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https://mathoverflow.net/questions/121076 | 5 | I'm looking for a definition of pseudo differential forms in ordinary differential
geometry. However searching the web gave me nothing. There are definitions in supergeometry
but that is not what I'm after.
Recently I read, that pseudo-differentialforms are the natural structure to integrate,
since integration work... | https://mathoverflow.net/users/21302 | Pseudo-Differentialforms | Pseudo-Forms:
Let $M$ be a topological manifold and $PM$ the frame bundle of $M$. If $dim(M)=n$ then $PM$
is a $Gl(n)$-principal bundle.
Let $\tau: Gl(n) \to \mathbb{R} \; ; \; A \mapsto abs(det(A))$ the map, that maps any linear
isomorphism $f \in Gl(n)$ to the absolute value of its determinant. This defines a lef... | 3 | https://mathoverflow.net/users/21302 | 121541 | 68,344 |
https://mathoverflow.net/questions/121540 | 3 | Fix an algebraic curve $C$ of genus $g$, and positive integers $d, r$. The variety $W^r\_d$ parametrizes complete linear series of degree $d$ and dimension at least $r$ on $C$ and the variety $G^r\_d$ parametrizes linear series of degree $d$ and dimension $r$ on $C$. I am trying to understand, what are the scheme struc... | https://mathoverflow.net/users/10626 | On the construction of the varieties parametrizing special linear series on a curve | For your first question, look up Grauert's theorem in Hartshorne. The vanishing of $R^1 \nu\_\* \mathcal L (\gamma)$ holds because the line bundle on the fiber has degree $\geq 2g-1$. Similarly $R^1 \nu\_\ast (\mathcal L(\Gamma)/\mathcal L) = 0$ since when this sheaf is restricted to fibers it has zero dimensional supp... | 3 | https://mathoverflow.net/users/7399 | 121544 | 68,346 |
https://mathoverflow.net/questions/121549 | 4 | If $G\subseteq\omega^{<\omega}$ is a computable clopen game, then $G$ has a winning strategy which is hyperarithmetic $(\Delta^1\_1)$, by an inductive ranking process. The key observation here is that the length of this induction is bounded above by the length of the Kleene-Brouwer ordering $G\_{KB}$, which is a comput... | https://mathoverflow.net/users/8133 | Complexity of winning strategies for open games (for open player) | The answer is yes.
The point is that if there is any winning strategy for a
designated player from a given position, then there is in a sense
a canonical winning strategy, which is to make the first move that
minimizes the game value of the resulting position, and for a
given winning position in a fixed game, I claim... | 5 | https://mathoverflow.net/users/1946 | 121551 | 68,349 |
https://mathoverflow.net/questions/121553 | 3 | Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\mathcal{F\_{t}})\_{0\le t\le T}$ a filtration generated by standard Brownian motion $W\_t$.
Let $f(x)$ be $C^1$ function such that $|f'(x)| > 0$. Then if $\xi$ is a random variable measurable w.r.t. $\mathcal{F}\_T$, process $M\_t=E(f(\xi)|\mathcal{F\_t})$ is ... | https://mathoverflow.net/users/19988 | On martingale representation theorem | No, but under some regularity conditions you might represent $a(t)$ in terms of Malliavin calculus by means of the Clark-Ocone formula (see e.g. the [Lecture notes](http://www.math.wisc.edu/~kurtz/NualartLectureNotes.pdf) by Eulalia Nualart, Section 1.5.3.)
| 4 | https://mathoverflow.net/users/20026 | 121560 | 68,352 |
https://mathoverflow.net/questions/116702 | 7 | Starting with the trivial model structure on the category of simplicial sets (that is the weak equivalences are exactly the isomorphisms and the cofibrations and fibrations are arbitrary maps), is it possible to get the usual Quillen model structure on simplicial sets by performing a number of explicit left and right B... | https://mathoverflow.net/users/1841 | The Quillen model structure on simplicial sets as a Bousfield localization | Such sequence of localizations/colocalizations does not exist, since [homotopy is not concrete](http://www.tac.mta.ca/tac/reprints/articles/6/tr6.pdf).
Suppose, for contradiction, that such sequence is constructed, then we obtain a corresponding sequence of reflections/coreflections on the level of homotopy categori... | 9 | https://mathoverflow.net/users/30641 | 121566 | 68,353 |
https://mathoverflow.net/questions/121563 | 3 | Let $T(t),t\geq 0$, be a $C\_0$-semigroup on a Banach space $X$. If $A$ is the infinitesimal generator of $T(t),t\geq 0$, then
$$T(t)x=\lim\_{n\infty}(I-\frac{t}{n}A)^{-n}x$$
for every $x \in X, t\geq 0$, and the limit is uniform on any bounded interval $[a,b]\subset [0,\infty)$. This is the exponential formula.
I wo... | https://mathoverflow.net/users/24060 | On exponential formula for $C_0$ semigroups | If you have an analytic semigroup (generated by a so-called [sectorial operator](https://en.wikipedia.org/wiki/Analytic_semigroup)), then the answer is well-known and yes, even convergence rates are possible. This and generalizations were proved by many, see for example the summary and references in
[M. Crouzeix, S. ... | 2 | https://mathoverflow.net/users/12898 | 121576 | 68,359 |
https://mathoverflow.net/questions/121585 | 3 | I decided to go for math at 36. Is this idea possible? I studied literature, political science and international relations and still I am not really sure what I am doing.
Since I was kid, I was not good at math and I remember, I had hard time to pass the college math courses, but with the help of short brush up course ... | https://mathoverflow.net/users/31378 | Math major at 36 | Follow your dreams!
One thing you should be aware of is that there is a disconnect between the math that non-mathematicians are familiar with --- everything in calculus and below --- and the stuff that comes after. A lot of people are surprised when they get there.
The stuff that comes after calculus is very abstra... | 11 | https://mathoverflow.net/users/29956 | 121590 | 68,364 |
https://mathoverflow.net/questions/121587 | 0 | In finitely presented groups, the question of the existence of a projective resolution $P\_i$ (with each $P\_i$ finitely generated) of $\mathbb{Z}G$ is equivalent to the existence of a $K(G,1)$ which has finitely many cells in each dimension. Polycyclic groups are finitely presented.
Are there polycyclic groups whic... | https://mathoverflow.net/users/18974 | Polycyclic group not of type $FP_\infty$ | Polycyclic groups are certainly of type $FP\_\infty$ since the integral group ring $\mathbb{Z}G$ is Noetherian ([<http://plms.oxfordjournals.org/content/s3-4/1/419>](http://plms.oxfordjournals.org/content/s3-4/1/419)) and so every finitely generated module over it has a resolution by finitely generated projective modul... | 5 | https://mathoverflow.net/users/345 | 121591 | 68,365 |
https://mathoverflow.net/questions/121592 | 1 | In MAGMA, I input the following:
G:=SmallGroup(20,3);
G;
E:=[xx:xx in G];
S:=[E[6],E[7],E[13],E[20]];
S;
S[1]^2;
S[2]^2;
S[3]\*S[4];
This gives the output:
GrpPC : G of order 20 = 2^2 \* 5
PC-Relations:
G.1^2 = G.2,
G.2^2 = Id(G),
G.3^5 = Id(G),
G.3^G.1 = G.3^2,
G.3^G.2 = G.3^4
... | https://mathoverflow.net/users/31382 | My output of a group and inverse-closed subset in MAGMA is no longer inverse-closed when entered as input to GAP. | The point is that in GAP, a^b means b^-1\*a\*b rather than b\*a\*b^-1. If you adjust
your relations accordingly, you get what you expect:
```
gap> rels:=[F.1^2*F.2^(-1),F.2^2,F.3^5,F.1^-1*F.3*F.1*F.3^(-2),
> F.2^-1*F.3*F.2*F.3^(-4)];;
gap> G:=F/rels;; S:=[G.2,G.2*G.3,G.1*G.3^2,G.1*G.2*G.3^4];;
gap> S[1]^2 ... | 7 | https://mathoverflow.net/users/28104 | 121596 | 68,368 |
https://mathoverflow.net/questions/121602 | 6 | A curve in Euclidean space $\mathbb{R}^n$, $n \geq 2$ is $analytic$ if the coordinates of its points $x= x\_{1},...,x\_{n}$ can be expressed as analytic functions of a real parameter $x\_{i}=x\_{i}(t)$, $i=1,...,n$ and $\alpha \leq t \leq \beta$ and the derivatives $x'(t\_{0})$ do not simultaneously vanish for any $t\_... | https://mathoverflow.net/users/31376 | Two different analytic curves cannot intersect in infinitely many points | Wrong: $t\mapsto \binom{t}{\sin(t)}$ and $t\mapsto\binom{t}{\cos(t)}$.
Edit: Grzegorz pointed out that $t$ is in a compact interval $[\alpha,\beta]$.
Under this assumption the statement is correct.
Proof: Let $f,g:[\alpha,\beta]\to \mathbb R^n$ be the two real analytically parameterized curves which intersect in $f... | 12 | https://mathoverflow.net/users/26935 | 121605 | 68,374 |
https://mathoverflow.net/questions/121606 | 0 | This question is a follow-up of the question I asked [here](https://mathoverflow.net/questions/104827/classification-of-long-exact-sequences).
Can you write down an explicit example of a monomorphism of finitely generated Abelian groups which is an indecomposable object in the category of two-step complexes of Abelia... | https://mathoverflow.net/users/1291 | Explicit indecomposable monomorphism of finitely generated non-indecomposable Abelian groups. | How about the map $f:\mathbb{Z}\to\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z})$ given by $f(n)=(2n,n)$?
Or for a finite example, the same idea works with a map $\mathbb{Z}/4\mathbb{Z}\to(\mathbb{Z}/8\mathbb{Z})\oplus(\mathbb{Z}/2\mathbb{Z})$.
