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https://mathoverflow.net/questions/60291 | 4 | For a image denoising problem (below):
<http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf>
the author has a functional E defined
$E(u) = \int\int\_\Omega F \\ d\Omega$
which he wants to minimize. F is defined as
$F = ||\nabla u ||^2 = u\_x^2 + u\_y^2$
Then, the E-L equations are derived:
$\frac{\... | https://mathoverflow.net/users/14074 | Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising | If the solution to
$
u\_t=u\_{xx}+u\_{yy}
$
reaches an equilibrium solution, then $u\_{t}=0$ at that equilibrium, so $u\_{xx}+u\_{yy}=0$. The author has shown that a necessary condition for $u$ to be minimizer of $E(u)$ is $u\_{xx}+u\_{yy}=0$.
This isn't steepest descent in the way that it is normally presented... | 2 | https://mathoverflow.net/users/9022 | 60294 | 37,396 |
https://mathoverflow.net/questions/60298 | 1 | Let $M$ be a non-compact matrix Lie group and $T\_e M$ its lie algebra.
Consider a point $x \in M $ and $ \triangle \in T\_e M$.
To move from $x$ to a point $y \in M$ along $\triangle$, below group operation seems to be commonly used in iterative optimization on Lie groups.
$y = x \exp(\triangle)$
Is this str... | https://mathoverflow.net/users/12573 | Explanation of $y = x \exp(\triangle)$ for a Lie Group | The answer of Theo basically says it all what the exponential is concerned, but I maybe can shed some light regarding the optimization perspective.
Let $M$ be your Lie group and suppose it is a subgroup of $\textrm{GL}\_n$. Now, if you want to solve
\[
\min f(x) \quad \textrm{s.t. $x \in M$}
\]
by an iterative meth... | 2 | https://mathoverflow.net/users/14039 | 60311 | 37,402 |
https://mathoverflow.net/questions/59939 | 67 | The standard version of this puzzle is as follows: you have $1000$ bottles of wine, one of which is poisoned. You also have a supply of rats (say). You want to determine which bottle is the poisoned one by feeding the wines to the rats. The poisoned wine takes exactly one hour to work and is undetectable before then. H... | https://mathoverflow.net/users/290 | Identifying poisoned wines | Each bottle of wine corresponds to the set of rats who tasted it. Let $\mathcal{F}$ be the family of the resulting sets. If bottles corresponding to sets $A$ and $B$ are poisoned then
$A \cup B$ is the set of dead rats. Therefore we can identify the poisoned bottles as long as for all $A,B,C,D \in \mathcal{F}$ such tha... | 87 | https://mathoverflow.net/users/8733 | 60312 | 37,403 |
https://mathoverflow.net/questions/60313 | 2 | I'm trying to solve the equation
$(1-|x|^2)T = 0$,
where $T$ is a tempered distribution. I know how to do this (it is a common exercise) in dimension $1$. How can I solve it in higher dimensions?
Thank you very much.
| https://mathoverflow.net/users/13127 | Division of distributions by polynomials. | The solution is the direct product of $\delta(|x|-1)$ and any distribution on the sphere.
| 2 | https://mathoverflow.net/users/12120 | 60316 | 37,405 |
https://mathoverflow.net/questions/60315 | 21 | What's the nicest place to see a list of the maximal connected subgroups of compact Lie groups? Is there anything on-line?
I looked at Tits' Bourbaki talk on Dynkin's and others' work, but he admits early on that the theory leads to huge tables, which he isn't going to include. Those are the sort of thing I'd like to... | https://mathoverflow.net/users/391 | Modern reference for maximal connected subgroups of compact Lie groups | I guess you are referring to the Tits Bourbaki seminar talk #119 [*here*](http://www.numdam.org/numdam-bin/fitem?id=SB_1954-1956__3__197_0) in the 1950s on subalgebras of semisimple Lie algebras (which translates the original question about compact Lie groups)? That's freely available online, but Dynkin's earlier paper... | 15 | https://mathoverflow.net/users/4231 | 60318 | 37,406 |
https://mathoverflow.net/questions/60247 | 15 | Suppose $G$ is a finitely generated discrete group and that there is a subset $E$ of $G$ such that if
$\mu$ is a finitely additive probability measure on $G$, then there is a
$g$ in $G$ such that $\mu(E \cdot g) \ne \mu(E)$.
Certainly $G$ is non amenable.
Can more be said about $G$?
Must $G$ contain $\mathbb{F}\_2$?
... | https://mathoverflow.net/users/10774 | When is non amenablity witnessed by a single non measurable set? | The answer is that the above is equivalent to non amenability.
Fix a group $(G,\*)$.
Since $(G,\*)$ is non amenable if and only if every finitely generated subgroup is non amenable,
we may assume that $G$ is finitely generated.
If $\mu$ and $\nu$ are finitely supported probability measures on $G$,
define
$$
\mu \* \n... | 12 | https://mathoverflow.net/users/10774 | 60319 | 37,407 |
https://mathoverflow.net/questions/60321 | 2 | Hi,
In Chang & Keisler "Model Theory" it is claimed that the theory of a one-to-one function of A onto A with no finite cycles is $\omega\_1$- categorical (page 140). Why is that, and is there a reference for this?
| https://mathoverflow.net/users/10708 | Omega_1 Categorical Theory | Let $A$ and $B$ be two models of size $\omega\_1$. In both A and B, being in the same cycle is an equivalence relation. Each equivalence class has size $\omega$. So there are $\omega\_1$ many equivalence classes in both A and B. Fix a bijection between the equivalence classes in A and B. You can use this to make an iso... | 7 | https://mathoverflow.net/users/27 | 60323 | 37,409 |
https://mathoverflow.net/questions/60297 | 6 | I am going to have to borrow the opening passage from Bombieri, Friedlander, Iwaniec${}^\*$ since they state this idea so well. In the following $\|f\|$ means $\big(\sum\_{n\leqslant x} |f(n)|^2\big)^{1/2}$.
>
> Given an arithmetic function $f(n)$, it is natural to study its distribution in residue classes $a\: (\t... | https://mathoverflow.net/users/12160 | Distribution of a function in an arithmetic progression | For individual $a$'s the best universal bound you can get is $\ll \frac{1}{\sqrt{q}} x^{1/2} \|f\|$, as shown by the example of the characteristic function of a reduced residue class mod $q$. I think the authors meant the following. We are interested in the sums $\sum\_{\substack{n\leqslant x, \ n\equiv a(\text{mod }q)... | 5 | https://mathoverflow.net/users/11919 | 60327 | 37,411 |
https://mathoverflow.net/questions/60326 | 3 | If f is a weight 2 cuspidal newform, then it is common for L(f,1) to vanish. Indeed, the sign of the functional equation of f can force such vanishing. However, if f has weight k>2, then there is no a priori reason why L(f,1) will vanish.
My question: are there known examples where L(f,1)=0 for a newform f of weight... | https://mathoverflow.net/users/14083 | Non-vanishing of L-series of modular forms (easy case?) | Based on your normalization, $L(s,f)$ is defined as an Euler product for $\Re(s)>\frac{k+1}{2}$, so $L(s,f)$ is non-zero in that right-half plane. Now Jacquet–Shalika [MR0432596](http://www.ams.org/mathscinet-getitem?mr=432596) showed that that non-zero region extends to the line $\Re(s)=\frac{k+1}{2}$ (for $\mathrm{GL... | 9 | https://mathoverflow.net/users/1021 | 60336 | 37,417 |
https://mathoverflow.net/questions/60334 | 2 | [some formatting tweaked, and the question copied from the title to the main body, by YC]
---
Hi,
I've been struggling a lot to calculate this integral.
$$ \int\_0^\infty \frac{k^{n-1}}{\prod\_{i=1}^n (k^2+ x\_i^2)}\; dk
$$
where $x\_i$ are constants and $n\geq 1$.
I did the calculation for n=1,2,3,4, with ... | https://mathoverflow.net/users/14085 | Integrate kˆ(n-1) / prod_{i=1...n} (kˆ(2)+x_iˆ{2}) dk between 0 and infinity, with x_i constants and n>=1? | You are correct. Use the partial fraction decomposition: <http://en.wikipedia.org/wiki/Partial_fraction> For example, if $n=4$, the decomposition is (over the rationals):
$$\begin{array}{l}
{\frac {ck}{ \left( {k}^{2}+c \right) \left( -c+a \right) \left( -c+
b \right) \left( -d+c \right) }}-\\\ {\frac {dk}{ \left( {k... | 2 | https://mathoverflow.net/users/nan | 60339 | 37,418 |
https://mathoverflow.net/questions/59895 | 7 | Let $U\_{d,n}\subseteq(\mathbb{P}^d)^n$ denote the locus of $n$-distinct points in projective space $\mathbb{P}^d$ that lie on a rational normal curve of degree $d$, and let $V\_{d,n}$ denote its topological closure. I would like to know if, for $d \le n-3$, the spaces $V\_{d,n}$ are normal. If they are, this implies t... | https://mathoverflow.net/users/10930 | Normality of a locus of points in projective space | Hi Noah,
Consider the forgetful morphism $(\mathbb P^d)^{n+1}\to (\mathbb
P^d)^n$. This restricts to a forgetful morphism $\pi:V\_{d,n+1}\to V\_{d,n}$. This is
kind of like the map $\overline{\mathscr M}\_{g,n+1}\to \overline{\mathscr
M}\_{g,n}$. The general fiber of $\pi$ is a $\mathbb P^1$ (the original degree $d$
... | 4 | https://mathoverflow.net/users/10076 | 60348 | 37,421 |
https://mathoverflow.net/questions/60296 | 5 | Let $N$ be a type $II\_{1}$-factor with trace $\tau$, and $B$ a von Neumann subalgebra. The existence of the semifinite trace on the Jones basic construction $\langle N, e\_{B} \rangle$ is reasonably easy to establish if $B$ is a subfactor of $N$, but appears not to be so easy in general.
>
> Question: What is the ... | https://mathoverflow.net/users/6269 | Is there a trivial construction of the trace on the Jones basic construction? | Perhaps I am missing some hypothesis, but I think the proof is just about the same whether or not $B$ is a factor. Here is the proof from Jones' original paper, and I believe it does not use factoriality of B (and not even facoriality of $N$?)
**Lemma.** *Let $J:L^2(N)\to L^2(N)$ be the modular conjugation. Then $\la... | 5 | https://mathoverflow.net/users/12660 | 60349 | 37,422 |
https://mathoverflow.net/questions/60345 | 0 | Is there anyone prove the results such like the follows?
