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https://mathoverflow.net/questions/95125 | 10 | Are there applications of algebraic geometry/commutative algebra to biology/pharmacology?
It might be that some Gröbner basis technique is used somewhere? I know there are some applications to robotics — in solving some complicated non-linear equations. Maybe something similar can happen in biology.
---
Related... | https://mathoverflow.net/users/10446 | Applications of algebraic geometry/commutative algebra to biology/pharmacology | 1. René Thom's theory of morphogenesis involves singularities, unfoldings, perturbations of analytic/geometric structures, etc., which, in its turn, involves (or, rather, should involve, as the whole theory is rather sketchy) a good deal of commutative algebra.
2. A conference "[Moduli spaces and macromolecules](http:/... | 9 | https://mathoverflow.net/users/1223 | 130331 | 72,338 |
https://mathoverflow.net/questions/130333 | 0 | EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained.
"let's consider a composite natural number $n$ greater or equal to $4$. Goldbach's conjecture is equivalent to the following statement: "there is at least one natural number $r$ such as $(n-r... | https://mathoverflow.net/users/13625 | A possible consequence of Dirichlet's theorem about primes in arithmetic progression | You seem to be using certain variable inputs in two different ways. First, $r$ is assumed to be less than $n-3$. Then, you define a notion of potential typical primality radius of $r$, which depends only on remainders of division by primes less than $\sqrt{2n-3}$. It is not clear in this definition that you are assumin... | 2 | https://mathoverflow.net/users/121 | 130346 | 72,346 |
https://mathoverflow.net/questions/130357 | 0 | Clearly true if n is a multiple of 2 or 3. Also, you can show it's true for n prime using Fermat's little theorem. You can also show that if n divides 3{a+b} - 2^{a+b} and n divides 3^a - 2^a then n divides 3^b - 2^b.
Not sure where to go from there though, or if I should be trying something completely different...
... | https://mathoverflow.net/users/33911 | Trying to solve: Show that n does not divide 3^n - 2^n for n greater than or equal to 2. | Take $p$ to be the smallest prime divisor of $n$. You have that $p$ divides $3^{p-1}-2^{p-1}$ and also $3^n-2^n$. So $p$ divides $3^{\operatorname{gcd}(p-1,n)}-2^{\operatorname{gcd}(p-1,n)}$. However it is easy to see that this gcd must equal $1$, so $p$ divides $3-2$, and we obtain the desired contradiction.
| 8 | https://mathoverflow.net/users/2384 | 130362 | 72,352 |
https://mathoverflow.net/questions/130355 | 28 | Every elliptic curve $E/\mathbf Q$ is modular, in the sense that there exists a nonconstant morphism $X\_0(N) \to E$ for some $N$.
It is tempting to extend this definition in a naïve way to an arbitrary projective curve over $\mathbf Q$; if $Y$ is such a curve, we might say that $Y$ is modular if there exists a nonc... | https://mathoverflow.net/users/6779 | Is there an algebraic curve over Q which is not modular? | One expects that the majority of algebraic curves over number fields having genus $> 1$ should not be modular in this sense.
For instance, take a sufficiently general genus 2 curve $C$ over $\mathbf{Q}$. Then its $\ell$-adic $H^1$ (which is just the $\ell$-adic Tate module of its Jacobian) will be a 4-dimensional Gal... | 32 | https://mathoverflow.net/users/2481 | 130372 | 72,354 |
https://mathoverflow.net/questions/130356 | 2 | What are some of the common popular stable theories that are known to be dp-minimal (or not dp-minimal)?
Some dp-minimal examples I am aware of are strongly minimal theories, superstable theories of U-rank 1, and infinitely many refining equivalence relations.
The particular examples I am interested in are:
$DCF\... | https://mathoverflow.net/users/38253 | dp-minimality and stability | A stable theory is dp-minimal if and only if all 1-types have weight 1. (See "On dp-minimality, strong dependence and weight" by A. Onshuus and A. Usvyatsov.)
A differentially closed field has 1-types of arbitrary large finite weight, hence neither $DCF\_0$ nor $DCF\_p$ are dp-minimal. Similarly, the generic type of ... | 6 | https://mathoverflow.net/users/19534 | 130394 | 72,367 |
https://mathoverflow.net/questions/130395 | 11 | Let $X$ be a smooth projective connected variety over the complex numbers with ample canonical bundle. If $X$ is generic and $\dim X \leq1$, the automorphism group of $X$ is trivial, see for instance
[Why is a general curve automorphism-free?](https://mathoverflow.net/questions/54994/why-is-a-general-curve-automorphi... | https://mathoverflow.net/users/30342 | Proving that a generic variety with ample canonical bundle has no automorphisms | It seems to me that this is not true and that a counterexample can be constructed as follows.
Take a double cover $\alpha \colon X \longrightarrow A$ of an abelian surface $A$, branched over a smooth divisor $B \in |2 L|$, with $L$ very ample. We have $$K\_X=\alpha^\* L, \quad \alpha\_\* \mathcal{\omega}\_X = \mathca... | 15 | https://mathoverflow.net/users/7460 | 130397 | 72,368 |
https://mathoverflow.net/questions/130400 | 5 | I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length.
Of course, the formula can easily be derived from the Gauss–Bonnet formula to be
$$A = \frac{1}{\kappa}(\alpha + \beta + \gamma - \pi... | https://mathoverflow.net/users/16702 | Triangle area on surfaces of constant curvature | M. Berger, Geometrie, vol. V. MR0536874
Edit. Let me sketch a proof for the spherical triangle. Let the sphere have area $4\pi$.
First you derive the area of digon. It is $2\alpha$, where $\alpha$ is the angle,
by completely elementary reasons. Now consider a triangle. Extend its sides to three
full great circles. Th... | 11 | https://mathoverflow.net/users/25510 | 130407 | 72,374 |
https://mathoverflow.net/questions/83004 | 8 | Suppose two elements of [the ring of periods](http://en.wikipedia.org/wiki/Ring_of_periods) are given by their systems of polynomial inequalities with rational coefficients. Is there a known algorithm deciding their equality? Is it known if their equality is algorithmically decidable at all?
It is quite obvious that ... | https://mathoverflow.net/users/9550 | Algorithmic decidability of equality in the ring of periods | There is no known algorithm for that, and it is unknown if such an algorithm exists.
| 7 | https://mathoverflow.net/users/33664 | 130430 | 72,385 |
https://mathoverflow.net/questions/130310 | 6 | My question was prompted by an earlier MO by @Daniel:
[***Duality map in strictly convex Banach spaces***](https://mathoverflow.net/questions/125765/duality-map-in-strictly-convex-banach-spaces/130226#130226)
I will even use his symbol $\phi$ below.
Let $B$ be an arbitrary Banach space. Let $S := \{... | https://mathoverflow.net/users/8385 | A characterization of Hilbert spaces? | Yes it is true. Let me show that the existence of $\phi$ implies that the norm of $B^\*$ is associated to an inner product. Then it follows easily that the spaces are Hilbert.
It suffices to verify that the norm is an inner product one on every two-dimensional subspace. (Indeed, this property is equivalent to the par... | 11 | https://mathoverflow.net/users/4354 | 130443 | 72,392 |
https://mathoverflow.net/questions/130413 | 2 | In the simplest asymmetric Colonel Blotto game with 2 players, dividing their given Ni soldiers (i=1,2) over 2 battlefields, what are their expected utilities, Ui (i.e., expected number of battlefield victories), in a Nash Equilibrium?
If the continuum Ni case is significantly easier to solve, you can consider the in... | https://mathoverflow.net/users/20757 | Non-Constant-Sum Blotto Game for Only 2 Players and 2 Battlefields | [O. Gross and R. Wagner, 1950. "A Continuous Colonel Blotto Game," *Rand Research Memorandum* RM-408](http://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM408.pdf) covers this case (pages 2-4) and much more.
If the players have equal resources, then it's always a tie regardless of strategy (since you d... | 1 | https://mathoverflow.net/users/2954 | 130451 | 72,394 |
https://mathoverflow.net/questions/130422 | 1 | We are given a graph $G=(V,E)$ with positive edge weights $w\_{i}$ and numerical {0,1,-1} labels $l$ for all vertices . We know that $G$ has a subset $G'$ with all vertices labeled 0(all vertices with 0 in $G$ are considered to in the subset called $G'$). The problem is to assign labels to the vertices in $G'$ in such ... | https://mathoverflow.net/users/33929 | Strategic vertex labeling | Now that I think I understand your problem, I think I also know the solution: There exists a polynomial-time algorithm to solve your problem.
For the sake of clarity, I am re-writing the problem statement (as I am now interpreting it):
The vertices of $G$ are each labeled as $0$, $1$ or $-1$. Let $G'$ denote the $0... | 1 | https://mathoverflow.net/users/29873 | 130452 | 72,395 |
https://mathoverflow.net/questions/120397 | 4 | A real $2n\times 2n$ Hamiltonian matrix has the general form
$$H=\begin{pmatrix}
A & B \cr
C & -A^T
\end{pmatrix}
$$
where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are symmetric. Are there any results regarding the eigenvalue distribution of an ensemble of such matrices? For example, the above con... | https://mathoverflow.net/users/25145 | Eigenvalues of random Hamiltonian matrices | In the course of a [physics project](https://arxiv.org/abs/1305.2924) in my group, I have had an opportunity to learn more about the eigenvalue statistics of Hamiltonian matrices. (Our physics problem actually involved skew-Hamiltonian matrices, so I made a small detour, joined by Jonathan Edge & Jan Dahlhaus.)
The e... | 1 | https://mathoverflow.net/users/11260 | 130458 | 72,397 |
https://mathoverflow.net/questions/130457 | 3 | It is known that for a fixed x $\in \{0,1,...,N-1\}$, the length of the cycle of x in a random permutation in $S\_N$ distributes uniformly in
$\{1, . . . ,N\}$.
My question is regarding the length of x in a random ***derangement*** (permutation without any fixed point).
Does the length distributes uniformly in $\{... | https://mathoverflow.net/users/33940 | The distribution of cycle length in random derangement | The number of permutations where $x$ is in a cycle of length $k$ is $(N-1)!$. In order to have a derangement, $k$ must be at least 2 and the remaining $N-k$ elements must be "deranged". When $N-k$ is not too small, the fraction of derangements will be very close to $1/e$, so the cycle length in a derangement will be as... | 6 | https://mathoverflow.net/users/14302 | 130459 | 72,398 |
https://mathoverflow.net/questions/129433 | 13 | If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois representations $\rho\_{E,l}: G\_{\mathbb Q} \rightarrow Gl\_2(\mathbb F\_\ell)$
on the $\ell$-torsion points of $E$ is surjective. If we defi... | https://mathoverflow.net/users/9317 | Best bounds toward Serre's uniformity conjecture | Since no specialist has replied to this question, I will add a long comment about the little I know.