Or if you want neither term to be indecomposable, the map $(m,n)\mapsto(2m,m+... | 1 | https://mathoverflow.net/users/22989 | 121609 | 68,376 |
https://mathoverflow.net/questions/121615 | 6 | Let $n\geqslant 5$ and let $E\_4(n)$ be a linear subspace of $(n\times n)$- real skew-symmetric matrices such that
$$
rank(A)=4,\text{ for all }A\in E\_4(n),A\neq 0.
$$
I'm curious about the following question:
QUESTION:
What can be said about the dimension of $E\_4(n)$? Of course, it is easy to check that $\opera... | https://mathoverflow.net/users/27832 | Subspace of Skew-symmetric Matrices of Rank Four | Partial progress: It's easy to achieve $n-3$. Consider matrices of the form
$$\begin{pmatrix}
0 & 0 & r\_1 & r\_2 & \cdots & r\_{n-3} & 0 \\
0 & 0 & 0 & r\_1 & \cdots & r\_{n-4} & r\_{n-3} \\
r\_1 & 0 & & & & & \\
r\_2 & r\_1 & & & & & \\
\vdots & \vdots & & & {\LARGE 0}& & \\
r\_{n-3} & r\_{n-4} & & & & & \\
0 & r\... | 2 | https://mathoverflow.net/users/297 | 121623 | 68,383 |
https://mathoverflow.net/questions/121571 | 43 | A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B$ and $M'\setminus \mathring B'$ along their boundary (an $(n-1)$-sphere) by an orientation-reversing homeomorphism. The construction depends... | https://mathoverflow.net/users/10696 | Connected sum of topological manifolds | In the topological category the proof that connected sum is well-defined depends on the Annulus Theorem, first proved by Kirby; the necessity of the Annulus Theorem is seen from Bruno Martelli's answer. So you are not likely to find a proof before Kirby's paper. Perhaps someone jotted a proof down, maybe someone who th... | 37 | https://mathoverflow.net/users/20787 | 121635 | 68,389 |
https://mathoverflow.net/questions/121610 | 6 | It is an apparently well known result that if one has a 1-parameter family of smooth Calabi-Yau 3-folds which acquires a node at a boundary point, then there is a vanishing lagrangian 3-sphere, and the geometrical monodromy is a Dehn-twist along this sphere whose effect on the homology of our base fiber is given by the... | https://mathoverflow.net/users/13139 | Analogue of Picard-Lefschetz formulas for more than one node? | In general, I think that the monodromy should be
$$\alpha\mapsto \alpha - \sum\_i (\delta\_i\cdot \alpha)\delta\_i $$
where $\delta\_i$ are the vanishing cycles corresponding to each node. I'll see whether I can
write down an outline of an argument later.
... Meanwhile later... Let me complement Tim's excellent comm... | 4 | https://mathoverflow.net/users/4144 | 121639 | 68,390 |
https://mathoverflow.net/questions/107188 | 7 | In “Algebras and Modules in Monoidal Model Categories” Schwede and Shipley introduced the monoid axiom. If a cofibrantly generated monoidal model category $M$ satisfies this axiom and some smallness hypotheses, then the category $Mon(M)$ of monoids inherits a model structure with fibrations and weak equivalences taken ... | https://mathoverflow.net/users/11540 | Does the Monoid Axiom hold for k-spaces? | It seems to me that there is a relatively simple answer to this question but perhaps I am overlooking something.
The category of K-spaces does satisfy the monoid axiom. (If I read the question correctly K-spaces are what is usually called "compactly generated spaces" and compactly generated spaces are what is usually... | 2 | https://mathoverflow.net/users/12547 | 121646 | 68,395 |
https://mathoverflow.net/questions/121552 | 5 | I want an example of a group $G$, a normal subgroup $H$, and a prime number $p$, such that:
* $G$ is powered over $p$, i.e., every element of $G$ has a unique $p^{th}$ root in $G$.
* $H$ is also powered over $p$, i.e., every element of $H$ has a unique $p^{th}$ root in $H$.
* The quotient group $G/H$ is not powered o... | https://mathoverflow.net/users/3040 | Normal subgroup that is invariant under powering such that the quotient group is not invariant | This is a corrected answer. I apologize for not posting a complete proof here.
Recall that a group $G$ is called *divisible* if for every $g\in G$ and $n\in \mathbb N$, there is $x\in G$ satisfying $x^n=g$. Recall also that there exist countable (and even finitely generated) torsion free divisible groups where every ... | 8 | https://mathoverflow.net/users/10251 | 121647 | 68,396 |
https://mathoverflow.net/questions/121633 | 4 | I'm sorry if this question is too elementary, but I asked it at [MathStackExchange](https://math.stackexchange.com/questions/301039/what-is-an-ideal-supporting-algebra) and it received no responses.
On the Wikipedia page for [congruence relation](http://en.wikipedia.org/wiki/Congruence_relation) it mentions how for g... | https://mathoverflow.net/users/29763 | What is an ideal-supporting algebra? | It looks like the term ‘ideal-supporting algebra’ was written by me and survived slightly more than a decade on Wikipedia without being altered. (Well, somebody added a hyphen, a change that I agree with.) Since I put brackets around it, I'm sure that I must have heard the term somewhere, but I couldn't tell you now. N... | 8 | https://mathoverflow.net/users/8508 | 121648 | 68,397 |
https://mathoverflow.net/questions/121649 | 0 | On the space $S=\{ 0,1,\ldots,m \}^{\mathbb{N}}$ for some $m\in \mathbb{Z}\_{+}.$ And given a probability $\mu$ on it. Is it true that $\mu$ is fully supported if and only if it is ergodic for the action of the shift?
| https://mathoverflow.net/users/31217 | Ergodicity with respect to the shift | No, there are many fully-supported non-ergodic measures. Just take a convex combination of a fully-supported measure and anything else. (Recall that an invariant measure is ergodic if and only if it cannot be writen as a non-trivial convex combination of two distinct invariant measures.)
Also there are many ergodic m... | 6 | https://mathoverflow.net/users/5701 | 121650 | 68,398 |
https://mathoverflow.net/questions/121128 | 3 | I have been reading the appendix in Serre's Local fields, to do with explicit computations of cup products (pg 176), but I'm stuck on one bit of lemma 4. It goes as follows
Let B be a $G$-module, $u: G \times G \rightarrow B$ a 2-cocycle and $\bar{u} \in H^2(G,B)$. Then for all $s \in G$ with $\bar{s} \in \widehat{H}... | https://mathoverflow.net/users/15566 | Computations of cup products in Serre's Local Fields | The critical point with your problem is to understand how the connecting homomorphism is computed. Let $P$ be a complete resolution . Then Tate cohomology is the cohomology of the cocomplex $Hom\_G(P,-)$. Let
$$0 \to B \xrightarrow{i}B' \xrightarrow{\rho}B''\to 0$$
be exact. We have a commutative diagramm with exact r... | 2 | https://mathoverflow.net/users/27895 | 121666 | 68,408 |
https://mathoverflow.net/questions/118562 | 7 | Suppose that $X\_1, X\_2, \ldots, X\_n$ are i.i.d random variables distributed according to [Weibull distribution](http://en.wikipedia.org/wiki/Weibull_distribution) with shape $0 < \epsilon < 1$ (it means that $\mathbf{Pr}[X\_i \geq t] = e^{-\Theta(t^{\epsilon})}$).
Now consider the random variable $S\_n = X\_1 + X\... | https://mathoverflow.net/users/3448 | Tails of sums of Weibull random variables | The affirmative answer can be found in [this](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tvp&paperid=1116&option_lang=eng) paper by A.V.Nagaev: essentially "the conjecture" is true for $\alpha = \varepsilon$ (which is clearly the best possible).
| 1 | https://mathoverflow.net/users/3448 | 121672 | 68,412 |
https://mathoverflow.net/questions/121636 | 5 | We have a well defined notion of complex orientation for a spectrum (coh. theory) $E$, that is, we have a class $x\_E\in \tilde{E}^2(\mathbb{C}P^\infty)$ which restricts to identity along the inclusion $\mathbb{C}P^1\hookrightarrow \mathbb{C}P^\infty$. Is there a connection between this notion of "complex orientation" ... | https://mathoverflow.net/users/11546 | Connection between complex orientations and R-orientations for a ring spectrum R? | When I wrote the comment above, my memory was blanking. The connection between ring maps $MU\to R$ and complex orientations that Mark describes goes back to Quillen's original work relating $MU$ to formal group laws. (Lemma 4.6, page 52, in Adams Stable Homotopy and Generalized Cohomology characterizes ring maps $MU\to... | 8 | https://mathoverflow.net/users/14447 | 121674 | 68,414 |
https://mathoverflow.net/questions/121607 | 2 | Let $U,V\subset \mathbb{C}P^2$ be complementary sets (ie $U\simeq V^c$) with $M=U\cap V$ a $3$-manifold. Assume further $M$ is orientable.
My question is, must it be the case that one of $i\_\*:H\_2(U;\mathbb{Q})\to H\_2(\mathbb{C}P^2;\mathbb{Q})$ or $i\_\*:H\_2(V;\mathbb{Q})\to H\_2(\mathbb{C}P^2;\mathbb{Q})$ are no... | https://mathoverflow.net/users/19176 | Complementary Sets in $\mathbb{C}P^2$ | Here is an argument using cup products, which shows that whenever $\mathbb{C}P^2=U\cup V$ for open sets $U$ and $V$, one of the inclusions $U,V\subseteq \mathbb{C}P^2$ must be non-trivial on second homology.
Working with coefficients in an arbitrary field $k$, cohomology is dual to homology. By the naturality of the ... | 2 | https://mathoverflow.net/users/8103 | 121683 | 68,416 |
https://mathoverflow.net/questions/121685 | 19 | Is the closed unit ball of the Hilbert space (or, for that matter, of the Hilbert cube, in some metric) homeomorphic to the unit sphere (viz., its own boundary) ? This is clearly uncharacteristic of finite-dimensional cubes. This question is motivated by general considerations in dimension theory. If there is such a ho... | https://mathoverflow.net/users/50650 | Is the closed unit ball of the Hilbert space homeomorphic to the unit sphere ? | The answer to the question in the title is yes.
In Bessaga and Pelczynski, *Selected topics in infinite-dimensional topology*, Chapter VI, §2 there is a proof of the following:
>
> **Theorem.** Each of the following sets is homeomorphic to the countable product $\mathbb{R^N}$ of the real line:
>
>
> 1. The sep... | 21 | https://mathoverflow.net/users/29555 | 121698 | 68,424 |
https://mathoverflow.net/questions/121568 | 0 | Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a ... | https://mathoverflow.net/users/10035 | On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients | $$
g(x)h(x)=(x^2-4x+5)(x^{14}+5x^{13}+16x^{12}+40x^{11}+81x^{10}+125x^9+96x^8
$$
$$
\qquad -x^7+6x^5+25x^4+71x^3+160x^2+286x+355)
$$
$$
=x^{16}+x^{!5}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+240x^9+484x^8
$$
$$
+ x^7 + x^6 + x^5 + x^4 + x^3 + 11x^2 + 10x + 1775.
$$
$g(2)=1$, $h(2)=378919$.
