parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/107472 | 5 | Consider a smooth probability density $\pi(x)$ on $\mathbb{R}^d$. I am looking for natural for the integral $\iint\_{u,v} \ \min\big(\pi(u), \pi(v) \big) \ du \ dv$ to be finite. If $\pi$ is a radially decreasing density, this is equivalent to the condition $\mathbb{E}\big[ \|X\|^{d} \big] < \infty$. Are there smooth d... | https://mathoverflow.net/users/1590 | minimum of two probability densities | If $\mathbb{E}\left[\lVert[ X\rVert^d\right]$ is finite then the integral in the question is necessarily finite. As mentioned, this holds whenever $\pi$ is radially decreasing. However, in the general case, you can swap regions with equal volume in $\mathbb{R}^d$ about in order to move the large probability regions clo... | 5 | https://mathoverflow.net/users/1004 | 107638 | 62,080 |
https://mathoverflow.net/questions/107653 | 4 | Suppose you have the set of all possible $n$ x $n$ square adjacency matrices where $n$={1,2,3,4...}. For each matrix, compute the logarithm of the largest eigenvalue. Is it true that the set of logarithms you obtain is dense in $\mathbb{R}$? How do you begin to prove/disprove this?
| https://mathoverflow.net/users/25414 | Prove log of eigenvalues are dense in R? | I think you mean dense in $[0,\infty)$, since the spectral radius of a nonnegative integer matrix must be at least 1 (the product of all nonzero eigenvalues must be a nonzero integer). You are effectively asking whether Perron numbers are dense in $[1,\infty)$, and this is easy to see. For example, let $A\_n$ be the co... | 10 | https://mathoverflow.net/users/8112 | 107655 | 62,090 |
https://mathoverflow.net/questions/107651 | 10 | It is well known that the group of diffeomorphisms of the circle contains free non-Abelian subgroups. Is it true (known) that the group of diffeomorphisms of the interval $[0,1]$ contains free subgroups? One approach to get a positive answer can be the following. Consider all functions $f\_a=\frac{\exp(ax)-1}{\exp(a)-1... | https://mathoverflow.net/users/nan | Free subgroup of Diff([0,1])? | Much more is true. The compactly supported diffeomorphism group of any (positive-dimensional, nonempty) manifold contains free subgroups of uncountable rank. In fact, there are such subgroups that are generated by sets which are arcwise connected! See the paper
MR0974661 (90b:58031)
Grabowski, Janusz(PL-WASW)
Free su... | 12 | https://mathoverflow.net/users/317 | 107656 | 62,091 |
https://mathoverflow.net/questions/107620 | 25 | Hello,
The smallest integer $n$ such that there exists two non-isomorphic simple groups of order $n$, is $n=20160$ (namely for the groups $\mathrm{PSL}\_3(\mathbb F \_4)$ and $\mathrm{PSL}\_4(\mathbb F \_2)$).
I read that there are infinitely many integer $n$ such that here exists two non-isomorphic simple groups of ... | https://mathoverflow.net/users/3958 | Non-isomorphic finite simple groups | Just to summarise the comments: the only nonisomorphic finite simple groups with the same orders are
1. $A\_8 \cong {\rm PSL}\_4(2)$ and ${\rm PSL}\_3(4)$ of order 20160.
2. The groups ${\rm P \Omega}\_{2n+1}(q)$ and ${\rm PSp}\_{2n}(q)$ for all odd prime powers $q$ and $n \ge 3$. These have order
$$(q^{n^2} \Pi\_{... | 22 | https://mathoverflow.net/users/35840 | 107660 | 62,093 |
https://mathoverflow.net/questions/66402 | 8 | Let $(e, h, f)$ be an $\operatorname{SL}\_2$-triple in $\mathfrak g$. My understanding is that $e + C\_{\mathfrak g}(f)$ is called a Kostant section only in case $e$ is regular; but I don't impose this restriction. (EDIT: It seems that, in general, it's called a Slodowy slice.) Given such a datum, there is a unique dec... | https://mathoverflow.net/users/2383 | Commutativity and Kostant sections | In some cases the answer to the weaker version of the question (involving the semisimple
part of $X\_2$) is YES. This will happen if $C\_g(e)$ is self-dual which is the case, for instance, when $g=gl\_N$, $N=nm$, and $e$ has $n$ Jordan blocks of size $m$.
To see this, one can use the fact that all maximal toral subalg... | 9 | https://mathoverflow.net/users/24386 | 107666 | 62,096 |
https://mathoverflow.net/questions/107663 | 1 | I believe that there might be an article by Jacques Tits somewhere in which he shows that a locally finite tree can be recovered from the topological group structure on its automorphism group (with the compact-open topology). Because the vertices can be identified with maximal compact subgroups, and one can give a crit... | https://mathoverflow.net/users/15482 | article by Jacques Tits about automorphism group of a locally finite tree | This is true if the action of the automorphism group is edge-transitive (so in particular the tree has to be biregular). This is precisely the statement of Lemma 2.6(vii and viii) of our paper "Simple locally compact groups acting on trees and their germs of automorphisms", P.-E. Caprace and T. De Medts, Transform. Gro... | 4 | https://mathoverflow.net/users/12858 | 107670 | 62,099 |
https://mathoverflow.net/questions/107664 | 4 | In this question [Hilbert class field of Quadratic fields](https://mathoverflow.net/questions/76628/hilbert-class-field-of-quadratic-fields) it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$.
Could someone point me to where the inters... | https://mathoverflow.net/users/16858 | Intersection of Hilbert class fields of imaginary quadratic fields | The generalization of the phenomena you see is *genus theory*. If $K = \mathbf{Q}(\sqrt{-d})$ and $H = K(j\_d)$ then $H$ contains the Genus field $G$.
If $d = \prod\_{i=1}^n p\_i$ is squarefree (and odd for convenience's sake) then $G = K(\sqrt{p\_i^\*})$ where $p\_i^\* = (-1)^{(p\_i-1)/2}p\_i$. In particular $p\_i^\... | 8 | https://mathoverflow.net/users/3384 | 107677 | 62,102 |
https://mathoverflow.net/questions/107678 | 7 | I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact"
Thank you in advance for the help!
| https://mathoverflow.net/users/24309 | Reference for : a Fréchet nuclear space is Montel | Have a look at Proposition 50.2 in
>
> F. Treves: *Topological Vectors
> Spaces, Distributions and Kernels*,
> Academic Press 1995 or Dover 2006
>
>
>
Statement (50.12) in that proposition is precisely what you need.
| 6 | https://mathoverflow.net/users/20302 | 107680 | 62,103 |
https://mathoverflow.net/questions/107679 | 6 | Let $G$ be a group with finite index subgroup $H$. Let $G^\prime = [G,G]$ denote the derived subgroup of $G$.
Is it true that $|G:H|<\infty$ implies that $|G^\prime: H^\prime|<\infty$.
If this is not true in general, is it true for a large class of groups? say, finitely generated.
Thanks to Mark Sapir for providi... | https://mathoverflow.net/users/26665 | Index of derived subgroup in derived group | Consider the infinite dihedral group $G=C\_2\*C\_2=\langle a,b \mid a^2=b^2=1\rangle$ where $C\_2$ is cyclic of order 2. It has a finite index (2) infinite cyclic group $H=\langle ab \rangle$ which is also (almost) the derived subgroup of $G$ ($G'=\langle (ab)^2\rangle$). Then $H'$ (trivial group) is of infinite index ... | 11 | https://mathoverflow.net/users/nan | 107684 | 62,106 |
https://mathoverflow.net/questions/107567 | 4 | I'm currently working on subwords of cube-free binary words.
A *binary* word is one composed of letters from a two-letter alphabet such as $\{0,1\}$. A word $y$ is a *subword* of $w$ if there exist words $x$ and $z$ (possibly empty) such that $w=xyz$. Thus, $01$ is a subword of $0110$, but $00$ is not a subword of $0... | https://mathoverflow.net/users/12357 | Subwords of cube-free binary words | I wrote a recursive program to find the words of each length with no cube, avoiding a given string. If I programmed correctly, there are only $230800$ cube-free binary words of length $30$.
$001$: The longest string is of length $17$:
```
11010110101101100
```
$010$: The longest $2$ are of length $23$, the one f... | 4 | https://mathoverflow.net/users/2954 | 107693 | 62,111 |
https://mathoverflow.net/questions/107597 | 4 | Let $M$ be a complete Riemannian manifold with bounded sectional curvature and $G$ a compact connected Lie group acts smoothly on $M$. Consider the fixed point set $F$, it is of course a submanifold of $M$ by the slice theorem. Let $\{F\_i\}$ be the connected components of $F$. Then for each $i$, is there a sequence of... | https://mathoverflow.net/users/13244 | Collapsing of Riemannian manifolds with a group action | As it was noted in the comments you probably wanted to say that the action is isometric and $M\_n$ is diffeomorphic to $M$ for all $n$. (Otherwise the question has no sense.)
In this case answer is NO. Consider $\mathbb S^1$ action on $\mathbb S^3$ with fixed point set $\mathbb S^1$ and note that simply connected spa... | 2 | https://mathoverflow.net/users/1441 | 107701 | 62,114 |
https://mathoverflow.net/questions/107708 | 9 | I'm teaching an introductory course in cryptography and explained the square-and-multiply algorithm to the class.
<http://en.wikipedia.org/wiki/Square-and-multiply_algorithm>
Someone asked who discovered the algorithm, which I didn't know, so after a short web search that gave no answers, I thought I'd ask on MO. ... | https://mathoverflow.net/users/11926 | Origin of square-and-multiply algorithm | This method is indeed over 2000 years old. The history, with references, is discussed by Donald Knuth in *Seminumerical Algorithms,* volume 2 of *The Art of Computer Programming*, page 441:
>
> The method is quite ancient; it
> appeared before 200 B.C. in Pingala's
> Hindu classic Chandah-sutra [see B.
> Datta a... | 12 | https://mathoverflow.net/users/11260 | 107709 | 62,120 |
https://mathoverflow.net/questions/107715 | -1 | If the derivative of a function is lipschitz,,,does it mean that the function itself is also lipschitz? Any proof for that?
| https://mathoverflow.net/users/26673 | Lipschitz condition on the first derivative of a function? | No. $f(x)=x^2$ on the whole real line is not Lipschitz, but the derivative $f'(x)=2x$ is.
However, if the function is defined on a bounded interval, then the statement is true.
Indeed, if $f'$ is Lipschitz, then it is continuous and thus bounded, and a function
with bounded derivative is Lipschitz.
| 4 | https://mathoverflow.net/users/25510 | 107717 | 62,123 |
https://mathoverflow.net/questions/107716 | 13 | One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for $\Lambda>\lambda$.