If $NP\not\subseteq BP(2^{\Omega(n)}),$ then $BPP\subseteq P^{NP}$
In summary, my question it that, can we get some derandomized results based on some nondeterminitic assumptions.
| https://mathoverflow.net/users/6326 | Derandomize on nondeterministic assumptions | As for derandomization under nondeterministic assumptions, you can basically relativize the usual results such as Impagliazzo-Wigderson. Directly, this gives: if some language in $E^{NP}$ requires exponential circuits with an NP-oracle, then $BPP^{NP}=P^{NP}$. There are similar results by [Miltersen and Vinodchandran](... | 1 | https://mathoverflow.net/users/12705 | 60356 | 37,425 |
https://mathoverflow.net/questions/60355 | 7 | In "Mackey - Unitary Group Representation in Physics, Probability and Number Theory" on page 326, George Mackey mentions a result of Ludwig Siegel, which was later generalized to semi-simple Lie groups by André Weil. In his words:
"Now in one formulation Siegel's main result takes the form of an identity between an t... | https://mathoverflow.net/users/10400 | Relation between Theta series and Eisensteinseries | Siegel showed that an Eisenstein series is a certain constant times the sum of weighted theta functions of all lattices in some genus. The lattices are weighted by 1/automorphism group. For example, for even unimodular lattices of dimension 8 there is only one such lattice, so Siegel's result says the Eisenstein series... | 15 | https://mathoverflow.net/users/51 | 60361 | 37,427 |
https://mathoverflow.net/questions/60359 | 6 | Let $K$ be a number field with integral basis $\{\omega\_1,\ldots,\omega\_n\}$. Then
$$ \Phi(X\_1, \ldots, X\_n) = N\_{K/{\mathbb Q}}(\omega\_1 X\_1 + \ldots + \omega\_n X\_n) $$
is a homogeneous polynomial of degree $n$ with integral coefficients, and the integral points on the affine variety
$$ \Phi(X\_1,\ldots,X\_n)... | https://mathoverflow.net/users/3503 | Parametrization of unit varieties | Thanks to Dror for pointing out a nice geometrical way to think of such varieties.
This paper shows that such varieties are not rational in general:
<http://www.math.jussieu.fr/~florence/norm_one.pdf>
However, any algebraic torus over a number field $K$ of dimension one or two is rational over $K$. Indeed, the case... | 8 | https://mathoverflow.net/users/5101 | 60362 | 37,428 |
https://mathoverflow.net/questions/60364 | 9 | Given a locally compact Hausdorff group $G$, one can construct several Banach star-algebras using $G$ (and its associated Haar measure): $L^1 (G)$, $M(G)$ (regular complex measures on $G$), $L^{\infty} (G)$, $C^\* (G)$, $C^\*\_r (G)$, $W^\* (G)$, etc. (see this [Wiki article](http://en.wikipedia.org/wiki/Group_algebra)... | https://mathoverflow.net/users/7392 | Which Banach algebras are group algebras? | A characterization of the Banach algebras which are isometrically \*-isomorphic to $L^1(G)$ for some $G$ was given by P. L. Patterson, Characterization of algebras arizing from locally compact groups, Trans. Amer. Math. Soc. 329 (1992), 489-506. This paper contains references to earlier ones dealing with special classe... | 13 | https://mathoverflow.net/users/12205 | 60374 | 37,433 |
https://mathoverflow.net/questions/60371 | 14 | Working in a problem the following family of graphs appears naturally. Consider the set $A\_{n}=\{1,2,3,\ldots,n\}$ and let $\mathcal{C\_{n}}$ be the set of all permutations of $A\_{n}$ of order $n$ (cycles of order $n$). Let $\sigma\_{1},\sigma\_{2},\ldots,\sigma\_{d}$ be random elements chosen uniformly and without r... | https://mathoverflow.net/users/13825 | Properties of Some Random Graphs | Yes, this model has been studied. You should look at Chapter 9 of Janson, Luczak and Rucinski's [Random Graphs book](http://rads.stackoverflow.com/amzn/click/0471175412), and in particular at Corollary 9.44. This corollary is in fact a rather well-known theorem, which I'll now explain.
Let $H\_n(d)$ be the distributi... | 17 | https://mathoverflow.net/users/3401 | 60377 | 37,434 |
https://mathoverflow.net/questions/60375 | 227 | The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \times X$ were homeomorphic to $\mathbb{R}$, then $X$ would be path connected. But then $X \times X$ minus a point would als... | https://mathoverflow.net/users/27 | Is $\mathbb R^3$ the square of some topological space? | No such space exists. Even better, let's generalize your proof by converting information about path components into homology groups.
For an open inclusion of spaces $X \setminus \{x\} \subset X$ and a field $k$, we have isomorphisms (the relative Kunneth formula)
$$
H\_n(X \times X, X \times X \setminus \{(x,x)\}; k)... | 244 | https://mathoverflow.net/users/360 | 60378 | 37,435 |
https://mathoverflow.net/questions/60322 | 10 | I'm interested in Lie theory and its connections to dynamical systems theory. I am starting my studies and would like references to articles on the subject.
| https://mathoverflow.net/users/14081 | References on Lie groups and dynamical systems | The connections between Dynamics and Lie Groups (or Algebraic groups) comes mainly in two flavours:
1. Smooth dynamics, like others have stated Hamiltonian dyanmics and differential equations.
2. Applications of Ergodic theory and Topological dynamics to Lie groups (or more generally, homogeneous spaces), or as Linde... | 9 | https://mathoverflow.net/users/8857 | 60391 | 37,443 |
https://mathoverflow.net/questions/60382 | 2 | Suppose $\Omega$ is a bounded domain in $\mathbb{R}^n.$ Let $w\in H^{1,2}$ (standard Sobolev space, order 1, integrability 2) and $L>0$ be given. Is it then true that the function $w\_L:=\min (L,w)$ is also in $H^{1,2}?$
I found this assertion in the book "Riemannian geometry and geometric analysis" of Jost, in the s... | https://mathoverflow.net/users/3509 | cutting off $H^{1,2}$-functions in the image | Note that $\min(w,L) = \frac{1}{2} (w+L) - \frac{1}{2} |w-L|$, so the main issue is to establish that the map $w \mapsto |w|$ is bounded on $H^{1,2}$. But this follows from the diamagnetic inequality $|\nabla |w|| \leq |\nabla w|$ (in the sense of distributions), which is obvious formally, but can be established rigoro... | 7 | https://mathoverflow.net/users/766 | 60397 | 37,448 |
https://mathoverflow.net/questions/40692 | 3 | I'm interested in representing homogeneous elastic deformations using Lie groups/algebras. Homogeneous deformations are those with a deformation gradient F which depends only on time (not position). If the velocity gradient L = (df/dt)F^-1 also is constant (independent of time or position), F = e^(Lt) where F is Lie (s... | https://mathoverflow.net/users/9624 | Physical Meaning of Constant Velocity Gradient | I'm not sure this is what you're after, but are you familiar with pseudo-rigid bodies? These are elastic media where the deformation tensor $F$ is constant in time and hence is an element of $GL(3)$. I'm a little vague on the details, but different continuum models are specified in terms of different subgroups of $GL(3... | 2 | https://mathoverflow.net/users/3909 | 60409 | 37,454 |
https://mathoverflow.net/questions/60408 | 3 | Hello,
Which manifolds in dimension five admit contact structures? I am not too familiar with
the contact realm so any references to look at would be much appreciated.
| https://mathoverflow.net/users/14097 | contact manifolds dimension five | Let M5 be a closed and oriented. A contact form α gives a 4-plane distribution with symplectic form dα, reducing the structure group of TM to U(2)×1; such a reduction is called an almost contact structure, and it exists iff the integral third Stiefel-Whitney class is zero (Gray, "Some global properties of contact struc... | 16 | https://mathoverflow.net/users/428 | 60410 | 37,455 |
https://mathoverflow.net/questions/60308 | 9 | What is the most computationally efficient way to check, given $x,y,D$ that they satisfy Pell's equation (positive or negative) ($x^2-Dy^2=1$)? (Obviously the question is concerned with very large values of $x,y,D$.)
I know (I think) that it'll have to be checking mod $p$ but I just can't find the right balance betwe... | https://mathoverflow.net/users/13753 | Most efficient checking algorithm for Pell's Equation | Based on the comments, it looks like this is not a question specific to Pell's equation, and that you just want to evaluate a single binomial with big inputs as quickly as possible.
If you check the equation directly using fast multiplication algorithms (e.g., [Schönhage-Strassen](http://en.wikipedia.org/wiki/Schonh... | 7 | https://mathoverflow.net/users/121 | 60411 | 37,456 |
https://mathoverflow.net/questions/60358 | 5 | A set of integers is said to be nonaveraging if it contains no three-term arithmetic progression. I call a nonaveraging subset of $\lbrace 1,2, \ldots ,n \rbrace$ optimal when it has maximal cardinality.
There is a regularly updated website on nonaveraging sets records at <http://www.math.uni.wroc.pl/~jwr/non-ave/DA... | https://mathoverflow.net/users/2389 | Structure of nonaveraging sets of integers | As to Conjecture 3, I strongly doubt it is true. If it were true, one could have decomposed any optimal progression-free subset of $[1,n]$ as $B\cup(B+t)\cup C$, where $|C|$ is small as compared to $|B|$. Now, $B$ is a progression-free subset of $[1,n-t]$ with the property that $t\notin B-B$ and $2t\notin\pm(B+B-2\ast ... | 3 | https://mathoverflow.net/users/9924 | 60414 | 37,457 |
https://mathoverflow.net/questions/60241 | 41 | Ideals are intimately related to quotients and congruence relations. They clearly play a very important role in ring theory and order theory. So do normal subgroups in group theory. (Enriched) category theory could be regarded as a common generalization of all these settings. Why is it that such important structures do... | https://mathoverflow.net/users/10368 | Why don't ideals and quotients work well for categories? | Here is a shortish answer that relates to several of the above replies: Yes! There is such a theory.
Ideals correspond to a particular type of internal category or groupoid in the category of rings, normal subgroups correspond to `dittos' in the category of groups. Quotients by an ideal/normal subgroup are the coequa... | 18 | https://mathoverflow.net/users/3502 | 60424 | 37,461 |
https://mathoverflow.net/questions/60417 | 1 | Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.
