parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/168705 | 10 | I was thinking a bit about distribution theory the last weeks and stumbled across the following question:
There are two natural locally convex topologies on the space of smooth functions of moderate growth $\mathcal{O}\_M(\mathbb{R}^n) := \{f\in C^\infty(\mathbb{R}^n) \mid \forall\alpha: \partial^\alpha f \,\text{is ... | https://mathoverflow.net/users/3041 | Are functions of moderate growth a bornological space? | Yes, $\mathscr O\_M$ is bornological. Grothendieck calls this "not trivial" and proves it in his thesis via a tensor product representation and quite general results about tensor products. Together with Julian Larcher, I recently gave an alternative proof which is still in the refereeing process. If you are interested ... | 17 | https://mathoverflow.net/users/21051 | 168712 | 87,339 |
https://mathoverflow.net/questions/168696 | 6 | Here the definition of complete symmetric polynomial $h\_{k}$ and elementary symmetric polynomial $e\_{k}$ are:
$$
e\_{k}=\sum\_{1\le i\_1<\cdots <i\_k\le n}x\_{i\_1}\cdots x\_{i\_k}, h\_{k}=\sum\_{1\le i\_1\le \cdots \le i\_k\le n}x\_{i\_1}\cdots x\_{i\_k}
$$
I know that they are "dual" to each other in the symmet... | https://mathoverflow.net/users/18850 | Why complete symmetric polynomials and elementary symmetric polynomials are dual to each other? | I think of all of the duality statements you wrote as a consequence of the fact that there is a ring involution of $\Lambda$ sending $e\_k$ to $h\_k$, so let me give a manifestation of that. First, the $e\_k$ and $h\_k$ are algebraically independent generators, so the existence of an automorphism given by $e\_k \mapsto... | 9 | https://mathoverflow.net/users/321 | 168717 | 87,342 |
https://mathoverflow.net/questions/168621 | 12 | I have been reading about cardinal invariants and I have a question about the independence number $\mathfrak{i}$. In Blass's paper (Combinatorial Characteristics of the Continumm) it is mention that in Sacks model, Eisworth and Shelah proved that $\mathfrak{i}= \aleph\_1$, unfortunately it mentions that the result is u... | https://mathoverflow.net/users/51436 | The independence number | This takes me back to Jerusalem in the late 90s. I've actually been asked about this a couple of times this year after 15 years of hearing nothing about it.
The proof consisted of looking at the maximal independent family constructed by Shelah in the course of his proof of Con($\mathfrak{i}<\mathfrak{u}$). The m.i.f.... | 9 | https://mathoverflow.net/users/18128 | 168725 | 87,345 |
https://mathoverflow.net/questions/168704 | 16 | I'd like to know what's the name (if any) of the following categorical structure, and also references where it has been considered.
Given a category $C$, let $O=\{O(n)\}\_{n\geq 0}$ be a sequence of functors $O(n)\colon C\times\stackrel{n}\cdots\times C\rightarrow C$ equipped with natural transformations $O(n)(O(p\_1... | https://mathoverflow.net/users/12166 | What is this operad-like structure called? | If I'm getting right what you mean by "satisfying the usual relations (as for operads)" then your structure is what Tom Leinster calls in his [Higher Operads, Higher Categories](http://arxiv.org/pdf/math/0305049v1.pdf) book an *unbiased lax monoidal category*.
They can be seen as (lax) algebras for a $\mathbf{Cat}$-... | 12 | https://mathoverflow.net/users/4315 | 168731 | 87,348 |
https://mathoverflow.net/questions/168722 | 3 | Where on the Internet can I find the article of Kleiman: "Algebraic cycles and Weil conjectures. Dix exposés sur la cohomologie des schémas" ?
| https://mathoverflow.net/users/nan | Where can I find the article of Kleiman: "Algebraic cycles and Weil conjectures. Dix exposés sur la cohomologie des schémas"? | the bad news is that this book "Dix exposés sur la cohomologie des schémas" is still under copyright with Elsevier (inherited from North-Holland), which is why you will not find it legally on the internet (it's volume 3 of their "Advanced studies in pure mathematics"); the Grothendieck circle only scanned the chapters ... | 7 | https://mathoverflow.net/users/11260 | 168743 | 87,350 |
https://mathoverflow.net/questions/168740 | 10 | Let $C$ be a non-hyperbolic (smooth quasi-projective connected complex algebraic) curve. That is, $C$ is isomorphic to $\mathbb P^1, \mathbb A^1, \mathbb G\_m$, or an elliptic curve.
Let $f:X\to C$ be a family of Fano varieties, i.e., $f$ is a smooth projective morphism whose geometric fibres are (smooth projective c... | https://mathoverflow.net/users/4333 | Families of Fano varieties over non-hyperbolic curves | Let $X = SO(10)/P\_5 \subset P^{15}$ be the spinor variety. It is projectively self-dual and has codimension 5, so its generic linear section of codimension 5 is smooth, and, moreover, generic pencil of codimension 4 is smooth. On the other hand, sections of codimension 4 are parameterized by $Gr(4,16)$ which has dimen... | 7 | https://mathoverflow.net/users/4428 | 168751 | 87,353 |
https://mathoverflow.net/questions/168744 | 0 | Let $\varphi\_1,\dots,\varphi\_k$ be a set of angles in $[0,\pi)$. For each $n\in\mathbb{N}$, let
$$ \rho(n):=\min\_{j=1,\dots,k} \{n\cdot \varphi\_j \mod \pi \}. $$
Is it true that for infinitely many $n$ we have $\rho(n) \geqslant \rho(1)$?
| https://mathoverflow.net/users/23862 | Is the following conjecture regarding rotational iterates of collection of points on a circle true? | Yes, the conjecture is true.
You can assume without loss of generality that $\varphi(1)=\rho(1)$ is smaller than the other angles. Pick $\alpha>0$ so that for all $i>1$ you have $\varphi\_i>\pi\alpha+\rho(1)$. It is also clear that it suffices to find one $n>1$ such that $\rho(n)\geq \rho(1)$.
Consider points $\{n\... | 3 | https://mathoverflow.net/users/38468 | 168754 | 87,356 |
https://mathoverflow.net/questions/168750 | 2 | Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix.
What is the distribution of $u^HAv$ ( or $||u^HAv||^2$)
where : u is a column vector of U. v is a column vector of V.
Simulations give that $||u^HAv||^2$ is exponential; is it true ? if yes, how to prove it ?
... | https://mathoverflow.net/users/51469 | Probability distribution of uAv… | a simple and exact answer follows if $A$ is **a unitary $n\times n$ matrix**; then $\tilde{v}=Av$ is a unit vector and since $u$ is a randomly oriented unit vector the distribution of the scalar product $u^HAv=(u,\tilde{v})$ does not depend on the orientation of $\tilde{v}$, so we may take it in the $x\_1$ direction, h... | 2 | https://mathoverflow.net/users/11260 | 168756 | 87,357 |
https://mathoverflow.net/questions/168755 | 2 | Suppose that $\langle\mathbb{P\_\alpha,\dot Q\_\beta}\mid \beta<\delta,\alpha\leq\delta\rangle$ is a system of iterated forcing.
Let $\dot a$ be a name in $\mathbb P\_\delta$, and let $G\_\alpha$ be a generic for $\mathbb P\_\alpha$ for $\alpha<\delta$. Is there a reasonable sense in which we can interpret $\dot a$ u... | https://mathoverflow.net/users/7206 | Partial interpretation of an iteration | $\newcommand\P{\mathbb{P}}\newcommand\Q{\mathbb{Q}}$
Yes. Perhaps it is easier to think about the case of two-step iterations, which contain the whole story. Namely, from any $\P\*\dot\Q$-name $\dot a$ we may in a canonical manner construct a $\P$-name for a $\dot\Q$-name, let us call it $\ddot a$, such that if $V\su... | 5 | https://mathoverflow.net/users/1946 | 168759 | 87,358 |
https://mathoverflow.net/questions/168607 | 8 | Let $X$ be a smooth proper algebraic variety over $\mathbb{C}$.
I know that in the analytic world, there is an isomorphism between the de Rham cohomology and the cohomology of the constant sheaf $\underline{\mathbb{C}}$:
$$ H^i\_{dR}(X^{an}/\mathbb{C}) \cong H^i(X^{an}, \underline{\mathbb{C}}). $$
The proof of this... | https://mathoverflow.net/users/1046 | Cohomology of a constant etale sheaf | The etale cohomology groups $H^i(X\_{et},V)$ vanish for $i > 0$ if $X$ is a quasi-compact, quasi-separated, normal scheme, and $V$ is a $\mathbb{Q}$-vector space. (In fact, one may replace "normal" with "geometrically unibranch.") To see this, recall the following presumably classical fact.
**Lemma**: If $X$ is a nor... | 11 | https://mathoverflow.net/users/51500 | 168763 | 87,360 |
https://mathoverflow.net/questions/168767 | 5 | Let $G$ be an abelian group. Let $A\subset G$ be a finite set. $\sum\_A$ is defined as: $$\left\{\sum\_{b\in B}b \mid B\subset A\right\}$$ Is there any result similar to Freiman's Theorem for $\sum\_A$? Can we say anything about $A$ if $\sum\_A$ is small?
| https://mathoverflow.net/users/18785 | Size of distinct sums in A | One way in which $\sum\_A$ can be small is if $A$ is a small subgroup of $G$. To exclude such examples, define $X$ to be *aperiodic*, if the only solution to $X+x=X$ is $x=0$. DeVos, Goddyn, Mohar and Šámal [proved](http://arxiv.org/pdf/math/0612045.pdf?origin=publication_detail) that if $\sum\_A$ is aperiodic, then $\... | 7 | https://mathoverflow.net/users/2233 | 168768 | 87,361 |
https://mathoverflow.net/questions/168748 | 1 | On a K3 surface $S$, a linear system $|C|$ is said to be hyperelliptic if the corresponding map is of degree 2 and the image is of degree $g\_a(C)-1$ in $\mathbb P^{g\_a}$.