How about sufficiently pinched, i.e. $\Lambda/\lambda=1+\epsilon$ for $\epsilon$ small?
Are they have vani... | https://mathoverflow.net/users/1190 | Are negatively pinched manifold locally conformally flat? | No in dimensions $\geq 3$. To be conformally flat in 3 dimensions, the [Cotton tensor](http://en.wikipedia.org/wiki/Cotton_tensor) must to vanish, and in dimensions $\geq 4$, the [Weyl tensor](http://en.wikipedia.org/wiki/Weyl_tensor) must vanish.
Maybe the point of your question though is to ask why is the Cotton o... | 15 | https://mathoverflow.net/users/1345 | 107721 | 62,124 |
https://mathoverflow.net/questions/107718 | 4 | I want to define Sobolev spaces for sections on a vector bundle, basically I want that a section will belong to the Sobolev space $W^{k,p}$ if its coordinates in any aceptable patch belong to the corresponding Sobolev space of functions. So here is my try:
Let $\mathcal{A}$ be the collection of atlases of the differe... | https://mathoverflow.net/users/26675 | Definition of Sobolev spaces as a space of sections of certain type | There is a fairly careful discussion of this in my book "Foundations of Global Non=Linear Analysis", Benjamin & Co. 1968
(Added later:) It occurred to me that this old book of mine is probably not easy to come by, so I have made a photo-copy of it available here:
<http://vmm.math.uci.edu/FGNLA.pdf>
I should add t... | 12 | https://mathoverflow.net/users/7311 | 107723 | 62,125 |
https://mathoverflow.net/questions/107730 | 8 | Hi, I am interested in the relationship between the pseudo-anosov map and volume of the hyperbolic 3-manifold.
Assume $H\_{1}$ and $H\_{2}$ are two handlebodies with $\partial H\_{1}=\partial H\_{2}=S$.
Question 1:For any pseudo-anosov homeomorphism $\psi: S\rightarrow S$, if the $n\in N$ is large enough, is $M\_... | https://mathoverflow.net/users/18496 | Pseudo-Anosov map, Heegaard splitting, hyperbolic 3-manfold | For Question 1:
Souto and Namazi ([pdf link](http://www.math.ubc.ca/~jsouto/papers/pseudo9-corrected.pdf)) showed that for a generic pseudo-anosov homeomorphism $\psi$ and $\epsilon >0$, there is $n\_\epsilon$ such that $M\_{\psi^n}$ admits a Riemannian metric with all sectional curvatures between $-1-\epsilon$ and $... | 7 | https://mathoverflow.net/users/4325 | 107737 | 62,131 |
https://mathoverflow.net/questions/104664 | 6 | Consider a finite state, irreducible Markov chain with a rate matrix $Q$ and a stationary distribution $\pi$. Suppose the chain starts with the initial distribution $p$ at time $0$, then at time $t$ the distribution is given by $$p\_t = p\*e^{Qt}$$
The relative entropy of $p\_t$ with respect to the $\pi$ (also known as... | https://mathoverflow.net/users/20062 | Convergence of Markov chains in terms of relative entropy | Turns out $D(p\_t||\pi)$ need not always be convex. The following paper demonstrates a counter-example in section $4.2$- <http://arxiv.org/abs/0712.2578>
| 4 | https://mathoverflow.net/users/20062 | 107744 | 62,134 |
https://mathoverflow.net/questions/107742 | 17 | $\DeclareMathOperator\Hom{Hom}$Is there an algebraic Künneth formula for cohomology?
More precisely assume $A\_{\*}, B\_{\*}$ are chain complexes of free $R$-modules ($R$ is a $PID$) and $M, N$ are $R$-modules. Then the map $\sum H^n(A\_{\*},M)\otimes H^m(B\_{\*},N)\rightarrow H^{n+m}(A\_{\*}\otimes B\_{\*}, M\otimes... | https://mathoverflow.net/users/26250 | Künneth formula for cohomology | **Edit:** The answer below assumes that $A$ and $B$ are bounded below (or above). This is not explicitly mentioned in the statement in Spanier, but seems to be necessary.
---
Yes, this is Theorem 5.5.11 in Spanier's "Algebraic Topology" text.
The conditions are that the torsion product $\operatorname{Tor}\_R(M,... | 19 | https://mathoverflow.net/users/8103 | 107745 | 62,135 |
https://mathoverflow.net/questions/107739 | 0 | Description
-----------
Let $\{e\_n\}$, $e\_n\in \mathbb{R}^p$ be a sequence of vectors, $\{U\_n\}$, $U\_n\in\mathbb{C}^{p\times p}$ be a sequence of **unitary matrices** (that is $U\_i^\*=U\_i^{-1}$, $^\*$denonts conjugate transpose). $Q\in\mathbb{R}^{p\times p}$ is a diagonal matrix with positive diagonal entries. ... | https://mathoverflow.net/users/26600 | Does this sequence converge to zero? | Here is a counterexample: Let $p=2$ and let $Q$ be the diagonal matrix with entries $(2,1)$. If $n$ is odd, let $e\_n$ be the first standard basis vector $(1,0)^t$ and if $n$ is even, the second $(0,1)^t$.
If $n$ is odd, let $U\_n$ be the identity matrix and if $n$ is even, let
$U\_n$ be the matrix with first row (0,1... | 1 | https://mathoverflow.net/users/nan | 107747 | 62,136 |
https://mathoverflow.net/questions/107732 | 4 | Let us say that a family $R$ of sets has the Finite Subcovering Property --- FSP --- if any subfamily of $R$ which covers the union $\cup B: B \in R$ has itself a finite subfamily which also covers. For example, take for $R$ the family of open balls in a compact metric space $M$. Clearly the family of finite intersecti... | https://mathoverflow.net/users/20300 | A combinatorial property implied by the Axiom of Choice | This is not really an answer but too long for a comment:
Let me point out that there might be a relation to the Alexander Subbase Theorem here:
A topological space is compact iff the topology has a subbase with the FSP.
What FIP gives you is this:
If $R$ has the FSP and is a subbase for the topology of a spac... | 8 | https://mathoverflow.net/users/7743 | 107750 | 62,137 |
https://mathoverflow.net/questions/107048 | 6 | Let $X$ be a one-dimensional one-step irreducible shift of finite type and let $\pi$ be a one-block factor code from $X$ to a sofic $Y$. Suppose $y$ is a right transitive point of
$Y$ and $\pi(u)=y$ for some $u\in X$. Given $u\_0=a$ and a block $B$ of $X$, is there a point $x\in\pi^{-1}(y)$ such that $x\_0=a$ and $B$ o... | https://mathoverflow.net/users/19639 | Relative irreducibility | Sorry that this is slightly technical... It's uses some concepts that Mahsa showed me in answering a more simple question that I asked her.
Using results from <http://arxiv.org/abs/1001.5323v1>, we may assume that there exists a 'magic symbol' $k\in Y$ satsisfying that for any
word $y\_m\cdots y\_n\in Y$ with $y\_m=y... | 3 | https://mathoverflow.net/users/24586 | 107751 | 62,138 |
https://mathoverflow.net/questions/107711 | 0 | Suppose $G\_1, \ldots, G\_k$ are unitary, Hermitian, and anti-commuting matrices, and assume the same for $F\_1, \ldots, F\_k$. Suppose these matrices are similar, i.e. there exists $T \in GL\_n(\mathbb{C})$ such that
$$
G\_i = T^{-1} F\_i T
$$
for all $i \in [k]$. Does there exist $V \in U\_n$, where $U\_n$ is the gro... | https://mathoverflow.net/users/26659 | Similarity about unitary matrices | Maybe this question deserves an answer which doesn't use decomposition into irreducible representations. Let $\pi$ and $\sigma$ be equivalent unitary representations. If $T$ intertwines, then so does $T^{\ast}$, because $\pi(g^{-1})=\pi(g)^{\ast}$ (same for $\sigma$). Now we can conclude that $|T|=\sqrt{T^{\ast}T}$ is ... | 3 | https://mathoverflow.net/users/24953 | 107757 | 62,141 |
https://mathoverflow.net/questions/107554 | -1 | `The "problem of quantization"`:
>
> Find a vector space $Obs$ (as large as possible) of real-valued functions $f(p, q)$ on $R^{2n}$, containing the coordinate functions $p\_j$ and $q\_j$ $(j = 1, . . . , n)$,
> and a mapping $Q : f → Q\_f$ from $Obs$ into self-adjoint operators on $L^2(R^n)$ such that (q1)–(q5)\... | https://mathoverflow.net/users/17780 | Problem of quantization: state of the art | Quantization is big area, so let me concentrate on some mathematical aspect which are close to me and somewhat related with representation theory and algebraic geometry. Undoubtedly this is biased and incomplete answer. Hope others would add more.
Morally quantization is a bridge between commutative and non-commutati... | 0 | https://mathoverflow.net/users/17780 | 107759 | 62,142 |
https://mathoverflow.net/questions/107572 | 11 | Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the *standard* Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie type over a field of $q$ elements and I note that all simple groups of Lie type can be obtained this way.
(I am usi... | https://mathoverflow.net/users/801 | Regular elements in the torus of a group of Lie type | This question is interesting as it forces one to look closely at rational points of $F$-stable tori in $G$.
If $T$ is such torus then $F$ acts the lattice of cocharacters $X\_\*(T)$ thus inducing an automorphism of the set of coroots. Let $\sigma$ be this automorphism and suppose it has order $m$ (this is all we need... | 5 | https://mathoverflow.net/users/24386 | 107763 | 62,143 |
https://mathoverflow.net/questions/107766 | 6 | I have two questions which are intuitively true.
Let $V$ be a Hilbert space. As usual we can turn $V\otimes V$ or $V\otimes V\otimes V$ into Hilbert spaces by intorducing the natural inner product and by performing completion.
Question 1. We have a sequence of simple tensors $f\_{i}\otimes g\_{i}$ that converges in... | https://mathoverflow.net/users/26516 | Limit of simple tensors | Under the natural identification of the completion of $V\otimes W$ with Hilbert-Schmidt operators $V\rightarrow W^\*$, monomial tensors give rank-one operators. A Hilbert-Schmidt-norm limit of rank-one operators is certainly rank-one. (This viewpoint gets away from the pitfalls of specific representations of the tensor... | 6 | https://mathoverflow.net/users/15629 | 107773 | 62,147 |
https://mathoverflow.net/questions/107526 | 17 | Rota and Shen's *[On the Combinatorics of Cumulants](http://www.researchgate.net/publication/222647995_On_the_Combinatorics_of_Cumulants)* ends with a conjecture which I'll restate as follows:
>
> Let $p \in \mathbb{R}[x\_1, x\_2, ...]$ be a polynomial such that, for any sequence $X\_1, X\_2, ...$ of i.i.d. random... | https://mathoverflow.net/users/290 | Reference request: a conjecture of Rota on positive functions of a random variable | I think your reformulation of the conjecture in the Rota-Shen paper is correct.