As time goes to infinity, will your position tend to infinity? If not, to what finite value does it converge?
Edit: To be a bit more precise, what ... | https://mathoverflow.net/users/9714 | Will a random walk on [0, inf) tend to infinity? | The symmetric random walk $(X\_k)$ on $\mathbb{Z}$ is recurrent. Therefore, with probability one, you will visit infinitely often $0$. The same is true for $(|X\_k|)$, which is more ore less your random walk (the more or less depends on what happens exactly at the origin).
In other words, with probability one, the $\... | 12 | https://mathoverflow.net/users/12088 | 60425 | 37,462 |
https://mathoverflow.net/questions/60430 | 18 | Consider the Hilbert polynomial for a projective scheme.
The degree, dimension and arithmetic genus extract information from the lowest term and the highest term in the polynomial. What about all other terms? It would seem they encode some more info about our scheme. I could not find any reference to these coefficients... | https://mathoverflow.net/users/14105 | What information Hilbert polynomial encodes other than dimension, degree and arithmetic genus? | For a curve, that's all of course, since the polynomial is linear.
Now let's say $X$ is a smooth surface with an ample divisor $H$ and canonical
divisor $K$, we have the Hilbert polynomial
$$\chi(\mathcal{O}\_X(nH))= \frac{1}{2}nH(nH-K) + \chi(\mathcal{O}\_X)$$
by Riemann-Roch. So the linear coefficient gives you the d... | 19 | https://mathoverflow.net/users/4144 | 60431 | 37,464 |
https://mathoverflow.net/questions/60427 | 1 | As we know, for $1<p<\infty$, the Fourier series of $f\in L^{p}(T)$ converges to $f$ in $L^{p}$-norm.
But is there any results concerning the convergence of Fourier series in $L^{\infty}$-norm?
Since $L^{\infty}(T)$ is not separable, the trigonometric system fails to form a Schauder basis of $L^{\infty}(T)$, this impli... | https://mathoverflow.net/users/13244 | Convergence of Fourier series in L^{\infty}-norm | If Fourier series of continuous functions would converge in $L^\infty$, then, by the Uniform Boundedness Principle, the operator norms in $C(\mathbb{T})$ of the partial Fourier series operators $S\_Nf(t):=\sum\_{n=-N}^N\hat{f}(n)e^{int}$ would be uniformly bounded. You can find, for example in Katznelson book, a proof ... | 3 | https://mathoverflow.net/users/1049 | 60435 | 37,466 |
https://mathoverflow.net/questions/60341 | 12 | I am looking for interesting applications of the [1/4-pinched sphere theorem](http://www.en.wikipedia.org/wiki/Sphere_theorem). The theorem says: A compact, simply connected riemannian manifold whose sectional curvature K satisfies $1/4 < K \leq$ 1 (possibly after multiplying the metric by a constant) is homeomorphic (... | https://mathoverflow.net/users/14082 | applications of the sphere theorem | The main theme of global Riemannian geometry is to derive topological conclusions from geometric assumptions. Sphere theorems provide various assumptions under which a manifold is (homeomorphic, diffeomorphic, or almost isometric) to a sphere.
The significance of sphere theorems is not in their applications or impli... | 16 | https://mathoverflow.net/users/1573 | 60438 | 37,468 |
https://mathoverflow.net/questions/60441 | 17 | [Artin's Conjecture](http://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots) says that any positive integer, which is not a square, is a primitive root modulo infinitely many primes. Christopher Hooley gave in
* Hooley, Christopher (1967). "*On Artin's conjecture.*" J. Reine Angew. Math. 225, 209-220.
... | https://mathoverflow.net/users/8176 | The multiplicative order of 2 modulo primes | The answer is "yes" - the order mod p of 2 is almost always as large as the square root of p (actually you get epsilon less than this in the exponent). If you take r multiplicatively independent numbers and ask for the group they generate mod p, the exponent is r/(r + 1). This is a paper of mine, and then in a paper of... | 18 | https://mathoverflow.net/users/6153 | 60444 | 37,470 |
https://mathoverflow.net/questions/60447 | 11 | It's probably a well known question, so it is just a reference question.
Let $G$ be a finite group and let $C[G]$ be a group algebra. Then we can define a bracket on $C[G]$ by $[f,h]=f\*h-h\*f$. What does $C[G]$ look like as a Lie algebra? When is it solvable?
| https://mathoverflow.net/users/4246 | How does the group algebra look as a Lie algebra | Assuming that your ground field $K$ has characteristic prime to the order of $G$. Then the group ring is a seminsimple algebra. Therefore, $C[G]=\bigoplus\_{i=1}^{r} Mat\_{n\_i}(R\_i)$ is a direct sum of matrix algebras, where $R\_i$ is a finite-dimensional division ring over $K$. All this is very classical and nicely ... | 21 | https://mathoverflow.net/users/9928 | 60450 | 37,474 |
https://mathoverflow.net/questions/60421 | 1 | let $J=S \cap D $,$G=S \cup D$,sort $G$,$a\_n \in G$.
Function $\gamma (n,s)=\frac{\Sigma\_{a\_i \in J}^n a\_i^s}{\Sigma\_{i=1}^n a\_i^s}$.
Given S ,a non computably enumerable set,is there a computably enumerable set D Such that $\lim\_{n \to \infty}\gamma (\infty,0)=1$? Or under what condition $\lim\_{n \to \inft... | https://mathoverflow.net/users/14024 | How to approximate non-computably recursive set by computably recursive set | If $S$ is selected randomly under the fair-coin (Lebesgue-Cantor) measure on $2^\omega$ then $S$ will be immune and $S$ will asymptotically contain $1/2$ of the elements of $D$, plus $1/2$ of the elements of $\omega\backslash D$. To get this to be true for all recursively enumerable $D$ it is enough that $S$ is 2-rando... | 4 | https://mathoverflow.net/users/4600 | 60454 | 37,476 |
https://mathoverflow.net/questions/60471 | -1 | Hello,
I read quite quickly an article in which it is shown that the unicity of factorization in the Selberg class S is equivalent to some kind of linear independence of distinct primitive functions of S. So my question is: is this unicity of factorization in fact equivalent to Selberg's orthonormality conjecture?
Th... | https://mathoverflow.net/users/13625 | Is the unique factorization in S equivalent to SOC? | Either the Selberg class is not family of automorphic $L$-functions on $GL(n)$, in which case all bets are off, or it is, and then both facts are very probably true, and therefore logically equivalent.
On the other hand, if we take as "model" for $L$-functions the representations of a group, even for a finite group, ... | 2 | https://mathoverflow.net/users/20038 | 60472 | 37,489 |
https://mathoverflow.net/questions/60437 | 2 | My function is $f:\mathbb{N} \rightarrow \mathbb{N},\ f(n)=2\uparrow ^n 3$ , the Ackermann(-Péter) function, with the second argument fixed to 3 (and "$\uparrow$" the Knuth up-arrow), which I believe is not primitive recursive, but which I could not prove - and that is what this question is about. Any ideas for the pro... | https://mathoverflow.net/users/14101 | final step(s) for a proof that a function is not primitive recursive | I believe one proof technique was to use a primitive recursive encoding of multiple argument functions, so that it would suffice to show a bound on a composition of unary primitive recursive functions. You might try first showing that your function bounds any unary primitive recursive function first.
Gerhard "Ask Me ... | 1 | https://mathoverflow.net/users/3402 | 60473 | 37,490 |
https://mathoverflow.net/questions/63 | 18 | Suppose $g,h:Z\to X$ are two morphisms of schemes. Then we say that $f:X\to Y$ is the *coequalizer* of $g$ and $h$ if the following condition holds: any morphism $t:X\to T$ such that $t\circ g=t\circ h$ factors uniquely through $f$. The question is whether it is possible for a coequalizer $f:X\to Y$ to **fail** to be s... | https://mathoverflow.net/users/1 | Can a coequalizer of schemes fail to be surjective? | Let $k$ be a field. Take $Y=\mathrm{Spec}\,k[[t]]$, and take for $X$ the disjoint sum of the closed subschemes $X\_n:=\mathrm{Spec}\,k[[t]]/(t^n)$ ($n>0$). Put $Z=X\times\_Y X$ with the two obvious maps to $X$. A coequalizer is just a direct limit of the system $X\_1\hookrightarrow\dots X\_n\hookrightarrow X\_{n+1}\hoo... | 17 | https://mathoverflow.net/users/7666 | 60477 | 37,493 |
https://mathoverflow.net/questions/60478 | 61 | In [Prospects in Mathematics (AM-70)](http://books.google.com/books?id=zK7jmvsI_2wC&lpg=PP1&ots=LpN2oUA88m&dq=hirzebruch%20%22prospects%20in%20mathematics%22&pg=PA7#v=onepage&q&f=false), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b\_1 x + b\_2 x^2 + \dots$ defining the Todd class must... | https://mathoverflow.net/users/6005 | Hirzebruch's motivation of the Todd class | Since you mention playing around with residues, I'm probably not telling you anything you don't already know. But there is a systematic way to extract the power series $f$ from
the coefficients of $x^{n-1}$ in $f(x)^{n}$, which goes by the name of the Lagrange inversion formula.
Assume that the constant term of $f$ i... | 104 | https://mathoverflow.net/users/7721 | 60481 | 37,496 |
https://mathoverflow.net/questions/60498 | 9 | If $L$ is a semisimple lie algebra then $L=[L,L]$. Is the opposite true?
| https://mathoverflow.net/users/14113 | Lie algebra semisimple if and only if perfect? | **No.** A Lie algebra satisfying that property is called perfect. For an example of a perfect Lie algebra that isn't semisimple, take a semisimple $L$ and a nontrivial irreducible representation $V$ of $L$, and define a bracket on $L \times V$ by
$$ [(X,v),(Y,u)] := ([X,Y],Xu-Yv). $$
This turns $L \times V$ into a perf... | 34 | https://mathoverflow.net/users/430 | 60500 | 37,511 |
https://mathoverflow.net/questions/60416 | 6 | [Pappus' Centroid Theorems](https://mathworld.wolfram.com/PappussCentroidTheorem.html) provide a slick way of computing the center of mass for plane curves and plane areas.
