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https://mathoverflow.net/questions/87345 | 22 | Is there some a priori reason why we should expect the Brauer group of real [complex] super vector spaces to be closely related to periodicity in real [complex] K-theory? By "a priori" I mean a proof that does not involve computing that both are Z/8 [Z/2] and does not involve noticing that both are related to Clifford ... | https://mathoverflow.net/users/284 | Brauer Groups and K-Theory | I'm a bit late to the party, but here's what I suspect the answer should look like. Forgive me for working somewhat to very heuristically throughout.
To star things off, here's a silly question: why is cohomology $\mathbb{Z}$-graded? Starting from a spectrum $E$ and a space $X$ we can consider the set $[X, E]$ of ho... | 17 | https://mathoverflow.net/users/290 | 177321 | 89,299 |
https://mathoverflow.net/questions/177026 | 1 | Consider the following scenario: one has 2 communication channels $C\_1$ and $C\_2$. Denote by $p(x)$ the input probability distribution.
The mutual information between the input and the output of $C\_1$ must be greater or equal than the mutual information between the input and the output of the composed channel $C\_... | https://mathoverflow.net/users/56299 | Mutual information staying constant under composition of channels | I only know the answer for the first one.
If you see the proof of data processing inequality in Cover page 34, it is easy to find out data processing inequality turns out to be equality only when the "double Markovity" is satisfies.
Let $X, Y$ and $Z$ are three random variables representing the input, output of the f... | 1 | https://mathoverflow.net/users/41666 | 177324 | 89,301 |
https://mathoverflow.net/questions/177297 | 8 | I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck site.
The definition for hypercovers in an $\infty$-topos that I am using is from *Higher Topos Theory*: In an $\infty... | https://mathoverflow.net/users/56531 | Hypercovers of sheaves in classical and quasi-categories | Local epimorphisms are precisely those morphisms in $\mathcal{P}\left(\mathcal{C}\right)$ which become effective epimorphisms after applying the sheafification functor. In particular, if $f$ is an effective epimorphism in $\mathfrak{X}=Sh\_\infty\left(\mathcal{C}\right),$ then when regarded as a morphism in $\mathcal{P... | 6 | https://mathoverflow.net/users/4528 | 177325 | 89,302 |
https://mathoverflow.net/questions/176592 | 1 | **Edite** According to the essential comment of Ian Agol I revise the question as follows
For a smooth manifold $M$, is there a non identity involution $\theta$ on the lie algebra $\chi^{\infty}(M)$ such that $X$ is topological equivalent to $\theta (X)$, for all smooth vec. field $X$ on $M$?
This question is motiv... | https://mathoverflow.net/users/36688 | A question on involutions on the Lie algebra of vector fields | The Lie algebra of vector fields of a smooth manifold determines the manifold with its smooth structure. Below I indicate a series of papers where this is proved in various settings. In particular, this implies that any Lie algebra involution of the Lie algebra of vector fields has to be induced by an involution of the... | 3 | https://mathoverflow.net/users/26935 | 177331 | 89,303 |
https://mathoverflow.net/questions/168281 | 1 | I am trying to understand meaning and importance of a g-natural metric. Since I do pure differential geometry for my research, I am not familiar with many notions which are needed for understanding a g-natural metric.
My motivation for this question is an example of a Kaehlerian manifold structure which can be constru... | https://mathoverflow.net/users/43143 | Natural bundle, g-natural metric, meaning | $F$ is a functor from the category of smooth manifolds of a fixed dimension whose morphisms are local diffeomorphisms into the category of smooth manifolds, mapping $M$ to a fiber bundle over $M$. All these are associated bundles to a higher order frame bundle for suitable actions of the jet group on a typical fiber.
... | 2 | https://mathoverflow.net/users/26935 | 177337 | 89,307 |
https://mathoverflow.net/questions/177323 | 5 | Let $R$ be a Poisson algebra (over $\mathbb C$, say) with Poisson center $Z = \{c \in R : \{c,R\} = 0\}$ and consider two types of ideals $I \leq R$:
1. $I = \langle (c\_i) \rangle$ is generated by Casimirs $c\_i$, i.e. each $c\_i\in Z$
2. $I$ is a Poisson ideal, i.e. $\{I,R\} \leq I$.
I'm having trouble distinguis... | https://mathoverflow.net/users/391 | Poisson ideals vs. ideals generated by Poisson central elements | I am not that familiar with the algebraic approach but the intersection on the rhs seems quite hard to me. Take the Poisson structure on the plane given by $\{x,y\}=xy$. Then $Z$ contains only constant functions and therefore for any Poisson ideal $I$ you have $I\cap Z$ equal to $Z$ or $0$ depending whether the ideal c... | 5 | https://mathoverflow.net/users/6032 | 177343 | 89,310 |
https://mathoverflow.net/questions/177055 | 4 | For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It seems that codimension $1$ irreducible subschemes of these models yield valuations of $F$. My question is: which valuati... | https://mathoverflow.net/users/2191 | Which valuations of a field yield codimension $1$ subschemes of their 'models' | Suppose you are given a ~~proper~~ noetherian integral scheme $X$ whose field of rational functions is $F$. Let $A$ be a discrete valuation of $F$ with residue field $k\_A$ and suppose it has a center $x\in X$. Then the existence of a model $Y$ (of finite type over $X$) of $F$ such that $A$ is induced by a codimension ... | 4 | https://mathoverflow.net/users/39387 | 177353 | 89,311 |
https://mathoverflow.net/questions/177359 | 5 | Let $p$ be a prime number and let $Y$ over $\mathbb F\_p$ be a Siegel modular variety, with minimal compactification $X$. It is well known that $X^{\operatorname{ord}}$, the ordinary locus of $X$ is affine (since it is cut out by the Hasse invariant, that is a section of an ample line bundle). But what about $Y^{\opera... | https://mathoverflow.net/users/54752 | Is the ordinary locus affine? | No. It's well known that the complement of a closed subset of codimension two or more in an affine variety is never affine. The minimal compactification has codimension *g*.
| 8 | https://mathoverflow.net/users/1310 | 177365 | 89,316 |
https://mathoverflow.net/questions/177347 | 1 | Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex variety $X^\sigma$ in the following way:
1. $X$ and $X^\sigma$ are equal as schemes.
2. If $p:X\longrightarrow\textrm{Sp... | https://mathoverflow.net/users/47136 | Conjugate surfaces: informations about the orbits | This problem is of a more arithmetic nature, than geometric.
For example, since every automorphism of $\mathbb{C}$ preserves $\mathbb{Q}$, we see that if $X$ can be defined over $\mathbb{Q}$, then $\Omega\_X$ consists of a single isomorphism class. Similarly if $X$ can be defined over a number field, then $\Omega\_X$... | 3 | https://mathoverflow.net/users/5101 | 177371 | 89,318 |
https://mathoverflow.net/questions/177314 | 4 | Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. [Rosenberg](http://books.google.com/books/about/The_Laplacian_on_a_Riemannian_Manifold.html?id=gzJ6Vn0y7XQC)). The heat operat... | https://mathoverflow.net/users/41626 | Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold | A lot is known about asymptotics of subelliptic heat kernels.
There are Minakshinsudaram-Pleijel expansions that hold for very general subelliptic operators and they apply in particular to the contact case:
[Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus](http://archive.numdam.or... | 2 | https://mathoverflow.net/users/48356 | 177372 | 89,319 |
https://mathoverflow.net/questions/177366 | 4 | What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?
| https://mathoverflow.net/users/nan | Connections between Standard, Hodge and Tate conjectures on algebraic cycles? | I think the list on Wikipedia is incomplete:
The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as conjecture D (for singular cohomology) over fields of characteristic 0.
The Tate conjecture also implies Lefschetz, Kunneth, and conjecture D (for etale cohomology) over all fields (... | 9 | https://mathoverflow.net/users/18060 | 177374 | 89,320 |
https://mathoverflow.net/questions/177348 | 9 | Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A\_1 + A\_2 + ... + A\_n = A\_0 ,$$
i.e. Horn's inequalities for n matrices?
Can such inequalities be obtained just from Horn's inequalities for 3 matrices if we couple matrices in this way
$$(((A\_1 + A\_2)... | https://mathoverflow.net/users/8381 | Horn's inequalities for n matrices | 1) Section 7 of [Knutson-Tao-Woodward], <http://arxiv.org/pdf/math/0107011.pdf>
2) Yes. To obtain the (minimal set of) inequalities, you can just glom the puzzles together.
| 12 | https://mathoverflow.net/users/391 | 177375 | 89,321 |
https://mathoverflow.net/questions/177358 | 11 | I [asked this question already on math.stackexchange](https://math.stackexchange.com/questions/869839), but maybe it is also useful to ask this here, since it was not answered there.
Suppose we have three directed sequences of $C^\*$-algebras, say $(A\_n,\varphi\_n)$,$(B\_n,\psi\_n)$ and $(C\_n,\theta\_n)$ and $\*$-h... | https://mathoverflow.net/users/56556 | Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras? | I may be showing my ignorance of category theory, but don't think this is true. Work on $l^2$. Let $\mathcal{A}\_n$ consist of the operators $A$ satisfying $\langle Ae\_i, e\_j\rangle = 0$ if $\max(i,j) > n$ (so $\mathcal{A}\_n$ is isomorphic to the $n\times n$ matrices). The direct limit of the $\mathcal{A}\_n$ is the... | 12 | https://mathoverflow.net/users/23141 | 177377 | 89,322 |
https://mathoverflow.net/questions/177378 | 7 | Suppose that $f$ is in $L^2(\mathbb{R})$ and consider the set of integer translates of this function, $V=\{f(x-k):k\in\mathbb{Z}\}$. This set is linearly independent: taking the Fourier transform of the finite sum $\sum a\_k f(x-k)$ one gets $p(e^{2\pi i\xi})\widehat{f}(\xi)$ for some polynomial $p$. If the sum is zero... | https://mathoverflow.net/users/5751 | Prove that ..., f(x-2), f(x-1), f(x), f(x+1), f(x+2),... is algebraically linearly independent without the Fourier transform | Suppose there is some linear dependence. If the set is linearly dependent, space $V$ should be finite dimensional.
Fix a finite subset $S$ of $\mathbb{Z}$ so that supposedly the set of $f(x-k)$ where $k\in S$ would span.
Note that as $k\_0 \rightarrow \infty,$ $\langle f(x-k\_0), f(x-k) \rangle \rightarrow 0,$ which ... | 11 | https://mathoverflow.net/users/32470 | 177379 | 89,323 |
https://mathoverflow.net/questions/177208 | 32 | Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange.