For $g\_a(C) > 2$, if $|C|$ is without fixed components, then it is hyperelliptic if and only if in the following cases:
$i)$ there exists an ... | https://mathoverflow.net/users/27125 | Linear system on an abelian surface | The situation is different for abelian surfaces: if $A$ contains no elliptic curves, a linear system $|C|$ on $A$ is very ample as soon as the genus of $C$ is $\geq 8$, see [this paper](http://www.cmi.ac.in/~ypandey/S.Ramanan/papers/85.pdf) of Ramanan, or §10 of Birkenhake-Lange.
| 1 | https://mathoverflow.net/users/40297 | 168771 | 87,362 |
https://mathoverflow.net/questions/74162 | 15 | In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite quandles and counting homomorphisms onto them, and so on. However, I have not yet come across any theorems that state forma... | https://mathoverflow.net/users/1392 | Formally undecidable problems on finitely presented quandles | The word problem for quandles was formulated by Rena Levitt and Sam Nelson. In Levitt's [research statement](http://pages.pomona.edu/~rml04747/Rena_Levitt/Research_files/RLevittResearchStatement.pdf) the problem was formulated as follows:
>
> **Word Problem for Quandles.** Given a finitely generated quandle $Q$, i... | 7 | https://mathoverflow.net/users/17845 | 168774 | 87,364 |
https://mathoverflow.net/questions/168776 | 2 | It was mentioned in the comments to <https://math.stackexchange.com/questions/517369/comparison-of-strong-operator-and-weak-topologies-on-bh> that continuous linear functionals on $\mathfrak{B}(\mathbb{H})$ are the same in the strong operator topology as in the weak operator topology. It doesn't seem obvious and I coul... | https://mathoverflow.net/users/51484 | Continuous linear functionals in strong operator and $\sigma$-strong topologies | The duals under the strong and the weak topologies can be identified with the finite rank operators, under the ultrastrong and the ultraweak with the trace class operators. This can be found in the classic "von Neumann algebras" by Dixmier.
Can you make precise what you mean by the $\sigma$-strong and $\sigma$-weak top... | 4 | https://mathoverflow.net/users/50767 | 168777 | 87,365 |
https://mathoverflow.net/questions/168778 | -4 | I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the Selberg class is continuous. It seems that Selberg's orthogonality conjecture (SOC for short) implies such a continuity condit... | https://mathoverflow.net/users/13625 | Is SOC known to imply the Grand Riemann Hypothesis? | It is not known that SOC implies GRH. In fact it is not even known that every automorphic $L$-function (which is the subject of GRH) belongs to the Selberg class (because the Ramanujan-Selberg conjecture is an open problem).
The rest of your post seems off-topic: MO is not for verifying mathematical arguments. For th... | 6 | https://mathoverflow.net/users/11919 | 168779 | 87,366 |
https://mathoverflow.net/questions/168762 | 3 | Conjecture - no natural number $k$ exists such that:
1. $P$ is the sequence of all primes starting from the $k$th prime
2. $A$ is a sequence of natural numbers such that:
* $\forall n : A\_n<P\_n<A\_{n+1}$
* $\forall n : F(A\_n) \leq F(A\_{n+1})$, where $F(x)$ is the number of primes in the factorization of $x$
... | https://mathoverflow.net/users/27456 | Conjecture about a sequence of natural numbers, such that, $\forall n : A_n<P_n<A_{n+1}$ | Indeed, as Tony Huynh expects, your conjecture follows from the prime triples conjecture. That conjecture imples that there exist infinitely many $k$ such that $k,6k-1,6k+1$ are all prime, and infinitely many $m$ such that $12m-1,12m+1$ are both prime. Whenever $P\_n$ is one of these $6k+1$ primes, you're forced to tak... | 6 | https://mathoverflow.net/users/5091 | 168782 | 87,368 |
https://mathoverflow.net/questions/168732 | 7 | I am looking for a proof of the equality in the title (where $q\in\mathbb N$, ($q\ge2$)).
Does anyone know such a proof?
Thanks in advance
| https://mathoverflow.net/users/33128 | $LCM(q-1,\cdots,q^n-1)=q^{\frac{3}{\pi^2} n^2+o(n)}$ | Niven, Fermat's theorem for matrices, Duke Math. J. 15 (1948) 823–826, MR0026672 (10,183e), proved, for any positive integers $a$ and $r$, $${\rm lcm}[x-1,x^2-1,\dots,x^r-1]\_{x=a}={\rm lcm}[a-1,a^2-1,\dots,a^r-1]$$ Now follow Greg Martin's comment.
| 9 | https://mathoverflow.net/users/3684 | 168788 | 87,371 |
https://mathoverflow.net/questions/168757 | 2 | I have the following, seemingly simple question:
Consider a stochastic process $(X\_t)$ satisfying $X\_t\le X\_s$ a.s. for all $t\le s.$ My question is: Does there exist a modification $\tilde{X}$ of $X$, which almost surely has increasing sample paths $t\mapsto\tilde{X}\_t(\omega)$?
I assume such a modification e... | https://mathoverflow.net/users/51498 | Increasing stochastic process | Yes, this works. As suggested by fedja, focus for a moment on the rational times. The function (on $\mathbb Q$) $f\_{\omega}(t)=X\_t(\omega)$ is almost surely increasing. Set $\widetilde{X\_t}=0$ for all $t\in\mathbb R$ on the exceptional set. Now if $t\in\mathbb R$ is arbitrary, then $f\_{\omega}(t-)\le X\_t(\omega) \... | 1 | https://mathoverflow.net/users/48839 | 168792 | 87,373 |
https://mathoverflow.net/questions/168793 | 5 | Let $G$ be locally compact group. Define group algebra as
$$L^1(G)=\{f\colon G\to\Bbb{C}\mid\int\lvert f(x)\rvert\, dx<\infty\}$$
with convolution product. When is the group algebra $L^1(G)$ semisimple?
| https://mathoverflow.net/users/51514 | When is the group algebra $L^1(G)$ semisimple? | Always, according to Naĭmark, *Normed Algebras*, VII p. 380.
| 10 | https://mathoverflow.net/users/19276 | 168794 | 87,374 |
https://mathoverflow.net/questions/168742 | 5 | Let $\phi$ be a stricly plurisubharmonic function on a domain in ${\Bbb C}^n$, and $S=\phi^{-1}(c)$ its level set. Consider $S$ as a Riemannian manifold equipped with a metric induced by $dd^c\phi$. I am interested in curvature restrictions on the Riemannian structure of $S$. In all examples I could check, its Ricci cu... | https://mathoverflow.net/users/3377 | Level set of convex and plurisubharmonic functions (Ricci curvature and other curvature conditions) | Well, I'm at the next airport and have a little time. While I don't have a complete answer worked out about the necessary and sufficient conditions on the curvature, I can definitely say that you don't get positivity of the Ricci curvature.
Here's an example: In the domain $D\subset\mathbb{C}^{n+1}$ defined by $|z\_0... | 4 | https://mathoverflow.net/users/13972 | 168799 | 87,376 |
https://mathoverflow.net/questions/168805 | 25 | It is known that no two distinct finite powers of the closed unit interval are homeomorphic:
$I^m$ is homeomorphic to $I^n$ iff $m=n$. (Brouwer, Lebesgue, 1911)
Is the analogous result for infinite powers of $I$ true?
That is, is it true that $I^\alpha$ is homeomorphic to $I^\beta$ iff $\alpha=\beta$, for cardin... | https://mathoverflow.net/users/50650 | Which powers of the closed unit interval are homeomorphic? | Yes. $I^{\alpha}=I^{\beta}$ does imply that $\alpha=\beta$. To see this, suppose that $\alpha$ is an infinite cardinal. Then each point in $I^{\alpha}$ is the intersection of $\alpha$ many open sets, but each point in $I^{\alpha}$ is not the intersection of less than $\alpha$ many open sets.
In greater detail, if $(... | 34 | https://mathoverflow.net/users/22277 | 168808 | 87,380 |
https://mathoverflow.net/questions/168812 | 2 | Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X.$ Let $F:X\to X$ be a continuos transformation that commutes with $T.$
Define $F\_{\*}\mu(A)=\mu(F^{-1}A)$ for every Borel set $A.$
I am looking for an example in ergodic theory ... | https://mathoverflow.net/users/39115 | Example of non-convergence of iteration of measures | Take $T$ to be the identity and $\mu$ the Dirac measure on a point $x \in X$. Obviously $\mu$ is $T$-invariant and $F$ commutes with $T$. Using well-known properties of the weak-\* topology, the sequence of measures $\frac{1}{N}\sum\_{k=1}^N F\_\*^k\mu$ converges if and only if for every continuous $f \colon X \to \mat... | 2 | https://mathoverflow.net/users/1840 | 168816 | 87,384 |
https://mathoverflow.net/questions/168817 | 1 | Let $T:X\to X$ be a continuous function on a compact metric space $X.$ Let $\mu$ be a $T$ invariant and ergodic probability measure on $X$ with strictly positive Sinai entropy $h\_{\mu}(T).$ Let $F:X\to X$ be a continuos transformation that commutes with $T.$
Define $F\_{\*}\mu(A)=\mu(F^{-1}A)$ for every Borel set $A... | https://mathoverflow.net/users/39115 | Non-convergence of ergodic measures with positive entropy | Let $T\_1, \hat{F} \colon X\_1 \to X\_1$ be commuting continuous transformations of a compact metric space and $\mu\_1$ a Borel probability measure on $X\_1$ which is invariant and ergodic with respect to $T\_1$, and such that $\frac{1}{N}\sum\_{k=1}^N {\hat{F}}\_\*^k\mu\_1$ does not converge, as in [your previous ques... | 2 | https://mathoverflow.net/users/1840 | 168821 | 87,386 |
https://mathoverflow.net/questions/168300 | 6 | This question is concerning the paper, particularly the proof of Lemma 2.1 in Section 2.1:
* Matas, A., Merker, J. *Existence of weak solutions to doubly degenerate diffusion equations*, Appl Math **57** (2012) pp43–69, doi:[10.1007/s10492-012-0004-0](https://doi.org/10.1007/s10492-012-0004-0), [EuDML](https://eudml.... | https://mathoverflow.net/users/49672 | Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof) | OK, I think I figured it out.$\newcommand\esssup{\operatorname{ess} \sup}$
First let me define a few things: for any given basis $\{w\_1\ldots w\_k\}$ of $A\subset L^{m'}$ let $\{w'\_1,\ldots w'\_k\}$ be the canonical dual basis of $A'$ (the finite-dimensional dual), i-e $\langle w'\_i,w\_j\rangle\_{A',A}=\delta\_{ij... | 5 | https://mathoverflow.net/users/33741 | 168825 | 87,390 |
https://mathoverflow.net/questions/168835 | 1 | Here is a "colourful" description of what I would like to count. Suppose you have one of those tables you see in a casino. I think they are for roulette, with $m$ squares, each of them with a number inside them. You have $c\_1$ chips of type 1, $c\_2$ chips of type $2,\dots,c\_n$ chips of type $n$. You have to place th... | https://mathoverflow.net/users/24478 | How is this combinatorial structure called? | The answer is the coefficient of $x\_1^{c\_1}\cdots x\_n^{c\_n}$ in $[(1+x\_1)(1+x\_2)\cdots(1+x\_n)-1]^m$. Here $(1+x\_1)\cdots (1+x\_n) -1$ corresponds to the chips placed on each square; zero or one of each color, but at least one chip. We can expand by the binomial theorem to get that this coefficient is
$$\sum\_{... | 2 | https://mathoverflow.net/users/10744 | 168836 | 87,393 |
https://mathoverflow.net/questions/168831 | 0 | Suppose $f: X \rightarrow Y$ is a morphism of schemes, and $y \in Y$ is closed point.