The first theorem states that the surface area $A$ of a surface of revolution generated by rotating a plane curve $\Gamma$ about an axis external... | https://mathoverflow.net/users/7144 | Are Pappus Theorems generalized? | There is a neat way of finding the centroid of surfaces/volumes of revolution. Suppose you have a planar figure $\Gamma$ and an axis $\ell$ in the plane that is disjoint from $\Gamma$. Now suppose that the centroid of $\Gamma$ projects on $\ell$ as $O$, and that it's distance from $\ell$ is $r$. Suppose we rotate $\Gam... | 7 | https://mathoverflow.net/users/2384 | 60502 | 37,512 |
https://mathoverflow.net/questions/3939 | 29 | Suppose $P(x)$ is a monic integer polynomial with roots $r\_1, ... r\_n$ such that $p\_k = r\_1^k + ... + r\_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the characteristic polynomial of a non-negative integer matrix?
(The motivation here is that I want $r\_1, ... r\_n$ to be th... | https://mathoverflow.net/users/290 | When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix? | This question is completely answered, and the result is that the condition involving
the Moebius inversion you mention is both necessary and sufficient! See
* K. H. Kim, N. Ormes, F. Roush. *The spectra of nonnegative integer matrices via formal power series*, J. Amer. Math. Soc. **13** (2000) 773–806. <https://doi.o... | 34 | https://mathoverflow.net/users/8112 | 60503 | 37,513 |
https://mathoverflow.net/questions/60510 | 8 | Let $R$ be a noetherian commutative ring of dimension $n$, and let $M$ be a faithful finite $R$-module. Let $I$ be a proper ideal of $R$, and let $x\in I$ be a non-zerodivisor on $M$.
When does multiplication by $x$ induce an injection $H^n\_I(M)\hookrightarrow H^n\_I(M)$?
| https://mathoverflow.net/users/1353 | Top degree local cohomology under action by a non-zerodivisor | Graham was right, the map is not necessarily $0$ as I wrote in the first comment. However, it is true that $H\_I^n(M)$ is $I$-torsion, so it will be injective if and only if $H\_I^n(M)=0$.
Amusingly, I will observe that the map is actually *surjective*.
Apply $\Gamma\_I(-)$ to the sequence:
$$ M \stackrel{x}{\to... | 6 | https://mathoverflow.net/users/2083 | 60521 | 37,525 |
https://mathoverflow.net/questions/60525 | 20 | For an abelian group $G$, let $G[\operatorname{tors}]$ be its torsion subgroup.
Consider the **torsion sequence**:
$0 \rightarrow G[\operatorname{tors}] \rightarrow G \rightarrow G/G[\operatorname{tors}] \rightarrow 0$.
>
> For which torsion abelian groups $T$ is it the case that for all abelian groups $G$ with... | https://mathoverflow.net/users/1149 | When is the torsion subgroup of an abelian group a direct summand? | These are the (torsion) [cotorsion groups](http://eom.springer.de/C/c110460.htm). The following follows from a theorem of Baer:
>
> A torsion abelian group is cotorsion if and only it is direct sum of a divisible torsion abelian group and an abelian group of bounded exponent.
>
>
>
The original paper of R. Bae... | 22 | https://mathoverflow.net/users/2384 | 60535 | 37,533 |
https://mathoverflow.net/questions/59884 | 17 | Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
>
> Under which conditions is the group $\operatorname{SL}\_n(R)$ generated by transvections?
>
>
>
(A transvection is a matrix with $1$ everywhere on the diagonal and exactly one other non-zero entry.)
This is certainly the case if $R$ is ... | https://mathoverflow.net/users/12858 | For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections? | I'm answering my own question based on the excellent reference given by Max and the additional comments of Jim Humphreys. There is nothing new in my answer, but I think it's useful to close the question in this way.
Following Hahn-O'Meara, we write $E\_n(R)$ for the subgroup of $SL\_n(R)$ generated by transvections (... | 9 | https://mathoverflow.net/users/12858 | 60536 | 37,534 |
https://mathoverflow.net/questions/60537 | 2 | Assume $G$ is a finite group and $H$ a subgroup. Is it true that the number of irreducible representations of $G$ is always larger than (or equal to) the number of irreducible representations of $H$?
| https://mathoverflow.net/users/14119 | Is the number of irreducible representations of G always bigger (or equal) to the number of irreducible representations of H, with H a subgroup of G? | Here is a simple example. As mentionned by Gjergji, this is a question about the number of conjugacy classes. Take $G=\frak A\_4$, which has $3$ classes (the identity, the double transpositions and the $4$-cycles). Now take $H$ the subgroup spanned by a $4$-cycle. Because $|H|=4$ and $H$ is abelian, it has $4$ classes.... | 1 | https://mathoverflow.net/users/8799 | 60543 | 37,537 |
https://mathoverflow.net/questions/60544 | 3 | In Mordell *Diophantine Equations* he says:
>
> In recent years it has been shown that there seems to be a close connection between the number of solutions of f(x,y) = 0 (mod $p^r$) and the existence of rational solutions f(x,y) = 0.
>
>
>
Does anyone know what observation this is referring to? Has it been tur... | https://mathoverflow.net/users/13121 | A remark of Mordell alluding to a local/global principle for cubic Diophantine equations | As Franz says, Mordell is talking about the conjecture of Birch and Swinnerton-Dyer. But I just wanted to add that in the modern formulation of the conjecture, it is not easy to discern the original heuristic "if $E$ has lots of points modulo each prime $p$, then it should have lots of points over $\mathbb{Q}$, more pr... | 9 | https://mathoverflow.net/users/35416 | 60546 | 37,538 |
https://mathoverflow.net/questions/60545 | 3 | Given a matrix $A\in GL\_n(\mathbb{Q})$. Can it be expressed as a product of two matrices $B,C$ with $B\in GL\_n(\mathbb{Z}[1/p])$ and $C\in GL\_{n}(\mathbb{Z}\_{(p)})$, where $ \mathbb{Z\_p}$ denotes the localization of $\mathbb{Z}$ at $(p)$ ?
| https://mathoverflow.net/users/3969 | decompositions of matrices over $\mathbb{Q}$ | Yes, there exists such a decomposition. Multiplying $A$ by a suitable integer, we may assume that $A \in M\_n(\mathbf{Z})$ and $\det(A) \neq 0$. By the theory of elementary divisors, we have $A=\gamma\_1 D \gamma\_2$ with $\gamma\_1,\gamma\_2 \in SL\_n(\mathbf{Z})$ and $D$ is a diagonal matrix with nonzero integral ent... | 7 | https://mathoverflow.net/users/6506 | 60547 | 37,539 |
https://mathoverflow.net/questions/60533 | 16 | This question is related to [another question](https://mathoverflow.net/questions/43124/conditions-for-smooth-dependence-of-the-eigenvalues-and-eigenvectors-of-a-matrix), but it is definitely not the same.
**Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices ... | https://mathoverflow.net/users/10095 | Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations? | A counterexample is given in Section II.5.3, p. 111 of T. Kato, Perturbation Theory for
Linear Operators, 2nd ed.
| 6 | https://mathoverflow.net/users/12120 | 60553 | 37,543 |
https://mathoverflow.net/questions/60387 | 19 | In [Non-vanishing of L-series of modular forms (easy case?)](https://mathoverflow.net/questions/60326) it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is normalized so that the center of the critical strip is given by $s=k/2$.) In partic... | https://mathoverflow.net/users/14092 | Non-vanishing of p-adic L-functions | So, I just talked to David and he pointed me to his paper *L-functions and Division Towers* ([MR0958262](http://www.ams.org/mathscinet-getitem?mr=958262)) whose Theorem 1 is the result (and proof) you're looking for. The proof doesn't care whether $p$ is good or bad or whatever. This takes care of the even weight case,... | 8 | https://mathoverflow.net/users/1021 | 60572 | 37,553 |
https://mathoverflow.net/questions/60534 | 5 | Hi,
My question seems to be closely related to that one : [What strict resolutions of singularities are needed?](https://mathoverflow.net/questions/31466/what-strict-resolutions-of-singularities-are-needed) (and it is perhaps even included in it, but I am not so sure).
Let $X$ be a Gorenstein projective variety wit... | https://mathoverflow.net/users/13841 | resolution of singularities and "permissible" blow-ups. | You can try the following.
**EDIT:** At this point, I don't think this can work in full generality. However, it will give you some bounds on the singularities, which is better than nothing.
Take a resolution of singularities obtained by blowing-up a sequence of smooth centers $X\_N \to X\_{N-1} \to \dots \to X\_1 ... | 1 | https://mathoverflow.net/users/3521 | 60576 | 37,556 |
https://mathoverflow.net/questions/59177 | 11 | Let $M$ be a Riemann surface (or a higher dimensional manifold) and let's assume that it's geodesically complete. Let $W(t)$ be a Brownian motion on the surface accordingly to the manifold's Laplacian and let $r>0$.
Define the Wiener sausage as:
$$
W\_{r}(t):=\{ x\in M: d(x,W(s))\leq r\quad\text{for}\quad 0\leq s\... | https://mathoverflow.net/users/13825 | Wiener Sausages in Riemann Surfaces | I just found out that the case $r$ fixed and $t\to\infty$ for simply connected symmetric manifolds of non-positive sectional curvature and dimension $d\geq 3$, and strictly negative curvature for dimension $d=2$, was solved by Chavel and Feldman in *"The Wiener Sausages and a Theorem of Spitzer in Riemannian Manifolds"... | 3 | https://mathoverflow.net/users/13825 | 60600 | 37,569 |
https://mathoverflow.net/questions/59513 | 11 | I have come across a limit of Gaussian integrals in the literature and am wondering if this is a well known result.
The background for this problem comes from the composition of Brownian motion and studying the densities of the composed process. So if we have a two sided Brownian motion $B\_1(t)$ we replace t by an ... | https://mathoverflow.net/users/10632 | Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)? | The expression you are interested in is of the form $\lim\_n T^n\psi\_t$ where $T$ is the integral operator
$$Tf(x)=2\int\_0^\infty \frac{e^{-\frac{x^2}{2y}}}{\sqrt{2\pi y}} f(y)dy$$
and $\psi\_t(x)= \frac{e^{-\frac{x^2}{2t}}}{\sqrt{2\pi t}}$. Note that $T$ is an operator with a positive kernel $T(x,y)=2 I[y>0] \frac{e... | 3 | https://mathoverflow.net/users/6781 | 60601 | 37,570 |
https://mathoverflow.net/questions/60607 | 2 | I know about
"Simple analytic proof of the prime number theorem" Newman, 1980
However, is there a proof of the Prime Number Theorem without the use of complex analysis? (Real analysis is fine).