I could never understand the intuition behind polarization of abelian varieties and how it arises. I know that there is an analogy roughly with a prequantum line bundle, but I think the concept ... | https://mathoverflow.net/users/40883 | Why polarization of abelian varieties? | Weil introduced the term "polarization" in connection with his study of abelian varieties with complex multiplication. His definition is slightly different from what one sees today; one might call it a polarization up to isogeny instead of a polarization. One can find a discussion in Weil's article "On the theory of co... | 27 | https://mathoverflow.net/users/56563 | 177380 | 89,324 |
https://mathoverflow.net/questions/177361 | 2 | This question is a follow up to [Why is the norm map dual to restriction under Tate local duality?](https://mathoverflow.net/questions/177204/why-is-the-norm-map-dual-to-restriction-under-tate-local-duality)
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider the Cartier... | https://mathoverflow.net/users/53197 | Why is the Tate local duality pairing compatible with the Cartier duality pairing? | To avoid notational confusion between translation in the derived category (of abelian fppf sheaves on the category of lfp $S$-schemes) and torsion in abelian schemes, I'll denote the $n$-torsion in $A$ as $A\_n$ rather than $A[n]$, and likewise for $B$.
The $n$-torsor Kummer sequence for $B:=A^t = \mathscr{Ext}^1\_S... | 6 | https://mathoverflow.net/users/52824 | 177384 | 89,326 |
https://mathoverflow.net/questions/177312 | 1 | Suppose we have a double Markov relation for three random variables $X$, $Y$ and $W$ as follows
$$X\to W\to Y,$$ and $$X\to Y\to W.$$
How to prove that there exist functions $f$ and $g$ such that
$$X\to f(Y)\to Y, W$$ and $$\Pr(f(Y)=g(W))=1?$$
| https://mathoverflow.net/users/41666 | Double Markovity | I think I could prove the existence of such functions but however, I can not show that we must have $X\to f(Y)\to Y, W$. However the proof for the first part might give some insight for the latter.
Suppose random variables $X$, $Y$ and $W$ are defined over alphabets $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{W}$.
... | 0 | https://mathoverflow.net/users/41666 | 177385 | 89,327 |
https://mathoverflow.net/questions/176803 | 12 | Let $G$ be a connected reductive group over a number field $F$, $G\_\infty=\prod\_{v\mid\infty} G(F\_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}\_f$ the finite adèles of $F$. Fix a maximal compact subgroup $K$ of $G\_\infty$. We have Hecke algebras $H\_\infty$ of $G\_\infty$ (relative to $K$) and $H\_f$ of $G(\mat... | https://mathoverflow.net/users/4351 | Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article | The definition on the top of page 195 in B-J's article (in Corvallis 1) is ok, because $f\ast\xi$ is well-defined for any smooth $f:G(\mathbf{A})\to\mathbf{C}$ and any $\xi\in H$.
It suffices to verify the claim for pure tensors $\xi=\xi\_\infty\otimes\xi\_f$, in which case $f\ast\xi$ is $f\ast\xi\_\infty$ convolved... | 4 | https://mathoverflow.net/users/11919 | 177387 | 89,328 |
https://mathoverflow.net/questions/177328 | 5 | Does anyone know reference for a theorem of the following sort:
Proposition: Let $K \subset\mathbb {R}^n$ be a compact convex set, and assume that
$$f(w):=\operatorname{argmax}\_{x\in K}w(x) $$ is unique for
each nonzero linear functional $w:\mathbb{R}^n\rightarrow \mathbb{R}$. Then the function $f$ is continuous ... | https://mathoverflow.net/users/56547 | Reference request: Continuity of unique maximizer of linear functional on convex set | 1. Let $\sigma(w) = \max\_{x\in K} w(x)$ be the support function of set $K$. This function is convex and its subdifferential is exactly your function $f$, i.e. $f(w) = \partial \sigma(w)$.
2. Your assumption that $f(w)$ is unique for all $w \neq 0$ is equivalent to the differentiablity of $\sigma$ on the set of nonzero... | 3 | https://mathoverflow.net/users/1184 | 177399 | 89,333 |
https://mathoverflow.net/questions/172185 | 2 | I'm currently reading the paper *Rectifiable Sets and the Traveling Salesman Problem* ([link](http://link.springer.com/article/10.1007%2FBF01233418#page-2)) by Peter Jones (Invent. math. 102, 1-15 (1990)), and am having trouble understanding an integral estimate made by rotating the dyadic grid. The estimate is made in... | https://mathoverflow.net/users/53152 | An integral estimate over rotations of the dyadic grid | I was able to complete this, and will post a solution in case anyone else finds it useful. Writing $J\_{n, j, \theta}$ for the line segment corresponding to $J^n\_j$ with everything rotated by angle $\theta$ (that is, the segment connecting $[j 2^{-n + 1} \pi + \theta, (j + 1) 2^{-n + 1} \pi + \theta]$, then all the es... | 1 | https://mathoverflow.net/users/53152 | 177402 | 89,334 |
https://mathoverflow.net/questions/177404 | 0 | This question comes from learning the paper "Existence of minimal models for varieties of log general type", where they define the log terminal model (See Definition 3.6.7). However, the question itself does not need that definition.
Let $\pi: X \to U$ be a projective morphism of normal quasi-projective varieties. Su... | https://mathoverflow.net/users/29730 | Pushforward of a log canonical pair | Yes, since
$$\phi\_\*K\_X=K\_Y.$$
Here is the reason. We take a common resolution of $X$ and $Y$, say
$$
p:W\rightarrow X, \\q: W \rightarrow Y.
$$
Then we can right
$$
K\_W-p^\*K\_X=E,\\
K\_W-q^\*K\_Y=F,
$$
where $E$ is exceptional over $X$ (and over $Y$ because "contraction") and $F$ is exceptional over $Y$. If app... | 2 | https://mathoverflow.net/users/42636 | 177407 | 89,335 |
https://mathoverflow.net/questions/177410 | 7 | Let $A$ be a $n \times n$ matrix over field $F$. Let $a\_1, \cdots, a\_n$ be the column vectors of $A$. For any subset $S \subseteq [n] = \{1, 2, \cdots, n\}$, let $a\_S = \sum\_{i \in S} a\_i$. Alon's celebrated [permanent lemma](http://lovelace.thi.informatik.uni-frankfurt.de/~jukna/EC_Book/exers_and_sols/node27.html... | https://mathoverflow.net/users/56187 | About an identity which gives immediate proof of the permanent lemma | This identity is a particular case of theorem 3 in ["A generalization of Combinatorial Nullstellensatz"](http://arxiv.org/abs/1302.4647) by Michał Lasoń. Indeed your proof is essentially reinventing the combinatorial nullstellensatz. :)
The usual proof of Alon's lemma looks at the polynomial $\prod\_{i=1}^n (\sum\_j ... | 8 | https://mathoverflow.net/users/2384 | 177414 | 89,338 |
https://mathoverflow.net/questions/177391 | 4 | I am looking for a reference for this question: given a branched surface in a 3-manifold, how we can construct a lamination fully carried by that branched surface.
any comments would be appreciated.
edit: maybe I should be more specific. I'm reading D.Gabai's paper "foliations and topology of 3-manifolds III" where h... | https://mathoverflow.net/users/56570 | laminations and branched surfaces | Tao Li <https://www2.bc.edu/~taoli/lbs.pdf> constructs an essential lamination for each branched surface satisfying the following conditions:
(1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after eliminating trivial bubbles in the bran... | 4 | https://mathoverflow.net/users/39082 | 177435 | 89,345 |
https://mathoverflow.net/questions/176073 | 3 | I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem).
**Restricted GA**
Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ pairs of nodes $(u, v)$ where $u \ne v$ ( all pairs $(u\_i, v\_i)$ are pair-wise disjoint and $0 \lt \epsilon \le 1$)... | https://mathoverflow.net/users/8784 | How hard is a variant of graph automorphism problem? | The problem is NP-hard, as far as I can see. Here is a reduction to it from a certain NP-complete satisfiability problem.
Let $e(x\_i,x\_j,x\_k,x\_l)$ denote a Boolean formula which is true if and only if exactly two of literals $x\_i,x\_j,x\_k,x\_l$ are true.
By [Schaefer's dichotomy theorem](http://en.wikipedia.org/w... | 6 | https://mathoverflow.net/users/nan | 177442 | 89,347 |
https://mathoverflow.net/questions/177417 | 7 | Let $G$ be a compact abelian group. Then we know, because of the Peter-Weyl theorem, that $L^2(G)$ decomposes as a Hilbert space direct sum of 1 dimensional representations of $G$.
Let $\mathbb{A}$ denote the adeles for $\mathbb{Q}$. Suppose we are given an automorphic funtion $\phi:GL\_2(\mathbb{A})\to \mathbb{C}$ ... | https://mathoverflow.net/users/11395 | Whittaker models for $GL_n$ and Fourier coefficients | It is indeed reasonable to wonder what's going on with "Fourier expansions" along non-abelian (sub-) groups... since, among other things, any one-dimensional representation has to factor through the maximal abelian quotient, so must lose some information.
For contrast: the unipotent radical $N$ of the standard minima... | 8 | https://mathoverflow.net/users/15629 | 177443 | 89,348 |
https://mathoverflow.net/questions/177421 | 4 | As we know there are patterns in simple continued fraction expansion of quadratic algebraic numbers,are there any patterns in simple continued fraction expansions of other algebraic real numbers?Or any law in them?or is there any universal algorithm to compute the integer sequence in simple continued fraction expansion... | https://mathoverflow.net/users/14024 | Are there any patterns in simple continued fraction expansions of algebraic real numbers? | For a "formula" for the continued fraction of algebraic numbers, in particular $2^{1/3}$, see [Bombieri and van der Poorten](http://maths.mq.edu.au/~alf/www-centre/alfpapers/a113.pdf). It's just not a simple pattern.
EDIT: Actually there's an error in the formula in the middle of page 152 there:
it should be
$$ \pm... | 13 | https://mathoverflow.net/users/13650 | 177444 | 89,349 |
https://mathoverflow.net/questions/177438 | 2 | My question is about the notion of exponential functors as they are frequently defined in the literature on (strict) polynomial functors, e.g., the paper "[General Linear and Functor Cohomology](http://arxiv.org/abs/math/9909194)" by Franjou, Friedlander, Scorichenko, and Suslin, or the paper "[Bar complexes and extens... | https://mathoverflow.net/users/7932 | When is an exponential functor a bialgebra? | In this situation one can use the language of monoidal categories. The terminology which I use below is standard in category theory.