The unconditional bound depending on the conductor which was used in the implementation in sage comes from Theorem 2 in
A.C. Cojocaru, On the surjectivity of the Galois representations associated to non-CM elliptic ... | 10 | https://mathoverflow.net/users/5015 | 130464 | 72,400 |
https://mathoverflow.net/questions/130461 | 1 | Hi:
When I read the book "An introduction to Homological algebra" by Weibel, the page 206, line 9 says that
"Shapiro's Lemma tell us that
$H\_q(S\_n(X)\otimes\_{Z}A)$ is zero if $q\neq 0$ and is isomorphic to $S\_n(X/G)\otimes\_ZA$ if $q=0$" ,
here $Z$ is the ring of integers, $G$ is a group, $X$ is a path-con... | https://mathoverflow.net/users/33943 | A computation by the Shapiro Lemma | It seems to me that "$H\_q(S\_n(X)\otimes\_{\mathbb{Z}}A)$" should read "$H\_q(G;S\_n(X)\otimes\_{\mathbb{Z}}A)$".
Then
$$H\_0(G;S\_n(X)\otimes\_{\mathbb{Z}}A)=(S\_n(X)\otimes\_{\mathbb{Z}}A)\_G = S\_n(X/G)\otimes\_{\mathbb{Z}}A$$
by 6.10.2, and Shapiro's lemma gives $H\_q(G; S\_n(X)\otimes\_{\mathbb{Z}}A)= 0$ for $... | 4 | https://mathoverflow.net/users/8103 | 130471 | 72,403 |
https://mathoverflow.net/questions/130483 | 1 | I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann\_R(r)\in {\rm Min}(R)$ we have $Ann\_R(P)=Rr$.
This question is a follow-up of my previous [question](https://mathoverflow.net/questions/129949/when-does-second-annihilator-of... | https://mathoverflow.net/users/21992 | An example of a ring $R$ with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$. | I believe the graded ring $k[x,y]/(xy)$ satisfies this property. The minimal primes are just $(x)$, $(y)$ and the annihilators are just the other ideal. The dimension is 1. This is not however local.
| 3 | https://mathoverflow.net/users/19642 | 130487 | 72,406 |
https://mathoverflow.net/questions/130491 | 1 | If the following question is not good enough, please close it.
Please give a elementary proof or reference of the following result:
Let $G$ be a finite groups of order $120$, and $|Z(G)|=2$. Suppose $G/Z(G) \cong A\_5$ and $G'=G$, then $G \cong SL(2,5)$.
Of course, using the Schur index of $A\_5$, I can get the p... | https://mathoverflow.net/users/22049 | about the non-solvable group of order $120$ | Since you know how to prove this already, it is difficult to know what you are looking for!
Here is a quick proof that there is a unique isomorphism class of groups $G$ with this property using the well-known fact that $A\_5$ has the presentation $\langle x,y \mid x^2=y^3=(xy)^5=1 \rangle$.
Let $t$ generate the cen... | 4 | https://mathoverflow.net/users/35840 | 130495 | 72,409 |
https://mathoverflow.net/questions/130438 | 6 | I'm trying to understand the standard $GL(3,\mathbb{R})$ action on the 15-dimensional space of possible values for the derivative of the Riemann curvature tensor of a 3-dimensional manifold $M$ at a point, thought of as the codimension-3 subspace of the space
$ S^2(\Lambda^2 T^\*M) \otimes T^\*M = \{r\_{ijkl,m} (dx^i... | https://mathoverflow.net/users/18048 | Invariants of a $GL(3,\mathbb{R})$ action | There's a 'quasi-normal form' on a dense open set that can be described without too much trouble. Here's one way to do it.
First, recognize that we are looking for a normal form for elements $Q\in W=\bigl[S^2(\Lambda^2(V^\ast))\otimes V^\ast \bigr]\_0$ under the action of $G=\mathrm{GL}(V)$, where $V$ is a vector spa... | 5 | https://mathoverflow.net/users/13972 | 130497 | 72,410 |
https://mathoverflow.net/questions/130306 | 21 | What is the role of the theory of Automatic Groups in the history of Geometric Group Theory?
Motivation:
When I read through the "Word Processing in Groups" I was amazed by the supreme beauty and elegance of the theory and of how robust it is. (That it started from conversations between (true artists) Cannon and Th... | https://mathoverflow.net/users/33828 | The role of the Automatic Groups in the history of Geometric Group Theory | I think there are a few different ways in which Automatic Groups affected the history of Geometric Group Theory.
One was mentioned by Derek Holt, which I will spin in a slightly different way: if you really want to know the group, and if it has an automatic structure, you had better know that structure. For an indivi... | 12 | https://mathoverflow.net/users/20787 | 130499 | 72,411 |
https://mathoverflow.net/questions/130492 | 1 | Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as $E(v)=$ { $x \in Vert(S)$ | $xv$ is an edge of $CH(S)$ }.
My question is best stated using an example in 2-D:
In this e... | https://mathoverflow.net/users/nan | Higher dimensional convex hull | I think the answer is yes. First observe that $CH(P) \subset CH(S)\cap H$: if $x\in CH(P)$ then $x$ is written as a convex combination of things which are convex combinations of vertices of $CH(S)$, so is a convex combination of vertices of $CH(S)$.
Then we note that $CH(S)\cap H$ has as its vertices the points in $P... | 2 | https://mathoverflow.net/users/3669 | 130504 | 72,414 |
https://mathoverflow.net/questions/130454 | 4 | There are a few equivalent definitions of Stein manifolds. As far as I know they were initially defined as holomorphically convex complex manifolds, and then the other definitions (e.g. complex manifolds which embed holomorphically in a complex Affine space) were proved to be equivalent to the first afterwards. What I ... | https://mathoverflow.net/users/31475 | Stein manifolds definiton | Compact complex manifolds are holomorphically convex. Stein's original definition had in addition to holomorphic convexity separation of points by holomorphic functions and local coordinates given by holomorphic functions. Stein himself called these holomorphically complete manifolds. He introduced it to study the solu... | 8 | https://mathoverflow.net/users/4696 | 130510 | 72,418 |
https://mathoverflow.net/questions/130515 | 1 | Let $G$ a simple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.
Let $\lambda$ a minuscule or quasiminuscule weight.
For which types and for which weights do we have that:
$\forall w\in W, \lambda-w\lambda$ is a multiple of a root?
For classical groups, we know that for types $A$, $B$, $C$ it's t... | https://mathoverflow.net/users/27398 | weights and exceptional root systems | EDIT: Reading the question more carefully, I think the difference between the highest weight and an arbitrary Weyl group conjugate will almost never be a single root or muiltiple of a root. (What's true is that the difference between "adjacent" weights in that orbit across a single reflecting wall will be 0 or else a r... | 2 | https://mathoverflow.net/users/4231 | 130518 | 72,420 |
https://mathoverflow.net/questions/130489 | 11 | This is a puzzle a colleague asked me recently.
Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn you take two balls at random out that have *different* colors and color one the color of the other. You then put them both back in the bag. What is the expected number of turns before all... | https://mathoverflow.net/users/33958 | Another colored balls puzzle | I think you can verify Greg Martin's answer using indicator variables and linearity of expectation.
Let the random variable $X\_i$ be the number of steps where one of the balls has the $i$th color before the $i$th color either disappears or becomes the only color.
The expected value of $X\_i$ is as in the Gambler's... | 15 | https://mathoverflow.net/users/19729 | 130524 | 72,421 |
https://mathoverflow.net/questions/130429 | 4 | I was reading the [Boneh-Franklin IBE paper](http://courses.cs.vt.edu/cs6204/Privacy-Security/Papers/Crypto/IBE-Weil-Pairing.pdf), and it seemed rather conspicuous to me that they
didn't address the question of how to find primes $p$ and $q$ satisfying what they need (on page 19).
Since one can [efficiently gen... | https://mathoverflow.net/users/nan | p such that p+1 has a large prime factor, effectively | Yes. G. Harman [has proven](http://www.ams.org/journals/mcom/2005-74-252/S0025-5718-05-01749-7/):
**Theorem:** Let $a \in Z$ and $\theta < .61$. Then there exists effectively computable constants $X\_{0}$ and $\delta>0$ such that if $x > X\_{0}$ then:
$$\sum\_{p\leq x : P(p+a)> x^{\theta} } 1 > \delta \frac{x}{\log... | 7 | https://mathoverflow.net/users/630 | 130536 | 72,424 |
https://mathoverflow.net/questions/130535 | 3 | Let $\mathcal{H}$ be an infinite dimensional separable (complex) Hilbert space. What is a natural space which parameterizes the choices of orthonormal bases for $\mathcal{H}$?
It seems like one option, in analogy to the finite-dimensional Stiefel manifold (something I know nothing about beyond what I just glanced at ... | https://mathoverflow.net/users/16107 | A space parameterizing the choices of orthonormal bases for a Hilbert space | You could look at the set of operators in $B(H)$ which are diagonalized by a given basis. For each choice of orthonormal basis, this gives you an atomic maximal abelian von Neumann algebra (atomic masa). Now there is a natural topology on the space of all von Neumann algebras sitting inside $B(H)$ called the Effros-Mar... | 3 | https://mathoverflow.net/users/23141 | 130538 | 72,425 |
https://mathoverflow.net/questions/129935 | 13 | The [Virasoro algebras](http://en.wikipedia.org/wiki/Virasoro_algebra) $Vir\_c$ are a family of infinite dimensional Lie \*-algebras parametrized by a real number $c$, called the central charge.¹
I hear that there exist two critical values of the central charge $c$, where qualitative changes happen to the structure o... | https://mathoverflow.net/users/5690 | What happens to Virasoro at c=25? | I have an incomplete understanding of this, but I will try to say what I know.