In fact, it is known that eac... | 6 | https://mathoverflow.net/users/17581 | 121705 | 68,426 |
https://mathoverflow.net/questions/121638 | 7 | Let $T$ be a commutative monoid, written additively. The set $T$ is equipped with a canonical pre-order, defined by $s \le t$ when there exists $s' \in T$ so that $s + s' = t$. Consequently, $T$ may be equipped with the [specialization topology](http://ncatlab.org/nlab/show/specialization+topology) for this pre-order, ... | https://mathoverflow.net/users/238 | Does every commutative monoid admit a translation-invariant measure? | I think the integers Z with max is a counterexample. First note the set $I\_n$ of all integers bigger than or equal to n is open. Thus each singleton is Borel by looking at $I\_n\setminus I\_{n+1}$. Hence by countability of Z the measure is a weighted counting measure. But the inverse image of n under translation by n ... | 4 | https://mathoverflow.net/users/15934 | 121708 | 68,429 |
https://mathoverflow.net/questions/121701 | 4 | Let $L/K$ be an extension of number fields, and $M/K$ the Galois closure of $L/K$ (everything happens inside a suitably large characteristic zero field $\Omega$). Let $p$ be a discrete prime of $K$.
Given the group $G={\rm Gal}(M/K)$, its subgroup $H={\rm Gal}(M/L)$, and the splitting of $p$ in $L/K$ can one find the... | https://mathoverflow.net/users/4800 | Decomposition of primes in Galois closures of number fields | **Changed answer:** The answer is *no*: Let $L$ be a root field of $X^3-2$ over $\mathbb Q$, and $M$ the Galois closure of $L/\mathbb Q$. Then the primes $2$ and $3$ are both totally ramified in $L$, yet in both cases there is only one prime $P$ above $p=2$ or $p=3$, with $e(P)=3$ if $p=2$, and $e(P)=6$ if $p=3$.
| 5 | https://mathoverflow.net/users/18739 | 121709 | 68,430 |
https://mathoverflow.net/questions/121710 | 6 | Let me ask a question which could be quite stupid, but still:
let $G$ be a group which is an amalgamated product of subgroups $A$ and $B$ over $C$:$\; \;$ $G = A \ast\_{C} B$ (subgroups are infinite!).
Question: How representation theory of $G$ is connected to representation theory of $A$ and $B$ (and $C$), in anot... | https://mathoverflow.net/users/29536 | amalgamated product of groups and representation theory | Let me expand on Mark Sapir's comment with a concrete example. Burger and Mozes famously [constructed a group](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=burger&s5=mozes&s6=products&s7=&s8=All&vfpref=html&... | 6 | https://mathoverflow.net/users/1463 | 121718 | 68,434 |
https://mathoverflow.net/questions/121620 | 9 | Does anyone know of a simple proof that the general linear group GL(N) does not admit spinor representations?
Thank you!
| https://mathoverflow.net/users/31386 | Why does GL(N) have no spinor representations? | The statement in GSW that you quote has to be interpreted properly. When they write, "Spinors form a representation of $\mathrm{SO}(N)$ which does not arise from a representation of $\mathrm{GL}(N,\mathbb{R})$", they really mean either "Spinors form a representation of ${\frak{so}}(N)$ which does not arise from a repre... | 17 | https://mathoverflow.net/users/13972 | 121720 | 68,435 |
https://mathoverflow.net/questions/120746 | 6 | Let $\phi$ be an acceptable programming system. For every recursive function $f$, let $(f)=\{x:\phi\_x=\phi\_{f(x)}\}$ the set of fixed points of $f$. Now, suppose that $f$ and $g$ are recursive functions such that $(f)$ and $(g)$ are disjoint sets. The question is: are there recursive functions $F$ and $G$ such that $... | https://mathoverflow.net/users/31149 | Disjoint sets of fixed points | This turns out to be false. I'll construct $f$ and $g$ that there are no $F$ and $G$ with the required properties.
Define $f$ such that for any $x$,
1. $\phi\_{f(x)}(0) = 0$
2. If $\phi\_x(0)$ has halted by $n$ steps then $\phi\_{f(x)}(n) = 0$
3. If $\phi\_x(0)$ has not halted by $n$ steps then $\phi\_{f(x)}(n)$ is... | 3 | https://mathoverflow.net/users/30790 | 121726 | 68,437 |
https://mathoverflow.net/questions/121725 | 3 | Let $\mathbb F\_p$ be the finite field of a prime order $p$, $f(x)\in \mathbb F\_p[x]$ an irreducible polynomial, $E = \mathbb F\_p[x]/\langle f(x)\rangle$ a finite extension of $\mathbb F\_p$, $\lambda\in E$ is a zero of $f$. Is there an efficient algorithm for computing the order of $\lambda$ in $E^\ast$ or, more gen... | https://mathoverflow.net/users/31419 | Order of an element in a finite field | Let $n$ be the degree of $f$, so the order is a divisor of $p^n-1$. As the computation of powers in $E$ is a cheap operation, I believe the following could be suitable: If $\alpha^m=1$, then check if there is a prime divisor $r$ of $m$ with $\alpha^{m/r}=1$. If not, then $m$ is the order, and if yes, then set $m=m/r$ a... | 0 | https://mathoverflow.net/users/18739 | 121728 | 68,439 |
https://mathoverflow.net/questions/121593 | 6 | Hi,
Is there a simple proof of the weight monodromy conjecture in the case of a curve over a mixed characteristic local field?
Thanks!
| https://mathoverflow.net/users/36285 | weight monodromy conjecture for curves? | It seems to me that you can find a proof in the paper of Raynaud: 1-motifs et monodromie géométrique. Astérisque No. 223 (1994), 295–319.
I don't know if this is "simple" but it is about curves and their Jacobian varieties. The argument involves the theory of 1-motives and the rigid uniformisation of abelian varieties... | 4 | https://mathoverflow.net/users/31421 | 121732 | 68,442 |
https://mathoverflow.net/questions/121735 | 9 | I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner.
In [Wikipedia](http://en.wikipedia.org/wiki/Triangulated_category#t-structures) it has been stated that since different abelian categories can give rise to equivalen... | https://mathoverflow.net/users/13351 | Recovering an abelian category out of its derived category | The derived category of an abelian category has a t-structure, so obviously that's the first thing you want.
To a t-structure corresponds a heart, which is an abelian category, whose derived category might be different than the one you started with.
To further complicate matters, as you noted, you could have a heart wh... | 12 | https://mathoverflow.net/users/25442 | 121752 | 68,452 |
https://mathoverflow.net/questions/121757 | 1 | hi,
I have a function $X(t)$ whose Laplace transform $\hat{X}(s)$ has a unique pole of largest real part $x\_0$, which is a real number. I am able to show that for each $t$, $X(t)$ is a convergent sum of the residues of $\hat{X}(s)e^{st}$, over its poles. If $x\_0$ is a pole of order $m$, is it true in general that $... | https://mathoverflow.net/users/20381 | dominant pole in the laplace transform | Yes, this is true. Just write the Laplace transform as a sum of $a/(x-x\_0)^m$ and the rest.
The inverse Laplace of the first summand is your asymptotic term, and the inverse Laplace transform
of the rest is small.
(I suppose that your expression "unique pole of the largest real part" means that the real parts
of the o... | 1 | https://mathoverflow.net/users/25510 | 121759 | 68,455 |
https://mathoverflow.net/questions/121715 | 10 | Let $G\_1$ and $G\_2$ be connected reductive algebraic groups defined over $\mathbb{C}$ and let $V\_1$ and $V\_2$ be irreducible representations of $G\_1$ and $G\_2$ respectively. I'm interested in general techniques for computing the (generators of the) invariant polynomial functions on the representation $V\_1 \otime... | https://mathoverflow.net/users/18871 | Invariant polynomials for a product of algebraic groups | There is a fruitful way of thinking about the algebra $\mathbb{C}[M\_{n,m}]^{{O\_n}\times O\_m}$ that originates from the $(GL\_n, GL\_m)$-duality. Namely,
>
> $$
> \mathbb{C}[M\_{n,m}]=\bigoplus\_{\lambda}V\_n^{\lambda}\otimes V\_m^{\lambda},
> $$
>
>
>
where the sum is over all partitions $\lambda$ with at ... | 9 | https://mathoverflow.net/users/5740 | 121760 | 68,456 |
https://mathoverflow.net/questions/121682 | 1 | If we have the result as the title, then I can solve my real question. The original question was stated as follows.
In a paper I found the following lemma：
Let $S$ be a nonsingular projective surface, $R\in PicS$ a divisor with $R^2>0$ . Let
$(E\_{i})$ be the family of distinct curves such that $R\cdot E\_{i}=0$... | https://mathoverflow.net/users/3525 | Does negative semidefinite intersection matrix of bunch of curves implies zero intersection with other curves? | The answer already given here seems to assume the E\_i are pairwise disjoint, which I didn't take to be the case from your question.
To get the result in general, I think you need to use the additional fact that an effective divisor can never be numerically trivial (since it will have positive intersection with an am... | 6 | https://mathoverflow.net/users/14848 | 121761 | 68,457 |
https://mathoverflow.net/questions/121769 | 13 | Can one develop a theory of étale homology via étale cosheaves? The hope is that this would, for example, return the Tate module (and not its dual) for an elliptic curve, and it would return group homology for $\rm Spec$ of a field.
The first step in this direction is to notice that you can define an étale cosheaf on... | https://mathoverflow.net/users/1355 | Etale homology via étale cosheaves | Sure thing! There's an equivalence of categories between cosheaves valued in profinite abelian groups and sheaves valued in torsion abelian groups, by Pontryagin duality one could say. So you can directly define the first etale homology of Z\_ell and it will give you the l-adic Tate module. But it's formally the same a... | 13 | https://mathoverflow.net/users/3931 | 121770 | 68,460 |
https://mathoverflow.net/questions/121753 | 5 | Let $V = \mathbb C^n$. Consider the plethysm $\bigwedge^k Sym^d V$ as a representation of $GL(V)$. In what special cases (e.g., for what $k$, $d$, and $n$) is this representation's decomposition into irreps known?
The only known nontrivial special case that I am aware of is when $k = 2$: in this case the decompositio... | https://mathoverflow.net/users/21193 | Known decomposition of $\bigwedge^k Sym^d \mathbb C^n$ in special cases? | When $d=2$, the decomposition is known for all $k$ and $n$. Given a partition $\lambda$ of $k$ with distinct parts, let $2[\lambda]$ denote the partition of $2k$ whose main-diagonal hook lengths are $2\lambda\_1, \ldots, 2\lambda\_k$, and whose $i$th part has length $\lambda\_i + 1$. Then
$$ \bigwedge^k {\rm Sym}^2 V... | 8 | https://mathoverflow.net/users/7709 | 121773 | 68,461 |
https://mathoverflow.net/questions/121751 | 7 | My question comes from trying to understand a technical step in [this paper](http://65.54.113.26/Publication/4171765/ruzsa-s-problem-on-sets-of-recurrence) by Bourgain.
Let $R,L$ be positive integers and let $f(x)=\sum\_{|n|\leq RL}a\_ne^{2\pi inx}$ be a trigonometric polynomial. Assume $f(x)>0$.