Tensor porduct of functors between categories $\mathcal{V} \rightarrow \mathcal{W}$ can be defined as soon as $\mathcal{W}$ is a monoidal category. The functor category $[\mathcal{V}, \... | 2 | https://mathoverflow.net/users/39004 | 177447 | 89,351 |
https://mathoverflow.net/questions/177440 | 4 | Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let
$$O\_n(k,f):=\{ g\in GL\_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$
and
$$SO\_n(k,f):=O\_n^+(k,f):=O\_n(k,f) \cap SL\_n(k)$$
be the associated (special) orthogonal group, i.e. linear transformations ... | https://mathoverflow.net/users/8338 | Automorphisms of SO_n(k,f) | This is at least a partial answer. There are two distinct viewpoints here: the concrete one involving forms and automorphism groups (which came first historically and usually requires characteristic $\ne 2$ to avoid tricky points) and the much more general one involving simple algebraic groups over a field $k$ (where B... | 6 | https://mathoverflow.net/users/4231 | 177454 | 89,353 |
https://mathoverflow.net/questions/177329 | 11 | Let $\{P\_i\}$ be a subset of $SU(n)$ such that for any $U$ in another subset (or perhaps subgroup) $H$ of $SU(n)$: $$P\_1UP\_2U\cdots P\_mU=I$$ where $I$ is the identity element. Is there a sequence $\{P\_i\}$ such that $H$ be enlarged to the whole $SU(n)$? Also (perhaps more interesting): how large can $H$ be made?
... | https://mathoverflow.net/users/1837 | Multiplicative Identity for all elements in SU(n) | I can show that it is impossible to achieve $H=SU(n)$. More generally, if $G$ is a compact connected Lie group, I will show that the map $U \mapsto P\_1 U P\_2 U \cdots P\_m U$ is surjective, and therefore the preimage of the identity can't be all of $G$.
**Proof** Such a $G$ is a compact connected orientable manifol... | 7 | https://mathoverflow.net/users/297 | 177468 | 89,357 |
https://mathoverflow.net/questions/177439 | 8 | Maybe not research level.
Let $Z\cong \mathbb{R}$ be the $z$-axis of $\mathbb{R}^3$. Clearly $\pi\_1(\mathbb{R}^3-Z)\cong \mathbb{Z}$. Now if $F\subset Z$ is a closed non-empty subset, then one easily sees that $\pi\_1(\mathbb{R}^3-(Z-F))=0$.
What is if $F$ is not closed (for example $F=\mathbb{Q}$)?
>
> Is fo... | https://mathoverflow.net/users/32972 | Fundamental group of $\mathbb{R}^3-F$ where $F\subseteq \mathbb{R}\times \{0\} \times \{0\}$ | Given a continuous pointed map $\gamma: (S^1,1) \to (\mathbb{R}^3 \setminus (Z-F), x)$, compactness of $S^1$ implies the intersection $\gamma(S^1) \cap Z$ is closed in $Z$. Thus, $\gamma$ represents an element of $\pi\_1(\mathbb{R}^3 \setminus (Z-F'), x)$ for some closed nonempty subset $F' \subset F \subset Z$, and as... | 7 | https://mathoverflow.net/users/121 | 177483 | 89,364 |
https://mathoverflow.net/questions/177461 | 24 | Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. [McCord](http://projecteuclid.org/download/pdf_1/euclid.dmj/1077376525) constructed a $T\_0$ topology on $B$ with the property that the inclusion $B \to X$ is a weak homotopy equivalence. [Clader](http://www.math.c... | https://mathoverflow.net/users/4362 | How much of homotopy theory can be done using only finite topological spaces? | Vidit, thanks for the advertisement; Paul I'll answer your email shortly.
As a minor point, there is a small but subtle mistake in Clader's work
that is corrected in [Matthew Thibault's 2013 Chicago thesis](http://search.proquest.com/docview/1424274072), which goes
further in that direction.
I do intend to finish t... | 18 | https://mathoverflow.net/users/14447 | 177485 | 89,365 |
https://mathoverflow.net/questions/177477 | 9 | Letting $d(m)$ be the number of divisors of $m$, is it the case that for $m=8n+6$,
$$ d(m) \equiv \sum\_{k=1}^{m-1} d(k) d(m-k) \pmod{8}\ ?$$
It's easy to show that both sides are 0 mod 4: the left side since two primes appear to odd order in the factorization of $m$, and the right side since $m$ is neither a squar... | https://mathoverflow.net/users/12878 | A divisor sum congruence for 8n+6 | The congruence you state is true for all $m \equiv 6 \pmod{8}$. The proof I give below relies on the theory of modular forms. First, observe that
$$
\sum\_{k=1}^{m-1} d(k) d(m-k) = 2 \sum\_{k=1}^{\frac{m-2}{2}} d(k) d(m-k) + d\left(\frac{m}{2}\right)^{2}.
$$
Noting that $d(m) \equiv d\left(\frac{m}{2}\right)^{2} \pmod{... | 17 | https://mathoverflow.net/users/48142 | 177486 | 89,366 |
https://mathoverflow.net/questions/177481 | 28 | The continued fraction
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance, is known explicitly as a ratio of Bessel function values and is (I believe - SS) known to be transcendental. Similarly, $[1,2,2^2,2^3,2^4,2^5,\dots] $ is surely transcendental (and likely related to Liouville numbers). Ar... | https://mathoverflow.net/users/14024 | Is any particular algebraic number known to have unbounded continued fraction coefficients? | As you indicate, real algebraic numbers of degree $\leq 2$ have this property in view of Lagrange's classical result characterizing them by the eventual periodicicty of the continued fractions expansion. It may be useful to know (if you don't already) that $\alpha \in \mathbb{R}$ having bounded continued fractions coef... | 37 | https://mathoverflow.net/users/26522 | 177491 | 89,368 |
https://mathoverflow.net/questions/177155 | 3 | I found a listing on Google books for a book containing the desired English translation, together with some biographical information on Wessel, and entitled *On the Analytical Representation of Direction*, but I cannot find any source that actually has a copy of this in stock. Needless to say, I can't find it anywhere ... | https://mathoverflow.net/users/14835 | Where can I find a translation of Caspar Wessel's "Om directionens analytiske betegning?" | There is an English translation of the first 10 sections of Wessel's paper in the anthology edited by Henrietta Midonick, *The Treasury of Mathematics*, volume 2 (Penguin Books 1968)
pp.321--329.
| 5 | https://mathoverflow.net/users/1587 | 177495 | 89,371 |
https://mathoverflow.net/questions/177499 | 3 | I ran into the following problem in a calculation involving permutations.
Let $[n] = \{1,...,n\}$, and assume that $[n]$ is partitioned into equivalency classes. That is, $[n]$ is the disjoint union of the nonempty sets $A\_1, ... ,A\_d$, and $a\_i = |A\_i|$ denotes the sizes of the classes.
Any ordering $\sigma \i... | https://mathoverflow.net/users/17599 | What is the probability of a given induced ordering of a random permutation? | Nice problem but actually its solution is simple.
$\Pr\{\tau = (1,2,3)\} = (a\_1/n)(a\_2/(a\_2+a\_3))$ and so on:
$$\Pr\{\tau = (1,2,3,4)\} = (a\_1/n)(a\_2/(a\_2+a\_3+a\_4))(a\_3/(a\_3+a\_4))$$
| 4 | https://mathoverflow.net/users/4600 | 177503 | 89,375 |
https://mathoverflow.net/questions/177506 | 6 | A bielliptic surface is a surface of type $S=E\_1 \times E\_2/G$ where $E\_1, E\_2$ are elliptic curves and $G$ is a finite group of translations of $E\_1$ acting on $E\_2$ such that $E\_2/G=\mathbf{P}^1$.
It is easy to classify these surfaces, and it turns out that there are $7$ families.
Is there a classification o... | https://mathoverflow.net/users/27125 | Linear systems on bielliptic surfaces | The structure of $\textrm{Num}(S)$ for a bielliptic surface $S$ is given in the paper by F. Serrano [*Divisors of bielliptic surfaces and embeddings in* $\mathbb{P}^4$](http://link.springer.com/article/10.1007/BF02570754), Mathematische Zeitschrift **203** (1990), 527-533.
First, Serrano proves that a basis for $\te... | 6 | https://mathoverflow.net/users/7460 | 177512 | 89,377 |
https://mathoverflow.net/questions/177505 | 2 | Given the Hopf map $h:S^3\to S^2$ and an inclusion $i:S^2\hookrightarrow S^3$, the map $h\circ i:S^2\to S^2$ has mapping degree zero. Therefore, it is homotopic to the constant map and the image of the equator $S^2\subset S^3$ can be contracted to a point. This gives a representative of the sum $[f]+[g]\in\pi\_3(S^2)$ ... | https://mathoverflow.net/users/25549 | Splitting the Hopf map in two | Here is an expanded version of my comment above.
Let $h: S^3\to S^2$ be the Hopf map (or any other pointed map for that matter). Choose a pointed null-homotopy $i\_t: S^2\to S^3$ of the standard inclusion $i:S^2\hookrightarrow S^3$.
The composition $h\circ i\_t$ is a null-homotopy of $h\circ i$, and therefore gives... | 6 | https://mathoverflow.net/users/8103 | 177513 | 89,378 |
https://mathoverflow.net/questions/177357 | 5 | The objective is as follows:
$\min\_{\mathbf{F}} a Tr(\mathbf{F} \mathbf{F}^H) - Re\{\mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}\}$
$s.t.\ \ \ Tr(\Sigma \mathbf{F} \mathbf{F}^H)<p$
where $a$ and $p$ are scalars, $\mathbf{b}$ is $1 \times N$ real vector, $\mathbf{d}$ is $N\times 1$ real vector, $\mathb... | https://mathoverflow.net/users/42511 | optimization problem, any solution? | there is a solution to your problem.
First of all, let us consider the problem
$$\mathcal{P}:\\ z : = \max\_{\mathbf{F}} Re\{\mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}\} {\rm ~~s.t.~~~} Tr(\mathbf{F} \mathbf{F}^H) \leq p. $$
Let $\mathbf{F}\_\star$ the solution to the above problem. It is clear that if... | 2 | https://mathoverflow.net/users/50560 | 177516 | 89,380 |
https://mathoverflow.net/questions/138025 | 5 | Let $C$ be the $2^k\times 2^k$-permutation matrix over $\mathbb{F}\_2$ of the $2^k$-cycle. We needed to know the structure of its centralizer in $\mathrm{GL}\_{2^k}(\mathbb{F}\_2)$, and we computed it - it was not too easy. It's an abelian group, and so we were able to compute the decomposition of the quotient of the c... | https://mathoverflow.net/users/11100 | centralizer of the order 2^k cyclic permutation matrix over F_2 | the details of our computation can now be found in <http://arxiv.org/abs/1405.0113>
| 0 | https://mathoverflow.net/users/11100 | 177544 | 89,391 |
https://mathoverflow.net/questions/177541 | 4 | It´s known that $J\_0(N) = J(X\_0(N))= \bigoplus\_f E(f)$ splits as a sum of abelian varieties parametrized by the Hecke eingenfunctions and that it´s an elliptic curve iff the Hecke eingenvalue is an ordinary integer. Of course, there are other induced splittings by the Fricke involution, however the finest one is by ... | https://mathoverflow.net/users/40883 | Reference or proof for the fact that $J(X_0(N))$ splits into abelian varieties with real multiplication | This is true. More precisely, any $E(f)$ has real multiplication by the field $K\_f$ generated by the coefficients of $f$, and this field is a totally real number field of degree over $\mathbb Q$ equal to the dimension of $E(f)$.