We know the first projection morphism $p\_1: X \ $ x$\_{Y} \ k(y) \rightarrow X$ is a homeomorphism onto $f^{-1}(y)$. Is it in fact a closed immersion of schemes?
| https://mathoverflow.net/users/51197 | Closed immersion of closed fiber? | It's the base change of a closed immersion, hence it is also a closed immersion.
| 2 | https://mathoverflow.net/users/46690 | 168839 | 87,395 |
https://mathoverflow.net/questions/168801 | 2 |
>
> A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which
> acts on $M$ and preserves complex structure then If $M$ has Kaehler
> manifold, then the symplectic quotient of $M$, i.e, $M//G$ has kaehler
> structure?
>
>
>
What about hyper-kaehler?
>
> B) If $M$ has hyper-kaehler structu... | https://mathoverflow.net/users/nan | If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler structure? | If $(X,\omega)=(M,I,\omega)$ is Kaehler, and $G$ is a compact Lie group, which acts freely on $M$ and preserves *both* $I$ and $\omega$, then the symplectic quotient *is* Kaehler. More precisely, $M//G\simeq X^{st}/G^c$, where $G^c$ is the complexification of $G$, and $X^{st}\subset X$ are the stable points for the act... | 9 | https://mathoverflow.net/users/6278 | 168842 | 87,396 |
https://mathoverflow.net/questions/168846 | 1 | I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ for every $1 \le q <2$. Does this imply that $f \in L^\infty(0, T; L^2(B))$?
I tried to build a counterexample, but at ... | https://mathoverflow.net/users/46298 | Interpolation and embeddings for parabolic function spaces | No. Take $n=2$ and define $$f(t,x) = \frac{1}{\sqrt{t^2 + \left| x \right|^2}}$$ Then for $1 \leq q < 2$
\begin{align\*}
\int\_{B\_1(0)} f(t,x)^q dx
&= 2\pi \int\_0^1 \frac{r}{(t^2 + r^2)^{q/2}} dr \\
&= 2\pi \left. \frac{1}{q+2} (t^2 + r^2)^{-q/2 + 1} \right|\_0^1
\end{align\*}
and so $f(t,x) \in L^\infty(0,T;L^... | 3 | https://mathoverflow.net/users/49404 | 168854 | 87,400 |
https://mathoverflow.net/questions/168863 | 2 | Let $(M^n,g)$ be a Riemannian manifold, and let $V$ be a $C^{\infty}$ vector field on $M$. Is it possible to locally approximate $V$ by gradient vector fields $\nabla f\_i$, such that the $L^{\infty}$-norm
$$\sup\_{B\_x(r)}|V-\nabla f\_i|\rightarrow 0?$$
for $r$ small enough. Thank you very much.
| https://mathoverflow.net/users/38600 | local approximation of a vector field on a Riemannian manifold | If you're asking the $f\_i$ to be smooth then the answer is no, unless $V$ is a gradient (in which case the statement is trivial).
Consider for example the Euclidean space $R^N$, with $B=B\_r(0)$ the standard ball centered at the origin. Assume by contradiction that there is such an approximating sequence for smooth ... | 6 | https://mathoverflow.net/users/33741 | 168865 | 87,403 |
https://mathoverflow.net/questions/168699 | 5 | This is a sort of follow up to [this MO question](https://mathoverflow.net/questions/165825/postnikov-towers-in-bounded-t-structures).
Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $... | https://mathoverflow.net/users/8320 | Homotopy factorization of morphisms of chain complexes | Consider the following diagram in which the rows are cofiber sequences:
$$\require{AMScd}\begin{CD}
\tau\_{\geq0}A @>>> A @>>> \tau\_{\leq-1}A @>>>\Sigma\tau\_{\geq0}A\\
@VVV @VVV @| @VVV\\
\tau\_{\geq0}B @>>> C @>>> \tau\_{\leq-1}A @>>>\Sigma\tau\_{\geq0}B\\
@| @VVV @VVV @|\\
\tau\_{\geq0}B @>>> B @>>> \tau\_{\leq-1}B... | 8 | https://mathoverflow.net/users/75 | 168868 | 87,405 |
https://mathoverflow.net/questions/168870 | 6 | Cross-post from <https://math.stackexchange.com/questions/808490/parking-functions-and-the-binomial-theorem>
A parking function is a function $f: \{1, \ldots n\} \rightarrow \{1, \ldots n\}$ which has the property that the list $(f(1), f(2), \ldots f(n))$ can be rearranged in some order $(a\_{1}, a\_{2}, \ldots a\_{n... | https://mathoverflow.net/users/45745 | Parking Functions and the Binomial Theorem | I'll use [this](http://math.mit.edu/~rstan/transparencies/parking3.pdf) lovely argument for the formula $\# PF = (n+1)^{n-1}$ that you mentioned as inspiration (see pg. 5 of the document).
First of all, we have to choose the $n- i$ slots for the $1$'s. So we really want to show that there are $(n-i)n^{i-1}$ ways to a... | 7 | https://mathoverflow.net/users/48839 | 168874 | 87,408 |
https://mathoverflow.net/questions/168871 | 0 | Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely using the exponential map by $(r,v)$, where $0<r< \pi/2$ is geodesic distance from the north pole and $v$ is a unit vect... | https://mathoverflow.net/users/49919 | Describe all differentiable functions on $\mathbb{S}^n \backslash S$ (S is the south pole) | I understand that one must prove that every smooth function $g : \mathbb S^n\setminus \{S\} \to \mathbb R$ can be expressed as $g = G \circ h$ where $G : U \to \mathbb R$ is a smooth function on some subset $U \subset \mathbb R^m$ for some $m$ and $h : \mathbb S^n\setminus \{S\} \to U,\ p \mapsto (h\_1(p), ..., h\_m(p)... | 1 | https://mathoverflow.net/users/51542 | 168875 | 87,409 |
https://mathoverflow.net/questions/168819 | 13 | Let $X$ be a smooth complex projective surface. Is the Hurewicz image $h(\alpha)\in H\_2(X)$ of a homotopy class $\alpha\in\pi\_2(X)$ algebraic?
| https://mathoverflow.net/users/9833 | Is $\pi_2$ algebraic? | A slightly better variant of this question is to ask: is the Hurewitz image of $\pi\_{2}(X)$ in $H\_{2}(X)$ a sub Hodge structure? This is in fact an old question of Philippe Eyssidieux. In section 4.3 of [this](http://arxiv.org/abs/math/0107129) paper we proved that if the fundamental group of $X$ is nice enough (more... | 16 | https://mathoverflow.net/users/439 | 168880 | 87,411 |
https://mathoverflow.net/questions/168866 | 7 | As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon\_0$, that is, $\omega^{\omega^\cdots}$ for $N$ layers of $\omega$. See e.g. the second answer to [Why do stacked quantifiers in PA co... | https://mathoverflow.net/users/38330 | Models of PRA/EFA with induction on $X$ but not $\omega^X$ | I suspect the paper you want is Avigad and Sommer, [A Model-Theoretic Approach to Ordinal Analysis](http://www.andrew.cmu.edu/user/avigad/Papers/alpha1.pdf). As the name suggests, they give an ordinal analysis rooted in the structure of models of arithmetic.
The main idea is that if $\alpha$ is an ordinal, we define ... | 7 | https://mathoverflow.net/users/8991 | 168886 | 87,413 |
https://mathoverflow.net/questions/39989 | 1 | I have been trying to find more information on Poisson and Lexis trials (generalizations of Bernoulli trials), but I have failed to find anything outside of MathWorld (I went through a number of statistics books and was also unable to find anything)
Does anyone know of references which are known to contain a discussi... | https://mathoverflow.net/users/6628 | References for Poisson and Lexis trials | Here is an old 1927 paper "On Poisson's and Lexis's problem of probability of repeated trials" by Jordan Charles: <http://arch.neicon.ru/xmlui/handle/123456789/1584111>
See also a short article "The Meaning of Binomial Distribution"
by A.W.F. Edwards:
<http://www.nature.com/nature/journal/v186/n4730/abs/1861074a0.h... | 1 | https://mathoverflow.net/users/32389 | 168898 | 87,417 |
https://mathoverflow.net/questions/168332 | 2 | Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$. It is know that $C$ is a set-theoretic complete intersection and that, if $n\geq 3$, is a not a scheme-theoretic complete intersection.
I would like to know if it is true that, if $n\geq 3$ then $C$ can not be written as a set-theoretic complete int... | https://mathoverflow.net/users/nan | Rational normal curves as set-theoretic complete intersections | If $\frac{2^{n-1}}{n}$ is an integer, that is if $n$ is a power of $2$, then $C$ is actually the set-theoretic complete intersection of $n-1$ quadrics.
This is a theorem of Perron (1941); to see the defining equations see formula (3) of my paper <http://www.dima.unige.it/~torrente/RationalNormalCurves.pdf>.
Furthermore... | 5 | https://mathoverflow.net/users/51558 | 168902 | 87,420 |
https://mathoverflow.net/questions/168688 | 12 | I have read parts of the motivation on nlab and all the posts on MO I could find on the subject, and by now there are a few questions on my mind. If they trivial for someone who understands the theory, I apologise beforehand. My questions are:
In Higher Differential Geometry, are we able to ...
1. ... generalise me... | https://mathoverflow.net/users/37072 | Learning higher differential geometry | You seem to be asking about my $n$Lab entries on [higher differential geometry](http://ncatlab.org/nlab/show/higher+differential+geometry) and related. I have now added there a
* [commented References section](http://ncatlab.org/nlab/show/higher+differential+geometry#References)
with some pointers. See there for a... | 18 | https://mathoverflow.net/users/381 | 168920 | 87,428 |
https://mathoverflow.net/questions/168832 | 12 | The following is stated without proof in Shelah's book "Cardinal arithmetic" (page 276), and is attributed to Uri Abraham:
>
> Suppose that $L[A], L[B]$ have no non-constructible reals and that $\aleph\_1^{L[A,B]}=\aleph\_1^L.$ Then $L[A,B]$ has no non-constructible reals.
