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https://mathoverflow.net/questions/167447 | 10 | This question is not precise, but I believe has a precise formulation.
Consider a mathematical theorem which gives an equivalency between two conditions. As an extreme example:
>
> **Theorem.**
> A compact 3-manifold is simply-connected if and only if it is homeomorphic to the 3-sphere.
>
>
>
The if direct... | https://mathoverflow.net/users/1345 | Proof complexity of two directions of equivalency? | I think that there are numerous trivial examples of this.
Take any implication $p\to q$ that is provable, but has no short proof. It follows that the equivalence $$q\leftrightarrow (p\vee q)$$
is also provable, and furthermore has a trivial proof in the forward direction, but no very short proof in the converse dir... | 9 | https://mathoverflow.net/users/1946 | 167450 | 86,826 |
https://mathoverflow.net/questions/167449 | 5 | I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f\_n)\_{n\in\mathbb{N}}$ is a sequence of continuous real valued functions on $[0,1]$ and the sequence of partial sums $S\_m(t)=\sum\_{n=1}^m f\_n(t)$ converges in $\mathrm{L}^2(... | https://mathoverflow.net/users/50925 | Equicontinuity and $L^2$ convergence imply uniform convergence | Since $S\_n(t)$ converges in $L^2$ to a function $S(t)$, it is bounded at least at a point. Since it is equicontinuous, every subsequence, by Ascoli-Arzelà, has a sub-subsequence that converges uniformly. The limit is the same function $S(t)$, hence $S\_n$ itself converges uniformly.
| 7 | https://mathoverflow.net/users/6101 | 167452 | 86,828 |
https://mathoverflow.net/questions/161428 | 1 |
>
> All the [subfactors](http://en.wikipedia.org/wiki/Subfactor) here are **irreducible** inclusion of hyperfinite II$\_1$ factors.
>
>
>
A subfactor $(N \subset M)$ is **Homogeneous Single Chain** ($HSC$) if its lattice of intermediate subfactors is a single chain: $N=P\_0 \subset P\_1 \subset \dots \subset P\_... | https://mathoverflow.net/users/34538 | Existence of homogeneous single chain compositions of a given maximal subfactor? | **Yes** by free compositions:
**Theorem**: Let $N \subset M$ be an irreducible finite index subfactor with $P$ an intermediate subfactor ($N \subset P \subset M$) such that $N \subset M$ is a free composition of $N \subset P$ and $P \subset M$.
If $L$ is another intermediate subfactor $N \subset L \subset M$, then... | 0 | https://mathoverflow.net/users/34538 | 167461 | 86,832 |
https://mathoverflow.net/questions/167467 | 4 | Are there examples of independence results over subsystems of true second order arithmetic that cannot be established using omega-models? To rule out trivial examples, let us assume that the base theory extends true first order arithemtic. A non example of such a statement would be Ramsey theorem for pairs since there ... | https://mathoverflow.net/users/2689 | Necessity of omega-models in second order arithmetic | Yes, of course there are. Any result that is equivalent to extra induction axioms will require nonstandard models for a semantic separation.
For example, Hirst proved that the pigeonhole principle $\mathsf{RT}^1\_{<\infty}$ is equivalent to $B\Sigma^0\_2$, and thus is not provable in $\mathsf{RCA}\_0$. But $B\Sigma^0... | 7 | https://mathoverflow.net/users/5442 | 167468 | 86,833 |
https://mathoverflow.net/questions/167463 | 11 | In [Characteristic numbers for 3-manifolds](http://www.ams.org/mathscinet-getitem?mr=500978) Milnor and Thurston define a characteristic number and this is cited in ch. 6 of Thurston's notes when discussing the Gromov approach to Mostow rigidity.
The paper opens with "This is a brief report on work which will be publ... | https://mathoverflow.net/users/14869 | Did Milnor and Thurston write anything else about characteristic numbers for 3-manifolds? | I think that some of their work was preempted by Gromov's (e.g. "Volume and Bounded Cohomology") (which then created a huge research area which I would not want to try to summarize here), but other directions in the paper were continued. See, for example, the beautiful papers by [Francaviglia, Frigerio, Martelli](http:... | 12 | https://mathoverflow.net/users/11142 | 167472 | 86,835 |
https://mathoverflow.net/questions/167456 | 0 | Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$).
There is a large amount of algorithms for symmetric problems (conjugate gradient) and square non symmetric ones (BICGstab).
But I have difficulties to find a method for both at o... | https://mathoverflow.net/users/47084 | Large scale least squares of non symmetric and non square problems | A widely used iterative method for large scale linear least squares problems that allows for regularization if you want/need it is the LSQR algorithm of Paige and Saunders. See
<http://www.stanford.edu/group/SOL/software/lsqr/>
| 1 | https://mathoverflow.net/users/9022 | 167474 | 86,837 |
https://mathoverflow.net/questions/167457 | 2 | Suppose $\lambda\not=0\in\mathbb{C}$. Does the following system have a non trivial solution in $L^2 [0,1]$?
\begin{array} {lcl} \int\_0 ^1 f(y)\log|x-y|dy=\lambda f(x) \\\int\_0 ^1f(x)dx=0& \end{array}
| https://mathoverflow.net/users/48438 | Solvability of a Fredholm system in $L^2$ | I will give a partial answer. If you don't like it I can delete it. I might also be restating what fedja is saying, in which case I apologize. It isn't clear to me that this is what he/she had in mind though.
For $|\lambda| > 1$ the answer is no. We can implement a fixed point argument to show this.
First, let $X =... | 2 | https://mathoverflow.net/users/49404 | 167477 | 86,838 |
https://mathoverflow.net/questions/167459 | 1 | It is well-known that non-weakly compact operators from $\ell\_\infty$ into any Banach space act as isomorphisms on some subspace of $\ell\_\infty$ isomorphic to $\ell\_\infty$. I have a question in this spirit.
Suppose that $T\colon \ell\_\infty \to c\_0$ is a bounded linear operator. Then $T$ is not weakly compact.... | https://mathoverflow.net/users/50928 | Operators from $\ell_\infty$ | Every bounded linear operator from $\ell\_\infty$ to a separable space is weakly compact. This follows, for example, from the fact that $\ell\_\infty$ is a Grothendieck space (use Google).
| 3 | https://mathoverflow.net/users/2554 | 167479 | 86,840 |
https://mathoverflow.net/questions/167482 | 17 | A line bundle is ample if some power of it is very ample. A line bundle is positive if the chern class in $H^2(X,\mathbb{Z})$ is represented by a Kahler metric in $H^{1,1}(X,\mathbb{Z})$.(Regarded as elements in $H^{1,1}(X,\mathbb{C})$ through the map $H^2(X,\mathbb{Z})\to H^2(X,\mathbb{C})$)
Are ample and positive t... | https://mathoverflow.net/users/nan | Are "ample" and "positive" line bundle the same concept? | Yes. For ample implies positive, use the fact that $c\_1(O(1))$ on projective space is the Kähler form of the Fubini-Study metric, and then restrict to $X$. For the converse, you need the Kodaira embedding theorem (in fact, this is more or less the content of that theorem).
| 21 | https://mathoverflow.net/users/4144 | 167486 | 86,841 |
https://mathoverflow.net/questions/167444 | 23 | As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a decomposition $K\_i(R[T,T^{-1}])\cong K\_i(R)\oplus K\_{i-1}(R)$ for $R$ regular.
Now turn to the philosophical part, the fi... | https://mathoverflow.net/users/50846 | Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$? | Let $G\_n := W(\tilde{A}\_{n-1})$. If I understand your description correctly, there is an extension
$$1 \to G\_n \to S\_{n} \wr \mathbb{Z} \overset{sum}\to \mathbb{Z} \to 1$$
and so a $\mathbb{Z}$-Galois cover $BG\_n \to B(S\_{n} \wr \mathbb{Z})$. There are exterior products
$$\mu\_{n,m} : G\_n \times G\_m \to G\_{n+m... | 13 | https://mathoverflow.net/users/318 | 167501 | 86,846 |
https://mathoverflow.net/questions/167471 | 9 | A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the groundstate problem in order to get the full spectrum and eigenfunctions afterwards by successively applying them to the prece... | https://mathoverflow.net/users/nan | Mathematical equivalent to ladder operators? | An explicit construction of generalized ladder operators $A^\pm=\mp d/dx+W(x)$ exists if the Hamiltonian can be factorized as
$$H=-\frac{d^2}{dx^2}+V(x)=A^+ A^- +E\_0,$$
with $E\_0$ the lowest eigenvalues of $H$. The function $W(x)$ satisfies the Ricatti equation,
$$W(x)^2-W'(x)=V(x)-E\_0.$$
A class of "shape-invariant... | 8 | https://mathoverflow.net/users/11260 | 167503 | 86,848 |
https://mathoverflow.net/questions/167508 | 5 | F.Kato has a statement said that a log smooth curve $f:(X, M,\alpha)\to (k,N,\beta)$ where $k$ is an algebraic closed field and $N$ is some fine log structure on $k$, is equvalent to pointed node curve in a paper "Log sooth deformation and moduli of log smooth curves". I am quite confused with the log structure $N$ on ... | https://mathoverflow.net/users/4504 | log smooth curve vs pointed node curve | You haven't really explained your notation, but it seems that you are covering the log curve $X$ with two neighborhoods $U$ and $V$, where $U$ contains a node, and $V$ contains a log point. I think the problem with your definition of $M\_V$ is that you forgot about the nontrivial log structure on the base log scheme. I... | 2 | https://mathoverflow.net/users/121 | 167510 | 86,851 |
https://mathoverflow.net/questions/167495 | 8 | Apparently, the closest thing I've found would be normal number <http://mathworld.wolfram.com/NormalNumber.html>
But requiring that every finite words occurs is weaker than this property. So I'm wondering if there are any study on this topic.
My original goal is to find a criterion for a Büchi automaton not to rec... | https://mathoverflow.net/users/50943 | Is there a name for infinite words containing every finite words? | One term that is used is [disjunctive sequence](http://en.wikipedia.org/wiki/Disjunctive_sequence). The linked article mentions some references, including an [overview](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.1370) (from 1997) by Calude, Priese, and Staiger.
| 7 | https://mathoverflow.net/users/47312 | 167517 | 86,854 |
https://mathoverflow.net/questions/167538 | 2 | We consider a slightly extended version of a nondeterministic finite automaton, call it a "propositional nondeterministic finite automaton". It is defined as follows. Consider a fixed propositional language $L\_A$ built over a finite set of propositional variables $A$. $L\_A$ is together with the usual logical connecti... | https://mathoverflow.net/users/50958 | QBF of exponential length? | $\DeclareMathOperator\prop{prop}$Allowing general poly-time properties $P$ unnecessarily blows up the complexity, it makes the problem complete for EXPSPACE = AEXPTIME. On the one hand, the definition of the problem itself is more or less a description of an exponential-time alternating Turing machine that solves it. O... | 1 | https://mathoverflow.net/users/12705 | 167542 | 86,861 |
https://mathoverflow.net/questions/167540 | 2 | In [my research](http://www.mathematics21.org/algebraic-general-topology.html) the following problem appeared (and if it is true, this solves positively several my conjectures):
Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set (usually $n$ is infinite).