This is due to Shimura and proven in his book "Introduction to the arithmetic theory of au... | 7 | https://mathoverflow.net/users/9317 | 177546 | 89,393 |
https://mathoverflow.net/questions/177496 | 11 | According to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalised via the Toda flow. More specifically, if $X=X(t)\in\mathbb R^{n\times n}$ is the solution of the initial value problem
($n^2\times n^2$)
\begin{equation}
\frac{dX}{dt} \,=... | https://mathoverflow.net/users/43681 | Diagonalization via the Toda flow | $\def\Tr{\mathrm{Tr}}$This proof is short enough that I thought I'd just write it out.
On a skim, this looks like the same proof that Christian Remling pointed you to, and which Deift-Li-Tomei say is the same as the proof of Moser.
Disclaimer: all signs in this argument have at best a 55% chance of being right.
Fir... | 17 | https://mathoverflow.net/users/297 | 177555 | 89,395 |
https://mathoverflow.net/questions/177556 | 0 | I have a function $f(n)$ that satisfies the following property: for any function $g(n) = o(n^{-2})$, we have $f(n) = \Omega(g(n))$ (the implied proportionality constant in the $\Omega$ expression will, naturally, depend on the function $g$). Is there any easy way to characterize this in this kind of notation, e.g. $f(n... | https://mathoverflow.net/users/56651 | Function that dominates everything in little o | Assume that $f(n)=o(n^{-2})$. Then $g(n):=n^{-1}\sqrt{f(n)}=o(n^{-2})$, so $f(n)=\Omega(g(n))$, so $\sqrt{f(n)}=\Omega(n^{-1})$, so $f(n)=\Omega(n^{-2})$, a contradiction. This proves that $f(n)=\Omega(n^{-2})$.
P.S. Here I am using the analytic number theory convention of the $\Omega$ notation, as discussed [here](h... | 1 | https://mathoverflow.net/users/11919 | 177557 | 89,396 |
https://mathoverflow.net/questions/177562 | 4 | Let $V$ and $W$ be topological vector spaces over $\mathbb{F}$ (with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$), and let $T:V \to W$ be a linear transformation. It is well-known that $T$ is not necessarily continuous. But is $T$ necessarily measurable (with respect to the Borel structures of $V$ and $W$)? Does the answer... | https://mathoverflow.net/users/15570 | Are all linear transformations measurable? | This question is answered by the well-known construction of a non-continuous linear form on an infinite dimensional Banach space using Hamel bases. Note also that there is a measurable graph theorem (L. Schwartz) which implies that all measurable linear maps, say between separable Banach spaces, are continuous. And the... | 6 | https://mathoverflow.net/users/53122 | 177563 | 89,400 |
https://mathoverflow.net/questions/177507 | 2 | Let $k$ be an algebraically closed field of characteristic $p>0$. Let $L$ be the extension
of $k(t)$ obtained by attaching a root of an irreducible polynomial $f\in k(t)[x]$.
Is there a way to tell from the form of $f$ when the extension $L/k(t)$ is unramified at all
the finite places of $k(t)$? (i.e. the places genera... | https://mathoverflow.net/users/9304 | Which polynomials define extensions of $k(t)$ unramified at the finite places | Presumably you know that $f$ is separable over $k(t)$ or else you wouldn't pose the question. Scale $x$ by $k(t)^{\times}$ so that $f$ becomes $x$-monic in $k[x,t]$ with $df/dx \ne 0$. Now the irreducible plane curve $C := \{f=0\}$ inside the $(x,t)$-plane over $k$ has projection to the affine $t$-line that is *finite*... | 0 | https://mathoverflow.net/users/52824 | 177565 | 89,402 |
https://mathoverflow.net/questions/177526 | 3 | Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P\_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X\_n=0\}$ and a hypersurface $X$. We want to study the image of $X$ under the projection from $Q$ to $H$. If $Q\notin X$ everything is clear and every point $(x... | https://mathoverflow.net/users/51272 | Projection of a hypersurface from a point | I agree that this question is perhaps not suitable for MO. Anyway, as you received no satisfactory answers on MSE, let me give you some examples that (I hope) can improve a bit your understanding of the situation. You are strongly encouraged to fill the details by yourself.
**Example 1.** Take a smooth quadric surfac... | 4 | https://mathoverflow.net/users/7460 | 177576 | 89,407 |
https://mathoverflow.net/questions/177569 | 1 | I have a sequence $(u\_k) \in L^2\_{loc}(\mathbb{R}^+; H^1\_0(\Omega) )$ and $u \in L^2\_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : $\int\_{[0;T]\times K} |u\_k(t,x)-u(t,x)|^2dtdx \rightarrow 0$ when $k \rightarrow \infty$
The book I am reading wants to pr... | https://mathoverflow.net/users/48151 | Weak convergence of a sequence | I guess the only delicate point is to retrieve (weak) convergence for the whole sequence, not only along some subsequence (which is trivial by the Banach Alaoglu theorem since $u\_k$ is bounded in $L^2(0,T;H^1\_0)$). A classical separation argument would do the trick here: if any converging subsequence converges to the... | 1 | https://mathoverflow.net/users/33741 | 177578 | 89,408 |
https://mathoverflow.net/questions/177591 | 2 | Let $F$ be a free finitely generated group, $H \leq F$ of infinite index. Let $c : F \rightarrow \hat{F}$ be the embedding in the profinite completion. Denote by $\tilde{F}, \tilde{H}$ the closure of $c(F), c(H)$ respectively. Is it possible that $[\tilde{F} : \tilde{H}] < \infty$?
Does it change anything if $H$ is f... | https://mathoverflow.net/users/38889 | Can a closure make the index finite? | If $H$ is fg then by Marshall Hall's theorem it is closed and so $\overline{H}\cap F=H$. But intersecting a finite index closed subgroup (=open subgroup) with $F$ gives a finite index subgroup. So this is impossible if $H$ is fg.
**Added**. If $H$ is infinitely generated then the closure could be finite index. Choos... | 5 | https://mathoverflow.net/users/15934 | 177593 | 89,414 |
https://mathoverflow.net/questions/177597 | 3 | Let $H$ be an $\infty$-dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
Let $\mathcal{A}$, $\mathcal{B} \subset B(H)$ be ${\rm II}\_1$-factors such that $\mathcal{A}'$, $\mathcal{B}'$ are also ${\rm II}\_1$-factors and $\mathcal{A} \cap \mathcal{B} = \mathbb{C}$.
*Examples*:
(1)... | https://mathoverflow.net/users/34538 | ${\rm II}_1$-factors with finite commutant and trivial intersection generate $B(H)$? | No, this is trivially false. Start with $\mathcal{A}, \mathcal{B} \subset B(H)$ that are not a counterexample and define $$\mathcal{A}^{(2)} = \{A \oplus A \in B(H \oplus H): A \in \mathcal{A}\}$$ and $$\mathcal{B}^{(2)} = \{B \oplus B \in B(H \oplus H): B \in \mathcal{B}\}.$$ They and their commutants are still $II\_1... | 3 | https://mathoverflow.net/users/23141 | 177609 | 89,418 |
https://mathoverflow.net/questions/177612 | 4 | Let $K$ be an extension of $\mathbb{Q}\_p$.
By local class field theory, the $p$-adic cyclotomic character $\mathrm{Gal}\_K \rightarrow \mathbb{Z}\_p^\times$ corresponds to a character $\chi : K^\times \rightarrow \mathbb{Z}\_p^\times$.
>
> Is there an explicit description of $\chi$?
>
>
>
When $K=\mathbb{Q}\_... | https://mathoverflow.net/users/39091 | Cyclotomic character in class field theory | If $K/L$ is an extension of fields, then the natural map $\operatorname{Gal}(K)^{ab} \to \operatorname{Gal}(L)^{ab}$ corresponds in class field theory to the norm map $K^\times \to L^\times$. So you just want to take the norm from $K$ to $\mathbb Q\_p$ and compose it with the character you describe.
| 8 | https://mathoverflow.net/users/18060 | 177615 | 89,422 |
https://mathoverflow.net/questions/177621 | 6 | Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic filters $\{ G\_r : r \in \mathbb{R} \}$?
If CH holds, then the answer is yes. Recursively on the countable ordinals, we... | https://mathoverflow.net/users/11145 | continuum many mutually generic filters | I think you can handle this by slightly modifying the construction in the last paragraph of your question. In addition to what you did there, enumerate all the dense subsets of $\mathbb P\times\mathbb P$, say as $E\_n$ ($n\in\omega$). Do this in such a way that every dense open set occurs infinitely often in the enumer... | 8 | https://mathoverflow.net/users/6794 | 177629 | 89,428 |
https://mathoverflow.net/questions/177626 | 7 | Let $G$ be a torsion-free, finitely-generated, nilpotent group of nilpotency class at least 3. Does there exist a normal subgroup
$N\leq G$ such that $G/N\cong \mathbb{Z}$ and $Z(G)=Z(N)$? (By $Z(H)$ I mean the center of the group $H.$)
The basic examples I've played with have this property, but I'm no group theori... | https://mathoverflow.net/users/34640 | Subgroups of Nilpotent groups with prescribed center | Here are two examples. I describe it as Lie algebras (over any field $K$).
(1) The 7-dimensional, 3-step nilpotent Lie algebra with basis $(X\_1,\dots,X\_7)$ and nonzero brackets
$$ [X\_1,X\_2]=X\_4,[X\_1,X\_3]=X\_5,[X\_2,X\_3]=X\_6,[X\_1,X\_4]=[X\_1,X\_5]=[X\_2,X\_4]=[X\_3,X\_6]=X\_7$$
(2) The 6-dimensional, 4-s... | 4 | https://mathoverflow.net/users/14094 | 177633 | 89,431 |
https://mathoverflow.net/questions/176698 | 15 | I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way close to the heat kernel in $\mathbb{R}^n$: that is, the probability $p\_t(\theta)$ that we have moved an angle $\theta$... | https://mathoverflow.net/users/56261 | heat kernel on n-sphere | Yes, it is true that $\theta$ on $S\_n$ is dominated by $\theta$ on $\mathbb{R}^n$.