>
>
>
How can we prove this result.
... | https://mathoverflow.net/users/11115 | Producing no non-constructible reals | (I don't know if this is Abraham's reasoning.) Let us write $\omega\_1$ for $\omega\_1^L$ throughout. Suppose for a contradiction the assumptions hold but that $L[A,B]$ has nonconstructible reals.
1) First note that the answer is easy if both $A,B \subseteq \omega\_1$:for then for some $\alpha < \omega\_1$ $L\_\alpha... | 9 | https://mathoverflow.net/users/6942 | 168925 | 87,431 |
https://mathoverflow.net/questions/168927 | 2 | Let's consider the domain $U=[-\pi,\pi]\times[-1,1]$. Assume that we have two functions $f\in H^2$ and $g\in H^{1/2}$.
I wonder if the following bound is true:
$$
\|f g\_{x\_1}\|\_{H^{-0.5}(U)}\leq C(\|f\|\_{H^2})\|g\|\_{H^{1/2}}.\quad (1)
$$
I tried using the duality pairing. Then, for a given $h\in H^{1/2}$ wi... | https://mathoverflow.net/users/33135 | A bound in Sobolev spaces of negative order | Yes, it is true. As you already found, it suffices to show that $H^2$ functions are multipliers in $H^{1/2}$. This follows by interpolation, since it is easy to show that $H^2$ functions are multipliers in both $L^2$ and $H^1$.
| 1 | https://mathoverflow.net/users/12120 | 168931 | 87,433 |
https://mathoverflow.net/questions/159936 | 3 | Let $A$ be a noetherian local ring with maximal ideal $m$. One says that an $A$-module $X$ is discrete if for every $x\in X$, there is a natural number $n$ such that $m^n.x=0$.
My question is: Given an $A$-module $M$, a submodule $N\subseteq M$ and a submodule $P\subseteq N$ such that $N/P$ is discrete, is there a s... | https://mathoverflow.net/users/11599 | The standard topology of a module over a noetherian local ring | The assertion is indeed true in a more general context. My references are B. Stenström: *Rings of Quotients*, and Lam: *Lectures on Modules and Rings*.
Let a (right) linear topology on a ring $A$ be given by a family of (right) ideals $\mathfrak{F}$ (i.e. $\mathfrak{F}$ consists of all open right ideals of $A$). On a... | 2 | https://mathoverflow.net/users/27465 | 168937 | 87,436 |
https://mathoverflow.net/questions/168938 | 5 | I cannot currently find the original, but if memory serves, Goedel once speculated that there might be a "large-cardinal completeness theorem for $V.$" This theorem would state:
**\*Theorem.** For every first-order sentence in the language of set theory, that sentence is decided by $\mathrm{ZFC}+\lambda$ for some lar... | https://mathoverflow.net/users/26080 | Is there a large-cardinal completeness theorem for $L$? | On the one hand, we cannot easily describe any such list of axioms, since if $\Lambda$ was computably enumerable, then by your third bullet point, the arithmetic consequences of ZFC+(V=L)+$\Lambda$ would be a c.e. completion of PA, contrary to the incompleteness theorem.
On the other hand, meanwhile, we can as a pure... | 11 | https://mathoverflow.net/users/1946 | 168939 | 87,437 |
https://mathoverflow.net/questions/168946 | 4 | It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model P-points (or Ramsey ultrafilters) are preserved by Sacks forcing (i.e. still generate an ultrafilter in the extension). Thus, ... | https://mathoverflow.net/users/13059 | Ultrafilters of weight $\aleph_2$ in Sacks model | I assume that by "the Sacks model" you mean the result of a countable-support iteration of Sacks forcing for $\omega\_2$ steps over a ground model satisfying GCH. On that understanding, think the answer to your question is no. If $U$ is a P-point in the final model, then, by a reflection argument, there will be ordinal... | 8 | https://mathoverflow.net/users/6794 | 168949 | 87,441 |
https://mathoverflow.net/questions/168955 | 4 | Suppose $k$ is a number field. I want to compute $H^\ast({\mathbb P}^n\_k,\mu\_l^{\otimes r})$ where $l,r\in {\mathbb N}$. I know that Milne has some computations, but he assumes throughout that his field is ${\mathbb C}$. I want to know if it differs when $k$ is number field.
Disclaimer: This is not a homework probl... | https://mathoverflow.net/users/51145 | etale cohomology of ${\mathbb P}^n_k$ | Let $G:=\mathrm{Gal}(\bar{k}/k)$. There is a Hochschild-Serre spectral sequence
$$E^{pq}\_2=H^p(G, H^q(\mathbb{P}^n\_{\bar{k}}\,,\mu \_l^{\otimes r})\ \Rightarrow\ H^\*(\mathbb{P}^n\_{k}\,,\mu \_l^{\otimes r})\ .$$
The cohomology of $\mathbb{P}^n\_{\bar{k}}$ is well known, but the Galois cohomology is rather difficult ... | 6 | https://mathoverflow.net/users/40297 | 168958 | 87,445 |
https://mathoverflow.net/questions/168952 | 35 | For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having **prime-power degree**.
>
> **Question**: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$?
>
>
>
My money is on "no", because I see no obvious reason why it should be true. If the answer is indeed ... | https://mathoverflow.net/users/6779 | Is my field algebraically closed? | $\def\QQ{\mathbb{Q}}$Building on YCor's construction, your field is $\overline{\mathbb{Q}}$. Let $L$ be any finite Galois extension of $\QQ$ with Galois group $G$; I will show that $L$ is contained in your field.
We can find an element $x$ in $L$ so that, for any nonempty subset $S$ of $G$, the product $\prod\_{\sig... | 34 | https://mathoverflow.net/users/297 | 168973 | 87,454 |
https://mathoverflow.net/questions/168986 | 5 | Let $L/K$ be a Galois extension with Galois group and $\mathfrak p$ be a prime of the ring of integers $\mathcal O\_K$.
I would like to prove that $H^1(G, I\_{\mathfrak p})=1$ where $I\_{\mathfrak p}$ is the free abelian group generated by the primes of $\mathcal O\_L$ above $\mathfrak p$.
Obviously, $I\_{\mathfrak p... | https://mathoverflow.net/users/33128 | $H^1$ and fractional ideals group | This vanishing is very different from Hilbert's theorem 90. I will only sketch a proof as it is rather standard and it will be good for you to fill in the details.
Your module is a permutation module, which means that it has a Galois invariant basis. This is a special kind of induced representation, and the cohomolog... | 6 | https://mathoverflow.net/users/5101 | 168992 | 87,463 |
https://mathoverflow.net/questions/168976 | 4 | **Set-up:**
Consider the trivial extension, where $p$ is the projection onto the $\mathbb{Z}\_2$ component,$$1\rightarrow N\rightarrow N\times\mathbb{Z}\_2\xrightarrow{p}\mathbb{Z}\_2\rightarrow 1$$
Define actions of $\mathbb{Z}\_2=\{1,T\}$ on a $\mathbb{Z}\_2$-module $A$ by$$\phi^1\_T:a\mapsto a\ \text{ (trivial)}\ ... | https://mathoverflow.net/users/51107 | spectral sequence with non-trivial action on coefficients | I guess here $\mathbb Z/2$-module means $\mathbb Z[\mathbb Z/2]$-module. So let's take $N=1$, $A=\mathbb Z$. $H\_0(\mathbb Z/2,A)$ is the coinvariant
$A/\mathbb Z/2$ so it is $A$ in the case of the trivial action, $\mathbb Z/2$ in the case of the sign action. So $E\_2$ term depends on the action.
| 4 | https://mathoverflow.net/users/43326 | 168998 | 87,467 |
https://mathoverflow.net/questions/168999 | 1 | Suppose $ X $ and $ T $ are complex algebraic varieties and let $ Y $ be a subvariety of $ X $ . If we have a coherent sheaf $\mathcal {F} $ on the product $ X\times T $ flat over $ X $, is it true that the restriction of $\mathcal {F} $ to $ Y\times T $ is also flat over $ Y $?
| https://mathoverflow.net/users/51601 | restrictions and flatness | Actually the answer is yes, and this is true in a much more general setting: any morphism $Z\rightarrow X$, $\mathcal{F}$ a quasi-coherent sheaf on $Z$ flat over $X$, any base change $Y\rightarrow X$; then the pull back of $\mathcal{F}$ to $Y\times \_XZ$ is flat over $Y$. This is EGA IV.2, Prop. 2.1.4.
| 3 | https://mathoverflow.net/users/40297 | 169003 | 87,469 |
https://mathoverflow.net/questions/168283 | 9 | Let $A\in\mathfrak{gl}(\mathbb{R},n)$ be an endomorphism, and think up to conformal factors (in particular, $\Lambda^n\mathbb{R}^n$ will be the same as $\mathbb{R}$). By the *total polarization* $\widehat{p}$ of a homogeneous polynomial $p\in S^k V$, where $V$ is a vector space, I mean the image of $p$ via the embeddin... | https://mathoverflow.net/users/22606 | What is the total polarization of the determinant? | Your brutal answer is slightly wrong. Namely, your map $(\mathbb R^n)^{\otimes k}\to \Lambda^k(\mathbb R^n)$ which is given by
$$
x\_1\otimes\dots\otimes x\_k \mapsto A\_1x\_1\otimes\dots\otimes A\_kx\_k\mapsto A\_1x\_1\wedge\dots\wedge A\_kx\_k
$$
does not factor over $x\_1\otimes\dots\otimes x\_k\mapsto x\_1\wedge\do... | 6 | https://mathoverflow.net/users/26935 | 169008 | 87,470 |
https://mathoverflow.net/questions/169009 | 10 | I was reading a paper by David Marker, whose main theorem was that if $T$ is a first-order theory which is not small, then $F\_2\leq\_B \cong\_T$. That's not especially relevant to the question at hand, except to point out that it comes with no set-theoretic hypotheses and no "it is consistent that..." hedging. So the ... | https://mathoverflow.net/users/15735 | A "suitably generic" set of Cohen reals without forcing? | A Cohen generic real, over a model $M$ of set theory, is an infinite binary sequence $g$ whose initial segments get eventually inside any dense set $D$ of finite sequences with $D\in M$. (The set $D$ is dense, if it contains extensions of any given finite sequence.)