For each $c \in \mathbb{C}$, we define $Verma\_c$ to be the category whose objects are Verma modules $V\_{h,c}$ of central charge $c$ and highest weight $h$ for some $h \in \mathbb{C}$, and whose maps are Virasoro-module maps. Feigin and F... | 5 | https://mathoverflow.net/users/121 | 130539 | 72,426 |
https://mathoverflow.net/questions/130540 | 3 | Hy guys. I'm doing some independent analysis which makes use of the tensor product of modules (over commutative rings with unit 1, and ring homomorphisms map $1 \mapsto 1$). Let $\pi: A' \to A$ be a surjective homomorphism of rings which identifies $A$ as an $A'$-algebra (and hence an $A'$-module). Let $M$ be any $A'$-... | https://mathoverflow.net/users/33975 | An example of a tensor product consisting of only simple tensors? | Your argument is right. In your situation, $A \cong A^{\prime} / I$ for some ideal $I$ of $A^{\prime}$, and the tensor product $M\otimes\_{A^{\prime}} A$ is isomorphic to $M / \left(IM\right)$, where $IM$ is the submodule of $M$ spanned by all products of the form $im$ with $i\in I$ and $m\in M$. This explains why it c... | 5 | https://mathoverflow.net/users/2530 | 130541 | 72,427 |
https://mathoverflow.net/questions/130278 | 1 | This question is related to [Lyndon-Hochschild-Serre spectral sequence and cup products](https://mathoverflow.net/questions/130008/lydon-hochschild-serre-spectral-sequence-and-cup-products).
I have the followin result by J.S Milne in his book Arithmetic duality theorems pg 105.
Let $$0 \rightarrow C \rightarrow W \... | https://mathoverflow.net/users/15566 | Cup-products and Transgression maps. | The homology analogue of the result stated by Milne is that the transgression in the Lyndon-Hochschild-Serre spectral sequence for group homology
$$
H\_2(G;M)\to H\_0(G;H\_1(C;M)) = H\_1(C;M)\_G
$$
is given (up to a sign) by *cap* product with the extension class $H^2(G;C)$. This makes sense since the cap product is a... | 2 | https://mathoverflow.net/users/8103 | 130566 | 72,439 |
https://mathoverflow.net/questions/130545 | 25 | What are the main ideas of Harald Helfgott's [proof](http://arxiv.org/abs/1305.2897) that all odd $n \geq 5$ is the sum of 3 primes?
| https://mathoverflow.net/users/13923 | Proof of the weak Goldbach Conjecture | I think this blog post of Terry Tao, as well as the comments following it (including some from Helfgott) answer this question as completely as one could reasonably hope.
<https://terrytao.wordpress.com/2012/05/20/heuristic-limitations-of-the-circle-method/>
| 21 | https://mathoverflow.net/users/5575 | 130571 | 72,441 |
https://mathoverflow.net/questions/129974 | 0 | Let
$$a\_{n,k}=\sum\_{s\_i \geq 1 \atop \sum\_{i=1}^{n-k} s\_i \leq n} \frac{2^{n}}{(2(n-\sum\_{i=1}^{n-k} s\_i)+1)!\prod\_{i=1}^{n-k} (2s\_i)! }$$
for $0 \leq k \leq n-1$. Prove for $1 \leq k \leq n-1$ that
$$b\_{n,k}=\sum\_{l=1}^k (-1)^{k-l} \sum\_{s\_i \geq 1 \atop \sum\_{i=1}^l s\_i =k} \prod\_{i=1}^l a\_{n,s\_i}>... | https://mathoverflow.net/users/31765 | positive expression | Found the answer on page 102 of
<http://web.iitd.ac.in/~maz088121/chui.pdf>
| -1 | https://mathoverflow.net/users/31765 | 130579 | 72,442 |
https://mathoverflow.net/questions/130557 | 9 | The theorem of Maschke tells us that every representation of a finite group is the direct sum of irreducible representation. More precisely:
Let $G$ be a finite group, $K$ a field whose characteristic does not divide the order of $G$, $V$ a $K$-vector space of finite dimension and $\rho:G\to {\rm GL}(V)$ a representa... | https://mathoverflow.net/users/33944 | A generalisation of the theorem of Maschke | Your expected generalization is true. Without loss of generality, you may assume that $G$ is a subgroup of $GL(V)$. A linear group whose elements have finite orders is locally finite, that's an old Theorem by Schur, and contained for instance in Wehrfritz's book *Infinite Linear Groups*. Now take a maximal linearly ind... | 16 | https://mathoverflow.net/users/18739 | 130584 | 72,445 |
https://mathoverflow.net/questions/130549 | 5 | What is the principal value of the integral $$\int \limits \_0^\infty \left( \frac {1}{x^2}-\frac{\cot(x)}{x} \right) dx ?$$ Maple finds $PV \int\_0^\infty \tan(x)/x dx = \pi/2.$ Such integrals arise in physics. I unsuccessfully asked it in SE.
| https://mathoverflow.net/users/32273 | Principal value of integral | formula 3.749.2 from Gradshteyn & Ryzhik gives:
$$\int\_0^{\infty}\frac{1-x\;{\rm cotan}x}{x^2+\epsilon^2}dx=\frac{\pi}{2\epsilon}-\frac{\pi}{e^{2\epsilon}-1}\quad{\rm for}\quad \epsilon>0.$$
taking the limit $\epsilon\downarrow 0$ gives your $\pi/2$; G&R do not explicitly say that their formula is a principal valu... | 4 | https://mathoverflow.net/users/11260 | 130586 | 72,446 |
https://mathoverflow.net/questions/130573 | 3 | As a member of boy scouts I was considering the following problem:
suppose you're organising some kind of olympic games...
\*You divide the boys in $2n$ teams (subsets of equal size)
\*There are $2n-1$ types of sports which all involve two-team matches
\*There will be $2n-1$ time slots: during every time sl... | https://mathoverflow.net/users/33927 | two boy scouts problems | Problem 1 is just asking for a Room square of size 2n-1. On the [Wikipedia page for Room squares](http://en.wikipedia.org/wiki/Room_square) or in Anderson's book "Combinatorial designs and tournaments", for example, you'll find that the cases n=2,3 are the only ones for which construction is impossible.
| 5 | https://mathoverflow.net/users/6771 | 130597 | 72,452 |
https://mathoverflow.net/questions/130532 | 4 | Consider a [Brownian bridge](http://en.wikipedia.org/wiki/Brownian_bridge) of length $r$ on the plane. What is the expected (non-signed) smallest area of the disc spanned by the loop? By "non-signed" I mean that if a loop goes around a unit square 8 times in the clockwise direction and then 3 times in the counterclockw... | https://mathoverflow.net/users/nan | Area of a Brownian bridge on the plane | If I understand your definition of area correctly, the Brownian bridge ends up having infinite filling area because of small regions with large winding number -- a [result of Yor](http://www.ams.org/mathscinet-getitem?mr=576898) shows that the expected area of the region with winding number $k$ is $\sim 1/k^2$. This co... | 7 | https://mathoverflow.net/users/25051 | 130598 | 72,453 |
https://mathoverflow.net/questions/130577 | 5 | I have two boundaries of two planar polygons, say, B1 and B2 of polygons P1 and P2 (with m and n points in Boundaries B1 and B2). I want to find out if the polygons overlap or not. If they overlap, then the area (approx) covered by the common region.
Is there any way to do this?
| https://mathoverflow.net/users/33984 | How to find overlap between two convex hulls, along with the overlap area |
>
> How to find overlap between two convex hulls
>
>
> I want to find out if the Polygons overlap or not
>
>
>
Convex hulls are convex, so you can use a convex polygon collision detection algorithm.
<http://content.gpwiki.org/index.php/Polygon_Collision#Separating_Axis>
---
>
> If they overlap,then t... | 8 | https://mathoverflow.net/users/27564 | 130600 | 72,454 |
https://mathoverflow.net/questions/128834 | 7 | (This is essentially a continuation of my previous question, [here](https://mathoverflow.net/questions/122859/).)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you can remove this assumption if you like) that each ball has positive finite ... | https://mathoverflow.net/users/nan | For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$? | For the uncentred maximal function, one can cook up examples by using the [star graph](http://en.wikipedia.org/wiki/Star_graph) with a suitable weight on the vertices. In more detail, if one takes a star graph (with the graph metric) with $n$ spokes, with the root vertex having measure one and the leaves having measure... | 5 | https://mathoverflow.net/users/766 | 130603 | 72,456 |
https://mathoverflow.net/questions/130522 | 1 | I am looking for literature that might contain the spherical representations of $GL(n, \mathbb{C})$. Here a spherical representation is an irreducible representation $\rho$ of $G$ on $\mathbb{C}$ such that $\rho\_{K}$, for $K$ a maximal compact subgroup, fixes a vector in $\mathbb{C}$. I realize my question is similar ... | https://mathoverflow.net/users/25794 | Finding spherical representations of $GL(n, \mathbb{C})$. | In the case of $GL(n, \mathbb{C})$, it is known that every unitary, irreducible, infinite-dimensional representation (the others are one-dimensional and factor through the determinant) is given as induced representation $\pi$ from a minimal parabolic associated to the Levi $M(\mathbb{C})$ (being the group of diagonal m... | 0 | https://mathoverflow.net/users/10400 | 130604 | 72,457 |
https://mathoverflow.net/questions/130607 | 4 | Are there other integers $n$ than even perfect numbers such that $\sigma(n)=\omega(n)n$ and $\omega(n)\vert n$?
Thanks in advance.
| https://mathoverflow.net/users/13625 | Integers n such that sigma(n)=omega(n)n and omega(n) divides n | For $n=120$ we have $\omega(120)=3$ and $\sigma(120)=360=3\cdot 120=\omega(120)\cdot 120$
with $\omega(120)\mid 120$. This is not an even perfect number.
| 5 | https://mathoverflow.net/users/32332 | 130614 | 72,462 |
https://mathoverflow.net/questions/130609 | 1 | Suppose we have the following differential inequality
$F''(w)\le \frac{p-1}{p}\frac{(F'(w))^2}{F(w)}$ on $w\in(w\_0,w\_0+\varepsilon)$, $w\_0>0$, $\varepsilon>0$, $p>1$. In addition, $F(w)>0$, $F'(w)>0$ for $w\in[w\_0,w\_0+\varepsilon)$.
