x|<\rho(x/|x|)\,\}\;.$$ The... | 3 | https://mathoverflow.net/users/70808 | 232600 | 108,050 |
https://mathoverflow.net/questions/232575 | 6 | The notion of a (left, say) Bousfield localization of a model category doesn't seem to be invariant under Quillen equivalence. There are a lot of things that could go wrong. But I don't know any examples. So if anyone could help me out with even one of the below questions, I'd appreciate it.
Let $M$ be a model catego... | https://mathoverflow.net/users/2362 | Bousfield Localization and Quillen Equivalence | For (1)-(2), look at work of Carles Casacuberta. He has lots of good examples. His paper with Chorny on the orthogonal subcategory problem has an example for your (2), on the last page. This paper of Casacuberta-Chorny goes into great depth about (3) as well, and it led to Chorny's work on class combinatorial model cat... | 7 | https://mathoverflow.net/users/11540 | 232604 | 108,051 |
https://mathoverflow.net/questions/232605 | 7 | Let $G$ be the group $[729,57]$, using GAP's notation. I have so far two descriptions of the group:
* a presentation
* an embedding (not surjective!) of the group into a Sylow $3$-subgroup of the unit group of a finite ring
Is $G$ isomorphic to a Sylow $3$-subgroup of some well-known group?
| https://mathoverflow.net/users/76083 | Is $[729,57]$ a Sylow $3$-subgroup of some well-known group? | Yes (to my surprise) it appears to be isomorphic to the Sylow $3$-subgroup of $3.J\_3$, the $3$-fold cover of the Janko sporadic simple group $J\_3$. Here is some Magma code:
```
> C:=MatrixGroup("3J3",1);
> P:=Sylow(C,3);
> IdentifyGroup(P);
<729, 57>
```
And here is GAP code for the same computation (thanks... | 10 | https://mathoverflow.net/users/35840 | 232608 | 108,053 |
https://mathoverflow.net/questions/232618 | 4 | Given a flat map $f: X \rightarrow Y$ such that $X$ is a projective variety and $Y$ is a smooth curve. Each generic fiber is isomorphic to an irreducible projective variety $A$ of dimension $d$.
The special fiber of $f$ must also be of dimension $d$ because of flatness, but does the special fiber have to be equi-dime... | https://mathoverflow.net/users/7780 | Equi-dimensionality of special fibers in a flat family | It must be set-theoretically equidimensional, but not scheme-theoretically. (Consider two lines in space colliding, developing an embedded point at the intersection.)
For the positive statement, let $d\leq e$ be the smallest and largest dimensions occurring among the components. Slice all fibers with the same general... | 6 | https://mathoverflow.net/users/391 | 232620 | 108,056 |
https://mathoverflow.net/questions/227597 | 10 | Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely
$$x(t)=a\_0+b\_0e^{-t}+\epsilon(a\_1+a\_0t+b\_1e^{-t}-b\_0te^{-t})+O(\epsilon^2).
$$
But the exact solution is obviously bounded uniformly in $t,\epsi... | https://mathoverflow.net/users/51484 | How to eliminate secular terms for perturbed non-oscillatory equations? | This is probably more a comment than an answer, but too long for the former.
A different approach might be to use normal hyperbolicity theory to first find a series expansion of an invariant manifold for your system. Then, you might be able to express solutions as a combination of an outer solution converging to the ... | 4 | https://mathoverflow.net/users/3928 | 232639 | 108,060 |
https://mathoverflow.net/questions/232447 | 4 | Let $\mathcal{M}\_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-Prokhorov metric. For $\mu \in \mathcal{M}\_1(\mathbb R)$, let $L^1(\mu)$ denote the Banach space of $\mu$-integrable functi... | https://mathoverflow.net/users/12978 | Is the following product-like space a Polish space? | *Yes, $\mathbb X$ is a Polish space, even a computable one, but it doesn't look like it has a nice metric.*
(I figured out the answer on my own question, but any other insightful answers or references would still be welcome.)
---
Suppose $\mathbb A$ is a Polish space and suppose that $\mathbb B(a)$ is a paramet... | 0 | https://mathoverflow.net/users/12978 | 232642 | 108,061 |
https://mathoverflow.net/questions/232616 | 14 | Let $K\_n$ be the field $\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]$ (the real subfield of the cyclotomic field $\mathbf Q[e^{\frac{i\pi}{2^{n+1}}}]$).