Let $F\_L(x)=\sum\_{... | https://mathoverflow.net/users/18698 | Inequality on Trigonometric polynomials | That is true for all non-negative trigonometric polynomials, though not entirely obvious unless you are a Fourier analyst yourself. To see it, just note that the convolution with $K\_{RL}=2F\_{2RL}-F\_{RL}$ recovers $f$ faithfully and $F\_{RL}\le RF\_L$. Of course, to Jean such things are as obvious as $2\times 2+1=5$ ... | 18 | https://mathoverflow.net/users/1131 | 121777 | 68,463 |
https://mathoverflow.net/questions/121745 | 2 | Is there a nice/known formula for the (some) principal specialization of a projective (P or Q) Schur polynomial? That is, I am talking about a projective analogue of Stanley's hook content formula $$s\_{\lambda} (1,q,\dots,q^{n-1}) = \prod\_{(i,j) \in \lambda} \frac{1-q^{n+j-i}}{1-q^{\lambda\_i + \lambda\_j'-i-j+1}}.$$... | https://mathoverflow.net/users/8541 | principal specialization of projective Schur functions | There is a very nice specialization when you consider *infinitely many* variables, due to an identity of Kawanaka:
$$Q\_{\lambda}(1,q,q^2,\dots)=\prod\_{i=1}^m \frac{(-1;q)\_{\lambda\_i}}{(q;q)\_{\lambda\_i}} \prod\_{1\le i < j \le m} \frac{q^{\lambda\_j}-q^{\lambda\_i}}{1-q^{\lambda\_i+\lambda\_j}}$$
where $\lambda$ i... | 5 | https://mathoverflow.net/users/2384 | 121782 | 68,465 |
https://mathoverflow.net/questions/121792 | 14 | In Riemannian geometry, there is a well-known notion of the scalar curvature on a Riemannian manifold $M$, which is a function on $M$ given by a suitable contraction the Riemannian curvature tensor. The scalar curvature gives a rough measure how much the geometry of $M$ differs locally from the flat Euclidean case, nam... | https://mathoverflow.net/users/1841 | Scalar curvature notion for Cartan connections | In the case that $H$ is trivial, $\omega$ is just a $\frak{g}$-valued coframing on the manifold $P$. The curvature $2$-form $\Omega = d\omega + \tfrac{1}{2}[\omega,\omega]$ can then be written in the form
$$
\Omega = R(\omega\wedge\omega)
$$
where $R:P\to {\frak{g}}\otimes\Lambda^2({\frak{g}}^\ast)$ is the curvature t... | 15 | https://mathoverflow.net/users/13972 | 121795 | 68,469 |
https://mathoverflow.net/questions/121797 | 4 | Let $\mathfrak C = \mathsf{Mes}$ be the category of meausurable spaces and measurable maps. For any object $X\in \mathfrak C\_0$ we assign a measurable space $\mathcal P(X)$ whose elements $\mu$ are probability measures over $X$. The space $\mathcal P(X)$ is endowed with the smallest $\sigma$-algebra that makes evaluat... | https://mathoverflow.net/users/11768 | A terminal coalgebra of a certain functor on Mes | [Final coalgebras for functors on measurable spaces](http://www.sciencedirect.com/science/article/pii/S089054010500204X#), Lawrence S. Moss and Ignacio D. Viglizzo:
"We prove that every functor on the category Meas of measurable spaces built from the identity and constant functors using products, coproducts, and the ... | 6 | https://mathoverflow.net/users/447 | 121801 | 68,471 |
https://mathoverflow.net/questions/121744 | 3 | Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F\_ t \right )\_ {t\geq 0}$ is filtration satisfying the usuual conditions, a non-negative continuous process $X = \left (X\_t \right)\_ {t\geq 0}$ satisfying $\mathbb {E}\left[ \bar ... | https://mathoverflow.net/users/24538 | Upper bound concerning Snell envelope | I think it is just a sequence of much stronger inequalities that goes as follows. All the inequalities below are $\Bbb P$-a.s.
1. Let $\tau\in \mathcal T\_{t,T}$ be an arbitrary stopping time, then
$$
(X\_\tau)(\omega) = (X\_{\tau(\omega)})(\omega)\leq \sup\_{t\leq s\leq T} X\_s(\omega)\tag{1}
$$
since $\Bbb P(\tau\... | 2 | https://mathoverflow.net/users/11768 | 121802 | 68,472 |
https://mathoverflow.net/questions/119847 | -1 | Let $C$ be the field of complex numbers. Polynomial $f \in K[x,y]$ is called closed or non-decomposible if
$C[f]$ is algebraically closed (definition for $f \in K[x1,...,xn]$ is the same).
Theorem.
Next conditions are equivalent:
(1) $f$ is closed;
(2) there does not exist any $F \in K[t]$ such that $f = F(h)$ f... | https://mathoverflow.net/users/29536 | product of two non-decomposible (closed) polynomials | As I mentioned above in the comment, the answer to my question (very easy question) is negative!
P.S. Of course I meant that $f$ and $g$ are algebraically independent!
| 0 | https://mathoverflow.net/users/29536 | 121803 | 68,473 |
https://mathoverflow.net/questions/121774 | 26 | Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold. Moreover, the core curve of the filling solid torus in these fillings is a geodesic in the resulting hyperbolic manifold... | https://mathoverflow.net/users/31434 | Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds | Sixty is an upper bound.
Hodgson and Kerckhoff's Universal Hyperbolic Dehn Filling theorem ("Universal bounds for hyperbolic Dehn surgery." Annals of Mathematics. 162(1), 367-421) says that, in a one-cusped manifold, you can push the cone angle up from zero all the way up to $2\pi$ as long as the normalized length of... | 15 | https://mathoverflow.net/users/1335 | 121806 | 68,475 |
https://mathoverflow.net/questions/121787 | 2 | Hello,
$g$ is a complex semisimple Lie algebra.
There is the result that $U(g)$ is free over $Z(g)$.
There is another result: If $E$ is a finite dimensional representation of $g$, then $Hom(E,U(g)^{ad})$ is a free $Z(g)$-module of rank equals the multiplicity of the zero weight in $E$ (here $U(g)^{ad}$ denotes $U... | https://mathoverflow.net/users/2095 | Kostant's theorem about U(g) being free over Z(g) and a corollary of it | The results here go back to a classic paper by Kostant: *Lie group representations on polynomial rings*, Amer. J. Math. 85 (1963), 327-404. A detailed exposition was given by Dixmier in his 1974 book, translated (with added misprints) into English as *Enveloping Algebras* and later reprinted by AMS (with some of the mi... | 6 | https://mathoverflow.net/users/4231 | 121807 | 68,476 |
https://mathoverflow.net/questions/121775 | 3 | Let $X$ be a topological space, and let $\operatorname{CL}(X)$ be its [hyperspace](http://en.wikipedia.org/wiki/Hyperspace_(topology)). That is, $\operatorname{CL}(X)$ is the set of closed subsets of $X$, equipped with the minimal topology so that the canonical map $$x \mapsto \overline{\{x\}}$$is a homeomorphism onto ... | https://mathoverflow.net/users/238 | What is a good example of a hyperspace where the base space is non-Hausdorff? | I've made use of this in some (unpublished) work in connection with a formalism for discussing so-called geometric limits of holomorphic dynamical systems. The details of the specific application are not so relevant, but I've copied the statement here to give a sense of how what is essentially Fell's Theorem yields a u... | 1 | https://mathoverflow.net/users/15819 | 121822 | 68,486 |
https://mathoverflow.net/questions/121771 | 0 | Let $T$ be the boundary of a solid torus in $S^4$. Are there any theorems or methods which would help one to compute $\pi\_2(S^4 -T)$? Or to say if, e.g., it had finite rank and no torsion?
More generally, suppose we have a closed manifold $X$ and we remove a submanifold $Y$ of at least codimension 2. Is there a gen... | https://mathoverflow.net/users/17913 | Second homotopy of a torus complement in the 4-sphere | The paper
Martins, João Faria The fundamental crossed module of the complement of a knotted surface. Trans. Amer. Math. Soc. 361 (2009), no. 9, 4593–4630.
looks relevant.
Note that for links in $\mathbb R^3$ it is standard to use the fundamental group and a Seifert-van Kampen Theorem. So for the case in questio... | 4 | https://mathoverflow.net/users/19949 | 121841 | 68,496 |
https://mathoverflow.net/questions/121804 | 9 | I consider all $k$ element subsets of the set $\{1,\ldots,n\}$ and define a partial order relation $\prec$ as follows: $\{a\_1,\ldots,a\_k\}\prec\{b\_1,\ldots,b\_k\}$, if and only if $a\_1<\cdots<a\_k$, $b\_1<\cdots<b\_k$ and $a\_i\leq b\_i$ for all $i=1,\ldots,k$.
What is this partial order commonly called in the li... | https://mathoverflow.net/users/31441 | Partial order relation on subsets | This order is sometimes denoted $L(k,n-k)$ and has many interesting properties. See for instance Chapter 6 of <http://math.mit.edu/~rstan/algcomb.pdf>. For the characteristic polynomial of the Hasse diagram of $L(k,n-k)$ (considered as a graph), see Remark 5.6 of <http://math.mit.edu/~rstan/papers/vac.pdf>.
| 6 | https://mathoverflow.net/users/2807 | 121842 | 68,497 |
https://mathoverflow.net/questions/121836 | 10 | Let $E$ be an elliptic curve over $\mathbb Q$, which does not have complex multiplication over the algebraic closure of $\mathbb Q$. For $x>0$, let $P(x)$ be the number of primes $p < x$ such that $E$ has good **super-singular** reduction at $p$. The Lang-Trotter's conjecture states that
$$P(x) = O (x^{1/2}/\log x).$$
... | https://mathoverflow.net/users/9317 | Best results regarding the Lang-Trotter conjecture | A couple of years after my Ph.D. thesis (whose main result is the infinitude of
singular primes, i.e. $P(x) \rightarrow \infty$ as $x \rightarrow \infty$),
Kaneko published a result[1] that let me obtain the *unconditional*
upper bound[2] $P(x) = O(x^{3/4} \log x)$ using some of the same ideas:
>
> [1] Masanobu Ka... | 19 | https://mathoverflow.net/users/14830 | 121845 | 68,499 |
https://mathoverflow.net/questions/121830 | 5 | I am reading [this introduction to enlargement of filtration](http://www.minet.uni-jena.de/Marie-Curie-ITN/EoF/talks/jeanblanc_introduction.pdf) and at the beginning of section 2.4 there is a claim that I cannot justify but seems like it should be well known. The author claims that if $\mathcal{F}\_t$ is the natural fi... | https://mathoverflow.net/users/43134 | If $\mathcal{F}_t$ is separable why is $\mathcal{F}_\infty$ generated by a random variable? | Let $(\Omega,\mathcal B,\mu)$ be a probability space and $\mathcal A$ a sub-sigma-algebra of $\mathcal B$. The following statements are equivalent:
1. $\mathcal A$ admits a countable set of generators.