Also your counterexample is correct. So I guess the conjecture was not stated properly
in that article. Perhaps one should restrict to translation invariant polynomials only.
By that I mean polynomials in the moments which remain invariant ... | 5 | https://mathoverflow.net/users/7410 | 107775 | 62,148 |
https://mathoverflow.net/questions/107768 | 9 | Let $G$ be a finite abelian group, and $g\_1, \ldots, g\_n \in G$ such that the cyclic groups that they generate are in direct sum $\langle g\_1 \rangle \oplus \cdots \oplus \langle g\_n \rangle$. Is it always possible to find elements $h\_1, \ldots, h\_n \in G$ and integers $a\_1, \ldots, a\_n$ such that the following... | https://mathoverflow.net/users/26691 | On the existence of a direct summand containing a fixed subgroup | I am not convinced that this is true, because the pure subgroup generated by the $g\_i$ might
not have the stipulated form as a direct sum of the $h\_i$.
Let $G = {\mathbb Z}/16{\mathbb Z} \oplus {\mathbb Z}/4{\mathbb Z}$, $n=1$, and $g\_1=(4,2)$.
What could $h\_1$ be?
We can prove that there is no such $h\_1$ as f... | 11 | https://mathoverflow.net/users/35840 | 107776 | 62,149 |
https://mathoverflow.net/questions/107761 | 5 | I would like to understand the following statement that I found in some of Sorger's lecture notes (Lectures on moduli of principal G-bundles over algebraic curves).
Let X be a projective and smooth curve and $G$ a reductive group.
We have the universal $G$-bundle $\mathcal{E}$ on $Bun\_G(X)\times X$ and we can form ... | https://mathoverflow.net/users/1328 | canonical bundle on the stack of G bundles on a curve | The tangent complex is a complex of quasi-coherent sheaves on $Bun\_G(X)$, i.e. a compatible family of complexes of quasi-coherent sheaves on every affine scheme $f:U\rightarrow Bun\_G(X)$ (i.e. a $G$-bundle $P\_0\rightarrow X\times U$) mapping smoothly.
The global sections $\Gamma(U, f^\* T\_{Bun\_G(X)})$ of the tan... | 7 | https://mathoverflow.net/users/18512 | 107784 | 62,152 |
https://mathoverflow.net/questions/107786 | 2 | Let $p$ be a rational prime and $\zeta\_p$ a primitive $p$th root of unity.
What do we know about the set $\{z\in\mathbb{Q}(\zeta\_p):|z|=1\}$?
| https://mathoverflow.net/users/26700 | elements of absolute value one in cyclotomic fields | For any $u \in L^\times = \mathbf{Q}(\zeta\_p)^\times$, letting $z=u/\bar{u}$, we have $|z|=1$. The converse is true : let $G=\mathbf{Z}/2\mathbf{Z}$ act on $L^\times$ by complex conjugation, then $G$ is the Galois group of $L$ over $K = L \cap \mathbf{R}$. By Hilbert 90, $H^1(G,L^\times) = \{1\}$, which says precisely... | 9 | https://mathoverflow.net/users/6506 | 107787 | 62,153 |
https://mathoverflow.net/questions/107735 | 2 | I am trying to say something about the asymptotics of
$$\int\_{\mathbb{R}} e^{cx - x^{4/3}}dx$$
as $c \to +\infty$, and need a sanity check. As I understand it, Laplace's method is to write
$$q(x) = x-c^{-1}x^{4/3}$$
and note that $q(x)$ has a global maximum at $x\_{0} = \frac{27c^{3}}{81}$. Then
it follows
$$\int\_{\... | https://mathoverflow.net/users/12968 | Valid use of Laplace's method? | The version of Laplace's Method I know uses that $q(x)$ does not depend on $c$. If you try to extend it, you need some extra uniformity condition to show that the rest of the function does not contribute. Here is a counterexample when you only assume that the maximum is always at $x\_0$ and the second derivative is con... | 1 | https://mathoverflow.net/users/2954 | 107794 | 62,157 |
https://mathoverflow.net/questions/107792 | 2 | Suppose that $S\_1,\dots,S\_n$ is a collection of disjoint shapes in the plane, and let $\mathcal{X}$ denote the set of all $n$-tuples of points $\lbrace x\_1,\dots,x\_n\rbrace$ such that $x\_i\in S\_i$ for each $i$. For any such tuple $X = \lbrace x\_1,\dots,x\_n\rbrace$, let $F(X)$ be defined as
$F(X) = \max\_{i} \... | https://mathoverflow.net/users/26702 | Worst-case nearest-neighbor distances between regions | If $p \in S\_i$ is not a possible $x\_{i^\*}$ for a given $r$, there must be some $j$ such that $S\_j$ is contained in the disk of radius $r$ around $p$. In particular a possible $r$ is half the second-largest diameter of the $S\_i$'s. And this is within some constant factor of optimality, if you consider a case where ... | 1 | https://mathoverflow.net/users/13650 | 107798 | 62,158 |
https://mathoverflow.net/questions/107791 | 2 | I am interested in a lower bound on the Bhattacharya distance between two independent multivariate Gaussian distributions. To be precise, consider zero-mean independent Gaussian distributions $p\_1\sim\mathcal{N}(\mathbb{0},\Theta\_1^{-1})$ and $p\_2\sim\mathcal{N}(\mathbb{0},\Theta\_2^{-1})$ (note that $\Theta\_1$, $\... | https://mathoverflow.net/users/26701 | Lower bound on Bhattacharya distance between independent Gaussian distributions ? | The KL divergence is given by
$$\hbox{Tr}(\Sigma\_0\Sigma\_1^{-1})- \ln(|\Sigma\_0\Sigma\_1^{-1}|)$$
Let $\mu\_{-}$ be the smallest eigenvalue of $\Sigma\_0\Sigma\_1^{-1}$ and $\mu\_{+}$ the largest, then
$$\mu\_{-} = ||\Sigma\_1 \Sigma\_0^{-1} ||\_2$$
$$\mu\_{+} = ||\Sigma\_0 \Sigma\_1^{-1} ||\_2$$
$$\hbox{Tr}(\... | 2 | https://mathoverflow.net/users/8737 | 107801 | 62,160 |
https://mathoverflow.net/questions/107774 | 25 | **Question:** What is the number of two-dimensional irreducible representations of a finite group ? How it can be expressed in groups-theoretic terms ? (Number of 1-dimensional irreps is |G/[G,G]| ).
The question is somewhat naive and actually I heard it from our teachers when I was an undergrad many years ago, it wa... | https://mathoverflow.net/users/10446 | Number of 2-dimensional irreducible representations of a finite group ? | This is not a complete answer in any sense, but I will make a few comments. The irreducible subgroups $G$ of ${\rm GL}(2,\mathbb{C})$ are the primitive ones, which have $G/Z(G)$ isomorphic to $A\_{4},S\_{4}$ or $A\_{5}$, and imprimitive groups, which have an Abelian normal subgroup of index $2.$ On the other hand, any ... | 24 | https://mathoverflow.net/users/14450 | 107805 | 62,162 |
https://mathoverflow.net/questions/107803 | 3 | I have a simple terminology request: recall that given sets $A$ and $B$, a *relation* $R$ from $A$ to $B$ is any subset of the product $A \times B$. Thus, one may view a relation as a function $A \times B \to \lbrace 0,1 \rbrace$ where $(a,b)$ maps to $1$ if and only if it lies in $R$.
What I'm looking for is the sui... | https://mathoverflow.net/users/24796 | Terminology for generalized relations | This is called an *$L$-valued relation*, when $L$ is the target of the function, which can be viewed as the collection of possible truth values.
Thus, a $2$-valued relation is just an ordinary relation of classical logic, where every instance has truth value either true or false. But for any Boolean algebra $\mathbb... | 4 | https://mathoverflow.net/users/1946 | 107807 | 62,164 |
https://mathoverflow.net/questions/76609 | 4 | Consider classical statement of Ito's formula: Let $X$ be a continuous
semimartingale and $F \in C^2(\mathbb{R}^d, \mathbb{R})$; then $F(X)$
is a continuous semimartingale and
$$F(X\_t) = F(X\_0) + \sum\_i \int\_0^t {\partial\_i F} dX\_s^i + \frac 1 2
\sum\_{i,j} {\partial^2\_{ij} F} d \langle X^i, X^j \rangle\_s.$$
In... | https://mathoverflow.net/users/5656 | Generalized Ito's formula | One can also use the Alexandrov-Bakelman-Pucci-Krylov-Tso estimates from parabolic PDE to show that Ito's Lemma holds for functions in $W^{2,p}$ when $X$ is a diffusion with uniformly positive definite covariance and $p$ is large enough. This result be found, for example, in Krylov's "Controlled Diffusion Processes" Ch... | 3 | https://mathoverflow.net/users/26707 | 107810 | 62,167 |
https://mathoverflow.net/questions/107804 | 0 | Intuitively, I want to construct the functional F in this way:
$$F(f)=\lim\_{x\rightarrow 0+}f(x)-\lim\_{x\rightarrow 0-}f(x)$$
for $f\in L^\infty$. I know this is not well defined so I'd like to find a way to use this idea. Maybe find an extension using Hahn-Banach, etc.
| https://mathoverflow.net/users/26704 | A continuous linear functional on $L^\infty(R)$ that vanishes on $C(R)$. | I am not sure what is your question. But you can construct a linear functional in this way.
The limit does not exist for some $f$ in $L^\infty$. But you can use the Banach limit.
Consider the bounded linear functional on $C(R)$ defined as the limit as $x\to 0$.
By Hahn-Banach this has an extension to a bounded linear f... | 5 | https://mathoverflow.net/users/25510 | 107811 | 62,168 |
https://mathoverflow.net/questions/107796 | 0 | By trying to find a marginal distribution I came accross integration of the product series. For the sake of generality, lets assume the integral is of following form:
$$\int \prod\_{k=1}^{n}\left ( x+a\_{k} \right )^{b\_{k}}dx.$$
$a\_{k}$ is a real coefficient and $b\_{k}$ is positive integers.