When one speaks informally about "sufficiently gen... | 10 | https://mathoverflow.net/users/1946 | 169012 | 87,472 |
https://mathoverflow.net/questions/169007 | 7 | Consider an extension\begin{equation}1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1\end{equation}Let $K$ act on a $K$-module $A$ by $\phi\_k: a\mapsto k\cdot a$. Define a $G$-action $a\mapsto g\cdot a:=\rho(g)\cdot a$; under this action, $A$ is a $G$-module. Now consider the Hochschild-Serre spectral sequ... | https://mathoverflow.net/users/51107 | Hochschild-Serre spectral sequence and non-trivial action on coefficients | The action of $K$ on $H\_q(N;A)$ may be non-trivial, even though $A$ is a trivial $N$-module. This may even happen if the extension is central, so that the conjugation action of $K$ on $N$ is trivial. You have to take into account the coefficient homomorphisms
$$
(\phi\_k)\_\ast : H\_q(N;A)\to H\_q(N;A),\quad k\in K.
$... | 3 | https://mathoverflow.net/users/8103 | 169015 | 87,473 |
https://mathoverflow.net/questions/168980 | 20 | In *"Standard conjectures of algebraic cycles"* Grothendieck says:
>
> "... These [Standard conjectures] are not really new, and they were worked out about three years ago independently by Bombieri and myself..."
>
>
>
Did Bombieri publish something of this work?
| https://mathoverflow.net/users/nan | Is there any publication of Bombieri about the standard conjectures on algebraic cycles? | Here's Bombieri's [publication list](http://www.unipi.it/ateneo/comunica/cerimonie/honoris/pubblicazioni.htm) and there seems to be just one paper that could qualify [1], unfortunately not available online. A later paper with Andreotti [2] is online, but does not quite seem to be the one.
[1] E. Bombieri, [Varieta a... | 12 | https://mathoverflow.net/users/11260 | 169020 | 87,474 |
https://mathoverflow.net/questions/149358 | 9 | If there is a proper class of Woodin cardinals, then Woodin showed (using stationary towers) that $(\Sigma^2\_1)^{\text{uB}}$ statements are generically absolute, where $\text{uB}$ denotes the pointclass of universally Baire sets of reals.
This generic absoluteness result has a more local version: if $\lambda$ is a lim... | https://mathoverflow.net/users/1682 | Are there trees for $(\Sigma^2_1)^{\text{uB}}$? | The answer is yes. Hugh Woodin showed me the following argument, which I post here with his permission.
Let $\varphi(v)$ be a formula in the language of set theory expanded by a unary predicate symbol. Given a pair of ordinals $(\alpha, \beta)$,
working in $V^{\text{Col}(\omega,\alpha)}$
we let $B$ be a universally B... | 4 | https://mathoverflow.net/users/1682 | 169021 | 87,475 |
https://mathoverflow.net/questions/168827 | 2 | Let $f\_n \to f$ on compact subsets of the real line (these are functions defined on the real line) satisfying some conditions: $f$ has linear growth (but is nonlinear function) and is continuous and increasing and passes through the origin. The $f\_n$ are smooth, satisfy the same linear growth condition, and $f\_n'$ a... | https://mathoverflow.net/users/51526 | A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$ | I agree with your edited question.
In fact, the information about $u\_n$ seems to be unnecessary.
Consider $g\_n := f\_n(u\_n)$; then your assumptions give
>
> $g\_n\rightharpoonup w$ in $L^2(0,T;L^2)$ and $\partial\_t g\_n$ is bounded in $L^2(0,T;H^{-1})$.
>
>
>
Indeed by the Lions-Aubin Lemma, for whic... | 1 | https://mathoverflow.net/users/16530 | 169047 | 87,480 |
https://mathoverflow.net/questions/168993 | 8 | Take generators $a\_{ij}$ for $1\le i,j\le n$. Impose relations which come from the matrix
\begin{eqnarray\*}
P=\left(\begin{array}{ccc}a\_{11} & \dots & a\_{1n} \\\vdots & \ddots & \vdots \\a\_{n1} & \dots & a\_{nn}\end{array}\right)
\end{eqnarray\*}
being a Hermitian projection, i.e. $a\_{ij}^\*=a\_{ji}$ and $\sum\_... | https://mathoverflow.net/users/29625 | Universal unital $C^*$ algebra generated by the relations of an n by n projection | This is example (12), pp. 253 in the paper of B. Blackadar, "Shape Theory for C\*-algebras", [Math. Scand. **56** (1985) 249-275](http://www.mscand.dk/article/view/12100/10116), which lays the general theory for universal C\*-algebras. The resulting universal C\*-algebra is called, as you may be expecting, the "noncomm... | 11 | https://mathoverflow.net/users/11211 | 169050 | 87,481 |
https://mathoverflow.net/questions/168912 | 7 | Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}\_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian $\mathcal{O}$-algebras with residue field $k$ and with local homomorphisms inducing identity map $k \to k$ on residue fields.
Denote b... | https://mathoverflow.net/users/10001 | formally smooth functor | The answer is affirmative, if one is careful about making the appropriate definitions (e.g., the test objects in the definition of formal smoothness, and the correct topology on the algebra of "formal power series" in an arbitrary set of variables). But I am not aware of any literature reference. This should have been ... | 8 | https://mathoverflow.net/users/43107 | 169051 | 87,482 |
https://mathoverflow.net/questions/169019 | 3 | I have a problem in which it would be helpful to know about the integral representations of some groups of small order (probably of fairly low degree). From what I've gathered so far, cyclic groups of order p and order p^2 are understood, as are some special dihedral groups. But, often, Krull-Schmidt does not hold maki... | https://mathoverflow.net/users/34884 | Integral representations of groups of small order | A couple of comments are combined here into a partial answer to the question:
As Geoff indicates, *Methods of Representation Theory I* by Curtis-Reiner has an extensive treatment of integral representations toward the end. In §34 there is a detailed discussion of examples, with the cautionary remark that a complete c... | 3 | https://mathoverflow.net/users/4231 | 169059 | 87,485 |
https://mathoverflow.net/questions/169052 | 3 | Let $R$ be a (Noetherian) ring, and $G$ a finite group acting on $R$. Consider the subring $R^G$. Is the map $R^G\rightarrow R$ faithfully flat?
If not, does this become true if we restrict to varieties?
Thanks!
| https://mathoverflow.net/users/4181 | Are quotients of affine schemes by finite groups faithfully flat? | The answer is no even for $G=\mathbb{Z}/2$ acting on $R=k[x,y]$ by swapping $x$ with $−x$ and $y$ with $−y$. In this case $R$ is finite, but not flat, over $R^G=k[x^2,xy,y^2]$, for example because the length of the fiber at 0 is 3, while the map has degree 2.
| 11 | https://mathoverflow.net/users/3847 | 169069 | 87,487 |
https://mathoverflow.net/questions/162168 | 6 | Let $f(x\_1, \dots, x\_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ is bounded.
I have the following questions:
1. Is it clear that $f$ is measurable (with respect to Borel sets on ... | https://mathoverflow.net/users/46852 | Multivariable monotonic function | $f$ need not be Borel measurable: Let $f(x,y)=0$ on $x+y<1$ and $f=1$ on $x+y>1$, and on the diagonal $x+y=1$, set $f=1/2$ for $x\in E$ and $f=0$ otherwise, where $E\subset [0,1]$ is not Borel. Then $f^{-1}(\{ 1/2 \})$ is not a Borel set in the square.
| 5 | https://mathoverflow.net/users/48839 | 169077 | 87,489 |
https://mathoverflow.net/questions/169083 | 5 | Suppose that there is a polynomial $P$ with integer coefficients such that $P(x\_i)=y\_i$ for $i=1,\ldots,n$. Is it true that the result of Lagrange interpolation through the data $(x\_i,y\_i)$ is a polynomial with integer coefficients?
Suppose there are $2$ cases :
$\bullet$ $x\_i$ are integers
$\bullet$ $x\_i$ ar... | https://mathoverflow.net/users/50068 | Lagrange Interpolation and integer polynomials | The answer is yes if the $x\_i$ are integers. Write
$$
B(X):=\prod\_{i=1}^n (X-x\_i),
$$
so that $B(X)$ is a monic polynomial in $\mathbf{Z}[X]$. For any $P(X)\in\mathbf{Z}[X]$, we can write
$$
P(X) = B(X) Q(X) + R(X)
$$
where $Q,R\in\mathbf{Z}[X]$ and $\deg(R)<n$. Here $R(x\_i)=P(x\_i)=y\_i$ for every $i$. Since there... | 5 | https://mathoverflow.net/users/30412 | 169084 | 87,493 |
https://mathoverflow.net/questions/169090 | 1 | One of the techniques used to quantifying the size of a point on an elliptic curve is the so called **canonical height** defined as follow: Let $R=(x,y)∈C(ℚ)$ where $x=(p/d),p,d∈ℤ$. Define the *naive* or *Weil height* of $R$ as
$$h(R)=log(max{|p|,|d|})$$
Then the canonical height $h$ is defined uniquely as map from... | https://mathoverflow.net/users/25947 | Can we use this formula to construct rational points on the curve $C$? | As ACL noted, $f^{(m)}(1)$ is supposed to be a multiple of the height regulator, not the height of a single point. Assuming Birch-Swinnerton-Dyer, one can in principle use the value of $f^{(m)}(1)$ to assist in searching for rational points (but not to write down a formula), as explained in
Computing rational points ... | 1 | https://mathoverflow.net/users/11926 | 169103 | 87,496 |
https://mathoverflow.net/questions/168964 | 2 | And is there a reference to which someone could point me for this?
| https://mathoverflow.net/users/51585 | Is the intersection of n-1 nef divisors in an n-dimensional algebraic variety guaranteed to be in the Mori cone? | The Mori cone $\overline{NE}(X)$ is the dual of $Nef(X)$. Let $C = D\_{1}\cap...\cap D\_{n-1}$ be the intersection of $n-1$ nef divisors. Since $Nef(X) = \overline{Amp}(X)$ we can find a sequence of divisors $D\_{i,\epsilon}$ with $\epsilon > 0$ such that $D\_{i,\epsilon}$ is ample for any $\epsilon >0$, and $D\_{i,\ep... | 1 | https://mathoverflow.net/users/14514 | 169109 | 87,499 |
https://mathoverflow.net/questions/169118 | 6 | It is a theorem of Steinhaus that for any finite measure $\mu$, the Banach space $L\_1(\mu)$ is weakly sequentially complete. Using the Radon-Nikodym theorem one can extend this easily to $\sigma$-finite measures. What about arbitrary measures. Is $L\_1(\mu)$ always weakly sequentially complete?
| https://mathoverflow.net/users/51659 | How does one prove that $L_1(\mu)$ is weakly sequentially complete for any measure? | Yes. You can reduce the general case to the separable case, and every separable $L\_1$ is clearly isometrically isomorphic to $L\_1(\mu)$ with $\mu$ a finite measure.