Thanks!
| https://mathoverflow.net/users/3609 | Prime Number Theorem w/o Complex Analysis | <http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf> explains the classic proof in context (there is what amounts to a priority dispute).
| 6 | https://mathoverflow.net/users/6153 | 60611 | 37,577 |
https://mathoverflow.net/questions/60567 | 10 | The paper "A note on the automorphic Langlands group" by J. Arthur,
<http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf>
discusses the mysterious `automorphic Langlands group'. This is the mysterious group whose complex representations should correspond to automorphic forms, 'generalizing' the wa... | https://mathoverflow.net/users/3513 | Why do we see SU(2,R) in the Local automorphic Langlands group? | Dear blt,
Just to rephrase Rob H.'s answer in a slightly different way: adding the $SU(2)$ is just another way of adding the $N$ operator, the explanation for which I think you're familiar with. In other words, the local Langlands group does the same job as the Weil--Deligne group, but just uses $SU(2)$ rather than $... | 3 | https://mathoverflow.net/users/2874 | 60621 | 37,583 |
https://mathoverflow.net/questions/60615 | 15 | Let $\{r\_i\}\_{i \in \aleph}$ be sequence of integers such that, for some $t \in \mathbb{N}$ and all $i \in \mathbb{N}$, we have $r\_i = r\_{i+t}$. My question:
Can $\displaystyle \sum\_{i=1}^n \dfrac{r\_i}{i}$ converge to $0$ if $n \rightarrow \infty$ for a non-trivial choice of the $r\_i$ and $t$? Or does $\displa... | https://mathoverflow.net/users/6698 | Can a certain sum converge to 0? | [Yes](http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WKD-4CRP4J8-DW-1&_cdi=6904&_user=145269&_pii=0022314X73900486&_origin=gateway&_coverDate=06/30/1973&_sk=999949996&view=c&wchp=dGLbVtz-zSkzV&md5=1b36190b46571879e3ff62c0490aa20e&ie=/sdarticle.pdf). All the $r\_i$ must equal $0$ if the period is prime, howev... | 22 | https://mathoverflow.net/users/6950 | 60623 | 37,585 |
https://mathoverflow.net/questions/60617 | 3 | Genus 0 curves are well understood in number theory. There is also are rich theory a bunch of conjectures about the arithmetic of elliptic curves.
This leads me to the question, what we know about higher genus curves.
The only thing that comes to my mind is Falting's theorem saying that
any such curve has only finit... | https://mathoverflow.net/users/3757 | The arithmetic of higher genus curves | There is a conjecture (or variants of one) due to Scharaschkin, Skorobogatov and Stoll predicting that Brauer-Manin or finite descent obstructions (these are fancy forms of a local-global principle) determine all rational points. This gives an algorithm to determine if the set of points is empty or not. See Stoll, Alge... | 12 | https://mathoverflow.net/users/2290 | 60626 | 37,587 |
https://mathoverflow.net/questions/60598 | 29 | I have searched for such a question and didn't find it. I recently had a presentation in which I introduced $p$-Sylow subgroups and proved Sylow's theorems. I will have another one soon, concerning applications of Sylow's theorem.
My question is:
>
> Are there any spectacular applications
> of Sylow's theorem in... | https://mathoverflow.net/users/13093 | Applications for p-Sylow subgroups theorem | If you are introducing Sylow subgroups and the Sylow theorems, then your audience likely does not have an extensive mathematical background (otherwise I imagine they would have seen the Sylow theorems at some point in their studies, at least in North America and Western Europe). When I taught the Sylow theorems in an u... | 29 | https://mathoverflow.net/users/3272 | 60628 | 37,588 |
https://mathoverflow.net/questions/60550 | 36 | What is a topological feature, that a (some) TQFT (e.g. in 3 or 4 dim) sees but homology/cohomology/homotopy groups don't? Or: what is an example where using classical theories is hard, but using a TQFT is comparatively easy?
| https://mathoverflow.net/users/14123 | Usefulness of using TQFTs | All the answers so far have focused on 3 dimensions, but the answer is much more striking in 4 dimensions. Freedman's theorem tells you that classical homology invariants give you complete information about topological, simply-connected 4-manifolds. These classical invariants cannot, however, distinguish between distin... | 27 | https://mathoverflow.net/users/5010 | 60629 | 37,589 |
https://mathoverflow.net/questions/60609 | 26 | Background/Motivation
---------------------
Let $R$ be a commutative ring with unit. If $G$ is a finite (or in general, discrete) group, let $R[G]$ be the [group $R$-algebra](http://en.wikipedia.org/wiki/Group_ring) associated to $G$. The isomorphism problem for group rings asks for condiions on groups $G$ and $H$ su... | https://mathoverflow.net/users/6950 | Strong group ring isomorphisms | If $G$ is a finite abelian group, then $\mathbb C[G] = \lbrace f \colon \hat G \to \mathbb C \rbrace$, where $\hat G$ is the Pontrjagin dual of $G$. The isomorphism $g \mapsto g^{-1}$ translates into the same map on the Pontrjagin dual (basically multiplication by $-1$ on $\hat G$), but now it is a bit easier to analyz... | 11 | https://mathoverflow.net/users/8176 | 60644 | 37,600 |
https://mathoverflow.net/questions/60641 | 36 | I have heard it said more than once—on [Wikipedia](http://en.wikipedia.org/wiki/Etale_topology), for example—that the étale topology on the category of, say, smooth varieties over $\mathbb{C}$, is equivalent to the Euclidean topology. I have not seen a good explanation for this statement, however.
If we consider the ... | https://mathoverflow.net/users/1770 | In what sense is the étale topology equivalent to the Euclidean topology? | Saying that the étale topology is equivalent to the euclidean topology is vastly overstating the case. For example, if you compute the cohomology of a complex algebraic variety with coefficients in $\mathbb Q$ in the étale topology, typically you get 0. On the other hand, it is a deep result that the étale cohomology o... | 33 | https://mathoverflow.net/users/4790 | 60652 | 37,606 |
https://mathoverflow.net/questions/60654 | 15 | How can I see that there is no lattice in $G=\mathrm{PSL}\_2( \mathbb{R})$ which contains $\Gamma\_1=\mathrm{PSL}\_2( \mathbb{Z})$ properly, or equivalently, that $X\_1 =\mathrm{PSL}\_2(\mathbb{Z}) \backslash \mathbb{H}$ is not a finite sheated nontrivial cover of a Riemann surface.
As a motivation, the eigenvalues ... | https://mathoverflow.net/users/10400 | There is no lattice in PSL(2,R) which contains PSL(2,Z) properly? | We can say something stronger.
>
> **Theorem:** (Helling 1976) Consider the family of subgroups of $SL\_2(\mathbb{C})$ that are commensurable with a conjugate of $SL\_2(\mathbb{Z})$. The maximal elements of this family under inclusion are precisely the conjugates of $\Gamma\_0(N)^+$ for $N$ a squarefree integer.
> ... | 12 | https://mathoverflow.net/users/121 | 60658 | 37,609 |
https://mathoverflow.net/questions/60660 | 4 | Hello!
If a compact Lie group $K$ acts smoothly on a smooth manifold $M$, then the set $M^K$ of fixed points under this action is a smooth submanifold of $M$. This is proved for example in Duistermaat's book on Lie groups, using the Bochner Linearization Theorem.
I am interested in knowing if some variant of this s... | https://mathoverflow.net/users/3108 | Fixed points of the action of an algebraic group | Linearly reductive groups, that is, linear algebraic groups, say over a field, in which the functor of invariants from finite dimensional representations to vector spaces is exact. I am not sure this is in the literature in this generality, but it is not so hard to prove with a formal scheme argument.
Also, I would c... | 10 | https://mathoverflow.net/users/4790 | 60664 | 37,612 |
https://mathoverflow.net/questions/60666 | 2 | Is there some other way to characterize the functions $f:\mathbb Z\times \mathbb Z\to \mathbb Z$ which are expressible as
$$f(x,y)=g(x)+g(y)-g(x+y)$$
for some $g:\mathbb Z\to\mathbb Z$?
**Easy facts:** (1) $f$ must satisfy $f(x,y)=f(y,x)$ and $f(x,0)=g(0)$ for all $x$. (2) Not all functions $f$ are expressible, since... | https://mathoverflow.net/users/5628 | solvability of an elementary functional equation | The necessary and sufficient condition is that
$$f(x,y+z)+f(y,z)=f(y,x+z)+f(x,z),\qquad\forall x,y,z\in\mathbb Z.\qquad (1)$$
On the one hand, this is obviously necessary. On the other hand, the function $g$, if it exists, is given by
$$g(k)=kg(1)-\sum\_{j=1}^{k-1}f(j,1).$$
Then $g$ is suitable if and only if
$$f(k,\e... | 4 | https://mathoverflow.net/users/8799 | 60675 | 37,617 |
https://mathoverflow.net/questions/60673 | 6 | If $X$ and $Y$ are Riemann surfaces (not necessarily compact), and $f:X\to Y$ is a holomorphic function, then it is obvious that the ramification points of $f$ in $X$ form a discrete subset of $X$. Is the same true of the branch points of $f$ (the set made up of the images of the ramification points)?
| https://mathoverflow.net/users/14143 | Basic question about branch points on Riemann surfaces | Dear Robert, there exists a holomorphic function $X\to Y $ having non discrete and even dense set of branch points, with $X=\mathbb C^\ast \setminus \{0\}$ and $Y=\mathbb C$.
Consider an enumeration $(q\_n)$ of $\mathbb Q$ and the polynomials $P\_n(z)=q\_n + (z-1/n)^2$.