Let $(B\_t)$ be a Brownian motion on the sphere. The radial process $\theta\_t=d(x,B\_t)$ is a Jacobi process, that is a Markov process with generator
$
L=\frac{n-1}{2} \text{cotan} (r) \frac{d}{dr} +\frac{1}{2} \frac{d^2}{dr^2 }
$
... | 7 | https://mathoverflow.net/users/48356 | 177655 | 89,440 |
https://mathoverflow.net/questions/177657 | 9 | Whilst reading Hartshorne's appendix C I came across the comparison theorem for etale cohomology and singular cohomology:
Let $X$ be a smooth projective variety over a number field $K$ and $\ell$ a prime number. Fix an embedding of $K$ into $\mathbb C$. Then there is a "natural" isomorphism of $\mathbb C$-vector spac... | https://mathoverflow.net/users/56682 | Comparison of etale and singular cohomology for varieties over number fields | Yes. In fact what Artin proves in SGA4 exp XI thm 4.4 is that étale cohomology and singular cohomology agree for smooth schemes over $\mathbb{C}$ with finite coefficients. The statement you want will follow from this by taking inverse limits to get to $\mathbb{Z}\_\ell$ and then extending scalars to $\mathbb{Q}\_\ell$.... | 9 | https://mathoverflow.net/users/4144 | 177658 | 89,442 |
https://mathoverflow.net/questions/177660 | 5 | Let $V\subset\mathbb{C}^{n\times n}$ be a linear space consisting of $n\times n$ complex matrices. Say that $V$ is *nilpotent* if every matrix $v\in V$ is nilpotent; denote by $V^k$ the subspace spanned by all possible products $v\_1\ldots v\_k$ with $v\_i\in V$. The conjecture is:
>
> Assume that both $V^k$ and $V... | https://mathoverflow.net/users/nan | Die hard nilpotent spaces | You conjecture is not true. Let $P$ be the $3\times 3$ matrix
$$
\left(
\begin{array}{ccc}
0&1&0\\
0&0&1\\
0&0&0\end{array}
\right)
$$
which is nilpotent with $P^2\neq 0$.
Consider the subspace of $9 \times 9$ uppertriangular matrices spanned by
$$
A=\left(
\begin{array}{ccc}P&0&0\\0&P&0\\0&0&0\end{array}
\right),~~... | 15 | https://mathoverflow.net/users/38468 | 177662 | 89,444 |
https://mathoverflow.net/questions/177650 | 1 | Do there exists a finite capable p-group of class two with property:
1. $G=\langle x, y, Z(G)\rangle$, $|x|=|y|=p^n$ ,
2. $Z(G)$ is not cyclic.
3. $Z(G)$ is not subgroup of $\Phi(G)$, Frattini subgroup of $G$,
4. $G'$ is cyclic of order $p^n$,
A group $G$ is capable if there exists a group $H$ such that $G\cong\dfr... | https://mathoverflow.net/users/56678 | Existence a finite capable p-group of class two | I am assuming $p$ is odd; a similar example is possible with $p=2$.
Let $ H = \langle x,y \mid x^{p^n} = y^{p^n} = [y,x]^{p^n} = [y,x,y] = [y,x,x] = 1\rangle$. Let $K = C\_p\times C\_p$. Then both $H$ and $K$ are capable, and hence so is $G=H\times K$. Capability if $K$ is trivial, and follows from the classical the... | 4 | https://mathoverflow.net/users/3959 | 177666 | 89,445 |
https://mathoverflow.net/questions/59906 | 9 | The abc conjecture asserts that whenever $a,b,c$ are pairwise coprime positive integers such that $a + b = c$ and $\epsilon > 0$, there exists a constant $C\_\epsilon > 0$ (which depends on $\epsilon$ but not on $a,b,c$) such that if $N(a,b,c) = \displaystyle \prod\_{p | abc} p$ is the radical of $a,b,c$, we have
$$\... | https://mathoverflow.net/users/10898 | A question related to the abc conjecture | For any $n$ the number of relatively prime $a,b<n$ such that
$n > ({\rm rad}(ab))^{1+\epsilon}$ is $o(n)$, indeed $o(n^{1-\epsilon'})$
for any $\epsilon' < \epsilon / (1+\epsilon)$. Therefore this bound
is true *a fortiori* of the number of such $a,b$ for which $a+b=n$.
We use the following lemma:
*For all $\delta ... | 10 | https://mathoverflow.net/users/14830 | 177670 | 89,446 |
https://mathoverflow.net/questions/177673 | 2 | Let $E$ be homotopy equivalent to a $k$-sphere. Let $q\colon E\to X$ be a map such that given any continuous $f\colon C\to X$ from a compact space $C$, there exists (a non-unique) $\tilde{f}\colon C\to E$ with $q\tilde{f}=f$. Assume also that $X$ is a connected CW complex, but possibly infinite-dimensional.
Can we sa... | https://mathoverflow.net/users/48208 | Map from homotopy sphere with lifting property induces surjections on homotopy groups. Is it weak equivalence? | Yes, we can conclude that either $q$ is an equivalence or $X$ is contractible. Since any cycle lives in a compact subset of $X$, $q$ will also induce surjections on homology. It follows that $X$ is a Moore space $M(\mathbb{Z}/n,k)$ for some $n$, and $q$ is homotopy equivalent to the unique map $S^k\to M(\mathbb{Z}/n,k)... | 7 | https://mathoverflow.net/users/75 | 177675 | 89,447 |
https://mathoverflow.net/questions/177690 | 1 | Let $G=H\times T$ such that $H$ is 2-generated p-group of class two and $H$ be abelian p- group. We know if $H$ and $T$ are capable groups, then $G$ is capable.
**Question:** Is the converse correct or no? i.e
If $G$ is a capable group then are $T$ or $H$( both $H$ and $T$) capable?
| https://mathoverflow.net/users/56678 | On direct product of capable groups | Let $H$ be extraspecial of order $p^3$ and exponent $p$ (for odd $p$), and $T=C\_p$. Then $G=H \times T$ is capable but $T$ is not.
We have $G = K/Z(K)$, where
$$K= \langle a,b,c,d,e,f \mid [a,b]=c,[a,c]=e,[b,c]=f, [a,d]=e,
[b,d]=[c,d]=1,$$ $$a^p=b^p=d^p=1, e,f {\rm\ central} \rangle.$$
| 2 | https://mathoverflow.net/users/35840 | 177698 | 89,455 |
https://mathoverflow.net/questions/177590 | 4 | The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal projectors on $\mathcal{U}$ and $\tilde{\mathcal{U}}$, and the norm is the Euclidean norm $\|M\|:=\sigma\_{\max}(M)$ (l... | https://mathoverflow.net/users/1898 | Sensitivity of the range of a matrix | Quick self-answer for future reference: there is a brief statement of this result on Stewart, Sun, *Matrix perturbation theory*, p. 154:
>
> If the hypotheses of Corollary 3.13 are satisfied (that is, when $\|A\_{11}^{-1}\|\_2\|E\_{11}\|<1$), then we may replace $\hat{\kappa}$ by $\kappa/\gamma$ in (4.1). Thus, $\k... | 3 | https://mathoverflow.net/users/1898 | 177699 | 89,456 |
https://mathoverflow.net/questions/177696 | 3 | By the Wiener algebra I mean the algebra of functions on the circle group having absolutely convergent Fourier series. Is there an analogue of Wiener algebra for nonabelian locally compact groups?
I am mainly interested in the compact case, but I will leave the question like this in case it may help someone else.
M... | https://mathoverflow.net/users/51431 | Wiener algebra for nonabelian locally compact groups | Yes there is an analogue, and yes it is what you describe in the compact case. It is generally accepted that the "correct" generalization of the Wiener algebra to the setting of locally compact groups is the *Fourier algebra*, as defined by Eymard following earlier work of Stinespring for unimodular groups.
See
> ... | 4 | https://mathoverflow.net/users/763 | 177703 | 89,458 |
https://mathoverflow.net/questions/177714 | 6 | Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$.
Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$.
Is $\limsup\_{n\rightarrow\infty} \frac{f(n)}{n} > 0$. It is less than $\frac{1}{2}$ by my previous statement, but I don't know how to proceed.... | https://mathoverflow.net/users/40983 | Probability that a positive integer is the euler phi function of another positive integer | See Erick Wong's response [here](https://math.stackexchange.com/questions/291334/values-taken-by-eulers-phi-function). In particular, Kevin Ford proved (in more precise form) that
$$ f(n) = \frac{n}{\log n} \exp\left(O(\log \log \log n)^2\right),$$
whence $f(n)/n$ tends to zero. The same consequence also follows from a... | 14 | https://mathoverflow.net/users/11919 | 177715 | 89,461 |
https://mathoverflow.net/questions/177720 | 0 | Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable?
What proportion of the plane does not have a particle on it after time t? Ie. pick n random points inside an open disc, as n approaches infinity, what fra... | https://mathoverflow.net/users/56718 | Brownian motion of every point in the plane | Presumably you mean you have continuum-many independent Brownian motions, one (call it $W\_p(t)$) with $W\_p(0) = p$ for each $p$. Unfortunately I'm pretty sure the number of $p$ for which $W\_p(t) = 0$ is not a measurable function, so your
question does not have an answer.
| 5 | https://mathoverflow.net/users/13650 | 177724 | 89,464 |
https://mathoverflow.net/questions/177736 | 0 | I've been reading [1] and attempting to prove statements given without proof. In the paper the authors construct a measurable space of measures over a base space, and as an aside show an elegant way to lift measurable functions from the base space into measurable functions in this new space. Perhaps a bit worrying is t... | https://mathoverflow.net/users/3627 | Measurable functions lifted onto a space of point measures are measurable | As you have shown the case of indicator functions you get the case of positive measurable functions $f$ by linearity of the integrals and Levi's theorem on monotone convergence: There is a sequence $0\le f\_n= \sum\limits\_{k=1}^{m(n)} a\_{n,k} 1\_{F\_{n,k}}$
such that $f\_n \le f\_{n+1}\to f$. Hence you get that
$$
[f... | 1 | https://mathoverflow.net/users/21051 | 177739 | 89,471 |
https://mathoverflow.net/questions/177748 | 7 | Is there any self-dual lattice $(X,\le)$ such that there is **not** any self-duality $f:X\to X$ such that $f\circ f = 1\_X$?
| https://mathoverflow.net/users/47958 | Self-duality in a lattice | Yes. Let $L$ be the lattice structure on $\mathbb Z$ with the following Hasse diagram:
```
-6 <----- -2 <---- 2 <---- 6 <---
\ / \ / \ / \
... -5 -3 -1 1 3 5 7 ...