Let's consider the solution of the Cauchy problem
$\begin{cases}G''(w)= \fra... | https://mathoverflow.net/users/33906 | Generalisations of the Gronwall's lemma | You can calculate the seconde derivative of $F^{1/p}$. you'll find that it is negative, and the seconde derivative of $G^{1/p}$ is 0, so we have $(F^{1/p})^{''}\leq (G^{1/p})^{''} $
Then by integrating twice you'll have $F^{1/p}\leq G^{1/p} $ and then $F\leq G$
| 1 | https://mathoverflow.net/users/30889 | 130617 | 72,464 |
https://mathoverflow.net/questions/130616 | 17 | Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves colimits separately in each variable. (What I'm about to say is true in greater generality, but let's just stick with these assumptions.)
Th... | https://mathoverflow.net/users/3049 | Reference for "lax monoidal functors" = "monoids under Day convolution" | This observation appears already in Day's thesis as Example 3.2.2. For some reason it is only stated for commutative monoids and symmetric (pro)monoidal functors and only as a correspondence of objects not an equivalence of categories, also no proof is given. (Day's thesis used to be available on [Street's homepage](ht... | 12 | https://mathoverflow.net/users/12547 | 130619 | 72,466 |
https://mathoverflow.net/questions/130519 | 6 | An bouneded linear operator $A \in L(X, Y)$ (here $X$, $Y$ are Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that
$$ \left( \sum\_{j=1}^N \| A x\_j\|\_X^2 \right)^{1/2} \leq C \cdot \sup \left\{ \left(\sum | \langle x\_j, \omega \rangle | \right)^{1/2} \mid \omega \in X', \|\omega\|\_{X'... | https://mathoverflow.net/users/16702 | Absolutely 2-summable operator on a Hilbert space | I answered the OP's question in a comment, but the right question is whether you can estimate the $2$-summing norm of an operator $T$ from a Hilbert space $H$ by taking $\sup (\sum \|Te\_n\|^2)^{1/2}$, where the sup is over all ON bases $(e\_n)$ for $H$. In fact, if $H$ is infinite dimensional, this sup is equal to the... | 4 | https://mathoverflow.net/users/2554 | 130620 | 72,467 |
https://mathoverflow.net/questions/130496 | 7 | $\newcommand{\bsV}{\boldsymbol{V}}$ $\newcommand{\bsE}{\boldsymbol{E}}$ $\newcommand{\bR}{\mathbb{R}}$ Suppose that $\bsV$ is an $N$-dimensional real Euclidean space. Denote by $\newcommand{\eA}{\mathscr{A}}$ $\eA$ the space of symmetric positive semidefinite operators $A:\bsV\to \bsV$. To each $A\in \eA$ we can associ... | https://mathoverflow.net/users/20302 | Continuous dependence of the expectation of a r.v. on the probability measure | This problem reduces quickly to Holder continuity of the operator square root. That is, there exists a $C > 0$ such that
$$
\begin{align}
\lVert\sqrt{A}-\sqrt{B}\rVert\le C\lVert A-B\rVert^{1/2}&&{\rm(1)}
\end{align}
$$
for any positive semidefinite operators $A,B$.[1]
Assuming (1), the proof of continuity with Hold... | 6 | https://mathoverflow.net/users/1004 | 130632 | 72,472 |
https://mathoverflow.net/questions/130543 | 11 | *(This question was [posted on math.SE](https://math.stackexchange.com/questions/375085/on-the-large-cardinals-foundations-of-categories) over two weeks ago, but received no answer. I am therefore posting it here as well.)*
It is well-known that there are difficulties in developing basic category theory within the co... | https://mathoverflow.net/users/7206 | On the large cardinals foundations of categories | Allow me to make some comments as someone who converted to the universeful approach recently; but take it with a pinch of salt, as I have only been studying category theory for 2½ years.
I should briefly mention the trigger that led me to the pro-universe camp: about 6 months ago, I started learning about quasicatego... | 12 | https://mathoverflow.net/users/11640 | 130639 | 72,474 |
https://mathoverflow.net/questions/130643 | 4 | I hope this question is not unreasonable.
We all know how to take products of numbers, this generalises to a huge amount of different types of products in mathematics. In a certain sense this notion is "on the surface".
On the other hand there is a different operation, called "co-product" that seem to be a less int... | https://mathoverflow.net/users/13441 | Is it true that Nature promotes products? | You need to distinguish between "coproduct" and "comultiplication." The categorical coproduct is just a generalization of addition and is intuitive in many contexts.
Comultiplications are more interesting. Actually it turns out that many familiar mathematical objects are canonically equipped with comultiplications. ... | 13 | https://mathoverflow.net/users/290 | 130645 | 72,476 |
https://mathoverflow.net/questions/130624 | 7 | A Pythagorean fraction is a number of the form $a/b$ or $b/a$ where $a$ and $b$ are the legs of a Pythagorean triple. Are there simple necessary and/or sufficient conditions for determining whether a rational number can be expressed as a ratio of two Pythagorean fractions? As an example, I would be interested to know i... | https://mathoverflow.net/users/18108 | Possible ratios of Pythagorean fractions | This is a sketch how to decide the question for $\frac{4}{9}$.
The question is if there are positive integers $a,b,d,e$ such that $\frac{4}{9}=\frac{a/b}{d/e}$ with $a^2+b^2$ and $d^2+e^2$ squares. Denoting $p:=\frac{9a}{b}=\frac{4d}{e}$, the question is if there is $p\in\mathbb{Q}^\times$ such that both $p^2+4^2$ an... | 5 | https://mathoverflow.net/users/11919 | 130651 | 72,481 |
https://mathoverflow.net/questions/130653 | 3 | Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seems obvious, but I haven't been able to prove this.
I would greatly appreciate if someone could supply a proof or reference. Thank you.
| https://mathoverflow.net/users/21277 | Simplex with edges of length at least s having smallest circumradius | Let $H$ be a Hilbert space. Let $e\_1\ \ldots\ e\_n$ be vectors such that $\forall\_{k=1\ldots n}\ \|e\_k\|\le 1$ for a natural $n>1$. I am addressing $(n-1)$-simplex for the sake of avoiding eyesores. Then:
$$ 0\ \le\ (e\_1+\ldots +e\_n)^2\ \ \le\ \ n\ +\ 2\cdot\sum\_{j\ne k}e\_j\cdot e\_k $$
It fol... | 4 | https://mathoverflow.net/users/8385 | 130664 | 72,487 |
https://mathoverflow.net/questions/130513 | 8 | The same colleague as in [Another colored balls puzzle](https://mathoverflow.net/questions/130489/another-colored-balls-puzzle) asked me the following variant which she called "part II".
Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn you take one ball uniformly at random from the b... | https://mathoverflow.net/users/33958 | Another colored balls puzzle (part II) | Ori Gurel-Gurevich pointed out that the transitions under Rule 1 are like the Hamming distances from $\vec{0}$ in a random walk on a hypercube $C$ (the [Ehrenfest urn model](http://en.wikipedia.org/wiki/Ehrenfest_model)). Further, the average number of steps for a random walk to return to a vertex in a regular connecte... | 2 | https://mathoverflow.net/users/2954 | 130678 | 72,493 |
https://mathoverflow.net/questions/130682 | 2 | Consider an analytic function $f : U \longrightarrow \mathbb{C}$ where $U$ is an open subset of the complex numbers which contains the closed unit disk. I have $|f(x)| \geq 1$ for any $ x \in [-1, 1]$. Is the following true ?
$$ |\int\_0^{2\pi} f(e^{it})\overline{f(e^{-it})}dt| \geq 4$$
| https://mathoverflow.net/users/nan | A New Analytic Inequality | Yes, indeed
$$\frac{1}{2\pi}\int\_0^{2\pi} f(e^{it})\overline{f(e^{-it})}dt= |f(0)|^2,$$
as you can check applying the Cauchy formula to the holomorphic function $f(z)\overline{f(\bar z)}$.
| 4 | https://mathoverflow.net/users/6101 | 130688 | 72,496 |
https://mathoverflow.net/questions/129830 | 7 | Is there an integration by parts formula for fractional laplacians in $L^p(\mathbb{R}^N)$, something like
$$
s\in(0,1),\qquad\int\limits\_{\mathbb{R}^N}f[(-\Delta)^sg] =\int\limits\_{\mathbb{R}^N}[(-\Delta)^{s}f]g
$$
or an intermediate formula involving "lower derivatives"?
Typically, I would like to know if
$$
... | https://mathoverflow.net/users/33741 | integration by parts for the fractional Laplacian | You can integrate by parts:
$$
\int\_{\mathbb{R}^d} (-\Delta)^s f(x) g(x)dx=\int\_{\mathbb{R}^d} (-\Delta)^s g(x) f(x)dx.
$$
Using Fourier and $L^2$ the equality is obvious. Let's do "by hand" in $d=1$ and $s=1/2$ (the other cases follow the same idea:
You have
$$
\int\_{\mathbb{R}} (-\Delta)^{1/2} f(x) g(x)dx=\in... | 9 | https://mathoverflow.net/users/33135 | 130700 | 72,503 |
https://mathoverflow.net/questions/130683 | 2 | In the heat equation:
$$\partial u(x,t)=D\partial\_{xx}u(x,t)$$
the diffusion coefficient $D$ is in general a constant or a given function of $u(x,t)$ in the nonlinear equation. Suppose I have a diffusion coefficient depending on the integral of $u(x,t)$. In this case I have:
$$\partial\_t u(x,t)=\left[\int\_{-L}^L u(x... | https://mathoverflow.net/users/21258 | Heat integro - differential equation | There is a simple way to manage this equation using a Fourier series. We assume a boundary at $0$ and $L$ and that exists the Fourier series for the solution
$$
u(x,t)=\sum\_{n=-\infty}^{\infty}u\_n(t)e^{i\frac{2\pi n}{L}x}
$$
then you note that
$$
D(t)=\int\_{-L}^L u(s,t)ds=\int\_{-L}^L\sum\_{n=-\infty}^{\infty}u\_n... | 2 | https://mathoverflow.net/users/19520 | 130703 | 72,505 |
https://mathoverflow.net/questions/130660 | 4 | The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the [Wikipedia page](http://en.wikipedia.org/wiki/Orientability) on the topic. While I was able to satisfy his curiosity, one statement on that page (which I did not even attempt to delve i... | https://mathoverflow.net/users/12301 | Picturing a Certain Torus and Klein Bottle | Jeff Weeks' "Shape of Space" book has several good examples of two sided non-orientable surfaces embedded in non-orientable manifolds. That book is written in a manner that is easy to approachable for an outsider, so it might provide a natural next direction for a conversation that started from discussing the wikipedia... | 7 | https://mathoverflow.net/users/27453 | 130704 | 72,506 |
https://mathoverflow.net/questions/42783 | 9 | I am interested in injective model structures on both symmetric spectra as exposed in [Hovey/Shipley/Smith](http://www.math.uiuc.edu/K-theory/0265/) and motivic symmetric spectra as in [Jardine's article](http://www.mathematik.uni-bielefeld.de/documenta/vol-05/vol-05.html). Both authors take a model structure on the un... | https://mathoverflow.net/users/733 | Is the injective model structure on symmetric spectra Bousfield localizable? | The answer is yes. As Lennart Meier points out, part of this answer is contained in Schwede´s book project. I´m writing to flesh out those references and discuss the topological injective case, which doesn´t appear in Schwede. Plus, old questions shouldn´t linger around without answers. Let´s start with the case where ... | 5 | https://mathoverflow.net/users/11540 | 130706 | 72,508 |
https://mathoverflow.net/questions/130710 | 0 | Let $A$ be a (semi-)abelian variety over an algebraically closed field $K$, and $X$ be a closed irreducible subvariety. Can $X$ have a non-trivial finite stabilizer ? By stabilizer, I mean the closed subgroup $S\subset A$ such that $X$ is stable by translation by any point in $S$.