Is there any method, tha... | https://mathoverflow.net/users/21753 | Integrating a product | Since the derivative of such a product has a similar form $$\left(\sum\_{k=1}^n\frac{b\_k}{x+a\_k} \right)\prod\_{k=1}^{n}\left ( x+a\_{k} \right )^{b\_{k}},$$ One might hope that something similar is true of some antiderivative. (I am implicitly assuming that the $b\_k$ are positive integers, although the formula is v... | 2 | https://mathoverflow.net/users/8008 | 107821 | 62,173 |
https://mathoverflow.net/questions/107797 | 2 | Let $H$ be a Hopf algebra, and $(M,\triangleleft)$ a $H$-module. Now for $m \in M$, and $h \in H$, then is it true in general that
$$
m \triangleleft h = 0 ~~~ \implies ~~~~m \triangleleft S(h) = 0?
$$
If not, do there exist conditions on $H$ or $M$ under which it is true.
Thanks in advance guys!
| https://mathoverflow.net/users/2612 | Zero Actions on a Hopf Module Preserved Under the AntiPode? | Not true. All you need is a right ideal, not closed under $S$. Then you will a cyclic module with a counterexample. The group algebra of $C\_3$ over a field with a primitive cubic root of 1 will be an example...
I cannot think of any conditions beyond obvious ones...
| 3 | https://mathoverflow.net/users/5301 | 107832 | 62,178 |
https://mathoverflow.net/questions/107824 | 4 | Hi, Let $M$ be a pseudo-Riemannian manifold and $G$ a (Lie) subgroup of $Iso(M)$ which acts on $M$ smoothly and properly. Suppose we know the orbits up diffeomorphism. Is there a systematic way to recognize the induced metric on an orbit of the action?
you may assume that $M$ is a space form of constant curvature and a... | https://mathoverflow.net/users/26148 | how to find the induced metric on an orbit? | The orbit is homogeneous, so it's enough to compute the metric at one point, call it $x$. The map from the Lie algebra to the tangent space to the orbit at $x$ is surjective. Concretely for a vector $X$ in the Lie algebra the corresponding tangent vector is $X\_M (x) := d/dt \exp (tX) \cdot x$ ( $\cdot $ denotes the ac... | 5 | https://mathoverflow.net/users/25355 | 107833 | 62,179 |
https://mathoverflow.net/questions/107836 | 3 | Let $C$ be an elliptic curve over the complex numbers. Consider a nontrivial extension
$$
0 \to \mathcal O\_C \to E \to \mathcal O\_C \to 0
$$
of rank 2 of the structure sheaf of $C$. This defines a ruled surface $X = \mathbb P(E)$ over $C$.
Is the automorphism group of $X$ transitive?
I ask because I'm looking for... | https://mathoverflow.net/users/4054 | Automorphism group of ruled surface | Any automorphisms of $X$ lies over an automorphism of $C$. It seems to me that there is a unique section $C \to X$ with trivial normal bundle, so this section should be carried to itself by an automorphism, which seems to show that the automorphism group is not transitive.
| 4 | https://mathoverflow.net/users/4790 | 107840 | 62,184 |
https://mathoverflow.net/questions/107839 | -2 | hi,
I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{U} \rightarrow \mathbb{R}$. Can this function be holomorphically extended to $U$ (maybe if we shrink $U$) in a unique ... | https://mathoverflow.net/users/26219 | holomorphic extension of a function | Yes, of course, after we shrink $U$. A convergent Taylor series at a real point converges in some complex
neighborhood of this point.
Added reply to your comment: you can apply identity theorem. Two real analytic functions
coinciding on an open set of $R^n$ coincide in a complex neighborhood of this set.
| 7 | https://mathoverflow.net/users/25510 | 107842 | 62,185 |
https://mathoverflow.net/questions/107841 | 1 | Let $X$ be a smooth projective variety (over an algebraically closed field; it could be the field of complex numbers); $Z$ is its hyperplane section. When there exists an etale $U/X$ such that:
1. $Z$ can be lifted to $U$.
2. There exists a morphism from $U$ to a curve $C$ such that $Z$ is the preimage of some point ... | https://mathoverflow.net/users/2191 | For a hyperplane section Z of X, when there exists its etale X-neighbourhood such that Z is a fibre of its morphism to a curve | If $X$ is a curve, then just take linear projection from a disjoint codimension 2 linear subspace in the spanning hyperplane of $Z$. If the dimension of $X$ is $2$ or more, then there never exists such an étale neighborhood and morphism. For $Z$ of dimension $n-1 \geq 1$, for the normal sheaf $\mathcal{N}\_{Z/X}$, the ... | 3 | https://mathoverflow.net/users/13265 | 107848 | 62,188 |
https://mathoverflow.net/questions/107851 | 5 | I encountered this issue recently, but do not know of any general results to deal with it, so I would appreciate any pointers.
>
> Let $\mathbb T=\{z\in\mathbb C\mid |z|=1\}$, and let $f:\mathbb T\to\mathbb C$ be continuous and injective, so its image $\mathbb T'$ is a Jordan loop. Under what (general) conditions ... | https://mathoverflow.net/users/6085 | Extending Jordan loops | What you're asking is equivalent to asking whether any homeomorphism $g : S^1 \rightarrow S^1$ can be extended to a homeomorphism of the disc. This is easy -- write the disc in polar coordinates $(t,\theta)$ with $\theta \in S^1$, and define an extension $G(t,\theta) = (t,g(\theta))$.
The question about whether this ... | 8 | https://mathoverflow.net/users/317 | 107855 | 62,191 |
https://mathoverflow.net/questions/107852 | 4 | An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid *equivalent* if one can be obtained from the other by rotation of space.
Now suppose that $E$ and $E'$ are two equivalent centric ellipsoids ... | https://mathoverflow.net/users/7599 | Can an ellipsoid be moved freely inside another ellipsoid? | The answer is yes.
You can connect your small ellipsoid to the big one by a continuous nested one-parameter family of ellipsoids, say $E\_t$, $t\in [0,1]$.
Nested means that $E\_{t\_1}\subset E\_{t\_2}$ if $t\_1 \le t\_2$.
Say if
$$E=\{\, x \mid \langle x,Ax\rangle \le1 \,\}\ \ \text{and}\ \ F=\{\, x \mid \langle x... | 7 | https://mathoverflow.net/users/1441 | 107858 | 62,192 |
https://mathoverflow.net/questions/107857 | 5 | Let A & B be two categories, the join A\*B is created by stipulating its class of object is the disjoint union of the objects of A & B, the morphisms remain the 'same', but we throw in an extra morphism for every object a in A, and b in B.
that is:
A\*B[a,a']=A[a,a'] if a,a' are in A
A\*B[b,b']=B[b,b'] if b,b' ar... | https://mathoverflow.net/users/22002 | is there a universal property that characterises the join of two categories? | It's a special case of what's called a *collage* or [cograph construction](http://ncatlab.org/nlab/show/cograph+of+a+profunctor). Recall that a profunctor or bimodule between categories $B$, $A$ is a functor $R: A^{op} \times B \to Set$. The cograph of $R$ is the category $\bar{R}$ where $Ob(\bar{R}) = Ob(A) \sqcup Ob(... | 11 | https://mathoverflow.net/users/2926 | 107859 | 62,193 |
https://mathoverflow.net/questions/107844 | 5 | Given a noncommutative ring $R$, and two (left) $R$-modules $M$ and $N$, how does one define a left action on the the vector space tensor product $M \otimes N$? Multiplying on just the first factor of the tensor product seems a little unnatrual, but I can't see what else to do.
| https://mathoverflow.net/users/11206 | Left-Module Structure on the Tensor Product ofTwo Left Modules | Let $R, S$ be two (unital and associative to be safe) algebras over a commutative ring $k$ and let $M, N$ be respectively a left $R$-module and a left $S$-module. Then we can define the tensor product $M \otimes\_k N$ by the usual universal property, and it is naturally a left $R \otimes\_k S$-module by functoriality. ... | 15 | https://mathoverflow.net/users/290 | 107861 | 62,195 |
https://mathoverflow.net/questions/107825 | 51 | The Thom class and Thom isomorphism theorem for oriented vector bundles are proven ( at least to my knowledge) by induction on the open covers and some manipulation with Mayer-Vietoris sequences.
What is the "actual reason" behind the existence of Thom class? It seems strange that such an interesting class would exis... | https://mathoverflow.net/users/26250 | Intuition behind Thom class | It is easy to understand the existence of a Thom class by considering cellular cohomology. Let the given vector bundle be $E\to B$ with fibers of dimension $n$. One can assume without significant loss of generality that $B$ is a CW complex with a single 0-cell. The Thom space $T(E)$ is the quotient $D(E)/S(E)$ of the u... | 55 | https://mathoverflow.net/users/23571 | 107864 | 62,196 |
https://mathoverflow.net/questions/106888 | 7 | From every [projective plane](http://en.wikipedia.org/wiki/Projective_plane) a coordinitisation can be constructed on a [planar ternary ring](http://en.wikipedia.org/wiki/Planar_ternary_ring), and conversely from every planar ternary ring a projective plane can be constructed. (For background see [Weibel's survey of no... | https://mathoverflow.net/users/26375 | When does a planar ternary ring uniquely coordinitise a projective plane? | The answer to the problem is well known; since :
all the PTRs coordinatizing a given projective plane are isomorphic iff it is a moufang plane iff all these PTRS are isomorphic alternative division rings. <http://www.math.uni-kiel.de/geometrie/klein/math/geometry/moufang.html>
However the problem to give a purely ... | 4 | https://mathoverflow.net/users/26722 | 107869 | 62,200 |
https://mathoverflow.net/questions/107874 | 7 | It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega\_\ast^U(X)$, is a coherent module over $\Omega\_\ast^U$ if $\Omega\_\ast^U(X)$ has projective dimension 0 or 1 over $\Omega\_\ast^U$. It is stated that in a series of le... | https://mathoverflow.net/users/11546 | Coherent MU_*-Modules | I believe that the result holds quite generally. The specific case of complex bordism is discussed in the following two papers:
* Larry Smith - On the finite generation of $\Omega\_\ast^U(X)$ (*J. Math. Mech.*, 1969)
* Pierre Conner & Larry Smith - On the complex bordism of finite complexes (*Publications Mathématiqu... | 8 | https://mathoverflow.net/users/1148 | 107880 | 62,203 |
https://mathoverflow.net/questions/107872 | 3 | I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, but for sub-populations of that population. I am trying to better understand the constraints that you place on your choice... | https://mathoverflow.net/users/26723 | When can you describe a population and its component subpopulations with the same parametric family of distributions? | There are so many possible answers and the subject is so large!
When the parametric family of distributions is some exponential family, there is this recent and very elegant paper:
* [Consistency under Sampling of Exponential Random Graph Models](http://arxiv.org/abs/1111.3054)
Although it is focused on studying ex... | 1 | https://mathoverflow.net/users/25326 | 107882 | 62,204 |
https://mathoverflow.net/questions/107875 | 3 | I know of the famous results on the Erdős-Szekeres empty convex polygon problem in the plane
(the [Happy-Ending Problem](http://en.wikipedia.org/wiki/Happy_Ending_problem)), and I know that there are higher-dimensional extensions.