\ / \ / \ / \
---> -4 -----> 0 ----> 4 ----> 8
```
where all the diagonal arrows go u... | 8 | https://mathoverflow.net/users/12705 | 177751 | 89,474 |
https://mathoverflow.net/questions/177617 | 12 | Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find?
In the adjoint representation, one can express the Killing form as a matrix and consider it as an inner product on $\mathfrak{su}(4)$. This matrix is some multiple of the identity (i.e. a scalar matrix). The geode... | https://mathoverflow.net/users/41654 | Geodesics on $SU(4)$ | In the OP's particular case, the situation is somehwat simpler than the general case that José discusses. That's because the family of left-invariant metrics on $\mathrm{SU}(4)$ that the OP wants to consider has special properties, although just how special does not become apparent until one looks at the problem from a... | 19 | https://mathoverflow.net/users/13972 | 177758 | 89,475 |
https://mathoverflow.net/questions/177538 | 3 | Let $T$ be a compact metrizable space. Consider a centered second order measurable process $(X\_t\colon t\in T)$ with continuous covariance function $c(t,s):= \mathbb{E}X\_t X\_s$.
Are there any known sufficient conditions ensuring that $t \mapsto X\_t$ lies almost surely in the reproducing kernel Hilbert space $\mat... | https://mathoverflow.net/users/46683 | When does a stochastic process have its sample paths a.s. in the reproducing kernel hilbert space (RKHS) induced by its covariance function? | If $T$ is an infinite set and the Gaussian measure is non-degenerate, then the RKHS (Cameron Martin space, in Bogachev's language) is infinite dimensional; hence $\operatorname{Prob}(X \in \mathcal H(c)) = 0$.
If $T$ is a finite set, then $\operatorname{Prob}(X \in \mathcal H(c)) = 1$.
For both results, see Bogachev,... | 6 | https://mathoverflow.net/users/22157 | 177762 | 89,476 |
https://mathoverflow.net/questions/177759 | 21 | Let $n$ be a natural number whose prime factorization is
$$n=\prod\_{i=1}^{k}p\_i^{\alpha\_i} \; .$$
Define a function $g(n)$ as follows
$$g(n)=\sum\_{i=1}^{k}p\_i {\alpha\_i} \;,$$
i.e., exponentiation is "demoted" to multiplication,
and multiplication is demoted to addition.
For example: $n=200=2^3 5^2$, $f(n) = 2 \c... | https://mathoverflow.net/users/6094 | Prime factorization "demoted" leads to function whose fixed points are primes | A way to get a non-trivial solution to $f(n) = p$ is that every odd number $\geq 7$ can be written as a sum of three primes (by Helfgott's recent work), so if $p \geq 7$ is prime, we can write $p = q + r + s$, and we have $g(qrs) = q + r + s = p$. (This is of course a bit overkill, we don't really need such a difficult... | 23 | https://mathoverflow.net/users/48142 | 177763 | 89,477 |
https://mathoverflow.net/questions/177753 | 6 | I am studying the non-commutative torus $ A\_{\theta} $.
When $ \theta $ is irrational, $ {K\_{0}}(A\_{\theta}) $ is generated by $ [1] $ and $ [p\_{\theta}] $.
(**Note:** $ p\_{\theta} $ is a projection in $ A\_{\theta} $, called the *Powers-Rieffel projection*, and satisfies $ \tau(p\_{\theta}) = \theta $, where ... | https://mathoverflow.net/users/47294 | Generators of the $ K_{0} $-group of the non-commutative torus $ A_{\theta} $ with $ \theta \in \mathbb{Q} $ (i.e. rational rotation algebra) | You should be able to find the construction in this paper by Rieffel
**"The cancellation theorem for projective modules over irrational rotation C∗-algebras, Proc. London Math. Soc. 47(1983), 285–302"**
For more general (higher dimensional) rotation algebras, look in Rieffel's paper **"Projective modules over higher ... | 5 | https://mathoverflow.net/users/22781 | 177766 | 89,478 |
https://mathoverflow.net/questions/177760 | 4 | Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism $X\to Y$ is finite (not necessarily etale of degree two)) with the property that $\# \mathrm{Aut}(Y) > \# \mathrm{Aut}(X)$.
... | https://mathoverflow.net/users/56731 | Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms | Sure: let $Y$ be your favorite hyperelliptic curve $u^2=f(t)$
with many automorphisms, and let $X$ be the curve $u^2=f(t(s))$
for some "random" rational function $f$ of degree at least $2$.
For example, let $Y$ be the genus-2 curve $u^2 = t^5-t$
(so $\#({\rm Aut}(Y)) = 2\#(S\_4) = 48$); and let $X$ be
$u^2 = P(s)^5 Q... | 5 | https://mathoverflow.net/users/14830 | 177769 | 89,479 |
https://mathoverflow.net/questions/177686 | 2 | Let $\mathcal{A} , \mathcal{B} \subset B(H)$ be ${\rm II}\_1$-factors such that $\mathcal{A}', \mathcal{B}' $ are also a ${\rm II}\_1$-factors.
**Question**: $\mathcal{A} \cap \mathcal{B} = \mathbb{C} \, \, \Rightarrow \, \, \mathcal{A}' \cap \mathcal{B}' $ [hyperfinite](http://en.wikipedia.org/wiki/Von_Neumann_alge... | https://mathoverflow.net/users/34538 | ${\rm II}_1$-factors with finite commutant: $\mathcal{A} \cap \mathcal{B} = \mathbb{C} \Rightarrow \mathcal{A}' \cap \mathcal{B}'$ hyperfinite? | The answer is no. If $N\_1$ and $N\_2$ are both finite index subfactors of a nonamenable ${\rm II}\_1$ factor $M \subset \mathcal B(L^2M)$ such that $N\_1 \cap N\_2 = \mathbb C$, then $N\_1'$ and $N\_2'$ are both finite and $N\_1' \cap N\_2'$ is nonamenable since it contains $M'$.
For an example of such a situation c... | 6 | https://mathoverflow.net/users/6460 | 177783 | 89,484 |
https://mathoverflow.net/questions/177757 | 3 | Suppose we have a finite group $G$ with subgroup $H$, a representation $\rho\_V$ of $H$ on a finite-dimensional vector space $V$, and an $H$-invariant inner product on $V$:
$$\forall x,y\in V, h\in H,\enspace \langle\rho\_V(h)x, \rho\_V(h)y\rangle = \langle x,y\rangle$$
We will write $V\_I$ for the direct sum of $\... | https://mathoverflow.net/users/23829 | Inclusion of copies of an irrep as orthogonal subspaces of an induced representation | I'm going to assume that the field is $\mathbb{C}$ (what I'm about to say will be wrong over $\mathbb{R}$. The space $Hom\_H(W,V)$ has an inner product given by $\langle f,g\rangle=\sum \langle f(e\_i),g(e\_i)\rangle$ for $e\_i$ an orthonormal basis of $W$. If you write $f,g$ as matrices in orthonormal bases of $V$ and... | 1 | https://mathoverflow.net/users/66 | 177784 | 89,485 |
https://mathoverflow.net/questions/177102 | 9 | I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real solutions. The equations were derived as the gradients of a sum-of-squares cost function, which I am attempting to find all ... | https://mathoverflow.net/users/19899 | Software tools for medium-scale systems of polynomial equations | In Maple you can just do
```
with(Optimization):
g := (your function):
Minimize(g,iterationlimit = 200);
```
On my machine this takes only about 1.5 seconds to return the following:
```
[2.35579022955789696*10^(-9),
[x0 = .696531801759957, x1 = .286105658731833, x10 = .342973444356395,
x11 = .72873251053287... | 6 | https://mathoverflow.net/users/10366 | 177787 | 89,487 |
https://mathoverflow.net/questions/177789 | 17 | Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.
| https://mathoverflow.net/users/38889 | Is $x^{n}-x-1$ irreducible? | This is true; it is due to Selmer. Ljunggren (*On the irreducibility of certain trinomials and quadrinomials*, Math. Scand. 1960) has obtained the complete list of reducible trinomials with $\pm 1$ coefficients. For more details see Gerry Myerson's answer to this question, which contains a review of Ljunggren's paper: ... | 29 | https://mathoverflow.net/users/26522 | 177793 | 89,491 |
https://mathoverflow.net/questions/177747 | 11 | I want to write a GAP program for checking the following question.
Let $G$ be a given finite group with order $n$. Is it true that for every factorization $n=ab$ there exist subsets $A$ and $B$ such that $|A|=a$, $|B|=b$ and $G=AB$?
We have many candidates for a counterexample such as $PSL(2,8)$, $PSL(2,11)$, $PSL... | https://mathoverflow.net/users/40520 | Factorization of a finite group by two subsets | There are subsets $A$ and $B$ of orders $21$ and $24$ of $G=PSL(2,8)$ with $G=AB$. A naive search of course does not work. However, in trying to find first $A$ with $A^{-1}A$ being small, a good candidate to work with is $A=G\_7G\_3$, where $G\_3$ and $G\_7$ are subgroups of order $3$ and $7$. The smallest possible siz... | 5 | https://mathoverflow.net/users/18739 | 177797 | 89,494 |
https://mathoverflow.net/questions/117948 | 8 | (A somewhat technical question, but maybe it is well known.)
Consider matrices over the ring $k[[x\_1,\dots,x\_n]]$, whose entries vanish at the origin (i.e. belong to the maximal ideal $\mathfrak{m}$). Denote by $J(A\_{k,l})$ the ideal of maximal minors of the matrix $A\_{k,l}\in Mat(k,l,\mathfrak{m})$.
Given two... | https://mathoverflow.net/users/2900 | When two determinantal ideals together generate a power of the maximal ideal? | Let me discuss the graded case, that is the ring is the polynomial ring and the matrices have general homogeneous entries of degree $1$. The local version should follows by taking "lowest order part" as you do in example $1$.
Consider first of the case of the polynomial ring $S$ in variables $x\_{ij}$ and $y\_{ij}$ ... | 3 | https://mathoverflow.net/users/48585 | 177808 | 89,499 |
https://mathoverflow.net/questions/177816 | 5 | I remember hearing some time ago that there is a locally compact Hausdorff space $X$ and a non-Borel subset $E$ which intersects every compact set in a Borel set. (This would contradict Lemma 13.9 of Royden, Real Analysis 3rd edition 1988, which is stated without proof).