For the reduction of the general case, let $X$ be the closed sublattice generated by your weakly null sequence. This is an abstract $L\_1$ space and s... | 5 | https://mathoverflow.net/users/2554 | 169125 | 87,503 |
https://mathoverflow.net/questions/169033 | 23 | Invariant Subspace Conjecture: A bounded operator on a separable Hilbert space has a non-trivial closed invariant subspace.
Can this conjecture be reformulated as an arithmetic statement, that is, $\Pi^0\_n$ statement for some n? (I tried to figure it out, but failed.)
EDIT: For what I understand from answers, it ... | https://mathoverflow.net/users/9833 | Is the Invariant Subspace Problem arithmetic? | I played with this a few years ago at <http://terrytao.wordpress.com/2010/06/29/finitary-consequences-of-the-invariant-subspace-problem/> ; in the language of the analytical hierarchy, I was trying to lower the complexity of the invariant subspace problem from $\Pi^1\_2$ to $\Pi^1\_1$. I didn't quite succeed, because I... | 15 | https://mathoverflow.net/users/766 | 169132 | 87,506 |
https://mathoverflow.net/questions/169128 | 5 | My research is somehow related to the following question :
Describe and classify all finite groups $G$ such that $G=HK$ with $H \cap K=1$, where $H \cong A\_m$ and $K \cong A\_n$ for some integers $m, n$ greater than 4. ($A\_n$ denotes alternating group of degree $n$).
In a paper of O. Kegel and H. Luneberg, (*Uber... | https://mathoverflow.net/users/13525 | Finite groups factorized into two simple alternating groups | For $m=5$ and $n$ arbitrary, all factorizations are classified here:
>
> W. R. Scott, *Products of $A\_5$ and a finite simple group*, J. Algebra
> **37** (1975), 165--171.
>
>
>
For $m=6$ and $m=7$ (again with $n$ arbitrary) all factorizations are classified by a series of papers by Darafsheh and various coau... | 8 | https://mathoverflow.net/users/801 | 169139 | 87,507 |
https://mathoverflow.net/questions/169159 | 23 | A [recent MO question about non-rigorous reasoning](https://mathoverflow.net/questions/115032/non-rigorous-reasoning-in-rigorous-mathematics) reminded me of something I've wondered about for some time.
The genus–degree formula says that genus $g$ of a nonsingular projective plane curve of degree $d$ is given by the for... | https://mathoverflow.net/users/3106 | Rigorous version of heuristic argument for genus-degree formula? | Yes, this argument can be made rigorous. One needs three steps.
**Step 1.** Show that there is *at least one* smooth plane curve of degree $d$ with the expected genus. Essentially, the proof is given by your heuristic topological argument (deform the union of $d$ lines in general position).
**Step 2.** Show that if... | 23 | https://mathoverflow.net/users/7460 | 169161 | 87,513 |
https://mathoverflow.net/questions/169127 | 6 | Let $(M,g)$ be a compact Riemannian manifold without boundary. Fix a point $x\in M$ and $N\ge 2$ large. Then there exists a metric $\tilde g$, conformal to $g$ such that $$ \det \tilde g=1+O(r^N)$$ for $r\le inj(x)$ the injectivity radius at $x$, where $r=|x|$ in the $\tilde g$-normal coordinates. References: Lee, Park... | https://mathoverflow.net/users/39507 | Taylor expansion of the determinant of a Riemannian metric | The answer to your question is yes. To see why, write $\det \tilde g = 1 + f$, where $f$ is a smooth function satisfying $f=O(r^N)$. Let $(u^i)$ be a smooth coordinate chart centered at $x$. Because $f$ is smooth, one version of Taylor's theorem says that near $x$, we can write
$$
f(u) = \sum\_{i\_1,\dots,i\_N} h\_{i\_... | 7 | https://mathoverflow.net/users/6751 | 169164 | 87,515 |
https://mathoverflow.net/questions/169170 | 6 | I have a sequence of real numbers. I want to know whether this sequence looks like the traces in the standard representation of a random sequence of elements of $G\_2$. (Here random is according to the Haar measure on the compact form.)
So I want to compare my sequence with the measure on $\mathbb R$ that is the push... | https://mathoverflow.net/users/18060 | How to estimate the Haar measure on $G_2$ | Here's a method that should work...
Step 0: Consider the octonions with basis $1, i,j, ij, \ell, \ell i, \ell j, \ell i j$, where $i,j,\ell$ are imaginary unit octonions, with $ij = - ji$, and $\ell$ orthogonal to $H = R + Ri + Rj + Rij$.
Step 1: Choose a random imaginary unit octonion, i.e. a random point $i'$ on ... | 6 | https://mathoverflow.net/users/3545 | 169175 | 87,520 |
https://mathoverflow.net/questions/169178 | 3 | I have been recently reading some papers on universality of spectral statistics of random matrices written by Terry Tao, Van Vu, L. Erdos, H.T. Yau and others, and I am puzzled by such a dichotomy, namely:
one should treat the bulk case and edge case separately (using possibly different techniques).
Could someone ... | https://mathoverflow.net/users/14390 | Random matrices: why to distingusish bulk and edge cases? | Intuitively, the fact that the density vanishes at the edge already hints that the spacing at the edge changes (which indeed it does), and that the asymptotics are different; this is confirmed in the Gaussian setup - the different scaling leads to a different scaling of the orthogonal polynomials (Hermite in this case)... | 3 | https://mathoverflow.net/users/35520 | 169184 | 87,523 |
https://mathoverflow.net/questions/169168 | 4 | I have a question regarding the following cited text from [1]:
>
> Let $F$ be a representation of the structure group $G$ of the principal bundle $P\_G\to M$ (a (semi-)Riemannian manifold), and let $F(P)$ denote the quotient of $P\_G\times F$ by the $G$ action.
> Let $P\_G\to M$ be a principal bundle with structure... | https://mathoverflow.net/users/nan | Equivalence of the construction of the Lagrangian in a book of Sternberg to the "usual" construction | There is no essential contradiction between the definitions in the two references that you gave, except that the one given in the nLab entry is more general.
Sternberg defines the Lagrangian (function) as a scalar-valued differential operator. One can easily change this definition to be volume form valued by simply r... | 5 | https://mathoverflow.net/users/2622 | 169185 | 87,524 |
https://mathoverflow.net/questions/169192 | 3 | I'm reading Ghrist's paper "[Configuration spaces and braid groups on graphs in robotics](http://www.math.upenn.edu/~ghrist/preprints/birman.pdf)".
In [this](https://mathoverflow.net/questions/139598/a-sufficient-condition-for-a-space-to-be-an-eilenberg-maclane-space) discussion, counterexamples are shown for both Theo... | https://mathoverflow.net/users/49267 | Configuration spaces of trees are Eilenberg-MacLane spaces | The paper
>
> Patty, C. W., *Homotopy groups of certain deleted product spaces,*
> Proc. Amer. Math. Soc. 12 (1961) 369–373.
>
>
>
contains a proof that for any connected, finite graph having essential vertices (of degree $\ge3$) the "deleted product" $$C\_2(\Gamma) = \Gamma\times\Gamma \setminus \Delta\Gam... | 4 | https://mathoverflow.net/users/8103 | 169194 | 87,526 |
https://mathoverflow.net/questions/169191 | 6 | I'm reading Katznelson's book "Harmonic Analysis" and there is an exercise that I can't solve :
>
> Show that if the sequence $\left\{ N\_j \right\} $ tends to infinity
> fast enough, then the Fourier series of the function $$ f(t)=\sum\_{j>0}2^{-j }K\_{N\_j}$$ does not converge in $L^1(\mathbb{T})$.
>
>
>
$K... | https://mathoverflow.net/users/51697 | A Fourier series that does not converge in $L^1$ | Let us give sufficient conditions on the growth rate of $N\_l$ for the Fourier series to diverge.
We consider $p = N\_l + 1$ and $q = \frac1M N\_{l+1}$. For $p < |n| \leq q$ we have that $$ \hat{f}(n) = \sum\_{j > l} 2^{-j}(1 - \frac{|n|}{N\_j + 1}) \geq (1 - \frac{1}{M}) \sum\_{j > l} 2^{-j} = (1 - \frac{1}M) \cdot... | 5 | https://mathoverflow.net/users/3948 | 169200 | 87,527 |
https://mathoverflow.net/questions/169199 | 2 | The following equation is considered:
$$
\frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f.
$$
I have difficulties in formulating boundary conditions for this equation.
If $\mathbf v$ is absent, the boundary conditions are
$$
a\frac{\partial u}{\partial n} + \beta(u - u\_b) = 0
$$
where $u\_b$ is... | https://mathoverflow.net/users/48757 | Heat transfer: boundary conditions with fluid velocity | your second boundary condition is missing a factor $u-u\_b$:
$$a\frac{\partial u}{\partial n} - \gamma(u-u\_b)(\mathbf v \cdot \mathbf n) + \beta(u - u\_b) = 0$$
the coefficient $\beta$ gives the strength of the heat transfer at the boundary; the coefficients $a$ and $\gamma$ are the same as in the diffusion-convec... | 3 | https://mathoverflow.net/users/11260 | 169207 | 87,529 |
https://mathoverflow.net/questions/169186 | 9 | While having lunch today with my advisor I tried to come up with a proof of the following fact:
>
>
> >
> > EGA 0-IV (17.3.3): Let $\phi : (A,\mathfrak{m}) \to (B,\mathfrak{n})$ be a flat local homomorphism of local Noetherian rings (necessarily faithfully flat). Then $B$ regular implies $A$ is regular.
> >
> > ... | https://mathoverflow.net/users/21278 | Descent of regularity under a faithfully flat morphism: Where does my proof fail? | (1) No. Use prime avoidance: if the maximal ideal is not associated, it is not contained in the union of the associated primes together with $\mathfrak{m}^2$.
(2) Yes, consider $k[[x^2]] \subset k[[x]]$.
| 16 | https://mathoverflow.net/users/460 | 169211 | 87,531 |
https://mathoverflow.net/questions/169203 | 6 | Let $M$ be a closed Riemannian manifold. I have several questions concerning the set of all minimal submanifolds (or immersion) in $M$.