2. There is a bounded random variable $X\colon \Omega\to \Bbb R$ such that $\sigma(X)=\mathcal A$.
3. There are cou... | 5 | https://mathoverflow.net/users/17118 | 121846 | 68,500 |
https://mathoverflow.net/questions/121824 | 15 | For each prime $p$, we have the algebraically closed field $\bar{\mathbb F}\_p$ with the Frobenius automorphism.
Given any first-order statement with no free variables using the symbols $0,1, +, \times, -, /, \sigma(),=$, we can interpret it in $\bar{\mathbb F}\_p$, interpreting the field operations to mean themselve... | https://mathoverflow.net/users/18060 | What sets of primes can we pick out with first-order statements? | You guessed the correct answer.
This is explained in the paper
of Mike Fried and George Sacerdote,
*[Solving Diophantine Problems Over All Residue Class Fields of a Number Field
and All Finite Fields](http://www.math.uci.edu/~mfried/paplist-ff/annals76.pdf)*,
The Annals of Mathematics, 2nd Ser., Vol. 104, No. 2. (Sep.,... | 11 | https://mathoverflow.net/users/10696 | 121847 | 68,501 |
https://mathoverflow.net/questions/121849 | 5 | For any (compact and connected) complex manifold $X$ and any line bundle $L$ on $X$ we have the well known inequalities $\kappa(X,L)\leq\alpha(X)\leq\dim(X)$ relating the Iitaka-dimension $\kappa(X,L)$ of $L$ over $X$, the number of algebraic independent meromorphic functions $\alpha(X)$ on $X$ and the dimension of $X$... | https://mathoverflow.net/users/15782 | Strictness of the inequality relating the Iitaka dimension and algebraic dimension | Given a compact complex manifold $X$, there exists a meromorphic map $F: X\dashrightarrow Z$ to a projective manifold such that every meromorphic function on $X$ is the pull back under $F$ of a rational function on $Z$. The pull-back of an ample line-bundle on $Z$ realizes the equality between algebraic and Iitaka's di... | 3 | https://mathoverflow.net/users/605 | 121852 | 68,503 |
https://mathoverflow.net/questions/121853 | 1 | Let $G$ be an algebraic group with closed normal subgroup $N$. Suppose that $N$ and $G/N$ are both unipotent. Does it imply that $G$ is also unipotent?
| https://mathoverflow.net/users/25701 | Extension of unipotent algebraic groups | By definition, G is unipotent if and only if every nonzero representation has nonzero fixed vectors. Consider a representation V of G. As N is unipotent, $V^N$ is nonempty. Because N is normal, $V^N$ is stable under G, hence under G/N, and hence has nonzero fixed vectors (because G/N is unipotent). Therefore G is unipo... | 2 | https://mathoverflow.net/users/31112 | 121854 | 68,504 |
https://mathoverflow.net/questions/121145 | 14 | Suppose that $M$ is a compact PL-manifold (possibly with boundary) and let $C^{PL}(M)$ denote the (simplicial) group of PL isomorphisms of $M \times I$ relative to $M \times \{0\} \cup \partial M \times I$, i.e. $PL(M\times I,M \times \{0\} \cup \partial M \times I)$. There is a stabilization map $\sigma: C^{PL}(M) \to... | https://mathoverflow.net/users/798 | What is the status of the PL-pseudoisotopy stability theorem? | As you say, Hatcher once argued that the map $\sigma\_M^{PL}:C^{PL}(M)\to C^{PL}(M\times I)$ is $k$-connected where $k$ is roughly $n/3$, but the proof was not all there.
And as you say Igusa later proved that the analogous map $\sigma\_M^{DIFF}:C^{DIFF}(M)\to C^{DIFF}(M\times I)$ is $k$-connected where again $k$ is ... | 13 | https://mathoverflow.net/users/6666 | 121863 | 68,508 |
https://mathoverflow.net/questions/121857 | 4 | Given a discrete group $G$ (**not** assumed to be abelian), is there a nice construction of a topological space which classifies bundles of $K(G,1)$'s? I guess I should take something like $B\operatorname{Aut}(K(\pi,1))$ where $\operatorname{Aut}(K(\pi,1))$ is the monoid of self homotopy equivalences of some fixed $K(\... | https://mathoverflow.net/users/35353 | What space classifies bundles of K(pi,1)'s? | For any space $F$ you can form the topological monoid $Aut(F)$ and take its classifying space. That will classify Hurewicz fibrations with fiber $F$. A little care is needed since $Aut(F)$ won't generally have the homotopy type of a CW complex unless $F$ is a finite CW complex, which $K(G,1)$ is usually not. For the pr... | 8 | https://mathoverflow.net/users/14447 | 121864 | 68,509 |
https://mathoverflow.net/questions/121861 | 7 | Recall the usual definition of a $k$-dimensional vector bundle (everything is assumed to be continuous/smooth/etc depending on the category):
>
> A $k$-dimensional vector bundle is a triple $(E,B,\pi)$, where
> $\pi\colon E \to B$, satisfying the following:
>
>
> a) The map $\pi$ is onto (I don't know if ev... | https://mathoverflow.net/users/36720 | Alternate definition of vector bundle? | No. Take $E=[0,1] \times \mathbb{R} \to [0,1]=B$ with the usual vector space structure at all fibers except for the fiber of $1/2$ where the vector space structure is twisted by any non-linear diffeomorphism of $\mathbb{R}$, for example $x \mapsto x^3+x$.
| 4 | https://mathoverflow.net/users/2841 | 121866 | 68,510 |
https://mathoverflow.net/questions/121859 | 5 | (This question was posted on MSE [here](https://math.stackexchange.com/questions/300276/when-is-a-homomorphism-between-multiplier-algebras-strictly-continuous) but didn't get any answers.)
The strict topology on the multiplier algebra M(A) of a C\*-algebra A is that generated by the seminorms
$$ x\mapsto \|ax\|\qua... | https://mathoverflow.net/users/11233 | When is a $*$-homomorphism between multiplier algebras strictly continuous? | Non-degenerate \*-homorphism from $A$ (or $M(A)$) to $M(B)$ are strict (where non-degenerate means that $\phi(A)B$ is total in $B$). An important property of strict maps $\phi:A\to M(B)$ is that they possess a unique strict extension $\tilde{\phi}:M(A)\to M(B)$.
A good reference is
E.C. Lance, Hilbert C∗-modules, L... | 5 | https://mathoverflow.net/users/30364 | 121877 | 68,514 |
https://mathoverflow.net/questions/121870 | 0 | In an article I am reading, they state a theorem from Victor Klee's "On Certain Intersection Properties of Convex Sets" and I am having trouble picturing this. The statement is:
Let $C$ and $C\_1, ..., C\_n$ be closed convex sets in a Euclidean space satisfying: (i) $C \cap \bigcap\_{i \neq j}^n C\_i \neq \emptyset$ ... | https://mathoverflow.net/users/31454 | Interpreting Set Notation on Theorem | Your question is ambiguous, but if you mean that you can't picture a nontrivial situation satisfying the assumptions, here's a quick example.
Four convex subsets of the plane giving an example to the assumptions. $C$ is the big square and $C\_1,\ldots,C\_3$ are the triangles. [Picture](http://postimage.org/image/xovw... | 0 | https://mathoverflow.net/users/26815 | 121882 | 68,515 |
https://mathoverflow.net/questions/121827 | 2 | I am trying to understand the semidirect product of groupoids, as defined in [this answer](https://mathoverflow.net/questions/23026/how-can-i-understand-the-groupoid-quotient-of-a-group-action-as-some-sort-of-p#23040) by Theo Johnson-Freyd. Part of my difficulty is that although the definition of a 2-group makes sense,... | https://mathoverflow.net/users/3993 | Semidirect product of groupoids | An action of a groupoid on another groupoid was defined in my paper "Groupoids as coefficients" Proc LMS (3) 25 (1972) 413-426, available [here](http://pages.bangor.ac.uk/~mas010/pdffiles/brown-gpdscoeff.pdf), which also uses methods of fibrations of groupoids. The term "coefficients" refers to nonabelian cohomology. T... | 4 | https://mathoverflow.net/users/19949 | 121891 | 68,521 |
https://mathoverflow.net/questions/121867 | 1 | Define a length $k$ arithmetic progression in $\mathbb{Z}\_p$ to be a set of the form
$\{ax+b : x \in [k]\}$ with $a \in \mathbb{Z}\_p^\*$ and $b \in \mathbb{Z}\_p$.
Let $HJ(k, c)$ be the Hales-Jewett number for length $k$ hypercubes and $c$ colors.
If $p > HJ=HJ(k, c)$ then every $c$-coloring of $\mathbb{Z}\_p$ conta... | https://mathoverflow.net/users/30994 | Van der Waerden's Theorem Over $\mathbb{Z}_p$ | A question that received a lot of attention over the last years are quantitative refinements of [Szemerédi's theorem](http://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem), a generalisation of van der Waerden's Theorem asserting that fixing any positive density $d$ one will find arithemetic progressions in a set of ... | 2 | https://mathoverflow.net/users/nan | 121894 | 68,523 |
https://mathoverflow.net/questions/121746 | 10 | Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H\_\ast(E;\mathbb{Z})$ the structure of a $G$-module. Also, the action automorphisms $F\to F$ and $B\to B$ give each module $H\_p(B;H\_q(F;\mathbb{Z}))$ the... | https://mathoverflow.net/users/25358 | Serre Spectral Sequence of Representations | Dylan, I am afraid that you are missing something. It is true that each element of the group gives a map of spectral sequences. But look at the fibers and my comment about isotropy groups above. If you construct the Serre spectral sequence in the usual way, there is a non-equivariantly inconsequential choice of base po... | 12 | https://mathoverflow.net/users/14447 | 121917 | 68,527 |
https://mathoverflow.net/questions/121915 | 2 | Let $F\_{i}$ be the fibonacci or a multinacci sequence. The number of representations of $N$ in the form
$ N=\sum\_{i=0}^{k}s\_{i}F\_{i}, s\_{i}\in ${0,1}
is known.
My question is what is known about sequence-based numeration systems given by other linear recurrences.
To make the question precise,
i am intereste... | https://mathoverflow.net/users/23542 | Linear numeration systems | Some results on the quantity in question can be found in
*J. M. Dumont, N. Sidorov and A. Thomas, Number of representations related to a linear recurrent basis, Acta Arithmetica 88 (1999), 371-394.*
We are mainly interested in the summatory function but there are also some upper bounds for the quantity itself. Our... | 4 | https://mathoverflow.net/users/8131 | 121918 | 68,528 |
https://mathoverflow.net/questions/121621 | 21 | A very soft question, but I hope not out of order here.