I call a *staroid* such an $n$-a... | https://mathoverflow.net/users/4086 | A conjecture about certain relations | The answer is no. Let $\mathscr{U}$ be an ultrafilter on $\mathbb{N}$. Let $A,B \in \mathscr{U}$ be such that $A \cap B$ is a proper subset of both $A$ and $B$. Let $\mathscr{F}$ be the collection of functions $f : \mathbb{N} \to \mathscr{U}$ such that either $f(n) \supseteq A$ for all but finitely many $n$, or $f(n) \... | 3 | https://mathoverflow.net/users/11145 | 167544 | 86,862 |
https://mathoverflow.net/questions/167511 | 6 | As the title says, I want to embed the genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing a nontrivial homology class.
I know that $H\_2(\mathbb{C}P^2 \# \mathbb{C}P^2; \mathbb{Z})\simeq \mathbb{Z} \oplus \mathbb{Z}$ where generators come from generators for $H^2$ of each copy of $\mathbb{C}P^2$, an... | https://mathoverflow.net/users/73791 | How to embed genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing nontrivial homology class | I'll expand a bit the comments by Igor Rivin and myself above.
The way I see it, there are two ways of constructing such a curve, and they both involved what I'd call "embedded connected sum". This is the construction you outlined above, and is an adaptation of the usual connected sum construction.
If you're given ... | 6 | https://mathoverflow.net/users/13119 | 167550 | 86,863 |
https://mathoverflow.net/questions/167549 | 7 | For measure-preserving dynamical systems, there exist several notions of mixing. The most basic ones are *strong mixing*, *weak mixing* and *ergodicity* (see the [wikipedia page](http://en.wikipedia.org/wiki/Mixing_%28mathematics%29), for instance), asserting different degrees of 'decay of correlation' between two arbi... | https://mathoverflow.net/users/9762 | What good is (strong) mixing in dynamical systems? | Furstenberg's proof of Szemerédi's Theorem seems to me a nice example:
<http://math.stanford.edu/~katznel/24812/bulletin.pdf>
| 10 | https://mathoverflow.net/users/49268 | 167552 | 86,864 |
https://mathoverflow.net/questions/167407 | 1 | Let $B(x,y) \geq 0$ be a function defined for $x, y \geq 0$ such that $B(x,0)=B(0,y)=0$ and $B''\_{xx}\leq 0, B''\_{yy}\leq 0$ (i.e. it is bicocncave function).
I am looking for the solutions among of such functions of the following differential equation:
$$
2B''\_{xx}B''\_{yy}-(B''\_{xy})^{2}=0
$$
Besides of the tri... | https://mathoverflow.net/users/50901 | Monge–Ampère type equation | I had a little time on a flight today to think about your problem, and so I applied the standard integration method to see whether or not your equation could be explicitly integrated (in the sense that the Monge-Ampère equation $u\_{xx}u\_{yy}-{u\_{xy}}^2=1$ can be integrated by transforming it to Laplace's equation). ... | 4 | https://mathoverflow.net/users/13972 | 167560 | 86,867 |
https://mathoverflow.net/questions/167562 | 6 | Let $\prod\_{n=1}^{\infty}\mathbb{Z}$ be the Baer-Specker group (infinite direct product of the additive group of integers) and $\bigoplus\_{n=1}^{\infty}\mathbb{Z}$ be the natural subgroup which is the infinite direct product with canonical basis $e\_n$, $n\geq 1$.
An abelian group $A$ is *slender* if every homomor... | https://mathoverflow.net/users/5801 | Nearly slender abelian groups | Yes. The displayed condition characterizes cotorsion-free groups. Every slender group is cotorsion-free but not every cotorsion-free group is slender. Take $A=\prod\_{n=1}^{\infty}\mathbb{Z}$. Cotorsion-free groups are important in the study of endomorphism rings, see <https://mathoverflow.net/a/117860/16678>
| 7 | https://mathoverflow.net/users/16678 | 167576 | 86,873 |
https://mathoverflow.net/questions/167555 | 3 | Hochschild cohomology can be used to characterise formal smoothness of unital associative algebras; in that such an algebra $A$ is formally smooth if and only if it is of Hochschild cohomological dimension at most $1$.
I was curious, is there a similar characterization of formal smoothness in the category of commuta... | https://mathoverflow.net/users/36886 | Hochschild cohomology and formal smoothness | Yes, the Hochschild-Konstant-Rosenberg theorem has a converse. More generally you have vanishing characterizations of smoothness in terms of Hochschild homology (one of them is e.g. Avramov, Luchezar L.; Vigué-Poirrier, Micheline, Hochschild homology criteria for smoothness, Internat. Math. Res. Notices 1992, no. 1, 17... | 7 | https://mathoverflow.net/users/36672 | 167586 | 86,875 |
https://mathoverflow.net/questions/167581 | 5 | Let $\sigma\_1$ and $\sigma\_2$ be two braids with $n$-strings.
Are there any formulas relating $J\_{\widehat{\sigma\_1\sigma\_2}}(q)$, $J\_{\hat{\sigma\_1}}(q)$, and $J\_{\hat{\sigma\_2}}(q)$?
Here, $J\_L(q)$ is the jones polynomial of a link $L$ and $\hat{\sigma}$ stands for the closure of a braid $\sigma$.
| https://mathoverflow.net/users/21694 | Jones polynomial of the concatenation of two braids | Here is one reason not to expect such a relationship (although I'm not sure if it can be completed to a proof). The Jones polynomial $J\_\sigma$ (roughly) comes from taking the trace of a linear map $A\_\sigma$ associated to the braid $\sigma$, so the question (roughly) asks about relations between $Tr(A\_\sigma)$, $Tr... | 16 | https://mathoverflow.net/users/2669 | 167592 | 86,877 |
https://mathoverflow.net/questions/167613 | 17 | It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F\_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F\_p$. (It is of course obvious that there are no linear factors.)
Are there any general irreducibility criteria which can help to prove such a result?
(M... | https://mathoverflow.net/users/4556 | Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$? | This is true. Pass to an extension field where the polynomial has a root $r$, notice that the other roots are of the form $r+1$, $r+2$, ..., $r+p-1$. Suppose that $x^p - x +1 = f(x) g(x)$, with $f, g \in \mathbb{F}\_p\left[x\right]$ and $\deg f = d$. Then $f(x) = (x-r-c\_1) (x-r-c\_2) \cdots (x-r-c\_d)$ for some subset... | 33 | https://mathoverflow.net/users/297 | 167614 | 86,885 |
https://mathoverflow.net/questions/163346 | 15 | We start with a finite dimensional chain complex over $\mathbb{F}\_2$, equipped with a basis. That is, we have finitely many finite dimensional $\mathbb{F}\_2$-vector spaces $C\_0,\dots,C\_k$ with bases $B\_0,\dots,B\_k$, and $\mathbb{F}\_2$-module maps $d\_i\colon C\_i\rightarrow C\_{i-1}$ with $d\_{i-1}d\_i=0$. I wan... | https://mathoverflow.net/users/48932 | Lift chain complex from $\mathbb{F}_2$ to $\mathbb{Z}$ | This is not always possible, even just with condition (1). Consider the complex $\mathbb{F}\_2^7 \to \mathbb{F}\_2^7 \to \mathbb{F}\_2^3$ where the basis of the first vector space is indexed by lines of the [Fano plane](http://en.wikipedia.org/wiki/Fano_plane), the basis of the second vector space is indexed by points ... | 12 | https://mathoverflow.net/users/297 | 167615 | 86,886 |
https://mathoverflow.net/questions/167605 | 4 | In this question I ask whether ambient spaces descend to models of varieties.
Let $k\subset K$ be a non-trivial extension of algebraically closed fields, e.g., $\overline{\mathbb Q}\subset \mathbb C$.
Let $X$ be a projective variety over $K$ which can be defined over $k$ (as an abstract scheme).
Assume that $X$ c... | https://mathoverflow.net/users/50996 | Minimal projective space containing projective variety independent of base field | Write $X=Y\times\_k K$ for some variety $Y$ over $k$. The closed immersion $i: Y\times\_k K\to\mathbb P^n\_K$ is defined by using finitely many coefficients in $K$, so it is defined over a finitely generated $k$-algebra $A$:
$$i\_S: Y\times\_k S\to \mathbb P^n\_S=\mathbb P^n\_k \times\_k S $$
where $S$ is the affine v... | 1 | https://mathoverflow.net/users/39387 | 167618 | 86,887 |
https://mathoverflow.net/questions/139542 | 6 | The real Stiefel manifold $V\_{n,k}$ of orthogonal $k$-frames in $\mathbb{R}^n$ can be viewed as the reductive homogeneous space $G/H=O(n)/O(n-k)$. If ${\frak{so}}(n)$ is the Lie algebra of $O(n)$, then we have the reductive decomposition
$$
{\frak{so}}(n)={\frak{m}}+{\frak{h}}
$$
where
$$
{\frak{m}}=\left \{
\begin{pm... | https://mathoverflow.net/users/14454 | Stiefel manifolds and polar decompositions | It is easier for me to examine this geometrically, rather than from the point of view of Lie groups and algebras.
First, the identity matrix $I \in O(n)$ represents the standard orthonormal basis $e\_1, \dots, e\_n$ of $\mathbb{R}^n$, and its coset $I\cdot O(n-k)$ represents the $k$-plane spanned by the first $k$ bas... | 8 | https://mathoverflow.net/users/613 | 167620 | 86,889 |
https://mathoverflow.net/questions/167612 | 3 | I'm studying methods of computation of Galois group of irreducible polynomials over $\mathbb{Q}$. In case of fifth degree there are 5 variants of Galois group:$S\_5,A\_5,AGL\_1(\mathbb{F}\_5).D\_5,\mathbb{Z}\_5$. I have troubles with determining wether Galois group is $\mathbb{Z}\_5$ or $D\_5$ if given that it is a sub... | https://mathoverflow.net/users/39304 | Computation of Galois group | Section 6.3 of Henri Cohen's "A Course in Computational Algebraic Number Theory" is about computing Galois groups using resolvents. Subsection 6.3.4 is specifically about the quintic case. Cohen gives an algorithm which addresses the case of $\mathbb{Z}\_{5}$ versus $D\_{5}$ in step 6 (once a 5-cycle contained in the G... | 4 | https://mathoverflow.net/users/48142 | 167621 | 86,890 |
https://mathoverflow.net/questions/158109 | 9 | Suppose I have a completely integrable system on a symplectic manifold $(M^{2n},\omega)$ with momentum map $H:M \rightarrow \mathbb{R}^n$ that has compact, connected fibers. Further, suppose I know the set of regular values is not simply connected.