A great source (albeit a decade out of date) is:
* Morris, W.; Soltan, V. (2000), "The... | https://mathoverflow.net/users/6094 | Empty convex polytopes for random point sets | Yes, the $2$-dimensional lower bound $\Omega( \frac{\log n}{\log\log n})$ implies a lower bound of that form in all higher dimensions by projecting to a plane.
If a set of points is not in convex position, then some convex combination of them equals another point in the set, and this is true for their projections. S... | 2 | https://mathoverflow.net/users/2954 | 107885 | 62,206 |
https://mathoverflow.net/questions/107879 | 8 | On a unit torus $T^n$ (or equivalently, on $\mathbb{R}^n$ with periodic boundary conditions), the linear Helmholtz equation:
$\nabla^2 \phi + k^2 \phi=0$
will have no non-trivial solutions for generic values of $k$, while for special values of $k$ it will have a finite-dimensional vector space of solutions, with a ... | https://mathoverflow.net/users/23829 | Space of solutions of nonlinear Helmholtz equation on a torus | Ignoring boundary conditions, the PDE has solutions $\phi = a\; \text{sn}\left(b x, c\right)$ where $\text{sn}$ is the Jacobi SN function (in Maple's parametrization: note that Mathematics uses a different convention), ${a}^{2}=2\;{\dfrac {{b}^{2}-{k}^{2}}{\lambda}}$, ${c}^{2}={\dfrac {{k}^
{2}}{{b}^{2}}}-1$. These are... | 6 | https://mathoverflow.net/users/13650 | 107889 | 62,207 |
https://mathoverflow.net/questions/107877 | 16 | Suppose I have an infinite discrete topological space $X$ of cardinality $\kappa$. Then I know some things about the Stone-Cech compactification, $\beta X$: it is Hausdorff and compact but not sequentially compact, has a basis of clopen sets, etc. My question is the following: is there a "nice" characterization of the ... | https://mathoverflow.net/users/8133 | Characterization of Stone-Cech compactifications | I confirme my comment :
$X$ is the stone-cech compactification of a discrete space if and only if $X$ is compact, haussdorf, extremally disconected, and has a dense set of open points.
here is a sketches of the proof :
If X is a stone-chech compactification of a discret set Y, then it is clear that X is compact, ... | 13 | https://mathoverflow.net/users/22131 | 107890 | 62,208 |
https://mathoverflow.net/questions/87257 | 13 | As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^\*(X)$; this is completely true for $\mathbb{Q}$-coefficients, and true up to bounded denominators for $\mathbb{Z}$-coefficients.
My question is: is there ... | https://mathoverflow.net/users/2191 | Motivic cohomology vs. K-theory for singular varieties | The precise relationship between K-theory and motivic cohomology for smooth schemes is the (analog of the) Atiya-Hirzebruch spectral sequence. This generalizes to non-smooth schemes if one uses K'-theory and higher Chow groups.
It is easy to see that motivic cohomology cannot be used to recover K-theory,
because it ... | 9 | https://mathoverflow.net/users/26735 | 107916 | 62,216 |
https://mathoverflow.net/questions/107891 | 1 | Is there any ways to compute the eigen vector without computing explicitly the associated eigenvalue?
Actually, I'd like to compute the largest eigenvalue of a positive matrix from its eigen vector, so I have to know its eigenvector first.
| https://mathoverflow.net/users/25747 | can eigenvector be found without computing the eigenvalue | If you pick a random vector $v$ and look at $v\_n=A^n v/\| A^n v\|,$ that will converge to the dominant eigenvector.
| 2 | https://mathoverflow.net/users/11142 | 107918 | 62,218 |
https://mathoverflow.net/questions/107913 | 2 | Let $F$ be an ultrafilter on some set $X$, $R$ an integral domain and $R\_F$ the resulting ultraproduct ring. For an element $(a)$ in the product ring of $R$ indexed by $X$, denote its equivalence class in the ultraproduct by $(a)\_F$.
It is a known result that if $X = \mathbb{N}$, and $R\_F$ satisfies the ascending ... | https://mathoverflow.net/users/26734 | Whether the result that an ultraproduct which satisfies ACCP is automatically a field generalizes to ultrafilters on larger indexing sets | The result you want is true for ultrapowers on an arbitrary set, provided that the ultrafilter is not $\sigma$-complete. This includes all nonprincipal ultrafilters on any set, unless there is a measurable cardinal, and in any case includes all nonprincipal ultrafilters on any set of size less than the least measurable... | 4 | https://mathoverflow.net/users/1946 | 107927 | 62,223 |
https://mathoverflow.net/questions/107902 | 5 | Can you show any finite subgroup of $SL\_2(R)$ is cyclic without using an invariant form?
| https://mathoverflow.net/users/25762 | Finite Subgroups of $SL_2(R)$ | Here's one approach with no geometry/integration, using finite groups. First prove it for a (nontrivial) finite $p$-group. If $p$ is odd, it contains a central element of order $p$, whose centralizer is $\mathbf{C}$-diagonalizable, hence embeddable as a $p$-subgroup of the nonzero complexes; this has to be cyclic. For ... | 6 | https://mathoverflow.net/users/14094 | 107935 | 62,226 |
https://mathoverflow.net/questions/107938 | 18 | A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) sources without proof suggesting this is an *if and only if* relationship, i.e. RH$\Longleftrightarrow\zeta'(s)\neq 0$. Howeve... | https://mathoverflow.net/users/20550 | A question about Speiser's 1934 result on the Riemann hypothesis | Yes, Speiser's theorem is an if and only if.
See Theorem 1 and "Corollary to Theorem 1" in Levinson and Montgomery's [Zeros of the derivatives of the Riemann Zeta-function](https://doi.org/10.1007/BF02392141). Acta Math. 133 (1974), 49–65.
---
Edit: An English language explication of Speiser's proof can be foun... | 24 | https://mathoverflow.net/users/630 | 107941 | 62,230 |
https://mathoverflow.net/questions/107905 | 3 | Let $\mathcal{F}\_i^j$ be a collection of sigma-algebras for
$i \in \mathbb{N}$ and
$j \in \{1,\ldots,n\}$ such that $\mathcal{F}\_{i+1}^j \subseteq \mathcal{F}\_i^j$ for all $i,j$. I would like to show that
$$\sigma\left(\cup\_j\cap\_i\mathcal{F}\_i^j\right) = \cap\_i\sigma\left(\cup\_j\mathcal{F}\_i^j\right)$$
... | https://mathoverflow.net/users/23661 | Finite union of tail sigma-algebras | $\newcommand{\cF}{\mathcal{F}}$
What you want to show is false. Let $x^j\_i$ and be a set of random variables, each taking values in $\{0,1\}$, where $j=1,2$ and $i=1,2,\ldots$. Let $\cF^j\_i=\sigma(x^j\_i,x^j\_{i+1},\ldots)$.
Consider the event $A=$ "$x^1\_i=x^2\_i$ for all but finitely many $i$".
This event clear... | 3 | https://mathoverflow.net/users/1061 | 107952 | 62,234 |
https://mathoverflow.net/questions/107782 | 5 | Suppose $\mathcal{A}$ is a unital associative algebra over $\mathbb{R}$. If we identify $\mathcal{A} = \mathbb{R}^n$ then the $\mathcal{A}$ multiplication corresponds to particular linear maps on $\mathbb{R}^n$. Of course any linear map on $\mathbb{R}^n$ corresponds uniquely to its standard matrix hence we obtain a cor... | https://mathoverflow.net/users/24854 | reference for list of left-regular representations of real associative algebras | Let $A$ be an algebra of dimension $d$ over an alg. closed field $k$, and $r\subseteq A$ its radical.
* If $d=1$, then of course $A\cong k$.
* If $d=2$, then either $\dim r=0$ and then $A\cong k^2$ because of Wedderburn's theorem, or $\dim r=1$. In the latter case, we must have $r^2=0$, so the ordinary quiver $Q$ of ... | 2 | https://mathoverflow.net/users/1409 | 107953 | 62,235 |
https://mathoverflow.net/questions/107933 | 8 | A question very close to this one was already asked: [Automorphisms of a weighted projective space](https://mathoverflow.net/questions/67363/automorphisms-of-a-weighted-projective-space)
But the answer given does not satisfy my needs. So avoiding having two questions that are identical, I am interested in a specific ... | https://mathoverflow.net/users/10898 | Automorphisms of a specific type of weighted projective space | I'm not sure how one deals with the general case, but your example is the cone in $\mathbb P^{N+1}$ over the $k$-th Veronese embedding of $\mathbb P^{n-1}(\mathbb K)$ in $\mathbb P^N$ ($n=$ number of 1's, $N=$ the appropriate dimension for the Veronese embedding).
So for $k>1$ the automorphism group $G$ is an extens... | 10 | https://mathoverflow.net/users/10610 | 107961 | 62,240 |
https://mathoverflow.net/questions/107945 | 41 | I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
* [Nelson (1987). Radically Elementary Probability Theory](https://web.math.princeton.edu/~nelson/books/rept.pdf)
* [Geyer (2007). Radically Elementary Probability and Statisti... | https://mathoverflow.net/users/25326 | Nonstandard analysis in probability theory | Non-standard analysis has been quite successful in settling existence questions in probability theory. [Hyperfinite Loeb spaces](http://www.jstor.org/stable/1997222) allow for several constructions that cannot be done on standard probability spaces. In particular, NSA was quite useful for the construction of certain ad... | 31 | https://mathoverflow.net/users/35357 | 107966 | 62,244 |
https://mathoverflow.net/questions/107989 | 4 | After a small search that I did I was unable to spot any answers here.What I am trying is to prove why the $2^{\sqrt2}$ is transcendental number. I know that this probably is a closed problem and probably many people have proved it already,but I want to reach the answer by myself after doing a research on this problem ... | https://mathoverflow.net/users/26755 | Guidelines to prove that $2^{\sqrt{2}}$ is a transcendental number? | By the [Gelfond–Schneider theorem](http://en.wikipedia.org/wiki/Gelfond%27s_theorem), $2^{\sqrt2}$ is transcendental.
$2^{\sqrt 2}$ is called the [Gelfond–Schneider constant](https://secure.wikimedia.org/wikipedia/en/wiki/Gelfond%E2%80%93Schneider_constant).