This question is equivalent to the one... | https://mathoverflow.net/users/25887 | Is the stabilizer of an irreducible subvariety of an abelian variety irreducible ? | No. Take a curve in its Jacobian and pull it back by multiplication by some n. The resulting pullback is a curve invariant by the n torsion.
| 2 | https://mathoverflow.net/users/2290 | 130714 | 72,512 |
https://mathoverflow.net/questions/130712 | 3 | Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows.
My favorite example of a groupoid is an action groupoid. If a group $G$ acts on the left on a set $X$,
we set
$$
A=\{(x,g,y)\mid x,y\in X, g\in G,\ y=g\*x\},
$$
then $A\rightrightarrows X$ with the evident maps is ... | https://mathoverflow.net/users/4149 | Equivalence and weak equivalence of groupoids | The answer to question 1 is “yes” since every weak equivalence is essentially surjective (let $y\in Y$, $[y]$ the connected component of $y$, which is in the image of $\pi\_0(F)$, thus there exists $y^\prime\in [y]$ which is an element of the image of $F$ and isomorphic to $y$) and full and faithful (since $F\_x$ is an... | 6 | https://mathoverflow.net/users/33842 | 130717 | 72,514 |
https://mathoverflow.net/questions/130716 | 0 | Hi all.
If $G$ is a finite group and $\varrho : G \to \text{GL}(V), \eta : G \to \text{GL}(W)$
are finite dimensional representations, $V\_0$ is a $G$-invariant subspace of $V$
and $f : V\_0 \to W$ is a homomorphism of representations, i.e. a homomorphism of vector spaces satisfying $f(\varrho(g)x) = \eta(g) f(x)$,
... | https://mathoverflow.net/users/20431 | Continuation of homomorphisms of representations... | You don't explicitly say your representation is complex but I think your example shows that this is the case you're interested in. If so, then $V\_0$ has a $G$-invariant complement $V\_1$ by Maschke's Theorem <http://en.wikipedia.org/wiki/Maschke%27s_theorem>, and the unique linear extension of $f$ whose kernel contain... | 1 | https://mathoverflow.net/users/345 | 130726 | 72,518 |
https://mathoverflow.net/questions/130725 | 0 | Let $Z$ be a compact, connected, orientable (**Edit: as Misha point out**) and locally Riemannian symmetric space. As a complete, simple connected, locally symmetric space is a global symmetric space. We can write $Z=\Gamma \backslash G/K$. One of such $(G,K)$ is $G=\mathrm{Iso}(\widetilde{Z})$,$K$ is the stablizer of ... | https://mathoverflow.net/users/16326 | locally symmetric space and global symmetric space | No, take $Z$ to be a non-orientable compact connected hyperbolic surface.
Edit: Here is an easy orientable example. Let $S$ be a non-orientable hyperbolic surface. Then take $Z$ to be the orientable 2-fold cover of $S\times S$. The point is is that fundamental group of $Z$ (regarded as a deck-transformation group) s... | 2 | https://mathoverflow.net/users/21684 | 130733 | 72,519 |
https://mathoverflow.net/questions/130731 | 1 | Given a transformation $t$ from the transformation semigroup $T\_{n}$, if you take powers of $t$ under composition you get a length $s$ stem followed by a cycle. Permutations by definition have a length zero stem.
Are the terms "stem" and "cycle" common, or is some other terminology used in the literature?
| https://mathoverflow.net/users/7359 | Transformation terminology question | This question seems a bit ill-posed to me. There are two possible interpretations here.
(1) If $s$ is an element of a finite semigroup, then $s^i=s^{i+p}$ where there is a minimal such $i$ and $p$. What is the common terminology for $i,p$?
The "official" terminology is that $i$ is the index and $p$ is the period. ... | 1 | https://mathoverflow.net/users/15934 | 130741 | 72,521 |
https://mathoverflow.net/questions/130740 | 4 | If $X$ is a compact oriented surface in a 4-dimensional oriented manifold $M$, then the self-intersection number $X^2$ of $X$ is given by the integral over $X$ of the Euler class of the normal bundle. In the case of $CP^1$ embedded in $CP^2$, the normal bundle is isomorphic to the Hopf bundle, therefore $X^2$ can be ob... | https://mathoverflow.net/users/29850 | Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings | You can find $\mathbb CP^1$ in a wide variety of $4$-manifolds having any Euler class you like. One really simple way is to take the connect-sum of $k$ copies of $\mathbb CP^2$. The idea is to embed $\mathbb CP^1$ in the connect sum so that you are simultaneously breaking the $\mathbb CP^1$ up as a connect sum in all t... | 3 | https://mathoverflow.net/users/1465 | 130743 | 72,523 |
https://mathoverflow.net/questions/130684 | 7 | Let $f$ be a function defined on $[-1,1]^d$. Assume that all partial derivatives of $f$ up to order $r$ are continuous; and the $\infty$-norm of these partial derivatives are uniformly upper bounded by a constant. Let $p^\*\_n$ be the best degree $n$ approximation polynomial of $f$. That is,
$ p\_n^\* = \mathrm{argmi... | https://mathoverflow.net/users/30853 | Multivariate polynomial approximation of smooth functions | Short answer: The estimate is similar to that for functions in one variable.
Longer answer: The estimates of best approximation of a real-valued smooth function (by algebraic as well as by trigonometric polynomials) in terms of moduli of continuity of its derivatives are known as Jackson theorems. They were first pro... | 7 | https://mathoverflow.net/users/14493 | 130747 | 72,524 |
https://mathoverflow.net/questions/130736 | 4 | Let $\mathcal{X}$ be an Artin stack. In "Abramovich, Graber, Vistoli - Twisted bundles and admissible covers", the authors describe a procedure to *rigidify* $\mathcal{X}$ by a central subgroup $H$ of the *generic stabilizer* and obtain an Artin stack $\mathcal{X}^H$. Roughly speaking, the objects of $\mathcal{X}^H$ ar... | https://mathoverflow.net/users/nan | Rigidification and good moduli space (morphism) in the sense of Alper | It is certainly not true that $\mathcal X \to \mathcal X^H$ is a good moduli morphism, unless $H$ is linearly reductive, because when you push forward the cohomology of $H$ will come into play.
On the other hand $\mathcal X^H \to X$ is a good moduli space, because the pushforward $QCoh(\mathcal X^H) \to QCoh(X)$ can ... | 12 | https://mathoverflow.net/users/4790 | 130748 | 72,525 |
https://mathoverflow.net/questions/130654 | 9 | I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$.
Where can I find this reference?
**UPDATE:** Some have questioned my use of the word "noteworthy." I really just want this i... | https://mathoverflow.net/users/29873 | A catalog of faithful representations of finite groups? | Dustin, if you know the character $\chi$ of a representation $\rho$ of a finite group, it is easy to see whether it is faithful or not.
Your representation is faithful if and only if **for every $g \in G$, $\chi(g)=\chi(e)$ implies $g=e$**. For if $g \in G$, and $\chi(e)=n$ is the dimension of your representation $\rho... | 14 | https://mathoverflow.net/users/9317 | 130751 | 72,526 |
https://mathoverflow.net/questions/130732 | 1 | Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation
$$ A = \sum\_{j=0}^\infty a\_j \langle \cdot, e\_j\rangle \cdot x\_j,$$
where $(e\_j)$ is an orthonormal basis of $H$ and $(x\_j)$ is a family of vectors in $X$ with $\|x\_j\| = 1$ and $(a\_j) \in ... | https://mathoverflow.net/users/16702 | Special kind of operators | See page 228ff of
Albrecht Pietsch: Operator ideals, Elsevier 1980.
[(pdf here)](http://gen.lib.rus.ec/book/index.php?md5=BAD44BB627E2D4148E9F042170671CDB)
Maybe, your operators are the $(\infty, p, \infty)$-summing operators there.
| 1 | https://mathoverflow.net/users/26935 | 130753 | 72,527 |
https://mathoverflow.net/questions/130718 | 6 | A generic abelian variety of dimension 2 or 3 is a jacobian of a curve.
Is there a canonical way to determine a curve whose jacobian is a prym variety of a unramified double cover of a curve of genus 3 or 4?
| https://mathoverflow.net/users/27125 | Prym varieties as Jacobian varieties | Yes, assuming you mean a way to go from the double cover to the curve, (rather than how to go from the Prym variety to the curve, which is just a constructive Torelli argument). This is based on the fact that curves of genus 3 and 4 are "trigonal", i.e. admit a degree 3 map to the projective line. The corresponding "tr... | 12 | https://mathoverflow.net/users/9449 | 130755 | 72,529 |
https://mathoverflow.net/questions/130768 | 9 | Is there a published reference for this ZF theorem?
Let $m,n\in\mathbb{N}$. If $a\_1,\dots,a\_m$ and $b\_1,\dots,b\_n$ are cardinals such that $a\_i\le b\_j$ for all $i$ and $j$, then there is a cardinal $x$ such that $a\_i\le x\le b\_j$ for all $i$ and $j$.