**Question** Short of computing local action-angle coordinates or th... | https://mathoverflow.net/users/7265 | Detecting Monodromy in Integrable Systems | After reading a bit into the problem, it seems that 'Cushman's principle' is the answer I am looking for. In the case of the spherical pendulum, one can observe with Morse theory that the energy level sets change topology as one passes through the isolated critical value. This indicates that a pull-back of the torus bu... | 2 | https://mathoverflow.net/users/7265 | 167623 | 86,892 |
https://mathoverflow.net/questions/167599 | 12 | The question in the title naturally breaks up in two parts, namely the torsion part and the rank part. I already read about some results on both the torsion and the rank part. And I want to know whether this is currently still the state of the art, or whether there has been some recent progress in these questions.
Fi... | https://mathoverflow.net/users/23501 | What is our current knowledge on the structure of J_0(N)(Q) and J_1(N)(Q) | For torsion subgroups, one can consider the "rational cuspidal subgroup", which is the subgroup of degree zero divisors generated by $\mathbb{Q}$-rational divisors coming from cusps. (The cusps themselves need not be $\mathbb{Q}$-rational, but linear combinations of them can be.) This gives a way of constructing lots o... | 6 | https://mathoverflow.net/users/48142 | 167629 | 86,895 |
https://mathoverflow.net/questions/167626 | 3 | Let $G$ be a group of order $n$ and its subgroup lattice be order-isomorphic to that of $\Bbb Z\_n$. Is $G$ cyclic?
| https://mathoverflow.net/users/47958 | a characterization for cyclic groups | Yes, finite cyclic groups are exactly the finite groups whose lattices of subgroups are distributive. The lattice of subgroups of $\mathbb{Z}/n$ is isomorphic to the dual of a divisibility lattice (which is distributive).
| 2 | https://mathoverflow.net/users/32332 | 167630 | 86,896 |
https://mathoverflow.net/questions/167575 | 4 | Let $A$ be a UHF-algebra of type $n^{\infty}$ and denote its unique and faithful trace by $\tau$. Let $L^2(A)$ be the Hilbert space of the GNS-representation associated to $\tau$. We have two commuting representations $L \colon A \to B(L^2(A))$ and $R \colon A^{\rm op} \to B(L^2(A))$ and by the universal property of th... | https://mathoverflow.net/users/3995 | von Neumann algebras generated by commutators | Let $\xi\_0 \in L^2(A)$ denote the cyclic vector corresponding to the identity in $A$, and let $P\_0$ denote the rank-one projection corresponding to $\xi\_0$. Then we clearly have $xP\_0 = P\_0 x = 0$ for all $x \in M$, and hence $M \subset P\_0^\perp B(L^2(A)) P\_0^\perp$. I claim that we actually have equality. This... | 5 | https://mathoverflow.net/users/6460 | 167634 | 86,898 |
https://mathoverflow.net/questions/167649 | 1 | The question is the following: how many subsets of size $5$ from a set $A$ of size $16$ do we need so that any subset of size 2 of $A$ is also a subset of one of the selected subsets of size $5$?
How does this the required number change as we change 16 to another number and if we change $5$ to another number? Perhaps... | https://mathoverflow.net/users/24478 | Constructive ideas behind "covering" a set with subsets of fixed size | You are looking for [covering designs](https://www.ccrwest.org/cover.html). In general, it is very hard to determine these numbers exactly. A lot of work has been done on bounding them ([Gordon, Kuperberg, Patashnik](http://arxiv.org/pdf/math/9502238): "Hundreds of papers have been written for particular values of $v$,... | 3 | https://mathoverflow.net/users/12674 | 167650 | 86,904 |
https://mathoverflow.net/questions/65935 | 5 | The (uncentered) Hardy-Littlewood maximal function $M(f)$ of (a locally integrable) function $f$ on $\mathbb{R}^{n}$ is defined by the rule $M(f)(x)=\sup\_{\delta>0,\left|y-x\right|<\delta} \text{Avg}\_{B(y,\delta)} \left|f\right|$, where $\text{Avg}\_{B(y,\delta)} \left|f\right| = \int\_{\left|z\right|<\delta} f(y-z) ... | https://mathoverflow.net/users/4842 | What is the $L^p$-norm of the (uncentered) Hardy-Littlewood maximal function? | Those are basic yet difficult questions. I don't know much about the uncentered case, but here is some information on the centered case.
A nonempty set $B \subseteq \mathbb{R}^d$ is *centrally symmetric with respect to $p \in B$* if $B$ is invariant under the affine transform $x \mapsto 2p - x$. We say that $B$ is a ... | 13 | https://mathoverflow.net/users/8452 | 167652 | 86,905 |
https://mathoverflow.net/questions/167660 | 5 | Let $G$ be a (countable) discrete abelian group and denote by $\hat{G}$ its Pontryagin dual, i.e. the compact abelian group of group homomorphisms $\chi:G \longrightarrow \mathbb{T}$. Recall that, for a subgroup $H \subset G$, the annihilator is given by $H^\perp = \{\chi \in \hat{G} \mid \chi(g) = 1~\forall g \in H\}$... | https://mathoverflow.net/users/51018 | Is the annihilator of the intersection of two subgroups of a (countable) discrete abelian group generated by the annihilators of the two subgroups? | The answer to both questions is yes. Indeed, by Pontryagin duality the inclusion $(H\_1\cap H\_2)^\perp \subset H\_1^\perp H\_2^\perp$ you want is equivalent to
$$
(H\_1^\perp H\_2^\perp)^\perp\subset H\_1\cap H\_2
\tag{$\*$}
$$
where for $\Sigma\subset\hat G$ we write $\Sigma^\perp=\{g\in G:\chi(g)=1 \text{ for all }\... | 2 | https://mathoverflow.net/users/19276 | 167666 | 86,913 |
https://mathoverflow.net/questions/167643 | 13 | Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all $a \in A$ and $m^2 = \sum\_{a \in A} a$.
Tambara and Yamagami showed that the non-identity associativity isomorphisms... | https://mathoverflow.net/users/51008 | Understanding the computation of the center of Tambara-Yamagami fusion categories when realized as C* categories | In order to describe the half-braiding we want to compare it to a fixed map
$$a \otimes m \rightarrow m \rightarrow m \otimes a.$$
For Gelaki, Naidu, and Nikshych, you start off by identifying these three objects so that $a \otimes m = m = m \otimes a.$ Since you can't simultaneously skeletonize and strictify, this f... | 8 | https://mathoverflow.net/users/22 | 167672 | 86,915 |
https://mathoverflow.net/questions/167670 | 9 | This question is inspired by the question: [Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?](https://mathoverflow.net/questions/104627/example-of-non-projective-variety-with-non-semisimple-frobenius-action-on-etale)
Let $K$ be a number field (or finitely generated field of ... | https://mathoverflow.net/users/21815 | Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology | Here's an example, if I'm not mistaken. Let $E / K$ be an elliptic curve and $x \in E$ a non-torsion $K$-point. Then the image of the divisor $\{x\} - \{\infty\}$ under the etale cycle class map is a nontrivial class in $H^1(K, H^1(E\_{\bar K}, \mathbf{Q}\_\ell)(1))$ and thus corresponds to a non-split extension of $H^... | 12 | https://mathoverflow.net/users/2481 | 167674 | 86,916 |
https://mathoverflow.net/questions/167617 | 3 | Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x\_0,\dots,x\_n\}$,
i.e., each $p=(p(x\_0),\dots,p(x\_n))\in \mathcal{P}$ is such that $p(x\_i)>0$ for all $i$ and $\sum\_{i}p(x\_i)=1$ and can be thought of a point in $\mathbb... | https://mathoverflow.net/users/7699 | Geodesic equation from Christoffel symbols | Let me first re-write your notations as little bit to make it easier for me.
Let $\xi\in \Xi \subset\mathbb{R}^n$. Define the functions $\pi\_k:\Xi \to\mathbb{R}$ by
$$ \pi\_k(\xi) = \begin{cases} \xi\_k & k \in \{1, \dots, n\} \\ 1 - \sum\_{1}^n \xi\_i & k = 0 \end{cases} $$
which to me is a more natural way t... | 9 | https://mathoverflow.net/users/3948 | 167677 | 86,918 |
https://mathoverflow.net/questions/167675 | 0 | let $M$ be a smooth compact complex manifold of dimension $m$ and $N\subset M$ a smooth complex submanifold of dimension $1\leq n \leq m-2$. Covering $N$ with well chosen open sets of $M$ we can always find $m-n$ holomorphic functions that locally cut $N$ in $M$.
My question(s) is (are) the following: can we extend ... | https://mathoverflow.net/users/4971 | Meromorphic extension of local defining equations of a complex submanifold | There are manifolds without non-constant global meromorphic functions, such as generic K3 or a torus. Among these K3 surfaces, there are ones with (-2)-curves, which give a counterexample to the question. It is not hard to see that a K3 admits a -2-curve if and only if it has a vector with square -2 in its Picard group... | 1 | https://mathoverflow.net/users/3377 | 167678 | 86,919 |
https://mathoverflow.net/questions/165651 | 11 | A doubly stochastic matrix that commutes with the adjacency matrix of a graph is a *doubly-stochastic automorphism* of that graph (definition by Tinhofer 1986). Each (classical) automorphism of a graph is clearly a doubly-stochastic automorphism.
A graph all of whose doubly-stochastic automorphisms are convex combina... | https://mathoverflow.net/users/26039 | doubly-stochastic isomorphisms of graphs | A way to compute some extremal points for the Petersen example might be as follows: take the complement $A$ of its adjacency matrix and form the linear function $\ell(X)$ given by $X\mapsto \langle A,X\rangle$ on the space of 10x10 matrices. Maximising $\ell$ on the polytope in question using a simplex method will give... | 1 | https://mathoverflow.net/users/11100 | 167682 | 86,922 |
https://mathoverflow.net/questions/167596 | 1 | Let $M$ be an SPD matrix and let $\Pi=QQ^T$ be the orthogonal projection onto the range of $Q$ (a "tall" matrix with orthonormal columns). I have an expression in the form
$$\tag{1}
K=\max\_{v}\frac{v^T(I-\Pi)v}{v^T(I-\Pi)M^{-1}(I-\Pi)v},
$$
which I would like to express as a maximum of something which has $M$ in the n... | https://mathoverflow.net/users/40734 | Maximising a Rayleigh quotient over a subspace II | After some playing with this problem, I think I've found a solution.