(1): Is there a general local theory for the set of minimal submanifold? What I mean is that, let $N \subset M$ be a minimal submanifold, then it corresponds to the critical point o... | https://mathoverflow.net/users/41094 | A general theory for local moduli space of minimal surface? | In general, asking whether or not all Jacobi fields on a minimal surface can be "integrated" to find a nearby minimal surface is a very difficult problem. For example, see Yau's remark [here](http://faculty.ccri.edu/joallen/M2910/Geometry%20and%20Analysis_Yau.pdf) (page 246):
>
>
> >
> > Unfortunately minimal sub... | 6 | https://mathoverflow.net/users/1540 | 169244 | 87,546 |
https://mathoverflow.net/questions/169250 | 2 | I have a problem in understanding the *inertia group* of an infinite extension. I am studying it in this context.
Let $K$ be a field, $v$ a discrete valuation on $K$, and $\mathcal{O}\_v$ the discrete valuation ring of $v$. Let $k$ be the residue field of $v$, we assume $k$ to be perfect. Let $K\_s$ be a separable cl... | https://mathoverflow.net/users/51717 | Maximal unramified extension and inertia group for separable closure | So it's been a while since I've thought about valuations, but this is too long for a comment anyway.
Firstly, I don't think $\overline{\nu}$ is a discrete valuation. For example, if $\pi$ is the uniformizer of $K$, then $\sqrt[n]{\pi}$ should have valuation $1/n$, so there isn't any "minimum positive valuation", whic... | 1 | https://mathoverflow.net/users/15242 | 169252 | 87,550 |
https://mathoverflow.net/questions/169243 | 2 | In [this paper](http://arxiv.org/abs/math/0411485) Vigeland shows how one can define a group law on subset of a tropical elliptic curve, so that this group is homeomorphic to $S^1$. It is not clear to me what is the relationship between the definition of tropical elliptic curve in the paper to tropicalization of (a tor... | https://mathoverflow.net/users/2234 | group structure on (subsets of) tropicalizations of Abelian varieties | Abelian varieties have tropicalizations, but the picture is not exactly as you guess.
Let me begin with the case of elliptic curves. So let $A$ be an elliptic curve over a non-archimedean field $K$, let $j$ be its $j$-invariant. Two possibilities arise.
If $|j(A)|\leq 1$, then, possibly after enlarging $K$, it is p... | 3 | https://mathoverflow.net/users/10696 | 169255 | 87,551 |
https://mathoverflow.net/questions/169231 | 0 | I'm interested in computing the direct limit of an arbitrary $2\times 2$ primitive matrix over $\mathbf{Z}$. That is for a fixed primitive matrix $M$, the colimit $\displaystyle\lim\_{\stackrel{\longrightarrow}{n}}(\mathbf{Z}^2,M)$ over a diagram of shape $\mathbf{N}$.
Is there an algorithmic way of computing these g... | https://mathoverflow.net/users/21271 | Direct limit of primitive integral matrices | Since you mention primitive, I guess you mean matrices with nonnegative entries such that some power of the matrix is strictly positive?
There is a large literature on this type of problem, especially when we take into account the ordered abelian group structure (when the matrices have only nonnegative entries). But... | 1 | https://mathoverflow.net/users/42278 | 169257 | 87,552 |
https://mathoverflow.net/questions/169240 | 10 | A finite group $G$ has a finite set of irreducible representations over the complex numbers. All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is abelian. Moreover, if the group $G$ is not abelian, those representations which *are* linear can be described by replacin... | https://mathoverflow.net/users/11124 | Which finite nonabelian groups have all their quaternionic representations of degree one? | **EDIT**: My previous answer was completely wrong. Here's a corrected version.
First, note that this property descends to subgroups: every irrep of a subgroup is a summand of a restriction from the bigger group. Thus if all the irreps of the bigger group are 1-d, all the irreps of the small group will be their restr... | 7 | https://mathoverflow.net/users/66 | 169259 | 87,553 |
https://mathoverflow.net/questions/169256 | 5 | Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of certain local factors which are rational functions in $p^{-s}$. The local factor at $p$ is the zeta function of the fibe... | https://mathoverflow.net/users/349 | Local factors of Hasse-Weil zeta function - what do they have in common? | This is an elaboration on ACL's answer, way too long for a comment, which highlights a technical ingredient (well-known to all experts) that underlies the precise sense in which the $\ell$-adic etale cohomology of the geometric generic fiber provides a "uniformity" in $p$: the good properties of constructible $\ell$-ad... | 23 | https://mathoverflow.net/users/43107 | 169274 | 87,559 |
https://mathoverflow.net/questions/169273 | 0 | Suppose $n$ values, $X\_1,...,X\_n,$ are generated by a random number generator with normal distribution $N(0,1).$ Suppose that the (sample) mean of $X\_1,...,X\_n$ is $\mu.$ What is known about the order statistics of $X\_1,...,X\_n$? (Eg. what is the expected value of $Y\_c=\#\{i: X\_i>c\}$ as an expression in $n,\mu... | https://mathoverflow.net/users/23935 | Expected number of samples above certain value of a normally distributed variable with a given sample mean | Hint: the conditional distribution of $X\_i$ given $X\_1 + \ldots + X\_n = \mu$ is
normal ...
EDIT:
Sorry for just giving a hint before: I was busy.
$X\_1$ and $S = X\_1 + \ldots + X\_n$ are jointly normal with means $0$ and
covariance matrix $C = \pmatrix{1 & 1 \cr
1 & n\cr}$. So they have joint density
$$f\_{X... | 1 | https://mathoverflow.net/users/13650 | 169275 | 87,560 |
https://mathoverflow.net/questions/169278 | 7 | Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has:
>
> Mathematics knows no minimum interval of continuous magnitudes [such as distance and time.] Attempts to supply a logical definition of such an infinitesimal min... | https://mathoverflow.net/users/nan | Survey of the history of calculus? | *A History of Analysis*, edited by Hans Niels Janhke, American Mathematical Society (2003) is a superb, all together scholarly collection of essays that cover a wide range of topics in the history of analysis.
Three other noteworthy works are:
*The Development of Newtonian Calculus in Britain, 1700-1800*, Niccolò G... | 11 | https://mathoverflow.net/users/18939 | 169280 | 87,563 |
https://mathoverflow.net/questions/169014 | 10 | Let $S^0\_p$ be the $p$-adic sphere spectrum. Let $GL\_1(S^0\_p)$ be the set of unit componen of $\Omega^{\infty}S^0\_p$. For any map $ X \to BGL\_1(S\_p^0)$ we get a Thom spectrum call it $Mf$. Now consider the identity map on $GL\_1(S^0\_p)$, the Thom spectrum associated to that map is what I call the universal Thom ... | https://mathoverflow.net/users/19186 | The homotopy of universal Thom spectrum | Assume that $R$ is a connective $E\_\infty$ ring spectrum. Typically $GL\_1(R)$ denotes the set of components in $\Omega^\infty R$ which span $GL\_1(\pi\_0 R)=\pi\_0 R^\times$. I would call the unit component $SL\_1(R)$. I will use this notation below.
In the formalism of Ando-Blumberg-Gepner-Hopkins-Rezk, the Thom ... | 5 | https://mathoverflow.net/users/8818 | 169282 | 87,564 |
https://mathoverflow.net/questions/169251 | 8 | Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$ is absolutely convergent in $\{s\in\mathbb{C}:\operatorname{Re}(s)>1\}$. I can only convince myself for the special... | https://mathoverflow.net/users/51720 | Absolute convergence of Rankin–Selberg series | This is an elaboration of Lucia's [comment](https://mathoverflow.net/questions/169251/absolute-convergence-of-rankin-selberg-series#comment428677_169251). Let us consider the Dirichlet coefficients of $L(s,\pi\times\pi')$, $L(s,\pi\times\tilde\pi)$, $L(s,\pi'\times\tilde\pi')$ at a prime power $p^k$. Following the proo... | 11 | https://mathoverflow.net/users/11919 | 169291 | 87,566 |
https://mathoverflow.net/questions/169235 | 4 | As defined by Karen Smith [here, beginning of section 3](http://projecteuclid.org/euclid.mmj/1030132733)? If $E$ is an elliptic curve, then it is when $E$ is ordinary. I wonder about higher dimension cases. Any references would be greatly appreciated.
| https://mathoverflow.net/users/2083 | Are Abelian varieties (sometimes) globally $F$-split? | The equivalence between ordinarity and Frobenius-splitting for abelian varieties (in fact smooth varieties with trivial tangent bundle) can be found in the paper by Mehta and Srinivas "Varieties in positive characteristic with trivial tangent bundle" (see <https://eudml.org/doc/89874> ). There are also many things abou... | 9 | https://mathoverflow.net/users/14440 | 169292 | 87,567 |
https://mathoverflow.net/questions/167954 | 18 | Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact and Hausdorff, the property of being equi-locally convex seems slightly arbitrary here, while the weaker property of be... | https://mathoverflow.net/users/51164 | Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex? | After some more digging I found a (somewhat non-explicit) counterexample to the original question. In his paper "un espace metrique lineaire qui n’est pas un retracte absolu" Cauty constructs a metric vector space $V$ which is not an absolute neighborhood retract. According to the characterization established in "une c... | 18 | https://mathoverflow.net/users/51164 | 169294 | 87,568 |
https://mathoverflow.net/questions/92194 | 5 | A distance-regular graph (DRG) is, in essence, a graph $\Gamma$ of diameter $d$ for which there are integers $c\_i, a\_i, b\_i, (0 \le i \le d)$ such that for *all* vertices $x$ of $\Gamma$ and for all vertices $y$ of distance $i$ from $x$, the number of vertices $z$ adjacent to $y$ and distance $i-1$ from $x$ is $c\_i... | https://mathoverflow.net/users/11124 | Graphs which are "distance-regular" with respect to a vertex (but not distance-regular) | The response by Chris Godsil is sufficient as an answer to this question. The general description of these graphs is collected in section 9.3 (Equitable Partitions) of Algebraic Graph Theory by Chris Godsil and Gordon Royle. As a family of graphs, the "equitable partition" requirement is not restrictive enough to get a... | 1 | https://mathoverflow.net/users/11124 | 169302 | 87,571 |
https://mathoverflow.net/questions/146553 | 21 | Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant curvature $-1$. Recall that, from Teichmüller theory, this is a finite-dimensional subspace of the space of all metrics... | https://mathoverflow.net/users/15743 | Avoiding integers in the spectrum of the Laplacian of a Riemann surface | I was informed by Sugata Mondal at the MPI that Scott Wolpert proved the following result in his 1994 Annals paper *Disappearance of cusp forms in special families*:
>
> **Theorem 5.14.** *The eigenvalues of the Laplacian above $\tfrac14$ on a closed hyperbolic surface vary nontrivially under analytic deformations.... | 18 | https://mathoverflow.net/users/15743 | 169308 | 87,572 |
https://mathoverflow.net/questions/169321 | 6 | If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a\_i x\_i$ with $a\_i \in A, x\_i \in X$ subject to some natural relations involving $0,+,$ and the basepoint of $X$. We topologize this space as a quotient of $\bigsqcup A^n \times X^n$ wh... | https://mathoverflow.net/users/37268 | Homotopy groups of linearization of a space | This result appears in M. C. McCord, [*Classifying Spaces and Infinite Symmetric Products*](http://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251719-4/S0002-9947-1969-0251719-4.pdf) published in Transactions of the American Mathematical Society, Vol. 146, (Dec., 1969), pp. 273-298.