It's enough if the proposition is stated for the case $m ... | https://mathoverflow.net/users/nan | Cardinals without choice: interpolation (reference wanted) | *The comments by The User and Joel David Hamkins refer to a previous version of the answer which contained a mistake. The current version is completely disjoint of the previous one, and the comments no longer apply.*
This appears in Tarski's book **Cardinal Algebras** as Theorem 2.28, called *Interpolation Theorem*, ... | 4 | https://mathoverflow.net/users/7206 | 130772 | 72,537 |
https://mathoverflow.net/questions/130668 | 3 | Does anybody have an electronic copy of Schaper's PhD thesis:
K.D. SCHAPER, ‘Charakterformeln fur Weyl-Moduln und Specht-Moduln in Primcharacteristik’,
Diplomarbeit, Bonn, 1981.
I would like to understand how to pass between Jantzen's formulation of the "Jantzen sum formula" for ${\rm GL}\_n$ in terms of reflection... | https://mathoverflow.net/users/19113 | The Jantzen-Schaper theorem | To expand my comment further, I do in fact have a copy of the 50+ page typewritten double-spaced document by Klaus-Dieter Schaper (dated June 1981). As usual this Diplomarbeit was not published, nor did Schaper himself apparently continue in mathematics. Though I'm not at all up-to-date on the problems surrounding Spec... | 5 | https://mathoverflow.net/users/4231 | 130773 | 72,538 |
https://mathoverflow.net/questions/130721 | 3 | Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - 1}{\xi-1}\right)^n\right| = 1$ (note the complex modulus). Actually, I think that the only solutions are the trivial $k=an$, with $a \... | https://mathoverflow.net/users/21700 | Diophantine equation with primitive nth root of unity | The expression $(- (\xi^k - 1)/(\xi - 1))^n$ is real: it equals the $n$'th power of $\sin(k \pi/n)/\sin(\pi/n)$ times $(-1)^{n+k-1}$, if I calculated correctly. So the thing you're taking absolute value of must be $\pm 1$. If it's $1$, you get the trivial solution. If it's $-1$, that tells you an $n$'th root of $\pm 2$... | 8 | https://mathoverflow.net/users/2698 | 130781 | 72,541 |
https://mathoverflow.net/questions/130780 | 3 | I am wondering if there is a general explanation for the following phenomenon. The partial sums of the geometric series $\sum\_{n\geq 0} x^n$ evaluated at a root of unity $\zeta\neq 1$ in the complex plane attain only finitely many values (since $1+\zeta+\ldots+\zeta^{n-1}=0$, where $\zeta^n=1$). The average of these v... | https://mathoverflow.net/users/4800 | Power series whose partial sums attain only finitely many values | The phenomenon you observe is a special case of a theorem of Frobenius (1880):
>
> If a series is Cesaro summable then it is also Abel summable, and the Cesaro limit is the same as the Abel limit.
>
>
>
In your case the series is $(1,\zeta,\zeta^2,\dots)$ which ensures Cesaro summability as the sequence of p... | 4 | https://mathoverflow.net/users/11919 | 130782 | 72,542 |
https://mathoverflow.net/questions/130687 | 14 | If $A$ is a commutative ring we have the estimate
$$
\dim (A)+1 \le \dim (A[x])\le 2\dim (A)+1
$$
for the Krull dimension, with $\dim (A)+1 = \dim (A[x])$ for Noetherian rings.
I am looking for nice examples of rings $A$ so that $A[x]$ has Krull dimension
$\dim (A)+2, \dim(A)+3,\ldots ,2\dim(A)+1$.
| https://mathoverflow.net/users/32332 | Examples of polynomial rings $A[x]$ with relatively large Krull dimension | Your question is essentially completely answered in the paper [*The dimension sequence of a commutative ring*](http://www.jstor.org/stable/2373549) by Arnold and Gilmer (Amer. J. Math. 96 (1974), 385--408).
**EDIT:** The aforementioned paper by Arnold and Gilmer treats the case of an arbitrary finite number of variab... | 8 | https://mathoverflow.net/users/11025 | 130790 | 72,543 |
https://mathoverflow.net/questions/130770 | 17 | The standard way to view the first and second group cohomologies is this:
### The Standard Story
Let $G$ be a group, and let $M$ be a commutative group with a $G$-action. Then the first cohomology has the following interpretation: $H^1(G,M)$ is bijective with sections (modulo conjugation by $M$) of the short exact ... | https://mathoverflow.net/users/5309 | Can group cohomology be interpreted as an obstruction to lifts? | In your setup, the extension
$$
C \to B \to B/C
$$
represents a cohomology class $u\in H^2(B/C;C)$. The homomorphism $\phi: A\to B/C$ admits a lift $\bar\phi : A\to B$ if and only if $\phi^\ast(u)\in H^2(A;C)$ is zero.
To see this, note that the induced map on second cohomology is given by pullback of extensions, so ... | 17 | https://mathoverflow.net/users/8103 | 130792 | 72,545 |
https://mathoverflow.net/questions/130554 | 1 | In ["Schrödinger Operator: Heat Kernel and Its Applications"](http://arxiv.org/pdf/1101.1792.pdf), Feng computes the heat kernels associated to Schrödinger operators with at most quadratic potentials.
I am trying to see how these work in one variable. So consider his formula for the heat kernel $K(x,y,t)$ associated... | https://mathoverflow.net/users/12968 | A heat kernel for Schrödinger operator with low-order terms | I think your formula is not correct. The right kernel is invariant by interchanging $x$ with $y$. This symmetry must be preserved.
Then, note that
$$
L=-\Delta+ax^2+bx=-\Delta+a\left(x+\frac{b}{2\sqrt{a}}\right)^2-\frac{b^2}{4a}
$$
and this operator is invariant for translations. This means that the kernel for $b\ne... | 0 | https://mathoverflow.net/users/19520 | 130796 | 72,549 |
https://mathoverflow.net/questions/130730 | 6 | As pointed out by David White in
[when mapping cone is contractible](https://mathoverflow.net/questions/73687/when-mapping-cone-is-contractible)
there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be the two loops in $X=S^1\vee S^1$ and glue in... | https://mathoverflow.net/users/8942 | Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible? | $\newcommand{\set}[1]{\lbrace #1 \rbrace}$I will assume that the notation $\Sigma X$ in the question denotes the *unreduced* suspension of the space $X$.
**Quick answer:** The notion of homotopy equivalence $\Sigma X\to I$ rel ends described in the question is actually equivalent to the contractibility of $\Sigma X$,... | 9 | https://mathoverflow.net/users/21095 | 130799 | 72,550 |
https://mathoverflow.net/questions/130802 | 4 | I'm looking for an example of the following situation:
* A group $G$ generated by finite subgroups $H$ and $K$,
* a non-trivial 3-cocycle $\omega \in H^3(G, \mathbb{k}^\times)$
such that
* the restrictions of $\omega$ to a 3-cocycle on each of $H$ or $K$ is a coboundary.
If such an example is possible with at l... | https://mathoverflow.net/users/3 | A group 3-cocycle, trivial on a pair of generating subgroups? | If $\mathbb{k}^\times$ is the multiplicative group of some field, the following works with $\mathbb{k}=\mathbb{F}\_3$:
The Quaternion group $Q\_8$ is generated by two cyclic subgroups $H,K$ of order 4 and
$$H^\ast(H;\mathbb{F}\_2)\cong H^\ast(K;\mathbb{F}\_2)\cong \mathbb{F}\_2[a,b]/(a^2)\; ,\quad|a|=1, |b|=2$$
$$... | 6 | https://mathoverflow.net/users/10194 | 130810 | 72,554 |
https://mathoverflow.net/questions/123794 | 6 | Let $X$ be a locally compact Hausdorff topological space, denote by $M\_n$ the $C^\*$-algebra of complex $n\times n$ matrices, by $C\_0(X,M\_n)$ the $C^\*$-algebra of continuous functions on $X$ with values in $M\_n$ vanishing at infinity, and by $C\_b(X,M\_n)$ the $C^\*$-algebra of bounded continuous functions on $X$ ... | https://mathoverflow.net/users/30364 | What are the sub $C^*$-algebras of $C(X,M_n)$? | Every $C^\*$-subalgebra $A$ of $C\_0(X,M\_n)$ has irreps of dimension $\leq n.$ (Just because every irrep of $A$ can be continued to an irrep of $C\_0(X,M\_n)$). Such $C^\*$-algebras are called $n$-subhomogeneous.
There is a complete (rather complicated) description of such $C^\*$-algebras in algebraic topological t... | 6 | https://mathoverflow.net/users/12081 | 130813 | 72,557 |
https://mathoverflow.net/questions/130805 | 6 | Consider the $\mathbb Z$-module $\mathcal Z$ obtained as the set of sequences of integers $\mathbb Z ^ \mathbb N$ modulo the relation that two sequences are deemed equivalent when their difference is $0$ almost everywhere. This space is the space of germs at $+\infty$ of sequences of integers. I think this is a quite n... | https://mathoverflow.net/users/24309 | Germs at infinity of sequence of integers | This abelian group, which can also be described as the quotient of the direct product $\prod\_{\mathbb N}\mathbb Z$ by the direct sum $\sum\_{\mathbb N}\mathbb Z$, is isomorphic to the direct sum of the following two pieces. The first piece is a torsion-free, divisible abelian group (so you can view it as a vector spac... | 11 | https://mathoverflow.net/users/6794 | 130826 | 72,563 |
https://mathoverflow.net/questions/130833 | 9 | I have encountered a certain generalization of the Lefschetz fixed point theorem as folklore, and I am hoping that someone out there knows its provenance or can otherwise refer me to a source where it is discussed or proved. The statement of the theorem should look vaguely like this:
Let $X$ be a smooth, complete var... | https://mathoverflow.net/users/34057 | Generalization of the Lefschetz fixed point theorem | One of the standard proofs of the Lefschetz formula proceeds by showing that the RHS of your equation computes the intersection number between the graph of $f$ and the diagonal inside $X \times X$. The usual case is when this intersection has expected dimension. The general case requires the excess intersection formula... | 9 | https://mathoverflow.net/users/1310 | 130837 | 72,565 |
https://mathoverflow.net/questions/129818 | 17 | Dear MO Community,
this is not a real maths question, but rather the hope that someone else has stored in his or her private archive some data I am interested in.
>
> I'd like to know some pairs of non-isogenous elliptic curves over $\mathbf Q$ possessing the same cyclic isogeny of degree $13$, i.e. they both ha... | https://mathoverflow.net/users/12668 | Elliptic curves over QQ with identical 13-isogeny | [**Edited** *mostly to include the second example, corresponding to*
$(t,X) = (3,-115/126)$]
Thanks to Jordan Ellenberg for
[calling attention to this nice question on his blog](http://quomodocumque.wordpress.com/2013/05/13/elliptic-curves-with-isomorphic-cyclic-13-subgroups).