Let $U$ be an orthonormal basis of the range of $I-\Pi$, that is, $[Q,U]$ is a square orthogonal matrix such that $U^TQ=0$ and $I-\Pi=UU^T$. Then
$$
\begin{split}
K&=\max\_v\frac{v^T(I-\Pi)v}{v^T(I-\Pi)M(I-\Pi)v}=\max\_v\frac{v^TUU^Tv}{v^TUU^TM^{-1}... | 0 | https://mathoverflow.net/users/40734 | 167684 | 86,924 |
https://mathoverflow.net/questions/166251 | 1 | Suppose that $G$ is an absolutely quasi-simple algebraic group defined over a non-archimedean local field $k$ of positive characteristic. Would there be any kind of reasonable sufficient condition for $[G(k),G(k)]$ to have nonempty interior in the strong topology? I have already asked this question in the anistropic ca... | https://mathoverflow.net/users/15482 | when the derived group of the group of $k$-rational points has nonempty interior in the strong topology | I claim the following is true:
Theorem: Let G be an isotropic (definition: contains a subgroup isomorphic to m) semisimple algebraic group over the nonarchimedean local field k. Suppose that the characteristic of k does not divide the order of the fundamental group of G. Then the commutator subgroup [G(k),G(k)] is op... | 1 | https://mathoverflow.net/users/425 | 167687 | 86,925 |
https://mathoverflow.net/questions/167577 | 17 | In [this](https://mathoverflow.net/questions/165405/category-theory-free-areas-of-pure-math-category-theory-loaded-areas-of-app%29%20have%20not%20been%20simplified%20by%20clever%20categorical%20arguments) question of mine in a comment to the accepted answer, [someone](https://mathoverflow.net/users/4362/paul-siegel) re... | https://mathoverflow.net/users/43263 | Expressing the Lebesgue integral using categories + the difficulty of describing estimates in category theory | Sorry to refer to my own work, but I think this answers your question directly: <http://www.maths.ed.ac.uk/~tl/glasgowpssl/>
That link is to a very short note, but I might as well repeat the result here. Let's agree that a "map" of Banach spaces is a map of norm $\leq 1$, and let's also agree that when $X$ and $Y$ ar... | 36 | https://mathoverflow.net/users/586 | 167693 | 86,928 |
https://mathoverflow.net/questions/167656 | 1 | I read about for any separable morphism of non-singular varieties $f:X'\to X$, one can define a homomorphism $\text{Tr}:f\_\*(\Omega\_{X'}^q) \to \Omega\_{X}^q$,so that the map $\Omega\_{X}^q \to f\_\*(\Omega\_{X'}^q) $splits. But I didn't find a reference about how is it done?
| https://mathoverflow.net/users/nan | Trace map for separated morphism of non-singular varieties | If you do not mind a reference that contains a serious mistake, you can use the following.
MR1716049 (2000h:13016) Reviewed
Zannier, Umberto
A note on traces of differential forms. (English summary)
J. Pure Appl. Algebra 142 (1999), no. 1, 91–97.
13N05 (14F10)
[article](http://www.sciencedirect.co... | 1 | https://mathoverflow.net/users/13265 | 167695 | 86,929 |
https://mathoverflow.net/questions/147086 | 8 | Some people use $\stackrel{\mathrm{def}}{=}$, $:=$ or $\stackrel{\Delta}{=}$ for definitions.
In more informal contexts, I have also seen $\stackrel{?}{=}$, for "I wish to prove this equality, which implies the thesis", used when working backwards, or even the less common $\stackrel{!}{=}$ (for which it is difficult to... | https://mathoverflow.net/users/1898 | Equal signs with fancy marks | One context in which the distinction between "equal by definition" and "equal because we proved it" has a precise mathematical meaning is intensional dependent type theory. This includes formalizing mathematics in a computer proof assistant based on ITT, such as [Coq](http://coq.inria.fr/) or [Agda](http://wiki.portal.... | 13 | https://mathoverflow.net/users/49 | 167704 | 86,931 |
https://mathoverflow.net/questions/167701 | 11 | Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?
| https://mathoverflow.net/users/48544 | Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups? | This is true more generally for finitely generated modules over a noetherian ring. Your question is equivalent to asking whether the sequence
$$0\rightarrow \operatorname{Hom}(B,A)\rightarrow \operatorname{Hom}(A\oplus B,A)\rightarrow \operatorname{Hom}(A,A)$$
is surjective on the right. To prove this, it suffices to ... | 26 | https://mathoverflow.net/users/10503 | 167706 | 86,932 |
https://mathoverflow.net/questions/167689 | 3 | Is there any closed form known for the expression $\sum\_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!
| https://mathoverflow.net/users/51031 | Sum of series $a^{i^2}$ | Calling your function $f(a),$ it is clear that $f(a)^{4} = \sum\_{n=1}^{\infty}r\_{4}(n) a^{n},$ where $r\_{4}(n)$ is the number of ways to express $n$ as a sum of four integer squares, as proved by Jacobi, who also gave an explicit description of $r\_{4}(n)$ in terms of the divisors of $n.$ I that sense ( and really r... | 6 | https://mathoverflow.net/users/14450 | 167708 | 86,933 |
https://mathoverflow.net/questions/150427 | 10 | Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be its Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am wondering if the derivative
$$
\frac{\mathrm{d}\operatorname{vech}\left(f(A)\right)}{\mathrm{d}\operatorname{vech}\left(A\r... | https://mathoverflow.net/users/2570 | The derivative of the Cholesky factor | The derivative can be found via implicit differentiation. That is,
$$
\frac{\mathrm{d}\operatorname{vec}\left(Y\right)}{\mathrm{d}\operatorname{vec}\left(X\right)} = \left(\frac{\mathrm{d} \operatorname{vec}\left(X\right)}{\mathrm{d}\operatorname{vec}\left(Y\right)}\right)^{-1}.$$
It is relatively easy to compute the ... | 9 | https://mathoverflow.net/users/2570 | 167719 | 86,939 |
https://mathoverflow.net/questions/167703 | 9 | Exact enumerations corresponding to the dimer model on a hexagonal grid, the dimer model on a square grid, and the four-vertex (aka square ice) model on a square grid are known, namely: lozenge tilings of hexagons, domino tilings of Aztec diamonds, and alternating-sign matrices. In each case, imposing appropriate bound... | https://mathoverflow.net/users/3621 | Exact enumerations from two-dimensional stat mech models | Jim already knows this very well but I'll state for the record that there are a number of nontrivial variants of the EKLP result, involving dimer placements on other two dimensional regular grids. Ciucu's [Perfect matchings and perfect powers](http://arxiv.org/abs/math/0501521) is a good survey and I added a few more t... | 3 | https://mathoverflow.net/users/297 | 167721 | 86,941 |
https://mathoverflow.net/questions/167697 | 1 | In my research, I ran into following types of improper integral
$\int^\infty\_0 e^{-a x^2} \cosh (b\sqrt{1+x^2})$
with real parameters $a>0,b>0$.
Mathematica cannot evaluate them. It also seems that a definite integral of sort
$\int \cos (\sqrt{1+x^2})$, $\int \cosh (\sqrt{1+x^2})$ ,… etc
cannot be evaluated... | https://mathoverflow.net/users/51035 | Improper integral $\int^\infty_0 e^{-a x^2} \cosh (b\sqrt{1+x^2})$ | Let us give only an expansion in $a,b$. Calling $I(a,b)$ the integral, we get easily
$$
I(a,b)=\sum\_{k\ge 0}\frac{b^{2k}}{(2k)!}\underbrace{e^{a}\int\_0^{+\infty} e^{-a (x^2+1)}(1+x^2)^k dx}\_{J\_k(a)}.
$$
We have
$
J\_k(a)=e^{a}(-\frac{d}{da})^k\bigl(J\_0(a)\bigr)=\sqrt π e^{a}(-\frac{d}{da})^k\bigl(e^{-a}a^{-1/2}\b... | 4 | https://mathoverflow.net/users/21907 | 167729 | 86,946 |
https://mathoverflow.net/questions/167728 | 8 | Let $G$ be a group given by a finite presentation.
On the one hand, it is easy to determine the abelian invariants of $G$, or in other words,
it is algorithmically decidable whether $G$ surjects to a cyclic group of prime order.
On the other, it is known to be algorithmically undecidable whether $G$ has a finite
quotie... | https://mathoverflow.net/users/28104 | For which series of finite simple groups is it algorithmically decidable whether they contain a homomorphic image of a given finitely presented group? | **UPDATE 22/10/17:** The question is answered for many classes of simple quotients in [this](https://arxiv.org/abs/1710.07183) preprint of Bridson--Evans--Liebeck--Segal.
---
Here's what I know.
Martin Bridson and I proved the [theorem](http://arxiv.org/abs/1401.2273) mentioned in the question, namely that it i... | 9 | https://mathoverflow.net/users/1463 | 167734 | 86,948 |
https://mathoverflow.net/questions/167685 | 47 | I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
hold for all complex numbers $a,b,c$ ?
For real values a case analysis will verify the inequality.
What is desired is a ... | https://mathoverflow.net/users/49117 | Absolute value inequality for complex numbers | It seems that your inequality is just an incarnation of [Hlawka's inequality](http://mathworld.wolfram.com/HlawkasInequality.html)
which says that for any vectors $x, y, z$ in an inner product space $V$ we have
\begin{equation\*}
\|x+y\| + \|y+z\|+\|z+x\| \le \|x\|+\|y\| + \|z\| + \|x+y+z\|.