A theorem due to Mittag-Leffler says that ... | 10 | https://mathoverflow.net/users/450 | 60681 | 37,623 |
https://mathoverflow.net/questions/60590 | 3 | Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as
$$
\mathrm{cap}(\Omega)=\inf \int\_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV}
$$
where $\varphi$ ranges over all continuous, compactly supported functions on $M\setminus\Omega$ which are $C^{\infty}$ on $M\setminus\overli... | https://mathoverflow.net/users/13825 | Capacity of Balls in Hyperbolic Space | The capacity of a set $\Omega$ it is known to be
$$
\mathrm{cap}(\Omega)=-\int\_{\partial\Omega}{\frac{\partial \Phi}{\partial \nu}dA}
$$
where $\Phi$ is an harmonic function with $\Phi|\partial\Omega=1$ and $A$ is the $(n-1)$ area of $\partial\Omega$ and $\frac{\partial}{\partial\nu}$ is the normal derivative along $\... | 3 | https://mathoverflow.net/users/13825 | 60683 | 37,624 |
https://mathoverflow.net/questions/60597 | 9 | Assume you have a simple, infinite graph $G$ with bounded degree (there is an upper bound for the degree of the nodes). Choose an arbitrary vertex $x\in V(G)$ and consider
$$
G\_{n}:=\{x\in G:d(x\_0,x)\leq n\}
$$
with the graph metric (hop metric). Assume that each pair of nodes is communicating a unit load of informa... | https://mathoverflow.net/users/13825 | Flow on Infinite Graphs | I think the following provides a counterexample.
The idea is to use the fact that on a graph $G$ with maximum degree $\Delta$ and diameter $D$ reasonably close to the natural lower bound $\log\_{\Delta-1}|V(G)|$ the traffic is almost uniformly distributed.
Explicitly, for a vertex $v$, let $S\_k(v):=\{x \in V(G) \:... | 4 | https://mathoverflow.net/users/8733 | 60692 | 37,627 |
https://mathoverflow.net/questions/60704 | 6 | Is there a smooth (noncompact) manifold $M$ such that for any Riemannian metric $g$ on $M$ there are $p\_i$ and unit tangent vectors $v\_i \in T\_{p\_i}M$ such that $Ricc(g)|\_{p\_i}(v\_i,v\_i) \leq -i$? This seems unlikely, but I'm not sure how to prove it.
---
Alternatively, what is the simplest example of a fi... | https://mathoverflow.net/users/1540 | Do manifolds with no Ricci lower bounds for any metric exist? | Any smooth manifold $M$ admits a metric with quadratic curvature decay.
In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K\_x|=o(|x\_0-x|\_g^{-2}),$$
here $|K\_x|$ is absolute bound for sectional curvature of $g$ at point $x$,
$x\_0$ is a fixed point and and $|x\_0-x|\_g$ denotes distance induced b... | 16 | https://mathoverflow.net/users/1441 | 60705 | 37,635 |
https://mathoverflow.net/questions/60569 | 17 | I recently attended an interesting seminar, where the concept of motivic Donaldson-Thomas invariants was explained ([0909.5088](http://arxiv.org/abs/0909.5088)).
Very roughly, the DT invariant is a generating function $\sum q^k e(M\_k)$ of a numerical invariant $e(\cdot)$ of a sequence of moduli spaces $M\_k$. The m... | https://mathoverflow.net/users/5420 | Is there a "motivic Gromov-Witten invariant"? | Let me give an answer from a slightly different point of view.
Let $M\_k$ be a moduli space as in your question; say it's a (compact) moduli space of sheaves on some (compact) Calabi-Yau threefold. In general, $M\_k$ is going to be very singular. However, it carries a so-called perfect deformation-obstruction theory... | 13 | https://mathoverflow.net/users/6107 | 60721 | 37,646 |
https://mathoverflow.net/questions/60687 | 13 | Is there any explicit computation of Conf($S^n$, $g\_{std}$), the group of conformal diffeomorphisms of the standard $n$-sphere?
| https://mathoverflow.net/users/14147 | The conformal group of $S^n$. | Let's say you want to find all locally conformal maps on some open subset of $\mathbb{R}^n$ where $n\geq 3$. The case of $n = 2$ is rather special, any holomorphic function with nonzero derivative is locally conformal.
Sticking to the case $n\geq 3$, unwinding the definitions leads to a system of PDEs which can be e... | 19 | https://mathoverflow.net/users/6818 | 60731 | 37,650 |
https://mathoverflow.net/questions/60736 | 0 | This may be the wrong forum, but are there any natural contexts (physics, economics, etc.) in which one might observe the relationship $y = ax/(bx+c)$ between a pair of variables $x$ and $y$? General linear fractional expressions come up all the time in optimization but I've had a hard time finding a really simple and ... | https://mathoverflow.net/users/13363 | Applications of linear fractional relationship | See the [Michaelis-Menten equation](http://en.wikipedia.org/wiki/Michaelis%25E2%2580%2593Menten_kinetics) in biochemistry.
| 1 | https://mathoverflow.net/users/13650 | 60739 | 37,655 |
https://mathoverflow.net/questions/60726 | 4 | Hi, I know that there are models in ZFC where don't exist p-points, but I can't find (neither on internet) a proof that I understand, some of you could help me please?
Another question: I know that assuming the continuum hypothesis then there exist Ramsey ultrafilters, but I don't know how to prove it. I found a proo... | https://mathoverflow.net/users/14155 | Questions on ultrafilters | I agree with the comments that you'll have to learn iterated forcing in order to understand the construction of models of ZFC without P-points. Your second question, though, is much easier. To prove the existence of Ramsey ultrafilters assuming CH, proceed as in the proof of Theorem 1.13 in the Diplomarbeit you cited, ... | 11 | https://mathoverflow.net/users/6794 | 60746 | 37,658 |
https://mathoverflow.net/questions/60722 | 6 | Let $n,t$ be positive integers and $p\_1,p\_2,\ldots,p\_n$ positive numbers summing to 1. Conjecture:
$$
\sum\_{i=1}^n p\_i (1-p\_i)^t \le \frac{n(1-1/n)^n}{t}
$$
always holds.
The motivation comes from my missing mass [question](https://mathoverflow.net/questions/60418/missing-mass-estimate); the quantity $\sum\_{i=... | https://mathoverflow.net/users/12518 | Missing mass conjecture | Since the single-variable optimization that David mentions still requires some work, I will present another solution. Let $f(p) = p(1-p)^t$. Define the function $g$ that is equal to $f$ on $[0, 1/t]$, and on $[1/t, 1]$ is a linear interpolation between the points $(1/t, f(1/t))$ and $(1, 0)$. Then we can check that $g$... | 7 | https://mathoverflow.net/users/3376 | 60747 | 37,659 |
https://mathoverflow.net/questions/60745 | 4 | Bateman & Horn [1], building on Bateman & Stemmler [2], give a conjectured formula for the density of numbers that produce simultaneous primes in a number of fixed polynomials.
The constant is
$$C=\prod\_p\left\{\left(1-\frac1p\right)^{-k}\left(1-\frac{\omega(p)}{p}\right)\right\}$$
where $k$ is the number of pol... | https://mathoverflow.net/users/6043 | Calculating the constant in the Bateman-Horn-Stemmler conjecture | See Pascal Sebah and Xavier Gourdon, Constants from number theory, available at <http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html>
If the formulas are unreadable with your software, as they are with mine, click on the link for the Postscript version, which looks fine on my machine. ... | 2 | https://mathoverflow.net/users/3684 | 60748 | 37,660 |
https://mathoverflow.net/questions/60750 | 10 | If we have two finitely generated residually finite groups $G$ and $H$, is there are relation between
the profinite completions $\hat{G},\hat{H}$ and the profinite completion of a semidirect
product $\hat{G \rtimes H}$
(and analogous question for pro-p completions)
| https://mathoverflow.net/users/7307 | Profinite completion of a semidirect product | Take a finite non-abelian simple group $A$ and consider the wreath product $G=A\wr \mathbb Z$. Let $N$ be any subgroup of finite index of $G$. Then $N\cap A^{\mathbb Z}\ne 1$. Let $g$ be a non-trivial element in the intersection. Suppose that the $i$-th coordinate $g\_i$ of $g$ is not $1$. Since $A$ has trivial center,... | 13 | https://mathoverflow.net/users/nan | 60753 | 37,662 |
https://mathoverflow.net/questions/60755 | 1 | I have a proper model category and in it two coequalisers, $A\_i \rightrightarrows B\_i \to C\_i$, $i=1,2$. I have a map of diagrams arising from maps $A\_1 \to A\_2$, $B\_1 \to B\_2$ where these two arrows are cofibrations. Can I reasonably expect the canonical arrow $C\_1 \to C\_2$ to be a cofibration? Are there addi... | https://mathoverflow.net/users/4177 | Cofibrations and coequalisers in a proper model category | No. Think of examples in which $A\_1$ is initial (so $B\_1=C\_1$). There's no reason why $B\_1\to C\_2$ should be a cofibration just because $B\_1\to B\_2$ is.
| 4 | https://mathoverflow.net/users/6666 | 60756 | 37,663 |
https://mathoverflow.net/questions/60742 | 3 | Hi, I know very little about the quantum cohomology (QC for short). I only got interested in the subject as the genus zero part may be relevant to a problem I'm working on. So I hope my question makes sense.
I understand QC defines a structure of an algebra over the operad $H^\*(\bar{M}\_{g,n})$ on the cohomology $H... | https://mathoverflow.net/users/1985 | Quantum cohomology for open varieties | I think that if you take an affine variety, all of its Gromov-Witten invariants
of degree $\neq 0$ are $0$ in any sense. So QC for affine varieties should coincide
with ordinary cohomology.
The following point of view might be useful: you might (and in fact, should) think
about small quantum cohomology as some kind o... | 4 | https://mathoverflow.net/users/3891 | 60759 | 37,664 |
https://mathoverflow.net/questions/60757 | 2 | I am a bit embarrassed to ask this question, but still: assume that I have
a finite morphism $\pi:X\to Y$ of affine algebraic varieties over a field (probably
finiteness is too strong an assumption, but it is true in my situation, so let
me assume it), which is an isomorphism
over an open subset $U$ of $Y$. Let $Z$ be ... | https://mathoverflow.net/users/3891 | Extending a polynomial function from an open subset | I don't think so. Let $Y$ be the plane curve $y\_1^2=y\_2^5$, let $X$ be the line with coordinate $x$, and define $\pi:X\to Y$ by $y\_1=x^5,y\_2=x^2$. It's an isomorphism outside the closed point $y\_1=0=y\_2$. The function $x^3$ vanishes on the scheme-theoretic preimage (that is, it belongs to the ideal of $k[x]$ gene... | 7 | https://mathoverflow.net/users/6666 | 60762 | 37,666 |
https://mathoverflow.net/questions/60200 | 4 | Hi,
In every textbook and at school, one can see the following way to solve for a Poisson equation using FEM:
- (1) start with $\Delta u = b$
- (2) obtain the weak formulation : $\int \Delta u~v~dx = \int b~v~dx$
- (3) integrate by parts to get : $-\int \nabla u \nabla v = \int b~v~dx$
then decompose $u$ an... | https://mathoverflow.net/users/8646 | FEM on a Laplacian | Step (3) is, essentially, a way of defining the weak version of the Laplacian. Given $ u \in H^1 $, the classical Laplacian $ \Delta u $ is generally not defined. However, for any test function $ v \in H^1 $, one can define $ (\Delta u, v ) = -(\nabla u, \nabla v) $. In other words, we have $ \Delta \colon H^1 \to H^{-... | 1 | https://mathoverflow.net/users/673 | 60774 | 37,674 |
https://mathoverflow.net/questions/60775 | 8 | Given a set $S$ of non-zero vectors in $\mathbb{R}^n,$ and a subspace $L,$ consider $f(S,L)=\max\_{s \in S}\frac{\|Ps\|\_2}{\|s\|\_2}$ where $Ps$ is the orthogonal projection of $s$ onto $L.$
Specifically, consider the set $X$ consisting of the $2^n-1$ (nonzero) vectors with all coordinates $0$ or $1$.