In the first edition of Elliott Mendelson's classic *Introduction to Mathematical Logic* (1964) there is an appendix, giving a version of Schütte's (1951) variation on Gentzen's proof of the consistency of PA. This is intriguing stuff, crisply and quite accessib... | https://mathoverflow.net/users/14111 | Mendelson's *Mathematical Logic* and the missing Appendix on the consistency of PA | Before posting this question I did search around a bit (probably inefficiently and certainly quite ineffectively) to see if I could find an email for Elliott Mendelson to ask him directly! But anyway, he picked up my similar query on FOM and very kindly wrote to me:
>
> I was intrigued by your comments about the co... | 17 | https://mathoverflow.net/users/14111 | 121930 | 68,532 |
https://mathoverflow.net/questions/121924 | 1 | I want to know if the following problem has been solved
max\_w w'Rw
where the entries of the vector w are binary (w\_i= {0,1} )
| https://mathoverflow.net/users/31474 | sparse binary vector | * After a Cholesky factorization $R=T'T$, your problem becomes the maximization of the Euclidean norm over the parallelotope formed by the points $Tw$. This problem is discussed in
H. L. Bodlaender, P. Gritzmann, V. Klee, J. van Leeuwen:
[Computational complexity of norm-maximization](http://link.springer.com/article/1... | 2 | https://mathoverflow.net/users/30800 | 121935 | 68,536 |
https://mathoverflow.net/questions/121914 | 2 | Let $X$ be a smooth variety over a field $k \subset \mathbb{C}$ and $Z$ a smooth subvariety. Let $U=X-Z$. I'm trying to understand what information do the Leray spectral sequences attached to the inclusions
$j: U(\mathbb{C}) \hookrightarrow X(\mathbb{C})$
and
$i: Z(\mathbb{C}) \hookrightarrow X(\mathbb{C})$
p... | https://mathoverflow.net/users/31470 | Leray spectral sequence of the inclusion of an open subvariety | Here are a few more details:
Suppose that $F=\mathbb{Q}\_X$.
In this case,
$$
R^qj\_\*F =
\begin{cases}
\mathbb{Q}\_X & \text{if $q=0$}\\
\mathbb{Q}\_Z & \text{if $q= 2 c-1$, $c=codim(Z)$}\\
0 & \text{otherwise}
\end{cases}
$$
To see this, first note that the computation is local, so we may replace the pair $(X,Z)$... | 0 | https://mathoverflow.net/users/4144 | 121936 | 68,537 |
https://mathoverflow.net/questions/119110 | 2 | I would like to know if there is a proof for the identity used in the superconformal index of 4d ${\cal N}=2$ gauge theory. In the [paper](http://arxiv.org/pdf/1104.3850v1.pdf) by Rastelli el al, it was discovered that Eq. (10) is equal to the right hand side in the previous equation.
\begin{equation}
\exp\left[\sum\_... | https://mathoverflow.net/users/17644 | Proof of generalized Cauchy formula | 1) In appendix E of [this paper](http://arxiv.org/pdf/1110.3740.pdf) there is an outline of a proof of this statement based on matching poles and residues on both sides of the identity.
2) One can also use the more generic arguments of [a more recent paper](http://arxiv.org/pdf/1207.3577.pdf) even for the more
ge... | 4 | https://mathoverflow.net/users/31476 | 121940 | 68,539 |
https://mathoverflow.net/questions/100033 | 183 | I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I am in charge of nourishing our departmental math library (PUCV) and I would like to add this kind of material in order t... | https://mathoverflow.net/users/11920 | Interesting mathematical documentaries | Leys, Ghys & Alvarez have also made a video series in a similar style about dynamical systems called "[Chaos](https://www.chaos-math.org/)". The nine chapters are all available under a Creative Commons license.
| 22 | https://mathoverflow.net/users/3121 | 121948 | 68,544 |
https://mathoverflow.net/questions/121903 | 3 | How does the idea of a "minimal polynomial" for a matrix (i.e. for a matrix $A$, the polynomial, $\mu (x)$, of least degree, such that $\mu (A) =0$) relate the the "minimal polynomial" for some element, $\alpha $ over a field (i.e. the minimal polynomial for $\sqrt {2}$ over $\mathbb{Q} $ is $x^2-2$)? In the previous e... | https://mathoverflow.net/users/31466 | How do minimal polynomials relate? | The general notion of minimal polynomial, as Roy Smith indicated in the comments, works whenever you have an algebra $A$ over a field $K$ and an element $a\in A.$ Namely, consider the unique algebra homomorphism $\pi:k[t]\to A$ that sends $a$ to $t.$\* Since $k[t]$ is a principal ideal domain, its kernel $\ker\pi$ is $... | 3 | https://mathoverflow.net/users/5740 | 121952 | 68,546 |
https://mathoverflow.net/questions/121140 | 11 | Theses are simple and natural questions, but I could not find anything about it. If anyone has an answer or a reference this would be very much appreciated.
Let $\mathcal{C}$ be an abelian category (possibly without enough injective nor projective).
(i) Let $A,B \in \mathcal{C}$. When are the $\mathrm{Ext}^n(A,B)$ ... | https://mathoverflow.net/users/24114 | Universality of Ext functor using Yoneda extensions | In their paper entitled "Extension categories and their homotopy", Neeman and Retakh define a spectrum of extensions $\operatorname{Ext}(A,B)$ for any two objects in an exact category $\mathcal E$ such that $\pi\_{-n}\operatorname{Ext}(A,B)=\operatorname{Ext}\_{\mathcal E}^n(A,B)$, in the sense of Yoneda, for any $n\ge... | 6 | https://mathoverflow.net/users/12166 | 121957 | 68,547 |
https://mathoverflow.net/questions/121958 | 4 | Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting representation of $H$ on $H\_{\*}(G;\mathbb{Z})$, the integral homology of $G$. Is anything in general known about this $H$-represen... | https://mathoverflow.net/users/25358 | Representations of Finite Subgroups on Homology | Since $H\leq G$ and $G$ is connected, each $h\in H$ is connected to the identity, hence its action on $G$ is homologous to the identity.
You're not going to get anything interesting here unless you either look at $N\_G(H)/H$ acting on $H\backslash G$, or (equivalently) $N\_G(H)$ acting on the $H$-equivariant cohomolo... | 7 | https://mathoverflow.net/users/391 | 121960 | 68,548 |
https://mathoverflow.net/questions/121785 | 2 | Hello,
Could anyone give a reference or proof for the following fact (which is, probably, not very difficult):
We work in category O for a semisimple complex Lie algebra. $M\_{\chi}$ denotes the Verma module with shifted highest weight $\chi$.
Fact: Let $\chi$ be dominant, $\lambda$ be such that $\chi - \lambda$ ... | https://mathoverflow.net/users/2095 | Covering of Verma modules by translation of a dominant Verma module | This is a consequence of Theorem 3.3 in Gelfand and Gelfand's [Tensor products of finite and infinite dimensional representations of semisimple Lie algebras](http://archive.numdam.org/ARCHIVE/CM/CM_1980__41_2/CM_1980__41_2_245_0/CM_1980__41_2_245_0.pdf); the projective cover of $M\_{\lambda}$ is obtained by translating... | 3 | https://mathoverflow.net/users/66 | 121962 | 68,549 |
https://mathoverflow.net/questions/121959 | 8 | Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers $m,n$ with $gcd(m,n)=1$.
What happens when the field $k$ is not algebraically closed? Can I find somewhere the classific... | https://mathoverflow.net/users/23758 | Classification of Tori of GL2, up to conjugation | In general, a group that is isomorphic to a $d$-dimensional torus over the algebraic closure of the ground field can be identified by the action of the Galois group on its character group / group of one-parameter subgroups, which is $\mathbb Z^d$, so you get a homomorphism $Gal(\bar{k}/k) \to GL\_d(\mathbb Z)$. In the ... | 14 | https://mathoverflow.net/users/18060 | 121969 | 68,552 |
https://mathoverflow.net/questions/121947 | 4 | I have seen Troelstra's Uniformity Principle stated as:
$\forall x \exists n R(x,n) \rightarrow \exists n \forall x R(x,n)$
where $x$ ranges over $\mathbb{P(N)}$ and $n$ ranges over $\mathbb{N}$.
Unless I'm misunderstanding something, this is obviously false in classical mathematics. For example, take $R$ to be the r... | https://mathoverflow.net/users/31478 | Understanding Troelstra's Uniformity Principle in Constructive Mathematics | You have stated Troelstra's Uniformithy Principle correctly (contrary to François's claim). There are two reasons why your counterexample does not work. First, we cannot show that every $X \subseteq \mathbb{N}$ is empty or not. Second, we cannot show that every inhabited subset of $\mathbb{N}$ has a minimal element. An... | 8 | https://mathoverflow.net/users/1176 | 121975 | 68,555 |
https://mathoverflow.net/questions/121993 | 20 | Let $\mathcal{F}$ and $\mathcal{G}$ be abelian categories. It is well-known that if a functor $\phi : \mathcal{F} \rightarrow \mathcal{G}$ has a right-adjoint (so $\phi$ is itself a left-adjoint to some other functor), then $\phi$ is right exact. Similarly, if $\phi$ has a left-adjoint (so $\phi$ is itself a right-adjo... | https://mathoverflow.net/users/31491 | Left/right exact functor "in nature" which is not a right/left adjoint | I would disagree that the hypotheses of the adjoint functor theorem are much stronger than exactness. Left exactness is equivalent to preserving all finite limits, and the hypotheses of the adjoint functor theorem are existence of all limits, preserving all limits, and a smallness condition that usually is easy to veri... | 25 | https://mathoverflow.net/users/75 | 121995 | 68,563 |
https://mathoverflow.net/questions/121979 | 3 | Let $\Phi(x,y,\theta)$ be a phase function defined on $X \times X \times (\mathbb R^n-0)$ where $X$ is some domain in $\mathbb R^n$, let $A(x,y,\theta)$ be an amplitude function. As usually an integral Fourier operator is defined by the formula
$$
(Fu)(y)=\frac{1}{(2\pi)^n} \int\limits\_{\mathbb R^n}\int\limits\_{X} e... | https://mathoverflow.net/users/17896 | Analogue of the integral Fourier operator with angle in some cone | Why don't you assume that the amplitude $A(x,y,\theta)$ is actually supported for $\theta$ in your cone $\Gamma$ along with the standard symbolic properties? You will always run into trouble with a definition like yours, which amounts to deal with singular amplitude. In particular any integration by parts will produce ... | 3 | https://mathoverflow.net/users/21907 | 122003 | 68,567 |
https://mathoverflow.net/questions/121973 | 4 | **DEFINITIONS:** Functions $c : \binom X3\rightarrow \{0\ 1\}$ are called 2-colorings of triangles in $X$. The $4$-element subsets $A\subseteq X$ are called tetrahedra. Each 2-coloring $c$ of triangles induces $(\alpha\ \beta)$-coloring of each tetrahedron $A$, where
$$\beta := \sum\_{T\subseteq... | https://mathoverflow.net/users/8385 | $2$-colorings of triangles, resulting in $(2\ 2)$-colorings of all tetrahedra | When $|X|=7$, no such coloring exists.