\end{equation\*}
Usin... | 53 | https://mathoverflow.net/users/8430 | 167741 | 86,952 |
https://mathoverflow.net/questions/167739 | 3 | I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant homology of the loop space of a simply connected space with the $S^1$ action on the loops, which is what I mean by String H... | https://mathoverflow.net/users/48544 | Are there analogs of String Homology structure in cyclic homology? | Yes, definitively, for example you can look at (my choices are completely arbitrary, I am sorry I am sure to forget plenty of very good references):
* Menichi's paper:
<http://math.univ-angers.fr/perso/lmenichi/BV_cyclic_Hopf_algebra.pdf>
corollary 1.7
* Abbaspour, Tradler and Zeinalian:
<http://xxx.lanl.gov/abs/0807... | 7 | https://mathoverflow.net/users/27816 | 167751 | 86,955 |
https://mathoverflow.net/questions/167714 | 2 | Let $\rho : G\to \mathrm{GL}\_n(\mathbb Z\_p)$ be a crystalline representation of $G=\mathrm{Gal}(\bar{\mathbb{Q}}\_p/\mathbb{Q}\_p)$. For any non zero element $a\in \mathbb{Z}\_p^n$ , it spans a rank one $\mathbb{Z}\_p$-submodule $L\_a$. Now consider the subgroup $H$ of $G$, $H=\{g\in G| g(L\_a)\subset L\_a\}$. Does $... | https://mathoverflow.net/users/4504 | Does the isotropic group of local galois representation have finite index? | If $\rho$ is a representation with this property, then by taking $a$ to run through a basis of $\mathbf{Z}\_p^n$ and intersecting the corresponding $H$'s, there must be a finite-index subgroup of $G$ whose image under $\rho$ is diagonal. Most crystalline representations will not have this property. There are even *unra... | 4 | https://mathoverflow.net/users/2481 | 167752 | 86,956 |
https://mathoverflow.net/questions/167742 | 10 | I'm studying Riemannian manifolds that admit an almost-complex structure, thus
$$3\tau+2\chi=c\_1^2,$$
where $\tau$ is the signature, $\chi$ is the euler characteristic and $c\_1$ is the first Chern class.
I know from Hizerbruch theorem that
$$3\tau=p\_1,$$
where $p\_1$ is the first Pontryagin class. Notice that $\... | https://mathoverflow.net/users/39997 | Chern-Weil Theory for $p_1$ | It follows from the definition $p\_k(E) := (-1)^kc\_{2k}(E \otimes \mathbb{C})$ that the formula $$p\_1 = c\_1^2 - 2c\_2$$ is valid for all complex vector bundles (rather than just the tangent bundle of an almost complex surface).
The problem in your Chern-Weil calculation is essentailly that you are confusing the tr... | 22 | https://mathoverflow.net/users/13061 | 167755 | 86,957 |
https://mathoverflow.net/questions/167756 | 0 | Let $q$ be a power of an odd prime.
Consider the affine curve $\mathcal C$ defined over $\mathbb F\_q$ by $y^2=\prod\_{\xi\in\mathbb F\_q}(x-\xi)$. I try to determinate the $\mathbb F\_q$-automorphism group of $\mathcal C$. I easily found these automorphisms:
$\sigma: (x,y)\mapsto (x+\xi,\pm y)$ with $\xi\in\mathbb F... | https://mathoverflow.net/users/33128 | Automorphism group of an affine curve | There are other $\mathbf{F}\_q$-automorphisms of $\mathcal{C}$. All such are
$(x,y)\mapsto (a^2x+b,ay)$ where $b\in\mathbf{F}\_q$ and $a\in\mathbf{F}\_q^{\times}$.
Here is a description of the automorphism group over the algebraic closure $\Omega$ of $\mathbf{F}\_q$ of the projective closure $\mathcal{D}$ of your aff... | 6 | https://mathoverflow.net/users/30412 | 167757 | 86,958 |
https://mathoverflow.net/questions/167725 | 2 | Given a positive (or completely positive map) $\phi:A\to B$ between C\* algebras, is there a way to construct an $A-B$ bimodule? This would more or less generalise the following construction: If $\phi$ was an algebra map, we could have ${}\_\phi B$, which is $B$ as a vector space, with $B$ product as the right $B$ acti... | https://mathoverflow.net/users/29625 | positive maps and bimodules | At least for unital C\*-algebras, the answer is yes. A proof plus explanatory comments are provided by Paul Skoufranis, <http://www.math.ucla.edu/~pskoufra/OANotes-HilbertC-Bimodules.pdf> - look for the theorem on page 11 (here the algebras are assumed to be unital, though I didn't check whether one can extend this to ... | 7 | https://mathoverflow.net/users/51018 | 167759 | 86,959 |
https://mathoverflow.net/questions/167694 | 6 | It's rather easy to notice that the operation of [join of categories](http://ncatlab.org/nlab/show/join+of+categories) reproduces the [ordinal sum](http://ncatlab.org/nlab/show/ordinal+sum) once restricted to act on (iso classes of) well-ordered set; it's rather easy to see that $\alpha\star [0]$ (as a category) equals... | https://mathoverflow.net/users/7952 | A categorical characterization of ordinal numbers | You're not going to learn much about the conceptual or categorical structure of the ordinals from their classical presentation, not just because Excluded Middle is needed at **every** stage but because of **normalformitis**: the systematic elimination of structure.
That being said, there is a paper by Peter Johnstone... | 6 | https://mathoverflow.net/users/2733 | 167761 | 86,960 |
https://mathoverflow.net/questions/167760 | 3 | I am reading [this](http://dx.doi.org/10.1016/0166-8641%2882%2990065-7) article in which two properties of open covers are described:
>
> **$\gamma$-property:** If $\mathcal U$ is an open $\omega$-cover of $X$, then there sequence $\{ G\_n : G\_n \in \mathcal U\} \subset \mathcal U$ such that $\underline{Lim} G\_n ... | https://mathoverflow.net/users/26238 | Properties of open covers | Two things:
* Suppose that $\{ \mathcal{U}\_n \}\_{n \in \omega}$ and $\{ \mathcal{V}\_n \}\_{n \in \omega}$ are sequences of open $\omega$-covers and each $\mathcal{V}\_n$ is a refinement of $\mathcal{U}\_n$. If there is a sequence $\{ G\_n \}\_{n \in \omega}$ such that $G\_n \in \mathcal{V}\_n$ and $\underline{\mat... | 3 | https://mathoverflow.net/users/13653 | 167775 | 86,966 |
https://mathoverflow.net/questions/96142 | 2 | Prove that the following function is decreasing (as a function of a) for a > 0 when 0 < r < 1:
$${K\_2(ar)I\_2(a)-I\_2(ar)K\_2(a)\over I\_2(a)}I\_2(ar).$$
The problem arose in the analysis of a model for yield stress fluids. We have numerical evidence, but I would be interested in an analytical proof.
| https://mathoverflow.net/users/12120 | Monotonicity of a combination of Bessel functions | This discussion makes me even firmer in my opinion that introducing fancy notation for special functions and making long lists of related formulae in reference books makes more harm than good and that we would know much more about and be at more ease with them if everybody had to start from the basics and deal with bar... | 12 | https://mathoverflow.net/users/1131 | 167777 | 86,968 |
https://mathoverflow.net/questions/166252 | 29 | I became motivated to ask this question after seeing the inspiring "© The Author(s) 2013 " in the header of [this very interesting article](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9053355), published in Compositio Mathematica.
Apart from open access math journals, which (**math!**) jo... | https://mathoverflow.net/users/15054 | Which journals allow authors to retain copyright...? | Based on the previous answers, it's not entirely clear how big a difference there is between retaining copyright but transferring an exclusive right to publish, and handing over the copyright. That said, here's a list of places that do, as a standard practice, allow the option of authors retaining copyright.
$\bullet... | 11 | https://mathoverflow.net/users/48142 | 167787 | 86,974 |
https://mathoverflow.net/questions/167736 | 5 | I am reading Andrew Granville's [Anatomy of Integers and Permutations](http://www.dms.umontreal.ca/~andrew/PDF/Anatomy.pdf) where it is argued the factorization of a permutation into disjoint cycles is analogous to the factorization of a number into prime factors.
In the [blog-sphere](http://terrytao.wordpress.com/2... | https://mathoverflow.net/users/1358 | Analogy between Integers and Permutations | The essential common feature that insures convergence to a Poisson Dirichlet distribution is explained in the book "Logarithmic Combinatorial Structures" by Arratia Barbour and Tavare. They do a great job, I think.
| 5 | https://mathoverflow.net/users/34435 | 167788 | 86,975 |
https://mathoverflow.net/questions/167631 | 7 | $X$ and $Y$ are Bernoulli random variables with weights $0 < \alpha < 1$ and $0 < \beta < 1$. Is it possible to construct a sampler for the Bernoulli random variable with weight $\min(\frac{\alpha}{\beta}, 1)$ given access to samplers for $X$ and $Y$ — i.e., without access to the weights themselves?
If not, are there... | https://mathoverflow.net/users/51005 | Constructing a Bernoulli random variable for ratio of Bernoulli weights | The obstruction that Bjørn Kjos-Hanssen describes can be made even worse-- it applies to any algorithm (not just von Neumann's trick) and also applies to randomized algorithms (i.e. even if the number of samples is not limited in advance, only the expected number of samples is limited).
Suppose we fix $\epsilon>0$ an... | 2 | https://mathoverflow.net/users/8938 | 167789 | 86,976 |
https://mathoverflow.net/questions/154431 | 100 | Recently, I have proved that Kazhdan's property (T) is theoretically provable
by computers ([arXiv:1312.5431](https://arxiv.org/abs/1312.5431),
explained below), but I'm quite lame with computers and have
no idea what they actually can do. So, my question is how feasible is it to
prove property (T) of a given group, sa... | https://mathoverflow.net/users/7591 | How feasible is it to prove Kazhdan's property (T) by a computer? | Using the $\Delta^2- \epsilon \Delta$ approach, Tim Netzer and I have verified Kazhdan's property (T) for ${\rm SL}(3,\mathbb Z)$. For the standard generators $e\_{ij}$ ($i\neq j$) we can show a spectral gap of the normalized Laplace operator of $1/120$. There is a lot of room for further improvement.
To my knowledge... | 59 | https://mathoverflow.net/users/8176 | 167797 | 86,978 |
https://mathoverflow.net/questions/167793 | 3 | I want to select M points on the N-sphere such that $min\_{i\neq j,i,j\in \{1..M\}} ||x\_i - x\_j||$ is maximized.