A [recent q... | https://mathoverflow.net/users/8008 | Projecting the unit cube onto subspaces | My answer concerns with the case $d=1$ only. Without loss of generality, we can focus on the subspaces, generated by a vector with all coordinates non-negative. It is easy to verify that for the subspace $L$, generated by the vector $(1,1/\sqrt{2},...,1/\sqrt{n})$, the projection onto $L$ of any non-zero vector $\epsil... | 4 | https://mathoverflow.net/users/9924 | 60778 | 37,675 |
https://mathoverflow.net/questions/60780 | 5 | I am looking for a generalization of Moise's theorem, which the few professors that I asked treat as a "known geometric fact" but none could find a reference to an article proving it.
The claim is like this:
Let $M$ be a compact 3 manifold (Riemannian but I do not think that helps), and let $X,Y$ be compact 2 manifol... | https://mathoverflow.net/users/14058 | Generalization of Moise's theorem | In fact it helps immensely that $M$, $X$, and $Y$ are all Riemannian, so much so that the question is both true and not at all a generalization of Moise's theorem. Instead, you are looking for a smooth triangulation of $M$ that supports $X$ and $Y$. A much better citation is to [Goresky's theorem](http://www.math.ias.e... | 9 | https://mathoverflow.net/users/1450 | 60785 | 37,679 |
https://mathoverflow.net/questions/60723 | 3 | The *separation* between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as
$$
\operatorname{sep}(A,B)=\min\_{X\neq 0}\frac{\left\Vert AX-XB\right\Vert}{\left\Vert X\right\Vert}
$$
for a suitable matrix norm (see e.g. the book by Stewart and Sun or ... | https://mathoverflow.net/users/1898 | enlarge the separation between two matrices | For the Frobenius norm the answer seems to be *no*. I haven't tested the operator-2 norm, but it might meet a similar fate.
Here's a simplistic argument. For the Frobenius norm, the separation as defined above reduces to
$$\Delta(S,U) := \text{sep}(S,U) = \sigma\_\min( I \otimes S - U^T\otimes I),$$
where $\sigma\_\m... | 3 | https://mathoverflow.net/users/8430 | 60786 | 37,680 |
https://mathoverflow.net/questions/60781 | 15 | Let $G$ be a connected linear algebraic group
over an algebraically closed field $k$ of characteristic 0.
Let $D\subset G$ be a closed diagonalizable subgroup of $G$
(a subgroup of multiplicative type).
Is it true that $D$ is contained in some torus $T\subset G$?
This is so for $G=\mathrm{GL}\_n$.
Is this true for an... | https://mathoverflow.net/users/4149 | Diagonalizable subgroups of a connected linear algebraic group | No. For example, $\mathrm{PGL}\_n$ contains a subgroup $G$ isomorphic to the product of two cyclic subgroups of order $n$, generated by the classes of the diagonal matrix whose entries are the powers of a fixed primitive $n^{\rm th}$ root of 1, and the permutation matrix corresponding to a cycle of length $n$. The inve... | 14 | https://mathoverflow.net/users/4790 | 60789 | 37,682 |
https://mathoverflow.net/questions/60104 | 3 | Similar questions have been asked elsewhere, but I think this is sufficiently different to warrant a new post. I have a particular matrix $A$ and would like to know when the system $Ax = 0$ has at least one non-negative solution (other than $\vec{0}$). The problem is underdetermined: in most cases I expect the number o... | https://mathoverflow.net/users/4974 | Existence of nonnegative solutions to an underdetermined system of linear equations | This scenario is explicitly handled by Gordan's theorem, which states
$$
\text{either} \quad
\exists x \in \mathbb{R}\_+^m\setminus\{0\} \centerdot Ax = 0,
\quad\text{or}\quad
\exists y\in\mathbb{R}^n\centerdot A^\top y > 0,
$$
where $\mathbb{R}\_+$ denotes nonnegative reals.
(Like Farkas's Lemma, this is a "Theorem ... | 2 | https://mathoverflow.net/users/2621 | 60797 | 37,686 |
https://mathoverflow.net/questions/60790 | 2 | Let $A\subseteq\mathbb N$, as usual we set $d^+(A)=\lim\sup\_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}$ and
$d^{-}=\lim\inf\_{n\rightarrow\infty}\frac{|A\cap[1,n]|}{n}$. It's very stardard that in general $d^+(A)\neq d^-(A)$. My question is little different: is there any $A$ for which $d^-(A)=0$ and $d^+(A)\neq0$? Eq... | https://mathoverflow.net/users/13809 | A set with lower density equal to $0$ and upper density different from $0$ | For an explicit example, you could use $A= \{ n: \lfloor \log\_2 \log\_2{n} \rfloor \text{ is even} \} $.
| 7 | https://mathoverflow.net/users/14102 | 60798 | 37,687 |
https://mathoverflow.net/questions/60794 | 12 | For any manifold $M$, the unordered configuration space of $k$ points is obtained as a quotient of ordered configuration space of $k$ points by the group action of symmetric group on $k$ letters. Does it induce some relation between the cohomology algebras of the two spaces?
| https://mathoverflow.net/users/14124 | Relation between cohomology of ordered and unordered configuration spaces? | If $F\_n(M)$ denotes the ordered configuration space and $C\_n(M)$ the unordered configuration space, the quotient map gives a map
$$H^\*(C\_n(M)) \longrightarrow H^\*(F\_n(M)).$$
If one takes rational coefficients, then this induces an isomorphism
$$H^\*(C\_n(M);\mathbb{Q}) \longrightarrow H^\*(F\_n(M);\mathbb{Q})^{\S... | 11 | https://mathoverflow.net/users/318 | 60799 | 37,688 |
https://mathoverflow.net/questions/59449 | 25 | I refer to definition of viscosity solution in [user's guide to viscosity solutions of second order partial differential equations](https://arxiv.org/abs/math/9207212) by Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions.
Viscosity solutions are generalized solutions which can be implied if the Sobolev theory... | https://mathoverflow.net/users/2082 | Why are viscosity solutions useful solutions? | Viscosity solutions are the "appropriate" notion of solutions for second-order elliptic equations in nondivergence form, and for some classes of first-order equations. Here is a brief summary of why.
From the point of view of applications, the viscosity solution is almost always the right solution. For example, in op... | 24 | https://mathoverflow.net/users/5678 | 60806 | 37,692 |
https://mathoverflow.net/questions/60743 | 7 | Say we are working over some $K=\overline{K}$, of characteristic $p>0$. Let $\phi: Y\rightarrow X$ be a nonconstant map of smooth projective curves. To this map we can associate a map $\psi: Z\rightarrow X$, where on the level of fields this is the Galois closure of $k(X)\subseteq k(Y)$. I would like to know about the ... | https://mathoverflow.net/users/3261 | Tameness for the Galois closure of a map of curves | Look at Lemma 2.1.3 i.v) from Grothendieck and Murre: "The Tame Fundamental Group of a Formal Neighbourhood of a divsors with Normal Crossings on a Scheme".
It says when given a tame field extension $L \supset K$, then its Galois closure will again be tame.
Here, tameness is just defined with respect to one valutio... | 6 | https://mathoverflow.net/users/5273 | 60807 | 37,693 |
https://mathoverflow.net/questions/60812 | 5 | Given an integer $n$ and let $1\leq m\leq n$ be such that $n$ and $m$ are coprimes define
$$
\mathcal{N\_{n,m}}:=\text{the set of primes $p$ such that $p\equiv{m}\hspace{0.1cm}\mathrm{mod}(n)$}.
$$
Let $\mathcal{P}$ be the set of all primes. I seem to recall that the following result is true:
$$
\varphi(n)^{-1}=\lim\_... | https://mathoverflow.net/users/13825 | Asymptotic Distribution of Primes | A good way to find the result you mentioned is to search for Dirichlet's (prime number) theorem; while Dirichlet only proved the infinitude of the set in question, nowadays one will frequently find the more precise assertion you mentioned when this result is discussed.
A more common way to state it is that the numbe... | 10 | https://mathoverflow.net/users/nan | 60815 | 37,698 |
https://mathoverflow.net/questions/60805 | 5 | Help me understand why
$\int\_{-\infty}^{\infty}\frac{1}{2}[1+\operatorname{erf}(\frac{\theta-x}{\sqrt{2q^2}})]\frac{1}{\sqrt{2\pi\sigma^2}}{\exp(-\frac{(x-\mu)^2}{2\sigma^2})}dx \approx \frac{1}{2}[1+\operatorname{erf}(\frac{\theta-\mu}{\sqrt{2(q^2+\sigma^2)}})]$
This transformation used by Mark E. Glickma in "Pa... | https://mathoverflow.net/users/14170 | Integral over error function and normal distribution | This is too long to be a comment.
Let $X$ and $Y$ be independent ${\rm N}(\mu,\sigma\_2)$ and ${\rm N}(0,q^2)$ rv's, respectively.