Is there anything known about the growth of the special values of the Dirichlet $\zeta$ function
$$\zeta\_{K\_n}(2)=\prod\_{\mathfrak p}\frac{1}{1-\frac{1}{N(\mathfrak p)... | https://mathoverflow.net/users/39552 | Growth of $\zeta_{\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]}(2)$ | Actually $\zeta\_{K\_n}(\sigma)$ is bounded for any fixed $\sigma > 1$.
Let $N = 2^n = [K\_n : {\bf Q}]$. Then all the local factors of
$\zeta(\sigma)$, other than the factor $(1-2^{-\sigma})^{-1}$ for
the prime above $2$, are of the form $(1 - q^{-\sigma})^{-g}$,
where $q$ is a prime power congruent to $\pm 1 \bmod... | 18 | https://mathoverflow.net/users/14830 | 232655 | 108,064 |
https://mathoverflow.net/questions/200343 | 1 | It is known that if $(S\_i= \sum\_{j \leqslant i }X\_i, \mathcal F\_i)$ is a martingale,
then for each
$
\beta>1$, $\delta\in (0,\beta-1)$ and $\lambda>0$, and each integer $N \geqslant 1$, the inequality
$$\tag{\*}
\mu\left\{\max\_{1\leqslant i\leqslant N}|S\_i|>\beta\lambda\right\}
\leqslant \frac{\delta^2}... | https://mathoverflow.net/users/17118 | Tail inequality for orthomartingales/martingale difference random fields | Actually, an extension of (\*) is a way to establish Burkholder's inequality
or [Nagaev's inequality](http://link.springer.com/article/10.1023%2FA%3A1025814306357). For general ortho-martingale random fields, this is a difficult task.
However, if we assume that the random field is strictly stationary and the filtra... | 0 | https://mathoverflow.net/users/17118 | 232667 | 108,067 |
https://mathoverflow.net/questions/232589 | 11 | I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
[http://www.amazon.com/Introduction-Intersection-Homology-Theory-Edition/dp/1584881844](http://rads.stackoverflow.com/amzn/click/1584881844).
Then, I read chapt... | https://mathoverflow.net/users/88427 | Examples of calculating perverse sheaves on algebraic varieties with easy stratification | Geordie Williamson has a very nice set of notes on perverse sheaves:
<http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf>
it deals with some examples on curves (section 10).
You can also look at De Cataldo and Migliorini's paper (section 2.2):
<http://www.ams.org/journals/bull/2009-46-04/S0273-0979-0... | 7 | https://mathoverflow.net/users/27816 | 232669 | 108,068 |
https://mathoverflow.net/questions/232652 | 14 | Define an arithmetic scheme $X$ to be a separated, integral scheme, flat and finite type over $\mathbb{Z}$. I am interested in obtaining examples of finite étale covers of arithmetic schemes. I am mainly interested in the case of arithmetic surfaces, but I would enjoy examples of finite étale covers of arithmetic schem... | https://mathoverflow.net/users/70019 | Examples of étale covers of arithmetic surfaces | You can, of course, use Bertini's theorem to make examples of a finite, flat, Galois extension with arbitrary finite Galois group $\Gamma$. Let $M$ be $\mathbb{Z}[\Gamma]$, the group ring of $\Gamma$ with coefficients in $\mathbb{Z}$. This is a finite, free $\mathbb{Z}$-module. Let $r>0$ be a positive integer. Then $M^... | 6 | https://mathoverflow.net/users/13265 | 232674 | 108,069 |
https://mathoverflow.net/questions/232664 | 6 | Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum\_{\left|t-\gamma\right|\leq1}\frac{1}{s-\rho}+O\left(\log\left(t\right)\right) \tag{1}$$ uniformly for $-1\leq\sigma\leq2$ (see for example... | https://mathoverflow.net/users/68301 | About the logarithmic derivative of the Riemann zeta function | I think your final goal follows by taking the logarithmic derivative of the functional equation:
$$\frac{\zeta'}{\zeta}(s)+\frac{\zeta'}{\zeta}(1-s)=\log\pi-\frac{1}{2}\frac{\Gamma'}{\Gamma}\left(\frac{s}{2}\right)-\frac{1}{2}\frac{\Gamma'}{\Gamma}\left(\frac{1-s}{2}\right).$$
Applying this with $s=it$ and using the fa... | 12 | https://mathoverflow.net/users/11919 | 232680 | 108,071 |
https://mathoverflow.net/questions/232115 | 5 | Fix a finite set $X$ and two natural numbers $d$ and $n$.