Is there a reference for this? Can this happen... | https://mathoverflow.net/users/20300 | non-Borel set which intersects every compact in a Borel set | If I understand the question, then you are correct, there is such a space. I'll sketch what I hope is a correct argument.
Take $X=\coprod\_AY$ for some fixed locally compact Hausdorff space $Y$ and some index set $A$. As long as $Y$ is sufficiently complicated (probably $Y=\mathbb R$ would work) and $A$ is sufficient... | 4 | https://mathoverflow.net/users/35353 | 177817 | 89,504 |
https://mathoverflow.net/questions/177574 | 11 | Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x\_0, \ldots, x\_k)$ be a $(k+1)$-dimensional *real* vector, and define
$$S(p,x) = -x\_0^2 + \sum\_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} \cdot (x\_i - p)^2.$$
Experiments show that for small values of $k$
$$\exists x \in \mathbb{R}^{k+1} \,.\, \forall ... | https://mathoverflow.net/users/1176 | Existence of solutions of a polynomial system | The solutions described via the link <http://winvector.github.io/freq/explicitSolution.html> (posted in one of the earlier answers) can be given by the following formula:
$$
x\_i=\frac{(k-2i)\sqrt{k}+(2i-1)k}{2k(k-1)}=\frac{1}{2(1+\sqrt{k})}+\frac{i}{\sqrt{k}(1+\sqrt{k})}.
$$
Note that (when $k$ is fixed):
* $x\_i$... | 6 | https://mathoverflow.net/users/1306 | 177820 | 89,507 |
https://mathoverflow.net/questions/177776 | 13 | There are plenty of popular NP-hard puzzles,
for example, generalized Sudoku ($n^2 \times n^2$-board), [Flow](http://html5games.com/2012/07/flow-free/) (I cannot give a source for this), Minesweeper, etc.
Recently, I read a bit about aperiodic tilings of the plane, and it is undecidable
whether a set of tiles can ti... | https://mathoverflow.net/users/1056 | Undecidable puzzles | Typing "undecidable" and "puzzle" into MathSciNet turned up a couple of candidates.
Baumeister, Dorothea and Rothe, Jörg, The three-color and two-color Tantrix rotation puzzle problems are NP-complete via parsimonious reductions, Inform. and Comput. 207 (2009), no. 11, 1119–1139, [MR2566946](https://mathscinet.ams.o... | 8 | https://mathoverflow.net/users/3684 | 177827 | 89,509 |
https://mathoverflow.net/questions/177768 | 6 | The question of knowing whether there are infinitely many Fibonacci primes is an open question. As $F\_p$ is prime only if $p$ is prime, one has $\pi\_{FP}(x)\le \pi(\log\_{\phi} x+0.5\log 5)$, but numerical computations seem to show that this quantity is roughly equal to $\log\_{\phi}\log\_{\phi}x$, where $\pi\_{FP}(x... | https://mathoverflow.net/users/13625 | is there any heuristics suggesting that the number of Fibonacci primes below $x$ is equivalent to $\log_{\phi}\log_{\phi}x$? | We need to find a reasonable-sounding answer for the following question: for a fixed prime $p$; what is the probability that $F\_p$ is prime?
A prime $q<F\_p$ divides $F\_p$ if and only if $z(q)$ divides $p$, where $z$ is the classical [Fibonacci entry point](http://oeis.org/A001177). Therefore, we must have $z(q)=p... | 9 | https://mathoverflow.net/users/nan | 177839 | 89,514 |
https://mathoverflow.net/questions/177847 | 3 | I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the (non-degenerate) quadratic forms on the $3$-dimensional $p$-adic vector space ${\mathbb Q}\_p^3$, one compact and one non-compact. T... | https://mathoverflow.net/users/3635 | $p$-adic analogues of $\mathrm{SO}(3)$ | If $q$ is a quadratic form over a field $k$, the orthogonal group $\mathrm{O}(q)$ doesn't change if you replace $q$ by $\lambda q$ for $\lambda\in k^\*$. In rank 3, since $\lambda ^3\equiv \lambda $ (mod. $k^{\*2}$), this implies that you can consider only forms of discriminant 1. Over $\mathbb{Q}\_p$ $(p>2)$ this leav... | 6 | https://mathoverflow.net/users/40297 | 177850 | 89,518 |
https://mathoverflow.net/questions/177857 | 2 | This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators.
We want to look at matrices that agree in most of their entries and want to investigate whether this implies convergence of the eigenvalues.
We start with two matrices $$ A\_1:=\left[ \b... | https://mathoverflow.net/users/nan | Alike looking matrices imply convergence of eigenvalues? | This is only a partial answer, but if you instead keep the diagonal constant, what you have is (essentially) a sequence of Toeplitz matrices,
determinants of such matrices satisfy linear recursions.
I gave a combinatorial proof of this statement [here](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i... | 2 | https://mathoverflow.net/users/1056 | 177858 | 89,521 |
https://mathoverflow.net/questions/177800 | 5 | I noticed something interesting studying this Sturm-Liouville Problem:
$$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + \lambda\right)f}{\sqrt{(1-x^{2})}}= 0,$$
where $\alpha \in \mathbb{R}$ with periodic boundary conditions on $[-1,1]$.
(Lambda is the eigenvalue)
Fr... | https://mathoverflow.net/users/nan | Spectrum of this ODE | Let's rewrite your equation as a Schrödinger equation, as follows: Introduce the new variable $t\in (-\pi/2, \pi/2)$ by $x=\sin t$. Then if $f$ solves your boundary value problem (with periodic boundary conditions), then $y(t)=f(x(t))=f(\sin t)$ satisfies
$$
-y'' +V(t)y = \lambda y , \quad\quad y(-\pi/2)=y(\pi/2),\quad... | 3 | https://mathoverflow.net/users/48839 | 177864 | 89,525 |
https://mathoverflow.net/questions/177829 | 4 | For an additive category $\mathcal C$ there is the notion of a Serre functor on $\mathcal C$, i.e. a an autoequivalence $S$ of $C$ such that there exist isomorphisms
$$Hom(A, S(B)) \cong Hom(B, A)^\*$$
natural in $A,B \in \mathcal C$.
If $\mathcal D$ is a full subcategory of $\mathcal C$ that is preserved by $S$, whe... | https://mathoverflow.net/users/459 | Serre functor of a subcategory (in particular parabolic category O) | It's not true that $D^b(\mathcal{O}^{\mathfrak{p}})$ is a full subcategory of $D^b(\mathcal{O})$. Think about the case $\mathfrak{g}=\mathfrak{p}=\mathfrak{sl}\_2$. The category $\mathcal{O}^{\mathfrak{p}}$ is finite-dimensional modules and thus semi-simple, whereas in $D^b(\mathcal{O})$, we have that $\mathrm{Ext}^2(\... | 8 | https://mathoverflow.net/users/66 | 177865 | 89,526 |
https://mathoverflow.net/questions/177849 | 11 | This is a second attempt (see [Primes $p$ such that $432 p +1$ is prime](https://mathoverflow.net/questions/177846/primes-p-such-that-432-p-1-is-prime))
Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite?
Fact: the number of such numbers $n$ such that $n\leq 10^6$ is... | https://mathoverflow.net/users/56775 | Squarefree numbers $n$ such that $432n+1$ is also squarefree | Here is a quick proof that the density in question exists and equals
$$ c:=\frac{2}{3}\prod\_{p\geq 5}\left(1-\frac{2}{p^2}\right)\approx 0.553087\ . $$
Let $f(d)$ denote the number of solutions of the congruence $n(432n+1)\equiv 0\pmod{d}$. Note that, for $p$ prime, $f(p^2)=1$ when $p<5$ and $f(p^2)=2$ when $p\geq 5... | 21 | https://mathoverflow.net/users/11919 | 177871 | 89,528 |
https://mathoverflow.net/questions/177867 | 0 | Let $\mu$ denote the Möbius function, and let's define the functions $\mu\_{-}$ and $\mu\_{+}$ as follows: $\mu\_{-}(n):=\frac{\mu(n)^{2}-\mu(n)}{2}$ and $\mu\_{+}(n):=\frac{\mu(n)^{2}+\mu(n)}{2}$. Let $M\_{-}$ be the summatory function of $\mu\_{-}$ and $M\_{+}$ the summatory function of $\mu\_{+}$. One has $M(x)=M\_{... | https://mathoverflow.net/users/13625 | Best error terms for functions related to square free numbers | As I say in the comment, the asymptotics for $M\_{+}$ and $M\_{-}$ follow directly from those for $M$ and $\hat{M}$. Therefore $M\_{+}(x) = \frac{1}{2 \zeta(2)} x + \frac{1}{2} M(x) + O(x^{1/2})$ and $M\_{-}(x) = \frac{1}{2 \zeta(2)} x - \frac{1}{2} M(x) + O(x^{1/2})$. These error terms will have the same order of magn... | 4 | https://mathoverflow.net/users/48142 | 177884 | 89,536 |
https://mathoverflow.net/questions/177877 | 4 | I am interested in the following situation: given a braid $B$, it induces a link $L$ in a pretty straightforward way ("glue" the endpoints, like [here](http://www.maths.ed.ac.uk/~jcollins/SeifertMatrix/braid.png)). For a braid $B$, we know how to represent it in a normal form, which provides in particular a complete in... | https://mathoverflow.net/users/48499 | Computable link invariants | There are two ways to interpret your question.
First -- since you are talking about braids, perhaps you are asking about the conjugacy problem for braids. This is the same as taking a braid closure, *with* an braid axis. In this case the ultra summit set (USS) of a braid $\sigma$ is a complete invariant for conjugac... | 8 | https://mathoverflow.net/users/1650 | 177885 | 89,537 |
https://mathoverflow.net/questions/177874 | 0 | Somewhere Colin M. Campbell noted:
>
> If $A$ is a semigroup defined as $$A=Sg(\pi)=\langle a\_1,\cdots, a\_d\mid u\_1=v\_1,\cdots,u\_e=v\_e\rangle $$ then the same generators with the same relations can also be interpreted as the presentation of the following group:
> $$A^\*=Gp(\pi)=\langle a\_1,\cdots, a\_d\mid ... | https://mathoverflow.net/users/13898 | Using group presentation for its corresponding semigroup? | You can do this in some examples. For example in braid groups or, more generally, Artin groups, you can just interpret the group presentation immediately as a semigroup (or monoid) presentation. In general, in that situation, the semigroup defined will not embed into the group. An obvious example is $\langle x \mid x^4... | 4 | https://mathoverflow.net/users/35840 | 177887 | 89,538 |
https://mathoverflow.net/questions/177889 | 3 | Is there a good reference for the proof that the cobordism group of pseudo-manifolds is isomorphic to the singular homology group?