Since $X+Y \sim {\rm N}(\mu,q^2+\sigma^2)$, it is equal in distribution to $Z + \mu$, where $Z \sim {\rm N}(0,q^2+\sigma^2)$. Hence,
$$
{\rm P}(X + Y \le \theta ) = {\rm P}(Z \le \thet... | 5 | https://mathoverflow.net/users/10227 | 60821 | 37,702 |
https://mathoverflow.net/questions/60833 | 1 | I'm reading "Abstract and Concrete Categories" and, in the Chap. I
(Definition 3.52 on Page 41), there's an ``Object-Free'' definition of
a Category which, through the isomorphism $A \rightarrow \textit{id}\_A$
turns out to be equivalent to the usual one. But all the other
definitions in the book are only given in the ... | https://mathoverflow.net/users/14178 | Definition of Initial&Terminal Objects in an ``Object-Free'' Category | In this view, objects are equated with morphisms that are identities, or "units" in their terminology. So a morphism $x$ is initial when it is a unit and for every unit $y$ there is a unique morphism $f$ for which $f \circ x$ and $y \circ f$ are defined. Similarly, a morphism $y$ is terminal when it is a unit and for e... | 2 | https://mathoverflow.net/users/10368 | 60838 | 37,710 |
https://mathoverflow.net/questions/60825 | 3 | I just read a quite sketchy proof, where some details were skipped and it appears to me that the proof uses some of the following things. However a (probably very easy) question came up:
Assume that we force with a Boolean algebra of the form $P(A) / I$, where $A$ is an arbitrary set and $I$ is an ideal over $A$; i.... | https://mathoverflow.net/users/4753 | Gluing functions together in the generic extension | The generic subset $G\subseteq P(A)/I$ adjoined by your forcing can be viewed as an ultrafilter on $A$ (disjoint from the ideal $I$). Your glued-together $g$ will (be forced to) be equal to $f$ on a set in this ultrafilter; that is, $g$ and $f$ will represent the same element of the generic ultrapower of $V$. But, as J... | 3 | https://mathoverflow.net/users/6794 | 60841 | 37,713 |
https://mathoverflow.net/questions/60744 | 3 | In some physical problem the following differential equation appears
$\dot{x}=F(x)+f(t)$,
where the dot denotes derivative with respect to $t$. $x$ is evidently a function of $t$. I'm wondering what the theory of differential equations knows about how to solve these kind of equation. Generic solution for arbitraril... | https://mathoverflow.net/users/5550 | Solving ODEs of the form $x'(t)=F(x(t))+f(t)$ | This is not a complete answer either but at [this page](http://eqworld.ipmnet.ru/en/solutions/ode/ode-toc1.htm) you can find many special cases of your equation for which the closed form solution *is* available.
| 2 | https://mathoverflow.net/users/2149 | 60848 | 37,716 |
https://mathoverflow.net/questions/60846 | 2 | Are there any articles/books/examples where a non-standard monomial order is used?
What are the applications of these monomial orders? In particular, uses in groebner bases and variable elimination.
(Nonstandard is by my definition something that is not mentioned in wikipedia,
and yes, all monomial orders can be spec... | https://mathoverflow.net/users/1056 | Nonstandard monomial orders? | Yes. An important concept involving non-standard monomial orderings is that of a universal Gr\"obner basis:
Let $k$ be a field and $I \subset k[x\_1, ..., x\_n]$ an ideal. Then, a finite subset $U \subset I$ is called a universal Groebner basis if $U$ is a Groebner basis of $I$ w.r.t. all monomial orders over $x\_1, ... | 10 | https://mathoverflow.net/users/4915 | 60850 | 37,717 |
https://mathoverflow.net/questions/60856 | 3 | Hi,
I am teaching this semester graph theory for undergraduate students.
Now, I am discussing with them about Hamilton Paths in finite graphs.
Last time we meet, I presented the following theorem:
>
> **Theorem.** For $n\geq 3$ the complete graph $K\_n$ is decomposable into edge disjoint Hamilton cycles iff n... | https://mathoverflow.net/users/2386 | Hamilton Paths in $K_{2n}$ | We can explicitly construct such a decomposition.
Label the vertices of the graph with $\{0,1,...,n-1\}$, take the first path to be $0, n-1, 1, n-2, 2,... ,n/2$ and generate the other paths by addition modulo $n$ (the $n$ paths come in pairs in which one is the reverse of the other).
More generally, a symmetric seq... | 10 | https://mathoverflow.net/users/14187 | 60859 | 37,722 |
https://mathoverflow.net/questions/60566 | 22 | Let $K$ be a number field with integral basis $\{\omega\_1,\ldots,\omega\_n\}$.
The affine variety $A\_K$ defined by
$$ N\_{K/{\mathbb Q}}(X\_1 \omega\_1 + \ldots + X\_n \omega\_n) = 1 $$
is an algebraic group, the group structure coming from multiplication
of units with norm $1$; in fact, $A\_K$ is a norm-1 torus. Fo... | https://mathoverflow.net/users/3503 | Weil Conjectures for Number Fields | Some of the relevant results can be found in the book "Algebraic Groups and their birational invariants" by V. E. Voskresenskii (Translations of Math Monographs 179, American Math Society 1998).
Specifically, Chapter 4, Section 9 is all about tori over a finite field, number of rational points, and zeta function; this... | 10 | https://mathoverflow.net/users/11786 | 60866 | 37,727 |
https://mathoverflow.net/questions/60864 | 7 | For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form.
My question is as follows. Let $M$ be a closed symplectic manifold which is homotopy equivalent to a complex projective space. Can we say th... | https://mathoverflow.net/users/11846 | Symplectic structures on a homotopy complex projective space | As far as I know your question is completely open. At the present moment no one knows if for $n>2$ there is a symplectic structure on any manifold homotopic to $\mathbb CP^n$ but not diffeomorphic to $\mathbb CP^n$. In fact our knowledge in these type of questions equals to zero. Namely, the following is open:
**Ques... | 9 | https://mathoverflow.net/users/943 | 60867 | 37,728 |
https://mathoverflow.net/questions/60861 | 4 | I'm sorry for the stupid question, but..
I need to check that zero-scheme of general global section of homogeneous vector bundle over G/P (G simple algebraic group over C, say Spin, P its maximal parabolic subgroup) is non-empty. Is there an easy way to do this?
For now it seems to me that the only way to do this i... | https://mathoverflow.net/users/12208 | how to compute chern classes of homogeneous vector bundles | It's easier to compute its $T$-equivariant $r$th Chern class $c\_r(E) \in H^\*\_T(G/P)$, and use the maps $H^\*\_T(G/P) \twoheadrightarrow H^\ast(G/P)$ and $H\_T^\*(G/P) \hookrightarrow \oplus\_{W/W\_P} H\_T^\*(pt)$.
The latter comes from restricting the equivariant Chern class to the $T$-fixed points on $G/P$ (here ... | 7 | https://mathoverflow.net/users/391 | 60879 | 37,734 |
https://mathoverflow.net/questions/60877 | 6 | It is not too hard to understand the geometric meaning of torsion in homology groups of CW complexes. However, I thought it would be interesting to hear how people describe/think of the geometric meaning of torsion in the homotopy groups of a CW-complex.
| https://mathoverflow.net/users/5450 | Geometric meaning of torsion in homotopy groups | Well, the silly answer is that $f:\mathbb{S}^k\to X$ represents a torsion element of order $p$ if $p \cdot f:\mathbb{S}^k\to X$ extends along $\mathbb{S}^k \hookrightarrow D^{k+1}$ to a map $\varphi: D^{k+1}\to X$.
The slightly less silly answer --- the slightly deeper answer, that is --- is that, equivalently, $f$ i... | 12 | https://mathoverflow.net/users/1631 | 60881 | 37,735 |
https://mathoverflow.net/questions/60849 | 4 | Given two free semicirculars X\_1 and X\_2 and a projection h in the von-Neumann algebra generated by X\_1, how does one show that the von-Neumann algebra generated by {X\_1, hX\_2(1-h)} is a factor? It is easy to show that the two elements in the generating set are free. But I am unable to see what kind of an object h... | https://mathoverflow.net/users/13670 | v-Na generated by | Let me first point out that $X\_1$ and $Y=h X\_2 (1-h)$ are *not* freely independent. This is most easily seen if $h$ has trace 1/2, in which case $Y$ has range and support projections $h$ and $(1-h)$, respectively. But since the support and range projections of $Y$ belong to $W^\*(Y)$, it would follow from the assumpt... | 6 | https://mathoverflow.net/users/12660 | 60886 | 37,738 |
https://mathoverflow.net/questions/60847 | 1 | I would like to know currently what's the best deterministic complexity for factoring polynomials over finite field (without the assumption of GRH)? I have searched on google, there are many source, but I can't find an explicit result. Some of them are only for some special cases. Some of them are used the assumption o... | https://mathoverflow.net/users/5482 | best deterministic complexity for factoring polynomials over finite field | First, let me echo Felipe Voloch's comment and the answer by (the other) unknown (google). Having done that, here are a few recent papers that might be of interest.
Mullin, Ronald C.; Yucas, Joseph L.; Mullen, Gary L.,
A generalized counting and factoring method for polynomials over finite fields.
J. Combin. Math. ... | 3 | https://mathoverflow.net/users/3684 | 60889 | 37,740 |
https://mathoverflow.net/questions/60888 | 8 | In my algebraic geometry class this semester, we've learned about [Leray's Theorem](http://en.wikipedia.org/wiki/Leray's_theorem), which states that for a sheaf $\mathcal{F}$ on a topological space $X$, and $\mathcal{U}$ a countable cover of $X$, if $\mathcal{F}$ is acyclic on every finite intersection of elements of $... | https://mathoverflow.net/users/1916 | Can we relate Cech cohomology and derived functor cohomology even when the cover we choose isn't nice? | There is a Mayer-Vietoris spectral sequence relating the two. This is a "direct" generalization of the Mayer-Vietoris long exact sequence, which is the special case in which your covering has just two open sets.
This is explained, if I recall correctly, in Bott-Tu.
| 8 | https://mathoverflow.net/users/1409 | 60891 | 37,741 |
https://mathoverflow.net/questions/60895 | 13 | On daily basis I need to check (and re-check and re-check...) some definitions and main theorems that are not in my research area. Usually I accomplish this by a Google-search and/or a visit to our library. Unfortunately this doesn't work too well as the local library is a small one and internet seems to be a contradic... | https://mathoverflow.net/users/nan | Suggestions for mathematics encyclopedia | The suggesstion by Peter Humphries in the comments is good. This book contains a nice overview of many fields in mathematics, although you might find that it does not contain the level of detail you're looking for.
There is also Springer's online Encyclopaedia of Mathematics at <http://eom.springer.de> that might be ... | 8 | https://mathoverflow.net/users/9545 | 60903 | 37,748 |
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