In this case there are 35 triangles and 35 tetrahedra. If $A$ is a tetrahedron, let $(\alpha\_A,\beta\_A)$ be its type. Each triangle in $X$ is a subset of exactly $4$ tetrahedra, so if
$$\sum\_{T\subseteq X, |T|=3}c(T)=n$$
then
$$\sum\_{A\subseteq X, |A|=4} \beta\_A = 4n$$
If... | 1 | https://mathoverflow.net/users/21193 | 122007 | 68,569 |
https://mathoverflow.net/questions/120853 | 7 | Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an apartment $A$ and corresponding root system $\Phi$ and split torus $T$; then for each $x \in A$ and $r>0$, we have a corres... | https://mathoverflow.net/users/31178 | When is a Moy-Prasad filtration subgroup the stabilizer of a subset of the building (up to center)? |
>
> Regretfully the answer to the first question is no.
>
>
>
For example consider a group of type $G\_2$. Normalize your roots so the short roots have length 1, for example, and let $x=0$. Write $\Phi = \Phi^l \cup \Phi^s$ as a union of long and short roots, respectively, and define the corresponding sets $\Ome... | 4 | https://mathoverflow.net/users/31178 | 122013 | 68,573 |
https://mathoverflow.net/questions/122006 | 8 | Suppose that a function $f$ on the line satisfies $|f(x+2h)-2f(x+h)+f(x)|\le |h|^{3/2}$ for all $x,h$ real. Is it true that $f$ is differentiable and its derivative satisfies
$|f'(x+h)-f'(x)|\le c |h|^{1/2}$ for all $x,h$?
| https://mathoverflow.net/users/31493 | Second order difference implies differentiability | As Gerald says, the answer is no without further hypotheses on f. But if one makes some minimal additional regularity hypotheses on f, such as continuity, then the answer is yes.
Write $D\_h f(x)$ for the difference quotient $D\_h f(x) := (f(x+h)-f(x))/h$, then the hypothesis is that
$$ D\_h f(x+h) = D\_h f(x) + O( |... | 14 | https://mathoverflow.net/users/766 | 122020 | 68,577 |
https://mathoverflow.net/questions/121965 | 3 | In reverse mathematics, $COH$ is a statement that there is a cohesive set for any uniform array of sets. Here uniform array of sets means that there exists a set $B$ such that $x\in B\_e \leftrightarrow (e,x)\in B$, while a set $A$ is cohesive for $\lbrace B\_e: e\in \omega\rbrace$ if and only if $A\subset^\* B\_e$ or ... | https://mathoverflow.net/users/23835 | $\Sigma_1^0-COH$? | I also don't recall this principle being directly adressed in the reverse math literature. However, known results can be pieced together to paint a decent picture for $\Sigma^0\_1$-COH. I will add to this answer if I find something more.
The results of Jockusch and Stephan, *A cohesive set which is not high*
[Math. L... | 2 | https://mathoverflow.net/users/2000 | 122023 | 68,579 |
https://mathoverflow.net/questions/122004 | 3 | Consider a $\mathbb{Z}$-graded chain complex $A^{\bullet}$, I know that a bounded below complex is one such that $A^i = 0$ for $i$ sufficiently small, and a bounded above complex is one such that $A^i = 0$ for sufficiently large $i$, every bounded above (or below) complex has a Cartan-Eilenberg resolution and every unb... | https://mathoverflow.net/users/22191 | Question on resolutions for arbitrary chain complexes. | Start with a (possibly bounded) sequence of maps satisfying $dd=0$. Per Fernando's comment, you can always add an infinite number of zeroes on the left and/or right to create a ${\mathbb Z}$-graded complex. You can then build a Cartan-Eilenberg resolution of that ${\mathbb Z}$- graded complex. Your question (I think) i... | 1 | https://mathoverflow.net/users/10503 | 122029 | 68,583 |
https://mathoverflow.net/questions/121951 | 3 | Consider a set of $2^n-1$ non negative integers $S= ${$ a\_{i,j}|1\le i\le n; 1\le j\le 2^{i-1} $} such that:
\begin{align}{}
1.\ \ &a\_{i,j}\le 2^{n+1-i} \\\\
2.\ \ &a\_{i,j}\le a\_{i-1,j} \\\\
3.\ \ &a\_{1,1}+\sum\_{i=2}^n\left(\sum\_{j=2^{i-2}+1}^{2^{i-1}}{a\_{i,j}}\right)\le 2^n \end{align}
Prove that: $$\sum\_{S... | https://mathoverflow.net/users/27041 | Combinatorial Inequality | The answer is *no*, this is false. Specifically, if we let $m$ be the largest integer such that $(m+2)2^m \le 2^n$, then the largest achievable sum is exactly $2^m (2m+2) + 2^n (n-m-1)$, which is not bounded by any constant multiple of $2^n$, since $n-m$ is not bounded.
Proof:
I'm going to denote the top element of... | 6 | https://mathoverflow.net/users/5583 | 122032 | 68,585 |
https://mathoverflow.net/questions/122031 | 14 | Let $B$ be some set. The problem is to find a set $A\subset\mathcal{P}(B)$ of subsets of $B$ which is totally ordered by inclusion and such that there exists a bijection $A\leftrightarrow \mathcal{P}(B)$.
This is an easy exercise if $B$ is countable where one can explicitly construct such a set (one identifies $B$ wi... | https://mathoverflow.net/users/30363 | totally ordered chain in the powerset with big cardinality | Let's think about the countable case like this: think of the
binary tree $2^{\lt\omega}$, which has size $\omega$, but has
$2^\omega$ many branches. Each branch describes a cut in the
natural lexical order on the nodes, and so we have a countable
linear order with $2^\omega$ many cuts.
So consider a cardinal $\kappa$... | 17 | https://mathoverflow.net/users/1946 | 122033 | 68,586 |
https://mathoverflow.net/questions/122060 | 0 | Let $G$ be a Hurwitz group, i.e the automorphism group of some [Hurwitz surface](http://en.wikipedia.org/wiki/Hurwitz_surface) $C$. Then [Hurwitz's automorphisms theorem](http://en.wikipedia.org/wiki/Hurwitz%2527s_automorphisms_theorem) shows that the quotient map of $C$ by $G$ has ramification points of indexes $2$, $... | https://mathoverflow.net/users/10626 | The description of Hurwitz groups | By the definition, the quotient $C/G$ is the orbifold $O=S^2(2,3,7)$: Sphere with 3 cone points with point-stabilizers $Z\_2, Z\_3, Z\_7$. Thus, the fundamental group $\pi$ of $O$ has the presentation
$$
\langle a, b, c| a^2, b^3, c^7, abc\rangle.
$$
(You can prove it by appealing to van Kampen theorem for instance, o... | 0 | https://mathoverflow.net/users/21684 | 122062 | 68,599 |
https://mathoverflow.net/questions/121998 | 11 |
>
> Are there any irreducible representations of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ over $\mathbb Q\_l$ that are unramified away from $l$ and crystalline at $l$ but are not known to arise from the etale cohomology of smooth and proper stacks over $\operatorname{Spec}\mathbb Z$?
>
>
>
Thee simplest e... | https://mathoverflow.net/users/18060 | Unramified Galois representations not from smooth and proper stacks | Here's a class of examples which isn't in your list as far as I know. Take an even unimodular lattice, e.g. the $E\_8$ lattice. The corresponding orthogonal group is a reductive group over $\mathbf{Q}$ which is split at every finite place and compact at $\infty$ (see Gross, "Groups over $\mathbf{Z}$", Inventiones 124 (... | 7 | https://mathoverflow.net/users/2481 | 122066 | 68,600 |
https://mathoverflow.net/questions/122063 | 0 | Hello, everyone!
Supposing that there is a unit vector in $n$-dimensional real space $\mathbf{x}\_1\in\mathbb{R}^n$, I want to get a group of $n-1$ vectors to form an orthogonal basis with $\mathbf{x}\_1$. One way to achieve this goal is to firstly randomly generate $n-1$ linear independent vectors and then orthogona... | https://mathoverflow.net/users/19399 | Is it possible to obtain the vectors orthogonal to a given one by orthogonal transformations? | Adams gave the negative answer; in fact, his theorem is stronger, since there is no requirement that the operators $A\_i$ be given by orthonormal matrices.
The positive answer is given by the theorem of Radon-Hurwitz, which is usually described using the theory of Clifford algebras. Their problem is to classify colle... | 4 | https://mathoverflow.net/users/437 | 122069 | 68,602 |
https://mathoverflow.net/questions/122070 | 8 | Do you know if the following statement is an equivalent form of the axiom of choice or not?
>
> If $X$ is a compact metric space, then every continuous function $f: X \longrightarrow \mathbb{R}$ is uniformly continuous.
>
>
>
If you know any references, please let me know.
| https://mathoverflow.net/users/30356 | Axiom of Choice and continuous functions | It seems to me that this is provable without using the axiom of
choice.
Suppose that $X$ is a compact metric space and $f:X\to\mathbb{R}$
is continuous. Let's show it is uniformly continuous. Fix any
$\epsilon\gt 0$. For each point $x\in X$, there is a small ball
$B$ centered at $x$ such that $f(y)$ is within $\epsil... | 12 | https://mathoverflow.net/users/1946 | 122075 | 68,604 |
https://mathoverflow.net/questions/121996 | 9 | Which applications of intersection (co)homology and of the (Topological) Decomposition Theorem have most chances to be understood by students?
| https://mathoverflow.net/users/2191 | Applications for intersection (co)homology and for the Decomposition Theorem for students? | I gave a course on the decomposition theorem (and its failure with positive char coefficients) in Bochum. Notes and exercises are here:
<http://people.mpim-bonn.mpg.de/geordie/bochum/>
I found the example of Weierstraß family of elliptic curves quite instructive. One considers the family
$Y^2 = (X-\lambda Z)(X-Z)X$ o... | 13 | https://mathoverflow.net/users/919 | 122082 | 68,608 |
https://mathoverflow.net/questions/122065 | 5 | **Edited** (this question contains two versions of a similar question)
Is there some finitely presented group $G$ generated by $g\_1,...,g\_n$ such that
there is an element $g\in G$ expressed as a finite word in $g\_i$'s so that it is impossible to prove neither $g=1$ nor $g\ne 1$?