I didn't remember an example in my "priv... | 16 | https://mathoverflow.net/users/14830 | 130844 | 72,569 |
https://mathoverflow.net/questions/130843 | 0 | How to prove that $ (\zeta\_{p^{n+1}}-1)^{p} $ = $ (\zeta\_{p^{n}}-1) $ as ideals where $ \zeta\_{n} $ is a primitive nth root of unity ?
| https://mathoverflow.net/users/30999 | Cyclotomic fields | Suppose $\zeta \_{p^{n+1}}=x$ say.Then, for every $\omega $ a $p$-th root of unity, $x\omega-1=x^r-1$ (for some $r$) generates the same ideal as $x-1$ (clearly, it is contained in the ideal generated by $x-1$; by switching $x$ and $x\omega$ we get the other statement).
Taking the product over all the $\omega $ we no... | 3 | https://mathoverflow.net/users/23291 | 130847 | 72,571 |
https://mathoverflow.net/questions/130849 | 4 | Let $f :X \to Y$ be a submersion between smooth projective varieties over $\mathbb{C}$ and let $\alpha \in Z^k(X)$ be an algebraic cycle of $X$. Is is true that for all odd numbers $p$ and $q$ such that $p+q = 2k$, $\alpha$ lives in the kernel of $H^{2k}(X,\mathbb{Q}) \to H^p(Y,R^qf\_\*\mathbb{Q})$?
The reason why I ... | https://mathoverflow.net/users/24290 | Hodge classes and Leray filtration | No, $\alpha$ doesn't have to lie in that kernel. To construct an explicit counterexample,
take a surface $X$ mapping to a curve $Y$, $k=1$, and let $\alpha$ be nontrivial divisor
on $X$ not supported on the fibres such that $\alpha\cdot (fibre)=0$.
Then $[\alpha]$ will be supported in $H^1(Y, R^1f\_\ast\mathbb{Q})$. T... | 4 | https://mathoverflow.net/users/4144 | 130853 | 72,574 |
https://mathoverflow.net/questions/130835 | 4 | Is there a good way to compute the ratio ( B[n] / n! ) that occurs so often in power series coefficients? Good in the sense that you get an answer that does not overflow a double; the largest n such that B[n] fits in a double is relatively small, but the ratio in question should never overflow, although I guess it will... | https://mathoverflow.net/users/12669 | computing Bernoulli numbers | these are socalled "scaled" Bernoulli numbers; an efficient algorithm is discussed by Brent & Harvey, [arXiv:1108.0286](http://arxiv.org/abs/1108.0286), Eq. 8.
alternatively, you can use any method that computes the Riemann zeta function (say via the Euler product), because of the identity
$$\frac{B\_{2n}}{(2n)!}=(... | 7 | https://mathoverflow.net/users/11260 | 130855 | 72,576 |
https://mathoverflow.net/questions/130856 | 2 | I am trying to read [this paper](http://www.optimization-online.org/DB_FILE/2010/01/2527.pdf) and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has Lipschitz constant $M$ and considers the scheme
$$ x(t+1) = x(t) - \frac{1}{M} [f'(x(t)]\_k e\_k$$ where $[\cdot]\_i$ ... | https://mathoverflow.net/users/21162 | On a version of gradient descent | Here is a simple argument. First, define $r\_t = \|x\_t - x^\ast\|$, where $x^\ast$ denotes an optimal point.
Since $f$ is convex, we have
\begin{equation\*}
f(x^\ast) \ge f(x\_t) + \langle f'(x\_t), x^\ast - x\_t \rangle.
\end{equation\*}
From this we conclude (where we use the fact that $f(x\_t) \ge f(x^\ast)$ a... | 2 | https://mathoverflow.net/users/8430 | 130864 | 72,580 |
https://mathoverflow.net/questions/130857 | 6 | How is the proof that
$$[L^2(0,T;X)]' = L^2(0,T;X')$$
looking like, where $X$ is a Hilbert space? I am asking for the proof that the dual space of $L^2(0,T;X)$ is the space $L^2(0,T;X^\*)$.
Is the proof much different to the $L^p(a,b)$ case?
I can't find any easy to understand proofs.
| https://mathoverflow.net/users/34064 | Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$ | To give you a reference: [Diestel-Uhl, Vector measures](http://books.google.hu/books?id=NCm4E2By8DQC&printsec=frontcover&hl=de#v=onepage&q&f=false), page 98, Chapter 4, Theorem 1:
$$L^p(\mu,X)^\ast = L^q(\mu,X^\ast)$$
if and only if $X^\ast$ has the Radon-Nikodym property with respect to $\mu$.
Here $\mu$ is a f... | 6 | https://mathoverflow.net/users/12898 | 130865 | 72,581 |
https://mathoverflow.net/questions/130635 | 7 | Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$
$$
\int\_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|}
$$
by Gauss-Bonnet and that this is saturated on the round circle -- here $|\Sigma|$ is the length of $\Sigma$ and $H$ the mean curvature.
Likewise, if $n=2$ we ... | https://mathoverflow.net/users/26801 | Lower bound on $L^2$ norm of mean curvature in general dimensions | I have no idea about the general case but in the convex case the sphere is indeed optimal. Moreover the $L^1$ norm of $H$ attains its minimum at the sphere (among the convex surfaces with the same area). To deduce the result for the $L^2$ norm, just apply Cauchy-Schwarz,
Let $A$ be the convex body bounded by $\Sigma$... | 4 | https://mathoverflow.net/users/4354 | 130871 | 72,583 |
https://mathoverflow.net/questions/130868 | 0 | This might be a very simple question, and that might be the reason that I could not find any reference on this.
My question is
Let $A$ be an abelian variety defined over a number field $k$, and $N$ the conductor. Let $m\geq 2$. Consider the division field $k(A[m])$. Let $\mathfrak{p}$ be a prime ideal in $k$ that... | https://mathoverflow.net/users/21090 | Ramification in Division field of Abelian Varieties | What if $m=pq$ with $\mathfrak p \mid p$ and $p\ne q$ and $k(A[p])=k$? Then the $p$-torsion doesn't cause ramification since its defined over $k$, and the $q$-torsion won't cause $\mathfrak p$ ramification (assuming $A$ has good reduction at the primes lying over $p$ and $q$).
It gets more interesting if you assume t... | 2 | https://mathoverflow.net/users/11926 | 130873 | 72,584 |
https://mathoverflow.net/questions/130470 | 6 | I have posted the following question also [here](https://math.stackexchange.com/questions/362457/existence-of-dominating-measure-for-weak-compact-set-of-measures) a longer time ago, but due to no answers I thought it might fit better to MO.
Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak\*-com... | https://mathoverflow.net/users/31306 | Existence of dominating measure for weak*-compact set of measures | There always exists a dominating measure.
First, given two finite measures $\mu,\nu$ on $(\Omega,\mathcal{F})$, the Lebesgue decomposition theorem says that there is an $A\in\mathcal{F}$ such that $1\_{\Omega\setminus A}\mu$ is absolutely continuous with respect to $\nu$ and $1\_A\mu,\nu$ are singular. This can also ... | 4 | https://mathoverflow.net/users/1004 | 130874 | 72,585 |
https://mathoverflow.net/questions/130872 | 2 | Let $X$ and $Y$ be algebraic varieties over $\mathbb{C}$. I am repeatedly encountering references to crepant morphisms $f:X\rightarrow Y$. I have found several definitions of such a morphism, one of which is the condition that $f^\*(K\_Y)=K\_X$. Is this generally accepted to be the meaning of a crepant morphism, or is ... | https://mathoverflow.net/users/25358 | Crepant Morphisms of Varieties | Crepant stands for *non-discrepant*. It's frequently applied to resolutions of singularities or birational maps (but can be applied more generally).
Let's start with the birational case, since that's where the history is.
If $f : X \to Y$ is birational, and $K\_Y$ is $\mathbb{Q}$-Cartier, then $f^\*(K\_Y)$ makes sens... | 7 | https://mathoverflow.net/users/3521 | 130877 | 72,587 |
https://mathoverflow.net/questions/130879 | 17 | Let $\Delta(\kappa, \mu)$ be the statement: "let $F$ be a family of cardinality $\kappa$ of sets of cardinality less than $\mu$. Then there is a family $G \subset F$ of cardinality $\kappa$ and a set $r$ such that $a \cap b=r$ for every $a,b \in G$".
We know that if $\kappa$ is a regular cardinal and $\lambda^{<\mu} ... | https://mathoverflow.net/users/11647 | Does the generalized $\Delta$-system lemma imply some weak version of the GCH? | It is a very nice question! The answer is yes, natural instances of the $\Delta$ system property, which hold under GCH, are in fact equivalent to the GCH.
**Theorem.** $\Delta(\omega\_2,\omega\_1)$ is equivalent to CH.
Proof: You've pointed out that CH implies the principle, since the
hypothesis you mention for th... | 21 | https://mathoverflow.net/users/1946 | 130885 | 72,589 |
https://mathoverflow.net/questions/130890 | 4 | A topological ("closed") knot is an embedding of a circle in $\mathbb{R}^3$. It's possible for a knot to be distinct from the unknot because there are no free ends to move around and untie the knot. An alternative notion of a knot is an embedding of a closed line segment in $\mathbb{R}^2 \times [0,1]$ such that the two... | https://mathoverflow.net/users/9021 | Can distinct open knots correspond to the same closed knot? | I will ignore many important questions about wildness and the like, and instead suppose that you have enough regularity to make work the following argument (which I learned from John H. Conway).
Take a closed knot. Place a small bead somewhere along it, and make the bead out of a very shiny material: silver, say. Cle... | 15 | https://mathoverflow.net/users/78 | 130902 | 72,597 |
https://mathoverflow.net/questions/130904 | 3 | Let $M$ be a smooth Riemannian manifold, let $R$ be the Riemannian curvature operator, and let $p$ be a point in the manifold. With respect to any orthonormal basis of the tangent bundle at the point $p$, the operator $R$ is a skew-symmetric matrix with entries that are two forms. Thus, $R = [R\_{ij}]$ where the transp... | https://mathoverflow.net/users/14839 | Can one (block) diagonalize the curvature matrix of 2 forms on a Riemannian manifold? | Let us start it from the other end.
Assume you can block-diagonalize the curvature tensor.
Then in the most of coordinate sectional directions the curvature is zero.