See also <https://math.stackexchange.com/questions/173804... | 9 | https://mathoverflow.net/users/532 | 107991 | 62,258 |
https://mathoverflow.net/questions/107993 | 3 | Generally, Quadratic Programming solves the problem
$$\text{Given }Q, c, A, b,\text{ choose }x \text{ to maximize } x^TQx + c^Tx \text{ subject to } Ax \le b$$
In this form, Quadratic Programming is NP-hard. For my purposes, I happen to know that $b$ and $c$ are $0$ and $Q$ is diagonal. Thus, the problem looks like... | https://mathoverflow.net/users/21816 | Does Quadratic Programming get easier when it's described by a diagonal matrix? | The problem is still NP-hard. The proof is by reduction from checking matrix copositivity, [which is co-NP-complete](http://www-personal.umich.edu/~murty/np.pdf). A symmetric matrix $Q$ is said to be copositive if $x^TQx\geq 0$ for all $x\geq 0$, so it is a weaker condition than positive semidefiniteness.
For a symme... | 2 | https://mathoverflow.net/users/5963 | 107994 | 62,259 |
https://mathoverflow.net/questions/107980 | 3 | Let for given real sequences $(a\_n)\_{n \in \mathbb Z}, (b\_n)\_{n \in \mathbb Z}$,
$c\_n:=\sum\_{k\in \mathbb Z} a\_k b\_{n-k}$ for $n \in \mathbb Z$ be the convolution of sequences $(a\_n)$, $(b\_n)$.
For classical convolution if one of two functions is in $L^p$, the second in $L^q$, where $1\leq p,q <\infty$ the... | https://mathoverflow.net/users/26382 | Convolution of sequences | Yes. True not only for $\mathbb Z$ but for for abelian (more generally unimodular) locally compact group. (20.18) in Hewitt & Ross, *Abstract Harmonic Analysis*.
| 5 | https://mathoverflow.net/users/454 | 107995 | 62,260 |
https://mathoverflow.net/questions/107887 | 10 | Hi, I'm looking for conditions on $G(t,x)$ such that
$$
\sup\limits\_{t\in [0,1]}E[G(t,X)]=E[\sup\limits\_{t\in [0,1]}G(t,X)]
$$
where $X$ is a random variable (it's easy to see that $\sup\limits\_{t\in [0,1]}E[G(t,X)]\leq E[\sup\limits\_{t\in [0,1]}G(t,X)]$).
Any suggestion or reference is greatly appreciated!
| https://mathoverflow.net/users/26726 | When do maximum and expectation commute? | This will require very strong conditions on $G$. The most general result I know of is an "almost upward-filtering" condition: Assume $G(t,\cdot)$ is measurable for each $t \in [0,1]$, and $\sup\_{t \in [0,1]}\mathbb{E}[G(t,X)] < \infty$; then
$$
\sup\_{t \in [0,1]}\mathbb{E}[G(t,X)] = \mathbb{E}[\text{ess}\sup\_{t \in ... | 14 | https://mathoverflow.net/users/26459 | 108007 | 62,262 |
https://mathoverflow.net/questions/108011 | 2 | Say I have a countably infinite number of iid random vectors $X\_i:i\in\mathbb{N}$, each uniformly distributed on $[0,1]^k$ with say, $k=2$.
I need to evaluate stuff like:
$E\_{X\_0,X\_1,\ldots}[\int\_{[0,1]^k}\inf\_{i\in\mathbb{N}}\|X\_i-y\|^2dy]$,
which I "believe" is equal to $0$ (this is indeed a random vecto... | https://mathoverflow.net/users/26763 | Countably many random vectors and related problems. | 1. Yes, you can define properly the first expectation,
see e.g. [PlanetMath](https://planetmath.org/infiniteproductmeasure)
2. Then you have with your notations
$$
\mathbb E\_{X\_i : i\in\mathbb N}\int\_{[0,1]^k}\inf\_{i\in\mathbb N}\|X\_i-y\|^2dy\leq \mathbb E\_{X\_i : i\in\mathbb N}\int\_{[0,1]^k}\min\_{1\leq i \leq... | 2 | https://mathoverflow.net/users/15517 | 108015 | 62,267 |
https://mathoverflow.net/questions/97261 | 18 | The [Rudin-Shapiro sequence](http://en.wikipedia.org/wiki/Rudin%E2%80%93Shapiro_sequence) (also known as the Golay-Rudin-Shapiro sequence) is defined as follows.
Let $a\_n = \sum \epsilon\_i\epsilon\_{i+1}$ where $\epsilon\_1,\epsilon\_2,\dots$ are the digits in the binary expansion of $n$.
$WS(n)$, the $n$th term of... | https://mathoverflow.net/users/1532 | Möbius Randomness of the Rudin-Shapiro Sequence | OK, it seems that a probabilistic swapping argument works (and is simpler than the two other suggestions I made above).
Firstly, by the Bourgain-Sarnak-Ziegler criterion (Proposition 1 in <http://terrytao.wordpress.com/2011/11/21/the-bourgain-sarnak-ziegler-orthogonality-criterion/> ), it suffices to show that
$$ \... | 18 | https://mathoverflow.net/users/766 | 108016 | 62,268 |
https://mathoverflow.net/questions/107998 | 4 | Let $f$ be a normalized newform of weight $k\geq 2$, $p$ a prime, and $V$ the associated Galois representation (with coefficients in a finite extension $K$ of $\mathbf{Q}\_p$ with ring of integers $\mathscr{O}$). Assume the residual representation attached to $V$ is absolutely irreducible, so that the choice of a $G\_\... | https://mathoverflow.net/users/4351 | Duals and Tate twists of Galois representations attached to modular forms. | There are several ways to normalize the Galois representation attached to a modular eigenform,
which differ by a Tate twist, but when you choose one normalization, no Tate twist of a
Galois representation attached to a modular form can ever be attached to another modular form.
One way to see this is the following: the... | 4 | https://mathoverflow.net/users/9317 | 108018 | 62,270 |
https://mathoverflow.net/questions/108031 | 6 | I'm reading AMS's book *Papers on Topology*, which collects Poincare's papers on topology.
However, the first paper stops me.
In the paper, he considered the group generated by transformations in $\mathbb{R}^3$, the generators are:
$(x,y,z)\rightarrow (x+1,y,z)$
$(x,y,z)\rightarrow (x,y+1,z)$
$(x,y,z)\rightar... | https://mathoverflow.net/users/18717 | Why additional constraint is need for this two groups to be isomorphic? | Poincare was correct. These are Abelian ($\mathbb{Z}^2$)-by-cyclic groups where the cyclic group acts on $\mathbb{Z}^2$ by the matrix $\left(\begin{array}{ll}\alpha & \gamma \\\ \beta & \delta\end{array}\right)$. These groups are isomorphic if and only if the matrices are conjugate in $SL(2,\mathbb{Z})$.
Here is mor... | 8 | https://mathoverflow.net/users/nan | 108034 | 62,275 |
https://mathoverflow.net/questions/108033 | 15 | Suppose $(M,g)$ is a compact Riemannian manifold with totally geodesic boundary. Its *double* is given by
$$
D(M) = M\cup\_fM
$$
where $f:\partial M\to\partial M$ is the identity map. In general $D(M)$ is not a $C^\infty$ manifold. Is $D(M)$ a $C^2$ manifold?
| https://mathoverflow.net/users/14579 | Double a manifold with boundary | It *is* a $C^\infty$ manifold if you define charts properly (e.g. using geodesics normal to the boundary as coordinate lines).
The *metric* of the double is $C^2$ (but not always $C^3$). Indeed, since the boundary is totally geodesic, the normal derivative of the metric tensor (in the above mentioned coordinates) van... | 21 | https://mathoverflow.net/users/4354 | 108039 | 62,279 |
https://mathoverflow.net/questions/108046 | 15 | I'm interested in publishing parts of my PhD thesis in advance and I'm wondering wether or not this will result in problems later on. One of the problems I'm thinking of is that usually the copyright is transferred to the journal/publisher but at our university you are required to publish your thesis online via the lib... | https://mathoverflow.net/users/26774 | Is it common practice to publish parts of a PhD thesis in advance? | I think a lot of journals will expressly allow this in their copyright policy. I just poked around, and I see for example that the AMS allows this:
[AMS copyright policies](http://www.ams.org/publications/authors/ctp)
and even Elsevier allows this in its journals:
[Elsevier copyright policies](http://www.elsevier... | 12 | https://mathoverflow.net/users/23408 | 108051 | 62,283 |
https://mathoverflow.net/questions/108042 | 6 | [VIZING’S CONJECTURE: A SURVEY AND RECENT RESULTS (2009) by Bostjan Bresar , Paul Dorbec , Wayne Goddard , Bert L. Hartnell , Michael A. Henning , Sandi Klavzar , Douglas F. Rall](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.159.7029) p.25:
>
> Conjecture 9.6. For all graphs $G$ and $H$,
> $$\gamma(G \s... | https://mathoverflow.net/users/12481 | Is this a counterexample to a conjecture about independent domination in cartesian graph products? | To summarize a bit:
* $\gamma(G)$ is defined as the usual [domination number](http://mathworld.wolfram.com/DominatingSet.html%20) of a graph $G$
* $i(G)$ is defined as the smallest cardinality of a dominating set that is also an independent set
Your graph $C$ is the disjoint union of $K\_{3,3}$ and $K\_1.$
Clearl... | 4 | https://mathoverflow.net/users/1737 | 108052 | 62,284 |
https://mathoverflow.net/questions/108053 | 2 | Methods that are usually adopted for time integration in transport phenomena problems are either:
Euler (explicit, first-order accurate)
$\frac{dY}{dt}=f(t,Y)$
$Y^{n+1}=Y^n+\Delta t f(t,Y^n)$
Backward (implicit, first-order accurate)
$Y^{n+1}=Y^n+\Delta t f(t+\Delta t,Y^{n+1})$
or Crank-Nicholson (implicit,... | https://mathoverflow.net/users/26776 | Gauss Legendre Method for Implicit Integration | 1. If I understand your first question, the answer is "no". $ k\_1 $ and $ k\_2 $ are defined implicitly in terms of each other, so it is not easy to write a simpler one-line expression. You could write \begin{align\*} k\_1 &= \Delta t f( t + (1/2 - \sqrt{3}/6) \Delta t, (1/2) Y^{n+1} + (1/2)Y^n - (\sqrt{3}/6) k\_2) \\... | 1 | https://mathoverflow.net/users/17113 | 108059 | 62,287 |
https://mathoverflow.net/questions/108048 | 2 | Are there conditions on the finite groups $A$, $B$, $C$ or the morphisms $A \leftarrow C\rightarrow B$ that restrict the number of infinite conjugacy classes in the amalgamated product $A \overset{C}{\*}B$?
| https://mathoverflow.net/users/23506 | Conjugacy classes of elements in an amalgam of finite groups | If your morphisms $A \leftarrow C \rightarrow B$ are injective (which I'm assuming you want, since you say amalgamated product, not pushout), then the group $A\ast\_C B$ is a virtually free group. In particular, there will be infinitely many conjugacy classes of infinite elements if the group is not finite or virtually... | 4 | https://mathoverflow.net/users/1345 | 108061 | 62,288 |
https://mathoverflow.net/questions/108044 | 4 | Let $G(N)$ be the congruence subgroup
$\big\{ \begin{pmatrix} a&b \\ c&d \end{pmatrix} \in SL\_2(\mathbb{Z}) \ \ | \ \ a \equiv d \mod N \textrm{ and } b \equiv c\equiv 0 \mod N \big\}$.