For a partition $\lambda$ and a number $d$ denote by $s\_\lambda^d(x\_1,\dots,x\_d)$ the Schur polynomial in $d$-many variables $x\_1,\dots,x\_d$. Denote by $\mathcal P\_n(X)$ the set of partition-valued functions on $X$ of total size $n$, i.e. the set of func... | https://mathoverflow.net/users/88153 | sum of squares of Schur polynomials indexed over partition valued functions on a set | assume $a$ is the constant function with value $1$.
as in the case of $X=\{\star\}$ we have the equality
$$(1-t^2)^{-d^2|X|}=\left(\sum\_{\lambda\in \{\mathrm{partitions}\}}s^d\_\lambda(t,\dots,t)^2\right)^{|X|}=\sum\_{\lambda\colon X\to\{\mathrm{partitions}\}}\prod\_{x\in X}s\_{\lambda(x)}^d(t,\dots,t)^2$$
and picki... | 2 | https://mathoverflow.net/users/88153 | 232682 | 108,073 |
https://mathoverflow.net/questions/232688 | 3 | The classical Shannon sampling theorem states that
a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))\_{i\in \mathbb{Z}}$ (The symbol $\hat f$ refers to the Fourier transform of $f$).
My question is as follows: Suppose that $f$ is not bandlimited but ... | https://mathoverflow.net/users/88489 | Sampling Theorem for non-bandlimited Functions | No as well to the edited question (if I understood it...): $f(x)=e^{-cx^2}\sin(\pi x/\alpha)$
| 3 | https://mathoverflow.net/users/75422 | 232697 | 108,075 |
https://mathoverflow.net/questions/232671 | 3 | Consider the first $2n$ steps of a simple random walk on the integers, starting at the origin. A simple binomial argument shows that regardless of $n$, the origin gets visited the most (in expectation).
**Question 1:** Does this fact generalize? that is, suppose we have a symmetric random walk on $\mathbb{Z}$ starti... | https://mathoverflow.net/users/74799 | Expected visits to the origin by a symmetric random walk on the integers | Here we prove that if steps are independent and identically symmetrically distributed, then Q1 has positive answer.
Let $p(k)$ be probability of step $k$, $p(-k)=p(k)$. Denote by $f(t)=\sum\_k p(k)t^k$ the generating function. Then average number of visits of 0 during first $n$ steps is $[1] g(t)$, where $g(t)=1+f(t)... | 5 | https://mathoverflow.net/users/4312 | 232702 | 108,077 |
https://mathoverflow.net/questions/232638 | 8 | I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a little to make it more transparent.
Let $f\_1, \dots, f\_n$ be $n$ functions whose Fourier transform is supported on ... | https://mathoverflow.net/users/37103 | The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$? | I checked with Jean and Ciprian about this, and there is indeed a small issue here; the bound indicated is "morally" correct, and may possibly even be true (using the weights $w\_B$ rather than a sharp truncation $1\_B$), but it does not quite follow from the standard device of representing a Fourier-localised function... | 12 | https://mathoverflow.net/users/766 | 232707 | 108,079 |
https://mathoverflow.net/questions/232700 | 1 | Consider a $\textbf{flat}$ surjective map $f: X \rightarrow \mathbb{A}^1$. The general fibers $F\_{\epsilon}$ are canonically isomorphic, and the special fiber $F\_0$ above $0 \in \mathbb{A}^1$ is not isomorphic to the general fibers.
Given a closed subscheme $B \subset X$, we define its special fiber limit $\wideti... | https://mathoverflow.net/users/7780 | Flat family: limit of intersection vs intersection of limits | Obviously $\widetilde{B\_1} \cap \widetilde{B\_2} \subseteq \widetilde{B\_1\cap B\_2}$. I'll discuss a sufficient condition for the reverse.