In fact your curvature tensor equals to a curvature of product of few surfaces and maybe the real line --- this is a very special case.
| 1 | https://mathoverflow.net/users/1441 | 130908 | 72,601 |
https://mathoverflow.net/questions/130893 | 1 | How to convert this to weiestrass form?
$x^{2}y^{2}-2\left( 1+2\rho \right) xy^{2}+y^{2}-x^{2}-2\left( 1+2\rho
\right) x-1=0$
| https://mathoverflow.net/users/24554 | Weiestrass Form | You can rewrite the form as
\begin{equation\*}
y^2=\frac{x^2+2(2\rho+1)x+1}{x^2-2(2\rho+1)x+1}
\end{equation\*}
so, for rational solutions (which I presume you want), there exists $z \in \mathbb{Q}$ with
\begin{equation\*}
z^2=(x^2+2(2\rho+1)x+1)(x^2-2(\rho+1)x+1)=x^4-2(8\rho^2+8\rho+1)x^2+1
\end{equation\*}
This qua... | 3 | https://mathoverflow.net/users/28043 | 130919 | 72,606 |
https://mathoverflow.net/questions/130920 | 1 | I am looking for a mathematical field that deal with this concept, I don't have a very formal definition of this and I'll have to count on these naive definitions I provided:
>
> 1. It's about the idea of an adaptative live system;
> 2. The system evolves with time cycles;
> 3. This system has $n$ properties;
>
> ... | https://mathoverflow.net/users/23600 | What mathematical field work with this kind of concept? | I think you are looking for dynamical systems theory.
In particular (especially discrete) cases you might be interested in automata theory, modal logic and game theory. But your “definition” sounds more like a general notion from dynamical systems theory.
Mathematical biology and modeling are “just” applications.
... | 2 | https://mathoverflow.net/users/33842 | 130924 | 72,609 |
https://mathoverflow.net/questions/130915 | 1 | Does anyone know a citeable reference which works out the properties (geodesics, geodesic distance, ect) of the Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, $m>n$, with the Hilbert-Schmidt inner product on the tangent space? (An isometry is a linear map A so that $A^\*A=1$.)
Even ... | https://mathoverflow.net/users/32938 | Reference request: Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$ | They are called Stiefel manifolds, and are principal $U(n)$-bundles over Grassmann manifolds.
They are homogeneous Riemannian manifolds, whereas the Grassmannian are symmetric spaces.
Riemannian geometry on homogeneous manifolds is governed by the Nomizu operator.
See page 364-367 of [here](http://www.mat.univie.ac.... | 3 | https://mathoverflow.net/users/26935 | 130925 | 72,610 |
https://mathoverflow.net/questions/130052 | 1 | Is the following statement true?
Assume that $(M^n, g, p)$ is a pointed complete manifold with metric $g$, $Rc(g)\geq 0$,
$\{s\_i\}$ is a positive sequence decreasing to $0$, and $(M^n, s\_i\cdot g, p)$ converges to metric space $(N, h, p)$, where $h$ is the metric on $N$. Then we always have that
$h= dr^2+ f(r)^2 dX... | https://mathoverflow.net/users/19555 | the tangent cone at infinity of manifolds with Rc\geq 0 | The answer is no. There are manifold with positive Ricci curvature and tangent cone at infinity is not polar. The example is constructed by Menguy "Examples of nonpolar limit spaces. Amer. J. Math. 122 (2000), no. 5, 927–937".
| 6 | https://mathoverflow.net/users/1190 | 130938 | 72,619 |
https://mathoverflow.net/questions/130957 | 3 | Let $G$ a connected reductive split group over $k=\bar{k}$, $(B,T)$ a split Borel pair. Let $F:=k((t)))$. Let $\tilde{W}$ the extended Weyl group, $\tilde{W}=N\_{G}(T(F))/T(O)$.
By Iwasawa decomposition,
$G(F)=\coprod\limits\_{w\in\tilde{W}}IwI$,
where $I$ is the Iwahori subgroup associated to $B$.
We denote by... | https://mathoverflow.net/users/27398 | affine weyl group and affine schubert cells | If $G$ isn't simply connected, then $G(F)$ isn't connected, so $G(F)/I$ isn't connected. Your question is about resolving those $\overline{IwI}/I$ that live in components other than the one containing the basepoint $I/I$.
On such a component $C$, there will still be a closed $I$-orbit $X\_C$, automatically smooth.
(I... | 2 | https://mathoverflow.net/users/391 | 130966 | 72,627 |
https://mathoverflow.net/questions/130960 | 7 | Dear all,
I am interested in residually finite, perfect groups. Are all of them known to be residually alternating? If not, how could one construct a counterexample?
A group $G$ is residually alternating if for every $g \in G$ there exists a finite alternating quotient $G/N$ such that $g \notin N$.
By a result by... | https://mathoverflow.net/users/23232 | Are residually finite, perfect groups residually alternating? | I think the question needs to be made a little more precise. One can take a finite simple group $G$ which is not any alternating group. Then $G$ is perfect, residually finite (!) and is not residually alternating.
If we ask for finitely generated infinite perfect groups which are not residually alternating, then thi... | 13 | https://mathoverflow.net/users/23291 | 130967 | 72,628 |
https://mathoverflow.net/questions/130980 | 14 | A website ( <http://www.math.unicaen.fr/~nitaj/abc.html#Consequences> ) says that the $abc$ conjecture implies that there are only **finitely** many solutions to the equation $x^n+y^n=z^n$ with $\gcd(x,y,z)=1$ and $n\ge 4$. This one I have proven.
Lang's Algebra (p. 196) says that the $abc$ conjecture implies that fo... | https://mathoverflow.net/users/31084 | Effect of abc conjecture on Fermat's Last Theorem | There is a slight ambiguity what "the ABC conjecture" is as there are some variation. However, the most common and what you likely mean is this fomulation (or something equialent to it):
For every $\epsilon >0$, there is a $C\_{\epsilon}$ such that: if $a+b=c$, with positive coprime intergers, then
$$c < C\_{\epsil... | 23 | https://mathoverflow.net/users/nan | 130981 | 72,634 |
https://mathoverflow.net/questions/130832 | 5 | (Base theory $RCA\_0$)The principle says there exists a function g such that g dominates any X-recursive function for any X in the model.
i.e. For any $f\le\_T X$, $\exists b\in M$ such that $g(a)>f(a) \forall a>b$. Here the second order structure (countable) is $\langle M,S\_M,+\_M,\cdot\_M,0\_M,1\_M \rangle$.
In t... | https://mathoverflow.net/users/23835 | First order consequence of a combinatorial principle | The simplest forcing to add a dominating function is Hechler forcing $\newcommand{\D}{\mathbb{D}}\D$. In set-theoretic circles, conditions in $\D$ are pairs $(s,f)$ where $s$ is a finite sequence of natural numbers and $\newcommand{\N}{\mathbb{N}}f:\N\to\N$, extension is defined by $(s,f) \leq\_{\D} (t,g)$ if $t \supse... | 7 | https://mathoverflow.net/users/2000 | 130983 | 72,635 |
https://mathoverflow.net/questions/130778 | 9 | This question is just a curiosity, but I'm really interested in the answer. It was originally posted on math.stackexchange (<https://math.stackexchange.com/questions/368897/inverse-problem-for-brauer-groups>), but hasn't received any responses despite some upvotes, so I'm posting it here.
Given a field $K$, we can fo... | https://mathoverflow.net/users/8133 | "Inverse problem" for Brauer groups | It is perhaps helpful to look at the following papers of Fein and Schacher:
>
>
> >
> > Fein; Schacher;
> > Brauer groups of fields algebraic over Q.
> > J. Algebra 43 (1976), no. 1, 328–337.
> >
> >
> > Fein; Schacher;
> > Divisible groups that are Brauer groups.
> > Commun. Algebra 7, 989-994 (1979).
> > ... | 8 | https://mathoverflow.net/users/30062 | 130984 | 72,636 |
https://mathoverflow.net/questions/130777 | 9 | Most of us know the Jacobian conjecture. Here's a version below for fixed positive integers $d$ and $n$:
$J(d,n)$: If $f: C^n \rightarrow C^n$ is a polynomial map of degree $d$, and if the Jacobian determinant $\vert Jf \vert$ is nowhere vanishing (hence constant), then $f$ is injective (hence bijective).
We know t... | https://mathoverflow.net/users/3545 | Could the Jacobian conjecture be undecidable? | There's no way to rule out *a priori* that the Jacobian conjecture is undecidable (in your favorite axiomatic system).
As I pointed out in my answer to [another MO question](https://mathoverflow.net/questions/74941/is-there-an-undecided-assertion-of-which-a-proof-that-its-not-undecidable-is-k), a proof that some stat... | 5 | https://mathoverflow.net/users/3106 | 130986 | 72,637 |
https://mathoverflow.net/questions/130985 | 4 | Dear friends,
I am only a theoretical physicist. However, the answer to this question is relevant for emergence of space-time from a quantum cellular automaton (in the future I will pose a much more interesting and difficult problem). But, for now:
Consider the Cayley graph of a finitely generated infinite group. I... | https://mathoverflow.net/users/31293 | Cayley graphs of finitely generated infinite groups quasi-isometrically embeddable in R^3 | Yes, the theorem is:
Suppose that $G$ is a finitely-generated group whose Cayley graph quasi-isometrically (qi) embeds in $R^3$. Then $G$ is commensurable to a free abelian group of rank $\le 3$. (The converse is, of course, also true.)
In other words, $G$ contains a free abelian subgroup $A$ of finite index, so t... | 5 | https://mathoverflow.net/users/21684 | 130994 | 72,641 |
https://mathoverflow.net/questions/130193 | 9 | What do you call a poset with this property? For any elements $a,b,c,d$ such that $\{a,b\}\le\{c,d\}$, there is an element x such that $\{a,b\}\le x\le\{c,d\}$. (Equivalently, for any finite sets $A\le B$, there is an element x such that $A\le x\le B$.)
For example, any upper or lower semilattice has this property. ... | https://mathoverflow.net/users/nan | Does this property of a partially ordered set have a name? | In the world of partially ordered abelian groups, this is the *interpolation property*.
These groups are called *partially ordered abelian groups with interpolation*, or simply *interpolation groups*. Intuitively, I think about them as "almost as nice as lattice ordered abelian groups".
A simple example of a non-latt... | 8 | https://mathoverflow.net/users/4814 | 130995 | 72,642 |
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