$G(N) \backslash \mathbb{H}$ seems to parametrize elliptic curves with a cyclic subgroup and some extra data (I think its automo... | https://mathoverflow.net/users/16858 | What does this quotient of the upper half plane parametrize? | Here's a very clunky answer:
For a particular elliptic curve $E$, the collection of bases $\{P,Q\}$ of the $N$-torsion $E[N]$ is acted upon (diagonally) by the group $(\mathbb{Z}/N\mathbb{Z})^\times$. This group has a subgroup $G$ consisting of elements $x$ with a lifting to an element $y\in (\mathbb{Z}/N^2\mathbb{Z}... | 3 | https://mathoverflow.net/users/12107 | 108064 | 62,290 |
https://mathoverflow.net/questions/108045 | 11 | There is a strange product that takes two square roots of unit matrix, say $A$ and $B$, $A^2=I$, $B^2=I$ to a square root again,
$$ A\star B=(A+B)^{-1}(A-B+2I), \qquad (A\star B)^2=I$$
Could anybody help me with identifying this structure? Where it comes from?
It was obtained from the Caley transform $C(A)=(1-A)^{-1... | https://mathoverflow.net/users/2052 | A product on the square roots of unit matrix | Let me denote $E\_\pm(M)$ the eigenspaces of $M$ associated with the eigenvalues $\pm1$. Let me assume that the characteristic of the scalar field $k$ is not $2$. Then your assumption is that
$$k^n=E\_+(A)\oplus E\_-(A)=E\_+(B)\oplus E\_-(B).$$
In addition, the assumption that $A+B$ is non-singular means $E\_+(A)\cap ... | 8 | https://mathoverflow.net/users/8799 | 108065 | 62,291 |
https://mathoverflow.net/questions/108058 | 5 | Let $A\subset \mathbb{R}^n$ be a (measurable) bounded set, and consider the following optimization problem: minimize $P(X)$, the perimeter of a set $X$, where $X$ ranges over all Caccioppoli subsets of $\mathbb{R}^n$ such that $X\supset A$.
If $A$ is convex, then it's "obvious" that the minimizer is $A$ itself (up to... | https://mathoverflow.net/users/8794 | Minimizing the perimeter around an obstacle | For any sensible definition of perimeter, an adaptation of following argument proves the claim. Consider the nearest-point projection map $p\_A:\mathbb R^n\to A$ (that is, for every $x\in\mathbb R^n$, let $p\_A(x)$ be the point of $A$ nearest to $x$). It is easy to see that $p\_A$ is Lipschitz-1, i.e., $|p\_A(x)-p\_A(y... | 8 | https://mathoverflow.net/users/4354 | 108079 | 62,296 |
https://mathoverflow.net/questions/108063 | -2 | What would be the bounds on the derivative of a function using its inverse fourier transform representation. Furthermore what would be the bounds on the absolute value of the function itself?
| https://mathoverflow.net/users/26673 | derivative of a function of time using its inverse fourier transform | The standard estimates are $|f(x)|$ is at most the $L^1$ norm of the Fourier transform,
and $|f'(x)|$ is at most the first moment of the Fourier transform. Is this what you are asking about?
| 0 | https://mathoverflow.net/users/25510 | 108104 | 62,312 |
https://mathoverflow.net/questions/108085 | 1 | Imagine I place a turtle on some desired vertex, $v\_i$, of a bounded $d$-dimensional integer lattice, $Z^d$, with dimensions $(l\_1, ..., l\_d)$. The turtle is able to travel from vertex to vertex along the edges of the lattice, but is forbidden from ever returning to a previously visited vertex (i.e. it "burns" the v... | https://mathoverflow.net/users/26528 | Covering a $d$-dimensional integer lattice by repeating a minimal set of deterministic moves | Yes your example is optimal provided that the longest side is longer than 2. Let me stick to dimension 2, the higher dimensions are similar. I assume the vertices have integer coordinates $(i,j)$, $1\le i\le M$, $1\le j\le N$.
The turtle changes its position by a fixed vector $v$ after each cycle of execution. Let $X... | 4 | https://mathoverflow.net/users/4354 | 108116 | 62,319 |
https://mathoverflow.net/questions/107973 | 5 | Does anyone know if there exists some natural way to interpret the Kazhdan-Lusztig C-basis in a categorification of the Hecke algebra ? The category of Soergel bimodules categorifies the C'-basis but it doesn't seem clear to me whether this or some other related structure (Rouquier complexes, sheaves...) could be adapt... | https://mathoverflow.net/users/26751 | Kazhdan-Lusztig C-basis and categorification | If you identify the Hecke algebra with the Grothendieck principal block of a graded lift of category $\mathcal{O}$ for the corresponding semi-simple Lie algebra (oddly, enough which of the two Langlands dual options you pick doesn't matter) such that $T\_y$ is the class of a Verma module with highest weight $-y\rho-\rh... | 5 | https://mathoverflow.net/users/66 | 108126 | 62,324 |
https://mathoverflow.net/questions/82786 | 15 | Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence
$$0 \to \mathcal{K}\_n(X) \to \bigoplus\_{x \in X^{(0)}} K\_n(k(x)) \to \bigoplus\_{x \in X^{(1)}} K\_{n - 1}(k(x))$$
where $\mathcal{K}\_n$ is the Zariski sheaf associated to $K\_n$?
... | https://mathoverflow.net/users/12914 | State of the art for Gersten's conjecture for K-theory? | Gersten's conjecture is known for regular local rings containing a field (Panin extended the result of Quillen for smooth schemes). In mixed characteristic, it is known that the statement for a discrete valuation ring implies the statement for smooth local rings over a discrete valuation ring (by work of Gillet-Levine)... | 12 | https://mathoverflow.net/users/26735 | 108134 | 62,328 |
https://mathoverflow.net/questions/107944 | 11 | Fix $n>4$. Is there a characterization of the set $T\_n$ of all natural numbers $t$ such that there is some graph on $n$ vertices with exactly $t$ distinct triangles? For example, it's clear that {$1,2,\ldots,n$} $\subseteq T\_n$ and $\binom{n}{3}-1 \notin T\_n$; what else can we say?
| https://mathoverflow.net/users/7967 | Is it possible to have t triangles in some graph on n vertices? | I will argue that the following are true:
* For any $\epsilon \gt 0$ and all large enough $n$, $T\_n$ contains all integers less than $\binom{n}{3}-\epsilon n^2.$
* For any fixed positive integer $c$ there is a finite list of expressions which accounts for all numbers larger than $\binom{n}{3}-cn-n/2$ provided that ... | 3 | https://mathoverflow.net/users/8008 | 108136 | 62,330 |
https://mathoverflow.net/questions/108128 | 1 | when investigating ZFC as a formal language a structure is a set, are we not engaging in circular logic here? Or is 'set' thought of in a more primitive sense?
| https://mathoverflow.net/users/22002 | How do we avoid circularity when we build a structure for ZFC? | From the point of view of mathematics, we are not in the circle. We are using the language of set theory for expressing our definitions, theorems and proofs, because we know how good and suitable this language is (although probably there exist also other possibilities). In order to be mathematically correct, we have fo... | 6 | https://mathoverflow.net/users/26799 | 108141 | 62,332 |
https://mathoverflow.net/questions/20251 | 29 | Suppose I have a sheaf $\mathcal{F}$ on the (small) étale site over $X$. By restriction, $\mathcal{F}$ is also a sheaf on $X$ (with the Zariski topology). When is it that the sheaf cohomologies (i.e. derived functors of the global section functors) agree in these two sites?
For example, in SGA 4 (Chapter VII, p355), ... | https://mathoverflow.net/users/362 | Cohomology of sheaves in different Grothendieck topologies | As you pointed out, your question (for general sheaves of abelian groups)
follows from the vanishing of the derived functors $R^if\_\*{\mathcal F}$ for $i>0$,
and that's all one can say.
A good example is the Beilinson-Lichtenbaum conjectures, which states that for a certain
complex of sheaves $\mathbb Z(n)$, the et... | 19 | https://mathoverflow.net/users/26735 | 108150 | 62,335 |
https://mathoverflow.net/questions/108123 | 6 | In short, I'm curious to know what modes of degeneration of metric might still keep the curvature bounded. More precisely, assume we are keeping the total volume of the manifold fixed and deform the metric, e.g by a conformal factor that preserves the volume. If we allow the metric to degenerate, namely allow it to bec... | https://mathoverflow.net/users/12019 | Degeneration of riemannian metrics with curvature bounds | The seminal work on how a Riemannian manifold can degenerate with pointwise bounds on curvature are Cheeger and Gromov's JDG papers, "Collapsing Riemannian manifolds while keeping their curvature bounded" as well as papers by Fukaya, which can be found in the references of the Cheeger-Fukaya-Gromov JAMS paper, "Nilpote... | 8 | https://mathoverflow.net/users/613 | 108159 | 62,338 |
https://mathoverflow.net/questions/108087 | 5 | Given some algebraic closed curve in the affine space $\mathbb{A}^3\_\mathbb{C}$, is there a way to decide whether its ideal (polynomials in $\mathbb{C}[X,Y,Z]$ vanishing on the curve) is generated by two elements?
I am mostly interested in the case where the curve is smooth and irreducible.
| https://mathoverflow.net/users/23758 | Ideals of affine space curves | There is a necessary condition: the element in the divisor class group of the affine curve
coming from the cotangent sheaf of the curve must be trivial. If $I$ is generated by two elements, then $I/I^2$ is a free sheaf of rank $2$ on the curve. By adjunction, this forces the cotangent sheaf of the curve to be trivial. ... | 7 | https://mathoverflow.net/users/13265 | 108162 | 62,339 |
https://mathoverflow.net/questions/108139 | 3 | Let's consider a Probabilistic Cellular Automaton on a one dimensional lattice $S$. Each site of the lattice can have two states, $0$ and $1$. The transition probability acting on each site is: $P(x\_i=1 | x\_{i-1}, x\_i, x\_{i+1}) = 1$ when $x\_{i-1}= x\_i = x\_{i+1} = 1$ and it is $P(x\_i=1 | x\_{i-1}, x\_i, x\_{i+1}... | https://mathoverflow.net/users/26798 | Ergodicity for a Probabilistic Cellular Automaton on a finite space | Define a random variable $Y\in \{0,1\}^N$ by $P(y\_i^{(n)}=1) = \epsilon$ for all $1\leq i\leq N$ and $n\in\mathbb{N}$ and observe that $x\_i^{(n)} \geq y\_i^{(n)}$ for all $i,n$ (as long as $X,Y$ are being driven by the same random process). With probability 1 there exists a time $n$ such that $y\_i^{(n)}=1$ for all $... | 5 | https://mathoverflow.net/users/5701 | 108163 | 62,340 |
https://mathoverflow.net/questions/108138 | 6 | [Russell's paradox](http://en.wikipedia.org/wiki/Russell%2527s_paradox) showed that naive set theory leads to a contradiction. This was something that was taken seriously and caused a lot of work.