Is such a group $G$ exists, what... | https://mathoverflow.net/users/13441 | An element $g$ in a group such that neither $g=1$ nor $g\ne 1$ can be proved. | Let me offer another way to explain a similar idea as in
Benjamin's answer.
The answer to the original question, as well as the updated
question, is Yes, there is such a finite group presentation.
Since you are asking whether or not a given assertion is
"provable", you should be explicit about the theory in which y... | 12 | https://mathoverflow.net/users/1946 | 122084 | 68,609 |
https://mathoverflow.net/questions/122045 | 8 | The following baby version of virtual fundamental cycle is well known:
Let $M\subset V$ be the zero locus of a section $s$ of a vector bandle $E \to V$, in general $s$ is not transversal to the zero section and $M$ does not have the expected dimension, then one uses excess intersection theory to define the virtual fu... | https://mathoverflow.net/users/26460 | Local model of virtual fundamental cycle | The answer is yes. Let $[E^{-1} \rightarrow E^0]$ be a perfect obstruction theory on $M$. After localizing in $M$ we can assume that the map $E^0 \rightarrow \Omega\_M$ is induced as $\mathcal{O}\_M \otimes \Omega\_{\mathbf{A}^n} \rightarrow \Omega\_M$ for some map $M \rightarrow \mathbf{A}^n$.(\*) As the map on differ... | 6 | https://mathoverflow.net/users/32 | 122086 | 68,610 |
https://mathoverflow.net/questions/121880 | 4 | Consider an elliptic surface $f :X \rightarrow C$ with $\chi(X) > 0$
or equivalently the fibration has a reduced singular fiber apart from possibly multiple fibers
(the field under consideration
is $\mathbb{C}$,$X$,$C$ are smooth and projective ).
Denote by $\Omega$ the cotangent bundle of $X$ and
$K$ the canonical bu... | https://mathoverflow.net/users/25576 | sections of the cotangent bundle of elliptic surfaces | This is a comment starting from a slightly more general context. Most of the following material can be found in [a paper of T. Saito and me](http://www.math.u-bordeaux1.fr/~qliu/articles/conduct-ineg.ps) (but most dealing with the positive characterisitc case).
Let $f : X\to C$ be a flat morphism of smooth (geometri... | 3 | https://mathoverflow.net/users/3485 | 122087 | 68,611 |
https://mathoverflow.net/questions/122014 | 10 | [Fáry's theorem](http://en.wikipedia.org/wiki/F%25C3%25A1ry%27s_theorem) says that every finite simple planar graph admits a planar embedding with straight line edges.
For which $(k,d)$ is it true that every finite $k$-dimensional simplicial complex embeddable in $\mathbb{R}^d$ has an embedding which is linear on eve... | https://mathoverflow.net/users/4558 | Higher-dimensional Fáry's theorem? | The answer is negative for all pairs $(k,d)$ with $k+1\leq d\leq 2k$, as long as $k \ge 2$.
Brehm [1] constructed a triangulation of the Möbius strip that does not admit a *geometric* (simplexwise linear) embedding into $\mathbb{R}^3$.
More generally, for every pair $(k,d)$ with $k+1\leq d\leq 2k$, Brehm and Sarkar... | 7 | https://mathoverflow.net/users/31511 | 122089 | 68,612 |
https://mathoverflow.net/questions/122083 | 7 | I would like to know the group cohomology of orthogonal groups $SO(n)$, which is the topological cohomology of the classifying space of the group:
$H^\*(BSO(n);\mathbb{Z}) = $ ? (for example for $n=10$)
I also like to know $H^\*(BPSU(n);\mathbb{Z})$ (say for $n=3$), where $PSU(n)=SU(n)/Z\_n$
and $Z\_n$ is the center... | https://mathoverflow.net/users/17787 | Group cohomology of orthogonal groups with integer coefficient | For a precise answer to your first question, see Theorem 1.5 of
Brown, Edgar H., Jr. *The cohomology of BSOn and BOn with integer coefficients.* Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.
For your second question, note that there is an isomorphism $PSU(n)\cong PU(n)$ for each $n$, and that the cohomology $... | 10 | https://mathoverflow.net/users/8103 | 122090 | 68,613 |
https://mathoverflow.net/questions/121987 | 5 | Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a corner of their convex hull under the restriction that comparing sums and/or differences of the distances is the only a... | https://mathoverflow.net/users/31310 | Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences | Here is a sketch of an answer that involves the *squared* Euclidean distances between the four points $A,B,C,D$ (and only the squared ones, not a mixture between squared and non-squared distances.)
Suppose $D$ is the origin, and consider the vectors $a=DA, b=DB, c=DC$. From
$$AB^2 = \langle a-b,a-b \rangle = DA^2 -2\... | 2 | https://mathoverflow.net/users/30800 | 122099 | 68,617 |
https://mathoverflow.net/questions/122097 | 8 | References for the definitions are Jech's Set Theory Definition 15.5, and Cummings paper in the Handbook of set theory Definition 5.15.
| https://mathoverflow.net/users/10708 | Is $\kappa$-distributive the same as $\kappa$-strategically closed? | Forcing with a Souslin tree is $\omega$-distributive, but it is not countably strategically closed. No $\omega\_1$-tree can be countably strategically closed, since if it were, we could construct a subtree of order type $2^{\lt\omega}$, which had all limits. This would give rise to an uncountable level in the tree, con... | 9 | https://mathoverflow.net/users/1946 | 122101 | 68,619 |
https://mathoverflow.net/questions/122106 | 7 | By the [Hurwitz's automorphisms theorem](http://en.wikipedia.org/wiki/Hurwitz%2527s_automorphisms_theorem) there is an upper bound $|\text{Aut}(C)|\leq 84(g-1)$ for all Riemann surfaces $C$ with $g(C)\geq 2$, but it is not sharp if $g=2$. What is the sharp upper bound for genus $2$ Riemann surfaces? Also, I'm asking fo... | https://mathoverflow.net/users/10626 | Upper bound for the order of the group of automorphisms of Riemann surfaces of genus 2 | One proof uses the fact that every genus $2$ Riemann surface is hyperelliptic.
If $C$ is a hyperelliptic Riemann surface, then there is a unique degree $2$ map to $\mathbb P^1$, which defines a homomorphism $Aut(C) \to Aut(\mathbb P^1)$ whose kernel is of order $2$, generated by the hyperelliptic involution. This giv... | 13 | https://mathoverflow.net/users/18060 | 122107 | 68,622 |
https://mathoverflow.net/questions/122105 | 3 | In Grayson's 'Higher Algebraic K-theory II', leading up to the categorical generalisation of the plus construction, he considers $\pi\_0(S) = \pi\_0(BS)$, where $S$ is a (small, symmetric) monoidal category and $BS$ is its classifying space. It is then tacitly assumed that $\pi\_0(S)$ is itself an abelian monoid... but... | https://mathoverflow.net/users/19313 | Path components of a monoidal category form a monoid? | The result will follow if you can show that $\pi\_0\colon Cat \to Set$ preserves finite products, because monoidal categories can be defined diagrammatically in the 2-category $Cat$, and these diagrams are sent, under the assumption of product preservation, to the diagrams defining a monoid in $Set$.
But $\pi\_0$ can... | 7 | https://mathoverflow.net/users/4177 | 122108 | 68,623 |
https://mathoverflow.net/questions/122104 | 4 | Let $X$ be a complete, simply connected Riemannian manifold satisfying a quadratic (coarse) isoperimetric inequality. (I.e., there is a constant $C\_{0}$ such that every loop of length $\ell$ has a filling disk of area $\leq C\_{0}\ell^{2}+C\_{0}$.)
For points $a,b,c\in X$, define $Area\_{X}(a,b,c)$ to be the minimal... | https://mathoverflow.net/users/9455 | Area of triangles vs. comparison triangles. | No.
Blow a bubble of the same size at each integer point of $\mathbb R^2$.
Clearly coarse isoperimetric inequality will hold.
On the other hand, the global metric on the plane can be made arbitrary close to the Manhattan metric, in particular there will be triangles which bound arbitrary large area
while its compar... | 5 | https://mathoverflow.net/users/1441 | 122113 | 68,626 |
https://mathoverflow.net/questions/122091 | 8 | Lel $B$ be a Banach algebra and give $B^{\*\*}$ one of the Arens products in order to make it a Banach algebra. Then the canonical embedding $\kappa\colon B\to B^{\ast\ast}$ is a homomorphic embedding wrt to either Arens product.
Suppose $M$ is a maximal abelian subalgebra of $B$. Can infer from this that $\overline... | https://mathoverflow.net/users/31512 | Masas in second duals of Banach algebras | It need not be so. The simplest example would be the Toesplitz algebra $\cal T$. It is the ${\rm C}^\ast$-algebra generated by the unilateral shift $S$ on $\ell\_2{\bf N}$, $Se\_n=e\_{n+1}$. The algebra ${\cal T}$ contains the ideal $\cal K$ of the compact operators, and the quotient is $\pi\colon{\cal T} \to {\cal T}/... | 8 | https://mathoverflow.net/users/7591 | 122116 | 68,627 |
https://mathoverflow.net/questions/122115 | 11 | Which of the problems from the [Scottish Book](http://en.wikipedia.org/wiki/Scottish_Book) ([pdf of English version](http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf)) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from Sta... | https://mathoverflow.net/users/30081 | problems from the scottish book | The [book version](http://rads.stackoverflow.com/amzn/click/3764330457) edited by Daniel Mauldin (from 1982) has commentaries on the problems as of that date.
| 9 | https://mathoverflow.net/users/454 | 122122 | 68,629 |
https://mathoverflow.net/questions/122118 | 4 | Suppose I have a ring $R$ defined by generators and relations;
for example,
$ C [A, b\_0, b\_1, b\_2, b\_3]$
with relations
* $b\_2\*A-b\_1\*b\_0=0$,
* $b\_3\*A-b\_1\*b\_2=0$,
* $b\_2^2-b\_0\*b\_3=0$.
---
Using Macaulay2 package "Depth" I'm able to check if ring is Cohen-Macaulay, but I also need to compute for... | https://mathoverflow.net/users/31520 | How to check if a commutative ring is Gorenstein. |
```
Macaulay2, version 1.3.1
i1 : S = QQ[A,b0,b1,b2,b3]
o1 = S
o1 : PolynomialRing
i2 : I=ideal(b2*A-b1*b0, b3*A-b1*b2, b2^2-b0*b3)
2
o2 = ideal (- b0*b1 + A*b2, - b1*b2 + A*b3, b2 - b0*b3)
o2 : Ideal of S
i3 : R = S/I
o3 = R
o3 : QuotientRing
i4 : k = coker vars... | 8 | https://mathoverflow.net/users/460 | 122123 | 68,630 |
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