Are there good upper bounds for this max-min distance?
| https://mathoverflow.net/users/48815 | Bound on maximum distance between points on a unit N-Sphere | Call the quantity in question $D(M, N),$ and let the volume of the spherical cap of dimension $N$ and radius $r$ $V(N, r),$normalized so that the volume of the whole sphere is $1.$ Since the caps of radius $2\arcsin D(M, N)/4$ are disjoint, you know that:
$$
M V(N, 2\arcsin D(M, N)/4) < 1.
$$
Since The function $V(N,... | 3 | https://mathoverflow.net/users/11142 | 167800 | 86,979 |
https://mathoverflow.net/questions/167811 | 1 | Let $R$ be a commutative ring with unity and $A,B\in M\_n(R)$ satisfying the property
(\*) All elements of the two-side ideal, in $M\_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if $R$ is an algebraically closed field, then $A,B$ are simultaneously triangularizable (noted ST). Else, McCoy, again... | https://mathoverflow.net/users/9091 | Simultaneous triangularizability over a commutative ring | When you write that all elements in the two-sided ideal $\langle AB-BA\rangle$ are nilpotent, what precisely do you mean by "nilpotent". Here is an example that I believe contradicts simultaneous diagonalizability. Let $R$ be the commutative, unital ring $\mathbb{C}[\epsilon]/\langle \epsilon^2 \rangle$, i.e., the ring... | 2 | https://mathoverflow.net/users/13265 | 167813 | 86,985 |
https://mathoverflow.net/questions/167822 | 3 | Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in mind are polynomial rings over $\mathbb{C}$.
Is every element of $\mathrm{SL}(n,R)$ of order $k$ diagonalizable?
| https://mathoverflow.net/users/23758 | Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable? | Yes. One way to see this is to decompose the group ring $R[G],$ where $G$ is a cyclic group of order $k$, into the product of simple rings. It follows from your assumptions that this is possible and since $G$ is cyclic, every simple component is isomorphic to $R.$ An $n\times n$ matrix $g$ of order $k$ with entries in ... | 9 | https://mathoverflow.net/users/5740 | 167826 | 86,992 |
https://mathoverflow.net/questions/167819 | 12 | The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ is a (bounded) function ($R < \infty$ is fixed), then there are (at most) countably many points $\{a\_j\}\_j$ such that ... | https://mathoverflow.net/users/40264 | Besicovitch Covering Lemma on Manifolds | In the Federer's book "Geometric Measure Theory",
there is a notion of "directionally limited" metric space.
He proves that the Besicovitch Covering Lemma holds for the directionally limited spaces.
It is easy to see that lower bound on sectional curvature plus upper bounds on dimension and diameter imply that the sp... | 9 | https://mathoverflow.net/users/10330 | 167829 | 86,993 |
https://mathoverflow.net/questions/167823 | 11 | Let $X$ be a compact metric space and $M(X)$ the set of all Borel probability measures on $X$.
It is know that $M(X)$ is a convex compact metric space endowed with the weak-\* topology i.e.
$(\mu\_n)\_n \subseteq M(X)$ converges to $\mu \in M(X)$ iff for all continuous function $f \in C(X)$ $\int\_X f d\mu\_n \to \int\... | https://mathoverflow.net/users/51088 | The Borel $\sigma$-algebra of the set of probability measures | Consider the set $\mathcal A$ of measurable subsets $A \subset X$, such that $\mu \mapsto \mu[A]$ is measurable. Obviously, $\mathcal A$ contains all open sets.
It's also easy to see that $\mathcal A$ contains an algebra of sets - say, sets $A \subset X$, such that their indicator is a pointwise limit of a sequence o... | 5 | https://mathoverflow.net/users/22758 | 167834 | 86,995 |
https://mathoverflow.net/questions/167831 | 5 | So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle $P$. Then, as far as I understand, we can construct the associated vector bundle to the one-dimensional representation $\... | https://mathoverflow.net/users/27828 | Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories? | I'm afraid that this answer will be somewhat physics-y. Apologies if this is deemed inappropriate for MO.
First of all, I think that it is slightly misleading to say that one has "fractional charge in $SU(N)$ gauge theory." A careful reading of the lectures linked in the question reveals that one actually has a Yang-... | 9 | https://mathoverflow.net/users/394 | 167838 | 86,998 |
https://mathoverflow.net/questions/167837 | 3 | I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}\_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results done by [Magma](http://magma.maths.usyd.edu.au/magma/), there are only 24 kinds of factorizations of different degrees... | https://mathoverflow.net/users/51099 | Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ | It is a special case of thm 2.3 in <http://arxiv.org/abs/1302.0625> (version 3, I've just updated it, it has a different number in version 2) that the factorization pattern of polynomials $x^q-(ax^2+bx+c)$ distributes the same as random permutation in $S\_n$ (unless $q$ is a power of $2$). In fact this theorem deals wi... | 10 | https://mathoverflow.net/users/2042 | 167845 | 87,000 |
https://mathoverflow.net/questions/167792 | 3 | There is a well-known construction of minimal idempotents in the group algebra of the symmetric group $\mathbb C[S\_n]$ using row symmetrizers and column antisymmetrizers. But these idempotents are not \*-idempotents with respect to the \*-structure on $\mathbb C[S\_n]$ which sends $g$ to $g^{-1}$. (In other words, the... | https://mathoverflow.net/users/284 | Minimal *-idempotents for the group algebra of the symmetric group | There is a well-known construction of the primitive idempotents for the symmetric group over the rationals that is due to Murphy (and possibly Jucys?). It can be found in his paper "A new construction of Young's seminormal representation of the symmetric group", J. Algebra, 69 (1981), 287-297.
To describe this for $... | 6 | https://mathoverflow.net/users/37373 | 167848 | 87,002 |
https://mathoverflow.net/questions/167849 | 11 | I'm writing a paper on orthogonal polynomials and I have to cite results by Chebyshev and Cholesky. I found several and different transliterations from Russian. I wonder if there is a standard and accepted way to spell them. Thanks
| https://mathoverflow.net/users/50721 | Correct spelling of names, Chebyshev and Cholesky | [Here](http://www.forvo.com/word/%D0%BF%D0%B0%D1%84%D0%BD%D1%83%D1%82%D0%B8%D0%B9_%D0%BB%D1%8C%D0%B2%D0%BE%D0%B2%D0%B8%D1%87_%D1%87%D0%B5%D0%B1%D1%8B%D1%88%D0%B5%D0%B2/) you can hear the pronounce by a Russian speaking person.
As to the romanization, which usually does have a standard form in any language, I'd use th... | 10 | https://mathoverflow.net/users/6101 | 167854 | 87,005 |
https://mathoverflow.net/questions/167855 | 3 | I have found a function to create the lowest common multiples for the first $n$ positive integers:
$$\text{lcm}(n):=\prod\_{m=2}^{n}C\_m(1),$$
where $C\_m(1)$ is the [cyclotomic polynomial](http://mathworld.wolfram.com/CyclotomicPolynomial.html) of order $m$ and $n\geq2.$
The *Mathematica* functions:
```
lcm[n_] ... | https://mathoverflow.net/users/16888 | New identity for lcm of the first n integers and the second Chebyshev function | Since $\log C\_m(1) = \Lambda(m)$ for $m \ge 2$, where $\Lambda$ is the Von Mangoldt function, this seems to be a restatement of the identity
$$\mathrm{lcm} \{1,\ldots,n\} = \exp \psi(n) = \exp\bigl( \sum\_{m=1}^n \Lambda(m) \bigr) = \exp\bigl( \sum\_{m=2}^n \log C\_m(1) \bigr).$$
| 5 | https://mathoverflow.net/users/7709 | 167859 | 87,008 |
https://mathoverflow.net/questions/167862 | 3 | Let $f:N\_1\to N\_2$ be a homotopy equivalence between two simply connected manifolds with boundary of the same dimension. Can it be extended to a homotopy equivalence between closed manifolds $f:M\_1\to M\_2$ such that $N\_i\subset M\_i$? (I suppose there are counterexamples, but what I am actually interested in are t... | https://mathoverflow.net/users/9833 | Extending homotopy equivalence between manifolds with boundary | The answer in general is no. Let $N\_1$ be $S^2\times D^2$, and $N\_2$ be the disk bundle of Euler class $1$ over $S^2$. Then both are homotopy equivalent to $S^2$. But no homotopy equivalence will extend to closed manifolds. For if $f: M\_1 \to M\_2$ is a homotopy equivalence, with $N\_i \subset M\_i$, then $N\_i$ car... | 6 | https://mathoverflow.net/users/3460 | 167864 | 87,011 |
https://mathoverflow.net/questions/167868 | 2 | In *G. B. Folland - A Course in Abstract Harmonic Analysis* we can read the following
" **(1.15) Proposition.** Let be $A$ a (complex) commutative unital Banach algebra with unit $e$, let $x\_0 \in A$ and suppose that one of the following holds:
(i) $A$ is generated by $x\_0$ and $e$.
(ii) $x\_0$ is invertible an... | https://mathoverflow.net/users/nan | Spectrum of a Banach algebra homeomorphic to the spectrum of one of its elements | The answer is yes for $C^\*$ algebra and this is essentially the Stone Weierstrass theorem:
The assumption that the spectrum of $x$ is isomorphic the spectrum of $A$ (under the canonical map) essentially mean that as a function from the spectrum of $A$ to $\mathbb{C}$, $x$ is an injection, hence $x$ separate the poin... | 3 | https://mathoverflow.net/users/22131 | 167870 | 87,013 |
https://mathoverflow.net/questions/167867 | 1 | Let $f:Y\rightarrow X$ be a birational morphism of smooth projective varieties, $F$ an effective divisor on $X$, $D=f^{-1}F\_{\mathrm{red}}+\mathrm{Ex}(f)$, $B$ a smooth subvariety of $Y$ contained in the non-snc locus of $D$, is $f\_\*(\mathcal{O}\_B(K\_Y+D))$ always nonzero on $X$? If the dimension of $B$ and $f(B)$ ... | https://mathoverflow.net/users/51119 | Pushforward of $K_X+D$ on the non-snc locus | This can be zero. For instance, let $X$ be $\mathbb{P}^3$. Let $q$ be a $k$-point. Let $G$ and $H$ be smooth hypersurfaces in $\mathbb{P}^3$ that contain $q$ and such that, as linear subspaces of the Zariski tangent space $T\_q(X)$, $T\_q(G)$ equals $T\_q(H)$. Call this common subspace $S$.
Let the effective divisor $\... | 3 | https://mathoverflow.net/users/13265 | 167872 | 87,014 |
https://mathoverflow.net/questions/167871 | 9 | If $\kappa$ is a strongly compact cardinal, then the singular cardinal hypothesis holds above $\kappa$. Hence the existence of large cardinals at the level of "strongly compact" or above is incompatible with even (apparently) mild [large powerset axioms](https://mathoverflow.net/questions/164673/why-isnt-there-more-int... | https://mathoverflow.net/users/26080 | Is there a "large powerset axiom" so extreme that it disproves the existence of strongly inaccessible cardinals? | Foreman's maximality principle is as you have requested, though it is not yet known if it is consistent or not.