I was looking for a more geometrical definition of homology and found [these notes](https://www.math.lsu.edu/~cohen/courses/PastSemesters/SPRING09/M7512/MacPherson.pdf) that give a more g... | https://mathoverflow.net/users/26709 | Pseudo-manifolds and homology | You should be aware of the book "A geometric approach to homology theory" by Buoncristiano, Rourke and Sanderson. However, I am not willing to claim that it is modern (1976) or easy to read.
| 7 | https://mathoverflow.net/users/10366 | 177891 | 89,540 |
https://mathoverflow.net/questions/177897 | 10 | Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove something about the rank of an elliptic curve is to look at the rank of the 2-Selmer group. Indeed, I was able to find a ref... | https://mathoverflow.net/users/56793 | Rank of Elliptic Curves | Cassels showed in "Arithmetic on curves of genus 1 VI" that there is no bound on the size of the 3-Selmer group for the curves of $x^3+y^3+dz^3=0$. Well, he actually showed that the 3-torsion of Sha can be arbitrarily large.
Similar results are known for other primes. [Tom Fisher](http://www.dpmms.cam.ac.uk/~taf1000... | 11 | https://mathoverflow.net/users/5015 | 177900 | 89,543 |
https://mathoverflow.net/questions/177899 | 1 | Let amenable groups $\Gamma$ and $\Gamma'$. They act outerly of *only one manner* on the hyperfinite ${\rm II}\_1$-factor $\mathcal{R}$.
**Question**: $(\mathcal{R} \subset \mathcal{R} \rtimes \Gamma) \simeq (\mathcal{R} \subset \mathcal{R} \rtimes \Gamma') \, \Rightarrow \, \Gamma \simeq \Gamma'$ ?
If $\Gamma$ i... | https://mathoverflow.net/users/34538 | Infinite amenable group subfactors | The answer to the first question is yes by a result of Herman and Ocneanu (MR1055223) or of Enock and Nest (MR1387518).
We call an inclusion of II$\_1$-factors depth 2 if $M\_0'\cap M\_2$ is abelian and $M\_0'\cap M\_3$ is a factor. (In fact, we only need to say $M\_0'\cap M\_3$ is a factor, and this works for arbit... | 3 | https://mathoverflow.net/users/351 | 177904 | 89,544 |
https://mathoverflow.net/questions/177777 | 5 | I am trying to understand the K-theory for the $C^\*-$algebra of the continuous functions on the $2-$dimensional torus $T^2$. In particular I am interested on the $K\_0-$group. I have read that the generators of the group $K\_0(C(T^2))$ are two elements: the unit $[1]$ and the Bott Projection $[Bott]$.
Unfortunately, I... | https://mathoverflow.net/users/47294 | K-theory for the $C^*-$algebra of the continuous functions on the $2-$torus and the Bott projection | If you want an explicit projection, you can form a variation of a Rieffel projection as follows. First take any function $f$ from $[-\pi/2,\pi/2]$ to $[0,1]$ that sends $-\pi/2$ to $1$, dips down to $0$ at $0$ and then goes back up to take value $1$ at $\pi/2$. Now define two more functions
$$
g=\begin{cases}
0 & x\in\... | 6 | https://mathoverflow.net/users/6133 | 177912 | 89,547 |
https://mathoverflow.net/questions/177653 | 16 | Let $M$ be a compact connected manifold.
Is there a chart $\Psi:U \to \mathbb{R}^n$ such that the closure of $U$ is $M$?
This is true for $S^n, T^n, K$, all compact surfaces, etc.
If it is not true in general, what is the obstruction?
| https://mathoverflow.net/users/16852 | Does every compact manifold exhibit an almost global chart | The exponential map for any Riemannian metric on your compact manifold $M$, based at any point $p$ of $M$, maps the tangent space $T\_p M$, an ${\mathbb R}^n$, onto $M$ and is a diffeo inside the cut locus. Back on the tangent space, this `inside' of the cut locus is a star shaped domain relative to the origin, so defi... | 6 | https://mathoverflow.net/users/2906 | 177913 | 89,548 |
https://mathoverflow.net/questions/177911 | 4 | I would like to know if Grothendieck published something about this conjecture?
Is there some book (or expository article) about this conjecture?
Is there any connection between this conjecture and others important conjectures (Tate, Hodge, Standadrd, Beilinson,...)?
| https://mathoverflow.net/users/nan | Reference request: Grothendieck´s period conjecture? | You'll find a detailed history of the conjecture, including a discussion of Grothendieck's original contribution, in [this paper](http://arxiv.org/pdf/1307.1045.pdf) of Bost and Charles.
| 9 | https://mathoverflow.net/users/40297 | 177914 | 89,549 |
https://mathoverflow.net/questions/177916 | 1 | In "The work of Tate" Milne says:
>
> "The relation between the two conjectures has been greatly clarified by the work
> of Deligne. He defines the notion of an absolute Hodge class on a (complete smooth)
> variety over a field of characteristic zero, and conjectures that every Hodge class on
> a variety over C ... | https://mathoverflow.net/users/nan | Reference request: Deligne's conjecture (cycles) | Deligne, *Hodge Cycles on Abelian Varieties*, in Hodge Cycles, Motives, and Shimura Varieties,
Lecture Notes in Math. 900, 1981, pp. 9-100. Maybe you could use Google to find these references by yourself?
| 6 | https://mathoverflow.net/users/40297 | 177917 | 89,550 |
https://mathoverflow.net/questions/177925 | 9 | Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.
In other words, if in a transitive permutation group, each element that fixes a point, fixes exactly one point, and there is at least one such eleme... | https://mathoverflow.net/users/42622 | Generalization of Frobenius groups | Yes, these groups exist for all $t$. To see that, let $G$ be a Frobenius group with complement $H$ of size $tu$ for some $u>1$, where $H$ has a normal subgroup $K$ of order $u$. We could, for example, choose $H$ to be a cyclic group of order $tu$, and we can do that for any $t$ and $u$. Then the image of the permutatio... | 14 | https://mathoverflow.net/users/35840 | 177926 | 89,552 |
https://mathoverflow.net/questions/177928 | 2 | Let $p: X \rightarrow T$ be a flat family of normal projective varieties over a variety $T$. Assume that $X\_{t\_{0}}=p^{-1}(t\_{0})$, for a $t\_{0}\in T$, has only canonical singularities of index $1$. Is there a neighborhood $U$ of $X\_{t\_{0}}$ such that any variety $X\_{t}$ belonging to $U$ has at worst canonical s... | https://mathoverflow.net/users/56818 | Deformation of Canonical singularities | When the base of the deformation is smooth the answer is *yes*. This was proven by Kawamata in his paper *[Deformations of canonical singularities](http://www.ams.org/journals/jams/1999-12-01/S0894-0347-99-00285-4/)*, Journal of the American Mathematical Society **12** (1999), 85-92.
For the reader's convenience, let... | 3 | https://mathoverflow.net/users/7460 | 177929 | 89,553 |
https://mathoverflow.net/questions/177939 | 3 | Let $p$ be a prime number, $m,n \in \mathbb{N}$, $F = F(p,m)$ be the free pro-$p$ group on $m$ generators. For which $(m,n)$ there is a continuous faithful representation (embedding) $\rho : F \rightarrow \text{GL}\_n(\overline{\mathbb{Q}\_p})$?
Note that every compact subgroup of $\text{GL}\_n(\overline{\mathbb{Q}\_... | https://mathoverflow.net/users/38889 | Faithful representations of free pro-p groups | It will only happen if $m=1$. See this paper:
<http://mlarsen.math.indiana.edu/~larsen/papers/2gen.pdf>
Indeed, the pro-$p$ groups that are linear over local fields of characteristic $0$ are just the pro-$p$ groups of finite subgroup rank.
Edit: As Ian Agol suggested, you should look at 'Analytic pro-p Groups' by... | 7 | https://mathoverflow.net/users/4053 | 177940 | 89,559 |
https://mathoverflow.net/questions/177932 | 2 | Let $(M, g)$ be a smooth Riemannian manifold, $p \in M$, and $\exp\_P$ the exponential map at the point $P$:
$\exp\_P: T(P) \to M$
It seems clear to me that $\exp\_P$ is smooth on $U \setminus \{0\}$, where $U$ is a neighborhood of the origin $0 \in T(P)$, because $\exp\_P$ is defined from the geodesics, which are so... | https://mathoverflow.net/users/56819 | Smoothness of the exponential map at the origin | As Thomas Rot already suggested: this follows directly from smooth dependence of ODEs on initial conditions.
Let $p \in M$ and denote by $\Phi^t\colon TM \to TM$ the geodesic flow. Then the exponential map at $p \in M$ is defined as the time one geodesic flow, restricted to $T\_p M$ and projected onto $M$, i.e.
$$
\... | 6 | https://mathoverflow.net/users/3928 | 177943 | 89,560 |
https://mathoverflow.net/questions/177947 | 5 | Let $G$ be a compact Lie group and $\mathcal{C}\_G$ the category of $G$-spaces (ie. topological spaces endowed with continuous left $G$-actions). Is there a model category structure on $\mathcal{C}\_G$ for which
(i) weak equivalences are the morphisms $f:X\rightarrow Y$ such that for all closed subgroups $H$ of $G$, ... | https://mathoverflow.net/users/25358 | Is the category of $G$-spaces a model category? | Yes, assuming that by "expected $G$-homotopy extension property" you mean the Serre $G$-cofibrations, not the Hurewicz $G$-cofibrations. A very explicit reference is Proposition A.1.18 in Schwede's [Global homotopy theory](http://www.math.uni-bonn.de/~schwede/global.pdf).
| 7 | https://mathoverflow.net/users/12547 | 177949 | 89,561 |
https://mathoverflow.net/questions/177893 | 2 | I have recently made use of the following generalization of a continuous function, which seems simple enough it ought to have been used before, but I cannot find any references.
We will say a function $f$ has a semi-continuity property if $f^{-1}(U)$ contains a non-empty open set whenever $U$ is a non-empty open set... | https://mathoverflow.net/users/50796 | Name of a generalized version of semi-continuity | This property has been studied and it goes by the name [**somewhat continuous**](http://www.google.com/search?q=%22somewhat+continuous%22+topology).
FYI, I was led to this discovery by its mention in the middle of p. 92 of Zbigniew Piotrowski's 1987 survey paper [*A survey of results concerning generalized continuity... | 3 | https://mathoverflow.net/users/15780 | 177950 | 89,562 |
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