Now, [Banach–Tarski paradox](http://en.wikipedia.org/wiki/Banach-Tarski_paradox) is arises from a result that a ball can ... | https://mathoverflow.net/users/nan | How to tell a paradox from a "paradox"? | Many paradoxes are first expressed in a semi-formal way, for example "the least number not describable by fewer than eleven words". They are warning signs that lead us to further analysis and can be resolved in different ways:
1. We can just get used to a "paradox" and accept it as "truth", e.g., there are infinite s... | 11 | https://mathoverflow.net/users/1176 | 108166 | 62,342 |
https://mathoverflow.net/questions/108090 | 1 | Let $A$ be a symmetric row-column increasing $n \times n$ matrix (i.e. $A(i, j) < A(i+1,j)$ and $A(i, j) < A(i, j+1)$) with integer entries $A(i, j) \in \{1, 2, ..., n^2\}$. Moreover, let us assume that $A$ contains $\Theta(n^2)$ distinct entries, so the set of entries has positive density.
Define a "row discrepancy... | https://mathoverflow.net/users/25905 | Is a "row discrepancy" of symmetric row-column increasing matrices unbounded? | This is my updated answer, probably still wrong somewhere...
If $A(i,j)=\lfloor (i+j)n/10\rfloor $, then the matrix is symmetric and satisfies strict inequalities if $n$ is at least $10$.
Also $A(a\_2,j)-A(a\_1,j)=(a\_2-a\_1)n/10$ (plus/minus 1, if you want integers and round the $A(i,j)$'s) and since $b\_2-b\_1\le... | 1 | https://mathoverflow.net/users/955 | 108168 | 62,343 |
https://mathoverflow.net/questions/107996 | 7 | What are some examples of finitely generated (finitely presented) elementary amenable groups which are not virtually solvable?
| https://mathoverflow.net/users/7307 | Examples of finitely generated elementary amenable groups which are not virtually solvable | Houghton groups are (non-split) extensions of the group of the finitely supported permutations of the integers by $\mathbf{Z}^d$ for $d\ge 2$ and are thus elementary amenable and not virtually solvable. They're finitely presented, as shown by K.Brown [here](http://math.cornell.edu/~kbrown/scan/1987.0044.0045.pdf) (*Fin... | 7 | https://mathoverflow.net/users/14094 | 108173 | 62,344 |
https://mathoverflow.net/questions/108160 | 0 | How to determine the number of $i$'s, as fast as possible, such that $1\le i \le L$ and $((a\*i+b)\mod p) \mod k = l$, where $1\lt a,b\lt p-1, p$ is a prime number, and $l \lt k \lt L \lt p$.
This problem seems to be too complex, let me begin with a simple one:
given two different primes $p\_1, p\_2$, integer $L$ a... | https://mathoverflow.net/users/24143 | count number of i such that ( (p_2*i) mod p_2) == l | Until a better idea strikes me, why not give this one a go?
Reduce the problem to $i + B \pmod p$ by the following. Find $f$ so that $fa\equiv1 \pmod p$.
then transform the set $ai+b$ to $i +fb \pmod p$. You will also have to transform the set
of numbers in $[0,p)$ of the form $tk+l$ to $tfk+fl \pmod p$, but if this ... | 1 | https://mathoverflow.net/users/3493 | 108174 | 62,345 |
https://mathoverflow.net/questions/108149 | 5 | Suppose I have an annulus $U\subset \mathbb{C}$ and a single-valued holomorphic function $V:U\to \mathbb{C}$.
I would like to know if there are (tractable) conditions on $V$ that ensure that the second order linear ODE
$$
\partial\_{zz}^2 f+ V f=0
$$
admits a *single-valued* holomorphic solution $f: U\to \mathbb{C}$ ... | https://mathoverflow.net/users/26801 | Periodic Holomorphic ODE | When the annulus is $0<|z|<1$, and $V$ has at most a second order pole at $0$, the answer is known,
and is given by Fuchs theory. This is the so-called regular singularity at $0$. The reference is any
book on analytic theory of linear differential equations, for example,
Ince, Ordinary differential equations. In all ot... | 4 | https://mathoverflow.net/users/25510 | 108187 | 62,352 |
https://mathoverflow.net/questions/107863 | 13 | Recall that rooted trees may be generated by starting with a trivial rooted tree (just a vertex), along with the operations of grafting a number of trees (identify their roots) and adding a new vertex to the tree to be a new minimum element. We will call this second operation "leafing".
Now let us define an invariant... | https://mathoverflow.net/users/14167 | Is the following invariant of rooted trees a complete invariant? | Chaudhary and Gordon ("Tutte polynomials for trees," J. Graph Theory 15, no. 3 (1991), 317-331) construct a couple of invariants that look very similar to yours. They prove that these invariants do in fact determine a rooted tree up to isomorphism.
Update: I think the answer to your original question is no.
The rel... | 17 | https://mathoverflow.net/users/8604 | 108190 | 62,353 |
https://mathoverflow.net/questions/108197 | 6 | Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1\_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1\_2$ and it is the largest countable $\Sigma^1\_2$ set of reals if $\omega\_1 ^L < \omega\_1$.
If $\phi$ is $\Sigma^1\_4$ then $V\models \phi$ iff $\mathcal M\_2 \models \phi... | https://mathoverflow.net/users/3859 | Sets of reals and absoluteness | At the projective level, there are nice level by level generalizations, and looking at Steel's paper in the Handbook should give you the proof and the pre-requisites to understand it fully. This is what is behind the relation between determinacy and large cardinals. On the other hand, $\Sigma^2\_1$ is never going to be... | 10 | https://mathoverflow.net/users/6085 | 108200 | 62,358 |
https://mathoverflow.net/questions/108164 | 5 | I'm studying about Graph Ramsey Theory now. Starting this study, I'm reading Chvatal and Harary's series of papers. In the second paper (V.Chvatal, F.Harary, Generalized ramsey theory for graphs,Ⅲ. Small off-diagonal numbers, Pacific Journal of Mathematics 41, No.2, 1972, pp.335-345), I can't understand the proof of $r... | https://mathoverflow.net/users/25386 | Small Ramsey numbers and Brooks' Theorem | It seems as if the Chvatal, Harary proof has a logical gap, and your proof seems to be missing some details.
Here is a proof that is based on Brook's Theorem. We plagiarize you and start by noting that $r(C\_4,K\_3)=7$, and so $G$ has minimum degree at least 3. We then plagiarize Chvatal, Harary and note that $G$ ha... | 5 | https://mathoverflow.net/users/2233 | 108204 | 62,359 |
https://mathoverflow.net/questions/97361 | 18 | A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants ([Generalized Fourier Series](http://en.wikipedia.org/wiki/Generalized_fourier_series), [Green's Function](http://en.wikipedia.org/wiki/Green%27s_function), [Pontryagin duality](http://en.wikipedia.org/wiki/Pontr... | https://mathoverflow.net/users/12178 | Explaining Mukai-Fourier transforms physically | You can think of line bundles and skyscraper sheaves as sheaf-theoretic analogues to exponentials and delta functions, respectively. The Fourier-Mukai transform on an elliptic curve takes one type of sheaf to the other (with a homological shift that I will ignore). In higher dimension, you get some mixtures of these ty... | 14 | https://mathoverflow.net/users/121 | 108214 | 62,362 |
https://mathoverflow.net/questions/107298 | 14 | My post is motivated at least in part by this [MO question](https://mathoverflow.net/questions/104183/order-increasing-bijection-from-arbitrary-groups-to-cyclic-groups).
Has there been any work done on realizable order sequences for finite groups? By an "order sequence" I mean a non-decreasing list of the orders of ... | https://mathoverflow.net/users/22971 | Realizable Order Sequences for Finite Groups | [Here's a quick list](https://docs.google.com/open?id=0B3X5C_7tdfR2VDhTRW9xRndVMkU) for groups up to order 512. Format is group order, followed by a list of possible order sequences for groups of that order, then an empty space, then a list of the number of groups with each respective order sequence (in the order they ... | 9 | https://mathoverflow.net/users/25494 | 108219 | 62,365 |
https://mathoverflow.net/questions/108183 | 9 | Let $k$ be a field. It is well-known that $A\otimes\_{k}B$ is not necessarily Noetherian even if $k$-algebras $A$ and $B$ are Noetherian. For example $\mathbb{R}\otimes\_{\mathbb{Q}}\mathbb{R}$.
1. When is the tensor $A\otimes\_{k}B$ Noetherian for Noetherian "commutative" $k$-algebras $A$ and $B$?
2. What if $A$ is... | https://mathoverflow.net/users/50973 | Strongly Noetherian property. When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian rings $A$ and $B$? | You could try having a look at Yekutieli and Zhang's paper Homological Transcendence Degree (<http://arxiv.org/abs/math/04120130>). They call a $k$-algebra $A$ "doubly Noetherian" if $A \otimes\_k A^{op}$ is Noetherian, and "rationally Noetherian" if $A \otimes\_k U$ is Noetherian for every division ring $U$.
There's... | 8 | https://mathoverflow.net/users/22460 | 108226 | 62,368 |
https://mathoverflow.net/questions/108211 | 11 | What is the Brauer group of the moduli space of principally polarized abelian varieties of a given dimension? I am primarily interested in the "open" moduli space, i.e. not a compactification. The question has several levels of generality depending on how general is the ground field (ring even?) but I know nothing, so ... | https://mathoverflow.net/users/2290 | What is the Brauer group of the moduli space of (p.p.) abelian varieties? | **Edit.** The main idea below is incorrect. I explain the mistake below the original post.
I am posting as an answer instead of a comment, even though I think this might be wrong, because I could not format the weblinks properly in comments.
Since the (orbifold) moduli space is a quotient of the Siegel upper half s... | 7 | https://mathoverflow.net/users/13265 | 108231 | 62,371 |
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