Also, this can be gene... | 8 | https://mathoverflow.net/users/644 | 170326 | 87,577 |
https://mathoverflow.net/questions/170330 | 3 | Let $(A,a)$ be a principally polarised (with indecomposable polarisation) Abelian variety over $\mathbb C$. Assume that End(A) contains an order $R$ of a totally real number field of degree $>1$ over $\mathbb Q$ and that $R$ respects the polarisation of $A$.
The units of $R^\*$ are therefore automorphisms of $(A,a)$... | https://mathoverflow.net/users/51764 | Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication |
>
> I presume I am making a very basic error somewhere but I don't see where.
>
>
>
Yes: you are confusing the automorphism group $\operatorname{Aut} A = (\operatorname{End} A)^{\times}$ with the automorphism group of the *polarized* abelian variety $(A,a)$. The former can be infinite as soon as $g = \operatorna... | 5 | https://mathoverflow.net/users/1149 | 170331 | 87,578 |
https://mathoverflow.net/questions/170334 | 2 | Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$.
Let $\mathbb{Z}\_{p}$ denote the ring of $p$-adic integers, and $\mathbb{Q}^{cyc}$ be the cyclotomic $\mathbb{Z}\_{p}$-extension of $\mathbb{Q}$ with $\Gamma=\text{Gal}(\mathbb{Q}^{cyc}/\mathbb{Q}) \cong \mathbb{Z}\... | https://mathoverflow.net/users/30999 | Some questions related to Iwasawa invariants of elliptic curves | 1) Iwasawa theory, as practiced by K.Iwasawa, is concerned with $\mathbb Z\_{p}$-extensions. There is only one $\mathbb Z\_{p}$-extension of $\mathbb Q$. Over more generally number fields, and in more general context, it is emphatically not true that only the cyclotomic extension is considered. The reason why we consid... | 4 | https://mathoverflow.net/users/2284 | 170337 | 87,581 |
https://mathoverflow.net/questions/170329 | 2 | **Question**
By Slodowy slice I mean a transverse slice at a subregular nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ (in particular I am not intersecting with the nilpotent cone). Consider the usual eigenvalue map
$$\mathfrak{g} \rightarrow \mathfrak{h}//W$$
which is equivariant with respect to the scaling ... | https://mathoverflow.net/users/6059 | A canonical G_m (or G) action on the Slodowy slice | Well, it's not completely canonical, but it's close: the $\mathbb{G}\_m$ action you're looking for is given by the Kazhdan grading (that's the induced grading on the polynomial ring $\mathbb{C}[\mathfrak{g}^\*]$), defined (I kind of assume not for the first time) in 4.1 of Gan and Ginzburg's paper *[Quantization of Slo... | 6 | https://mathoverflow.net/users/66 | 170339 | 87,583 |
https://mathoverflow.net/questions/170361 | 1 | Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it?
to clearly express the problem assume that
$$
z(\mathbf{a,B,c})=\mathop {\inf} \left\{ {\bf{a^Tx}
|\quad\bf{Bx-c}\le0} \right\}
$$
is any closed form (rather linear) function $y(\mathbf{a,B,c})$ ve... | https://mathoverflow.net/users/51780 | a closed form lower bound solution for linear programming | Take the dual. Any dual feasible solution will give you a lower bound, by weak duality (assuming your problem is feasible and not unbounded).
| 2 | https://mathoverflow.net/users/51757 | 170362 | 87,590 |
https://mathoverflow.net/questions/170349 | 11 | Is there a nice way to partition the edges of the complete $5$-uniform hypergraph
on $11$ vertices into $7$ copies of the Steiner system $S(4,5,11)$? If this is
obvious or elementary, I apologize in advance.
| https://mathoverflow.net/users/nan | On the Steiner system $S(4,5,11)$ | Unfortunately, no. It is known that the maximum number of mutually disjoint $S(4,5,11)$s on the same point set is $2$. Any such pair are always isomorphic. So, you can't find $7$ disjoint copies of an $S(4,5,11)$ in the complete $5$-uniform hypergraph on $11$ vertices (or partition it into copies); you can find only tw... | 13 | https://mathoverflow.net/users/27829 | 171360 | 87,591 |
https://mathoverflow.net/questions/171358 | 3 | Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}\_2{\mathbb C}$?
Equivalently, what is a good reference (for mathematicians) for the representation theory of the Lorentz group?
| https://mathoverflow.net/users/39082 | R-linear representations of sl(2,C) | The real-linear representations are sums of irreducibles, and the irreducibles are parametrized by pairs of nonnegative integers: see Knapp's §2.3 [here](http://books.google.com/books?id=QCcW1h835pwC&pg=PA31).
Explicitly the irreducible parametrized by $(m,n)$ acts on the polynomials $f$ in $(z\_1,z\_2,\bar z\_1,\bar... | 10 | https://mathoverflow.net/users/19276 | 171361 | 87,592 |
https://mathoverflow.net/questions/171368 | 7 | The *cofinality* of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ there is a larger element $q\in T$ such that $p\leq q$.
A *$\sigma$-ideal* $I$ of $\mathbb{R}$ is a collection of su... | https://mathoverflow.net/users/52782 | Cofinality of a $\sigma$-ideal of $\mathbb{R}$ | $\textbf{A counterexample when $2^{\aleph\_{0}}$ is regular}$.
This holds if $2^{\aleph\_{0}}$ is a regular cardinal. In fact, it holds for any regular cardinal. If $\kappa$ is a regular cardinal, then the ideal of non-stationary sets in $\kappa$ cannot be generated by $\kappa$ many elements. To prove this fact, supp... | 8 | https://mathoverflow.net/users/22277 | 171373 | 87,598 |
https://mathoverflow.net/questions/141100 | 14 | Let $p\_1\ p\_2\ \ldots$ be the sequence of all natural prime numbers. There is a slight (just slight) but clear tendency for imitating the number of primes in an interval $(p\_k;\ p\_n)$ by the number of primes in the double interval $(p\_k\!+p\_{k+1};\ p\_{n-1}\!+p\_n)$; possibly by $(2\cdot p\_k; 2\cdot p\_n... | https://mathoverflow.net/users/8385 | An interaction between prime numbers | The answer to P1 is negative, thanks to the [recent work of Maynard](http://arxiv.org/abs/1311.4600) on bounded gaps between primes.
What Maynard shows is that given any $d$, there exists a k-tuple $h\_1,\dots,h\_k$ such that for infinitely many $n$, at least $d+1$ of $n+h\_1,\dots,n+h\_k$ are prime. In fact, the arg... | 23 | https://mathoverflow.net/users/766 | 171374 | 87,599 |
https://mathoverflow.net/questions/169285 | 2 | Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple?
As noted in the comments, since $R$ itself is not semisimple this can not be true.
What if $R$ is semisimple (and has charact... | https://mathoverflow.net/users/38889 | A semisimple group ring | In any case, whether the associative $R$ is commutative or not ( as long as $R$ has $1$) if every finitely generated unital $RG$ module is to to be completely reducible, then $|G|1\_{R}$ has to be invertible in $R$. The proof I give may seem laboured, but I don't want to make unjustified assumptions.
For let $A$ be t... | 4 | https://mathoverflow.net/users/14450 | 171384 | 87,603 |
https://mathoverflow.net/questions/171383 | 8 | Let $G$ be a connected, simply-connected complex semisimple group. We have the famous Springer resolution $$T^\*(G/B)\rightarrow\mathcal{N}$$ of the closure of the regular nilpotent orbit. My understanding is that the closures of nilpotent $G$-orbits $\mathcal{O}$ admit symplectic resolutions $$T^\*(G/P)\rightarrow\ove... | https://mathoverflow.net/users/25358 | Which nilpotent orbit closures admit Springer resolutions? | This is answered (except for a few cases in $E\_7$ and $E\_8$) in:
Baohua Fu, [Symplectic Resolutions for Nilpotent Orbits](http://arxiv.org/abs/math/0205048).
Maybe it's worth mentioning: orbits which are the image of $T^\*(G/P)$ for some $P$ are called **Richardson** and are relatively easy to understand. The tric... | 9 | https://mathoverflow.net/users/66 | 171385 | 87,604 |
https://mathoverflow.net/questions/169063 | 10 | Let $\mathcal{J}\_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $j'\_{n+1/2,1}$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}\_{n+1/2}$.
---
My question: Given a non-negative integer $n$, is it true that $|\mathcal{J}\_{m+1/2}(j'\_{n+1/2,1})|<|\ma... | https://mathoverflow.net/users/50891 | Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum? | We suppose $0<m<n$. It is known, from Landau's paper quoted above that
$
n\to n^{1/3}J\_n(x\_n)
$
increases (to some universal constant $b$). Thus
$$
J\_n(x\_n)>\left(\frac{m}{n}\right)^{1/3}J\_m(x\_m)>\left(\frac{m}{x\_n}\right)^{1/3}J\_m(x\_m).
$$
It is also known that for $x>n$,
$
x\to \sqrt{x^2-n^2}((J\_n(x))^2+(Y\... | 4 | https://mathoverflow.net/users/40120 | 171413 | 87,617 |
https://mathoverflow.net/questions/171418 | 3 | I read about Du Val singularities on surface are classified by equations of ADE type. For example, $x^2+y^2+z^{n+1}=0$ for A type. As not every surface can have a neighbourhood embedded in $\mathbb{A}^3$. Why do we use analytic coordinates to characterize singularity?
| https://mathoverflow.net/users/nan | Why do we use analytic coordinates to characterize singularity? | The point is that, *locally*, any Du Val singularity (that is, any isolated surface singularity that arises by contracting an $A$-$D$-$E$ curve) can be realized as a double cover of a nonsingular surface. This means that there exist local analytic coordinates such that the germ of singularity has the form $$x^2=f(y, \,... | 3 | https://mathoverflow.net/users/7460 | 171421 | 87,618 |
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