If $B\_1\cap B\_2$ is equidimensional, then so is $\widetilde{B\_1\cap B\_2}$. It seems like you know that its dimension is that of $\widetilde{B\_1}\cap \widetilde{B\_2}$. By a... | 4 | https://mathoverflow.net/users/391 | 232714 | 108,081 |
https://mathoverflow.net/questions/232709 | 1 | I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as
$diag(x)Ax=1$
$x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal elements of $A$ are strictly positive, other elements of $A$ are arbitrary, $1$ is a vector of ones and $diag(x)$ is just s ... | https://mathoverflow.net/users/88499 | Positive solutions to simultaneous real quadratic equations | Your system says $(Ax)\_i = x\_i^{-1}$. Suppose you had two distinct, positive solutions $x, y$. Then
$$ \eqalign{(x-y)^T A (x-y) &= \sum\_{i} (x-y)\_i (A (x-y))\_i\cr & = \sum\_i (x\_i-y\_i)(x\_i^{-1} - y\_i^{-1})\cr
&= - \sum\_{i} \dfrac{(x\_i - y\_i)^2}{x\_i y\_i} < 0} $$
which contradicts the assumption that $A$ ... | 5 | https://mathoverflow.net/users/13650 | 232716 | 108,082 |
https://mathoverflow.net/questions/231328 | 7 | **Edit:** According to comment of Pace Nielsen, I remove question 2 of the previous version:
Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows:
We say $a M b$ iff $a=xy,\;b=yx$ for some $x,y\in R$. (This is inspired by the usual Murray Von Neumann equivalent relation in K theory,... | https://mathoverflow.net/users/36688 | The saturation of Murray von Neumann relation | In answer to the second question, yes this is true. Say $x^k=0$. Let $x=v|x|$ be the polar decomposition of $x$ in $A^{\*\*}$ (the bidual of $A$). Let $a=v|x|^{\frac 1 2}$ and $b=|x|^{\frac 1 2}$. Then clearly $x=ab$.
Both $a$ and $b$ belong to $A$. In $b$'s case, by functional calculus. It is a well-known property of ... | 3 | https://mathoverflow.net/users/13381 | 232729 | 108,088 |
https://mathoverflow.net/questions/232678 | 1 | How do I prove that gauge-equivalence classes of $U(1)$ connections on a line bundle $L\to M$ are determined uniquely by pairs $(\alpha,F)$, where
$$\alpha\in\text{Hom}(\pi\_1(M),U(1)),~~~~F\in \Omega^2(M)?$$
| https://mathoverflow.net/users/69531 | Is a non-flat hermitian connection determined uniquely by its holonomy and curvature? | The 2-form $F$ has to be closed. Then you choose an open covering $U\_i$ on which $F$ has a primitive $\theta\_i$, i.e. $d \theta\_i = F$. Now you try to glue together the trivial bundles $U\_i \times U(1)$ with connection $\theta\_i$. In order to do this, the curvature form has to be integral.
Finally, equivalence ... | 2 | https://mathoverflow.net/users/17047 | 232758 | 108,096 |
https://mathoverflow.net/questions/232737 | 5 | This question is a followup to two of my previous questions, see [here](https://mathoverflow.net/q/219572/61522) and [here](https://mathoverflow.net/questions/224566/if-k-is-an-algebraically-closed-field-of-any-characteristic-then-the-fundamen).
>
> 1. Let $A$ be an abelian variety over a field $k$ of characteristi... | https://mathoverflow.net/users/nan | Property of bundles with connections on abelian variety doesn't hold for additive or multiplicative group? | No, it fails for the additive and multiplicative group as they are not compact. Consider the differential equation of the Airy function $d^2 f/dx^2 = x f$. We can write this in first order form as $df/dx=u$, $du/dx = x f$. This becomes a vector bundle with connection by taking the vector bundle to be a rank $2$ free bu... | 2 | https://mathoverflow.net/users/18060 | 232774 | 108,101 |
https://mathoverflow.net/questions/232555 | 0 | This is my first question on this community. I am a applied scientist, not a mathematician.
I have the following simplified problem:
Let $u: [0,1] \rightarrow \mathbb{R}\_+$ a real valued function and $k\in \mathbb{R}$.
The function $u(\cdot)$ is decreasing and may be continuous or not.
Let $x^\*(k)$ the value tha... | https://mathoverflow.net/users/88406 | Need help with computational and numerical methods for solve equations | squaring a function to find its zeroes is generally no good idea; first you can't exploit sign changes of the function values to conclude that a certain intervall must contain a zero (or discontinuity with sign change for left and right limit); another problem is that numeric precision gets worse because the slopes van... | 2 | https://mathoverflow.net/users/31310 | 232787 | 108,105 |
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