**Foreman's maximality principle:** Any non-trivial forcing notion either it adds a real or colapses some cardinals.
It follows from it that:
1) $GCH$ fails everywhere,
2) there are no inaccessible ca... | 12 | https://mathoverflow.net/users/11115 | 167874 | 87,015 |
https://mathoverflow.net/questions/167873 | 1 | Let $(R,\frak{m})$ be a hypersurface (i.e., $R=Q/(f)$, where $Q$ is a regular local ring, $0\not=f\in Q$). If $M$ is a MCM $R$-module, is it possible for the injective dimension of $M$ over $R$ to be $\infty$? What about the Gorenstein injective dimension in this case?
Since a noetherian local ring is regular iff it ... | https://mathoverflow.net/users/36703 | Is it possible for a MCM module over a hypersurface to have infinite injective dimension? | If $M$ is not free it will have infinite injective dimension. This is because $R$ is Gorenstein (since a hypersurface), so the projective dimension of a finitely generated module is finite iff its injective dimension is. By Auslander-Buchsbaum, finite projective dimension for a MCM module is equivalent to freeness.
| 1 | https://mathoverflow.net/users/460 | 167878 | 87,016 |
https://mathoverflow.net/questions/167885 | 3 | Let
$ E $ - Elliptic curve defined over $ {\mathbb{Q}} $.
$G\_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$.
$ E[3] $ - $3$-torsion points of $ E $.
Suppose $ \rho$ denotes the $ G\_{\mathbb{Q}} $-representation associated to $ E[3]$.
If $ E $ has a $3... | https://mathoverflow.net/users/44637 | Galois representation attached to $3$-torsion points of an elliptic curve | Assuming that $E$ has a rational $3$-torsion point, that point must be fixed, and hence your representation takes the form
$$ \rho \sim \left( \begin{matrix} 1 & \eta\_{i} \\ 0 & \chi \end{matrix}\right)$$
as you say. It's a general fact that the determinant of the $G\_{\mathbb{Q}}$ representation attached to an ellipt... | 9 | https://mathoverflow.net/users/48142 | 167889 | 87,019 |
https://mathoverflow.net/questions/167545 | 5 | Let $\pi:\mathcal{X} \to B$ be a flat family of projective varieties. Assume that $B$ is irreducible. Suppose that $\mathcal{X}$ is smooth except for a closed subscheme, say $Y$ which is isomorphic to $B$ and the composition $Y \hookrightarrow \mathcal{X} \to B$ is flat. Does there exist a smooth scheme $\tilde{\mathca... | https://mathoverflow.net/users/43198 | Simultaneous resolution of singularities in special cases of flat families of projective varieties | Here is a counterexample. Let $k$ be a field. Let $A$ be an Abelian $k$-variety of dimension $d \geq 2$ together with a projectively normal closed immersion, $$i : A \to \mathbb{P}^n\_k,$$ (these do exist). Let $C \subset \mathbb{P}^{n+1}\_k$ be the projective cone over $i(A)$. Denote by $v$ the vertex of this cone. Si... | 3 | https://mathoverflow.net/users/13265 | 167891 | 87,020 |
https://mathoverflow.net/questions/167904 | 3 | In one the the answers to this thread " [Can one embedd the projectivezed tangent space of CP^2 in a projective space?](https://mathoverflow.net/questions/119850/) " it was mentioned that " $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the variety of complete flags in the vector space $\mathbb{C^3}$ ".
I'm having a har... | https://mathoverflow.net/users/33518 | Why is $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the space of complete flags $GL_3/B$? | Let $\pi:\mathbb{P}(T\_{\mathbb{P}^2})\rightarrow\mathbb{P}^2$ be the projectivized tangent bundle. The point $x:=(p,[L])\in\mathbb{P}(T\_{\mathbb{P}^2})$ corresponds to the point $p = \pi((p,[L]))\in\mathbb{P}^2$ and to the class of the line $L\subset\mathbb{P}^2$ passing through $p$. Now, the point $p$ is a line thro... | 2 | https://mathoverflow.net/users/14514 | 167913 | 87,029 |
https://mathoverflow.net/questions/167916 | 7 | I thought of this question the other day and have not been able to get any traction on references or results along its lines, so I finally caved and decided to ask it here. I am no expert on Galois Theory or the Inverse Galois Problem (IGP), but have more interest in the representation theory side of this.
>
>
> >... | https://mathoverflow.net/users/12301 | Incomplete Failures of the Inverse Galois Problem | Given a group $G$, the regular IGP for $G$ and the strong IGP for $G$ are equivalent. More precisely, if $L/K$ is a Galois extension with ${\rm Gal}(L/K) \cong G$, and $\pi\_{G} : G \to S\_{n}$ is a faithful, transitive, permutation representation, let $M$ be the fixed field of a stabilizer of a point. Choose $\beta \i... | 14 | https://mathoverflow.net/users/48142 | 167917 | 87,031 |
https://mathoverflow.net/questions/167940 | 13 | What is it, in Mihailescu's proof of Catalan conjecture, that uses explicitly the fact that there is a 1 on the right hand side of $x^p - y^q = 1$? In other words, why can't we extend his argument to prove stuff about, say, $x^p - y^q = 2$?
| https://mathoverflow.net/users/51156 | Can we extend the proof of Catalan's conjecture? | [This](http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf) article by Tauno Metsänkylä gives a good explanation of Mihăilescu's proof. There are some crucial steps in the proof that could not be done in the case $x^p-y^q=2,$ say. The idea is to write
$$\frac{x^p-1}{x-1}(x-1)=y^q... | 18 | https://mathoverflow.net/users/23008 | 167945 | 87,035 |
https://mathoverflow.net/questions/167931 | 8 | Let $f$ be a Modular form/Maass form on $GL(2)$ with level $N$ and character $\eta$ and Fourier coefficients $a(n)$.
The Rankin-Selberg convolution
$$L(s,f\times\bar f)=\sum \frac{a(n)\overline{a(n)}}{n^s}$$ has a pole at $s=1$.
---
My question:
What's the analytic conductor/level for $L(s,f\times\overline{f... | https://mathoverflow.net/users/22170 | Functional equation and conductor for a Rankin-Selberg convolution | First, a few things about normalization. The expression $L(s,f \times \overline{f})$ needs to be multiplied by $\zeta(2s)$ in order to have a functional equation. Also, if $f$ is a holomorphic modular form of weight $k$, we should have the $n$th Fourier coefficient be $a(n) n^{\frac{k-1}{2}}$ in order for the functiona... | 6 | https://mathoverflow.net/users/48142 | 167962 | 87,040 |
https://mathoverflow.net/questions/167960 | 4 | The following should be known, but I could not find an example.
Let $\kappa$ be an uncountable cardinal. Find a model $M$ of size $\kappa$ which has $\ge\kappa$ many automorphisms, but for some $m\in M$, $(M,m)$ has only the trivial automorphism (I call this element rigid on the title. Not sure if there is a standard... | https://mathoverflow.net/users/13694 | A model with $\kappa$ many automorphism and a rigid element. | Let $(F,+,\cdot)$ be a rigid field, and put $M=(F,+,R)$, where $R(x,y,u,v)\iff xy=uv$. Then $(M,1)$ is interdefinable with $(F,+,\cdot)$ and therefore rigid, but $M$ has $|F|$ automorphisms $x\mapsto ax$ for each $a\in F^\times$.
There exist rigid fields of any infinite cardinality, see e.g. <https://mathoverflow.net/a... | 7 | https://mathoverflow.net/users/12705 | 167964 | 87,041 |
https://mathoverflow.net/questions/167958 | 41 | Edit on March 2, 2018: I just noticed that this is almost identical to a question asked on MO by David Harden in 2011, and that I had even given an (incomplete) answer to that one. I would delete the question, but I think this can't be done when an answer has been accepted ( I did try).
The title says it all really. ... | https://mathoverflow.net/users/14450 | Does a finite simple group of order divisible by $60$ have $A_{5}$ as a subgroup? | Let me make my comment into an answer just get things off the ground. I claim that ${\rm PSL}(n,q)$ contains $A\_5$ whenever its order is divisible by $60$.
Clearly ${\rm SL}(n,q)$ contains ${\rm SL}(m,q)$ for all $m \le n$, and hence ${\rm PSL}(n,q)$ contains some central quotient of ${\rm SL}(m,q)$ (which is someth... | 43 | https://mathoverflow.net/users/35840 | 167968 | 87,043 |
https://mathoverflow.net/questions/167965 | 12 | I found the following definition.
A Weil divisor $D = \sum\_{i}D\_i \subset X$ on a smooth variety $X$ is simple
normal crossing if for every point $p \in X$ a local equation of $D$
is $x\_1\cdot...\cdot x\_r$ for independent local parameters $x\_i$
in $O\_{p,X}$. A log resolution of the pair $(X,D)$ is
a birational ... | https://mathoverflow.net/users/nan | Simple normal crossing divisors | You gave the definition of normal crossing divisor. The definition of simple normal crossing is the following.
A Weil divisor $D = \sum\_{i}D\_i \subset X$ on a smooth variety $X$ of dimension $n$ is simple
normal crossing if any component $D\_i$ is smooth and for every point $p \in X$ a local equation of $D$ is $x\_... | 9 | https://mathoverflow.net/users/14514 | 167969 | 87,044 |
https://mathoverflow.net/questions/167961 | 9 | We wish to find the set of natural numbers that cannot be expressed as a difference between a square and a prime.
e.g.
$1 = 2^2 - 3$
$2 = 3^2 - 7$
$3 = 4^2 - 13$
and so on.
The smallest such number is $16$. The proof that $16$ cannot be expressed as a difference of a square and a prime:
Let $r^2 - p = 16... | https://mathoverflow.net/users/51167 | Natural numbers that cannot be expressed as a difference between a square and a prime? | We have a representation $m=x^2-p$ where $m$ and $x$ are positive integers and $p$ is a prime if and only if there is a prime of the form $x^2-m$. Little is known about primes of the form $x^2-m$; it has not been proved for any fixed value of $m$ that there are infinitely many such primes. However, [Bunyakovsky's conje... | 20 | https://mathoverflow.net/users/23008 | 167970 | 87,045 |
https://mathoverflow.net/questions/167967 | 11 | In what follows, we have a level $N \geq 3$, and the modular curve $X(N)$, and the invertible sheaf $\omega$ on $X(N)$ such that the global sections of $\omega^{\otimes k}$ correspond to modular forms of weight $k$ and level $N$.
In this letter, Serre looks at an exact sequence of sheaves:
$0 \rightarrow \omega^{k ... | https://mathoverflow.net/users/15899 | Serre's 1987 letter to Tate about mod p modular forms | This kind of reasoning is now standard in the subject (that's a fact) but not easy (at least that's my opinion -- each time the word "canonical" is used in an essential way in an argument, things are not "easy").
For question 1, you understand well what Serre meant, and I have nothing to add. Concerning question 2, ... | 11 | https://mathoverflow.net/users/9317 | 167971 | 87,046 |
https://mathoverflow.net/questions/167952 | 9 | Given $f \in L^2([0,1])$, $f \neq 0$, we can consider the orthogonal complement $f^\perp$ . The smooth functions $C^\infty([0,1])$ are dense in $L^2([0,1])$. Is the intersection $f^\perp \cap C^\infty([0,1])$ dense in $f^\perp$? If not, can we find a counterexample, and find conditions on $f$ such that the statement is... | https://mathoverflow.net/users/40707 | For which $f \in L^2([0,1])$ is $f^\perp \cap C^\infty$ dense in $f^\perp$? | $f^\perp\cap C^\infty([0,1])$ is dense in $f^\perp$.
Indeed, given a closed finite codimensional subspace $M$ of a normed space $X$ and a dense subspace $V$ of $X$, the intersection $V\cap M$ is dense in $M$.
See IV.2.8 Lemma in S. Goldberg "Unbounded linear operators". McGraw-Hill, 1966.
| 7 | https://mathoverflow.net/users/39421 | 167973 | 87,048 |
https://mathoverflow.net/questions/167943 | 11 | There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([[1]](https://mathoverflow.net/questions/96289/extensions-of-the-modularity-theorem), [[2]](https://mathoverflow.net/questions/12416/the-difficulties-in-proving-modularity-lifting-theorems-over-non-t... | https://mathoverflow.net/users/43108 | Modularity theorem for abelian varieties | Abelian varieties over the rationals are modular if and only if they are of "$GL\_2$"-type, which is a notion introduced by Ribet who proved that this statement is a consequence of Serre's conjecture which, as you know, has since been proved. Here is a link to Ribet's paper:
<http://math.berkeley.edu/~ribet/Articles/... | 15 | https://mathoverflow.net/users/2290 | 167978 | 87,050 |
https://mathoverflow.net/questions/167881 | 0 | Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by
$$x\_0^2+x\_1^2+...+x\_k^2 =0.$$
If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$.
* If $n = 3$, $k=2$, then $Q\subset\mathbb{P}^3$ is a quadric cone. If $\pi:X\rightarrow\mathbb{P}^3$ is the blow-up of ... | https://mathoverflow.net/users/nan | Singular irreducible quadrics | In the case $n=4$, $k=2$, one can perform an explicit computation in charts of the blow-up. Consider the quadric $Q$ defined by
$$\{x\_1^2+x\_2^2+x\_3^2=0\}\subset\mathbb{A}^4.$$
The singular line is given by $L={x\_1=x\_2=x\_3 = 0}$. Let us consider the points $p = (0,0,0,0)$ and $q = (0,0,0,1)$ on $L$. We can look at... | 1 | https://mathoverflow.net/users/14514 | 167986 | 87,053 |
https://mathoverflow.net/questions/166155 | 2 | I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is not necessarily monoidal. (Imagine left-tensoring with an algebra $A$ internal to $\mathcal{C}$, so $T = A \otimes -$.)
... | https://mathoverflow.net/users/13767 | When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple? | tetrapharmakon's hint is excellent. The article he refers to (and the earlier article <http://arxiv.org/abs/math/0604180> by some of the same authors) define "quasi-triangular Hopf monads" that are an abstraction of quasi-triangular Hopf algebras. For these monads, the Eilenberg-Moore category is indeed braided.
(I did... | 2 | https://mathoverflow.net/users/13767 | 167990 | 87,054 |
https://mathoverflow.net/questions/167892 | 1 | Let $\mathbf{a}\_k\in\mathbb{C}^n$ for $k=1,2,\ldots,m$ be i.i.d. standard complex normal random vectors with distribution $c\mathcal{N}(0,\mathbf{I})$. I am interested in a tight upper bound on the following quantities with high probabilities (say with probability at least $1-\frac{1}{n}$ or something similar):
\begin... | https://mathoverflow.net/users/34919 | maximum of certain Gaussian processes | I actually found a simple counter example. Setting $x=\frac{\mathbf{a}\_1}{\|\mathbf{a}\_1\|\_{\ell\_2}}$ already rules out my claim.
| 2 | https://mathoverflow.net/users/34919 | 167995 | 87,057 |
https://mathoverflow.net/questions/168003 | 9 | I'm in particular interested in understanding Grothendieck's argument for this in SGA 1 (page 232 in <http://arxiv.org/pdf/math/0206203v2.pdf>)
Let $G$ be $\text{GL}\_n$ over a scheme $S$ for some integer $n$, and let $P/S$ be a principal $G$-bundle. Then we know that there is an fpqc morphism $S'\rightarrow S$ such ... | https://mathoverflow.net/users/15242 | why are principal GL(n)-bundles (Zariski-)locally trivial? | I would argue using the associated fiber bundle construction in Exp. XI just after corollary 4.3 (a few pages back from your spot). For any $GL\_n$-torsor $P$, you may use the action of $GL\_n$ on $\mathbb{G}\_a^n$ to construct a canonical vector bundle $E = P \times^{GL\_n} \mathbb{G}\_{a,S}^n$, and there is a canonic... | 8 | https://mathoverflow.net/users/121 | 168004 | 87,062 |
https://mathoverflow.net/questions/165835 | 1 | **Information:**
**a-)** $X$ and $Y$ are two continuous random variables on $\mathbb{R}$ having continuous distribution functions $F$ and $G$ with $G(y)\geq F(y)$ for all $y$.
**b-)** $S^X\_n=\sum\_{i=1}^n X\_i$, $S^Y\_n=\sum\_{i=1}^n Y\_i$, $A>0$, and $B<0$; where $X\_i$ and $Y\_i$ are **i.i.d.** replicas of $X$ a... | https://mathoverflow.net/users/36356 | Comparing the expected stopping times of two stochastically ordered random processes (Added:(14.05.2014)) | I found a counter example for the claim with mean shifted Gaussian distributions. So the claim is not true even with the condition in $d$.
| 0 | https://mathoverflow.net/users/36356 | 168007 | 87,065 |
https://mathoverflow.net/questions/167939 | 24 | I know that number fields have been the object of many statistical experiments.
Is there some kind of heuristics for the following?
Fix a degree $d$ and fix a bound $N$ on the coefficients of a monic integral degree $d$ polynomial $P$ (a more natural choice would probably be to bound the discriminant of $P$). Among t... | https://mathoverflow.net/users/5239 | Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$ | I summarize the first two pages of Kedlaya, [A construction of polynomials with squarefree discriminants](http://arxiv.org/abs/1103.5728)
>
> When $P$ is irreducible and the discriminant $\Delta(P)$ is square free, the number field $\mathbb{Q}[x]/P(x)$ has ring of integers $\mathbb{Z}[x]/P(x)$... When the coefficie... | 17 | https://mathoverflow.net/users/297 | 168018 | 87,068 |
https://mathoverflow.net/questions/168015 | 5 | A useful "abstract nonsense" construction in ergodic theory takes a measure-preserving transformation
$T$ of a probability space $(X,\mathcal B,\mu)$ and extends it to an *invertible* measure-preserving transformation $\bar T$ of a probability space $(\bar X,\bar{\mathcal B},\bar\mu)$.
One description of this is in ... | https://mathoverflow.net/users/11054 | Natural extensions in ergodic theory / Measurability question | I think the answer is yes.
Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0,1]$ and $\mathcal{L}$ the Lebesgue $\sigma$-algebra. Suppose $T$ is $(\mathcal{L}, \mathcal{L})$-measurable and measure preserving. In particular $T$ is $(\mathcal{L}, \mathcal{B})$-measurable (the usual sense of "Lebesgue measurable") ... | 4 | https://mathoverflow.net/users/4832 | 168022 | 87,069 |
https://mathoverflow.net/questions/158292 | 7 | Given two separable (infinite dimensional) Banach spaces $X$ and $Y$, it is not difficult to show that there exists an injective (bounded linear) operator $T:X\to Y$ with range dense in $Y$. See S. Goldberg and A.H. Kruse. "The Existence of Compact Linear Maps Between Banach Spaces$. Proc. A.M.S. 13 (1962), 808-811.
... | https://mathoverflow.net/users/39421 | Existence of injective operators with dense range | In
Argyros, Spiros A.; Arvanitakis, Alexander D.; Tolias, Andreas G.
Saturated extensions, the attractors method and hereditarily James tree spaces.
Methods in Banach space theory, 1–90, London Math. Soc. Lecture Note Ser., 337, Cambridge Univ. Press, Cambridge, 2006
the authors construct a separable space $Z$... | 6 | https://mathoverflow.net/users/2554 | 168027 | 87,070 |
https://mathoverflow.net/questions/168033 | 7 | In the theory of Coxeter groups there is the notion of so called *parabolic subgroups*. I'm wondering is this term just a random name, or there are some historical reasons? Why *parabolic*?
Thanks.
| https://mathoverflow.net/users/35603 | Coxeter groups - Parabolic subgroups | The history is definitely somewhat convoluted. Note first that the term "Coxeter group" itself was introduced by Bourbaki in their 1968 volume containing chapters 4-6 of *Groupes et algebres de Lie*. The first section of Chapter 4 studies Coxeter systems $(W,S)$ (with $S$ typically finite) in great generality, inspired... | 13 | https://mathoverflow.net/users/4231 | 168035 | 87,074 |
https://mathoverflow.net/questions/168012 | 4 | We can think hyperbolic 5-space as, $$\mathcal{H}^5=SO^+\_{5,1}(\mathbb{R})/SO\_5(\mathbb{R})=SL\_2(\mathbb{H})/Sp^\*\_2(\mathbb{H}),$$$\mathbb{H}$ is real quaternion algebra. By Iwasawa Decomposition the orientation preserving isometry group of $\mathcal{H}^5$ is $$G=PSL\_2(\mathbb{H})=NAK,$$where $$N=\left\lbrace n(x... | https://mathoverflow.net/users/36735 | Action of the isometry group of the hyperbolic 5-space | The "rule" for the generalized linear fractional transformation in these coordinates is simply the re-Iwasawa-decomposition of $g\cdot n(x)a(y)$.
I think it is just at this point that the writing of $x,y$ as $z=x+\*y$ becomes much less tenable. Indeed, for hyperbolic $n$-space as $SO(n,1)/O(n)$ there seems to be no u... | 1 | https://mathoverflow.net/users/15629 | 168037 | 87,075 |
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