parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/158998 | 3 | Let $a(n)$ for $n \geq 1$ be the number of powers of two $2^m$ that lie between $3^{n-1}$ and $3^{n}$:
$$a(n) = 1, 2, 1, 2, 1, 2, 2, 1, ...$$
It represents the increments between successive terms of allowable dropping times in the Collatz ($3x+1$) problem (see [oeis A022921](http://oeis.org/A022921)).
Is there an... | https://mathoverflow.net/users/35419 | Collatz dropping times aperiodicity | $a(n)=\lfloor \alpha n\rfloor-\lfloor \alpha(n-1)\rfloor$, where $\alpha=\log\_23$. Sequences of this type are called Sturmian and are well known (and easily shown) not to be periodic. Basically: for any $x,x'$ whose difference is not an integer, there is a multiple of $\alpha$ such that $\lfloor x+n\alpha\rfloor-\lflo... | 7 | https://mathoverflow.net/users/11054 | 158999 | 83,864 |
https://mathoverflow.net/questions/158838 | 5 | Let $X$ be a smooth irreducible algebraic curve in $\mathbb{P} V$.
The general position lemma states that the points given by general hyperplane section of $X$ are "in general position". I'm considering a similar situation for tangent lines. For each point of the general hyperplane section, the tangent line at that po... | https://mathoverflow.net/users/47552 | general position lemma for tangent lines of an algebraic curve | No, that is not true. Let $V$ be the $4$-dimensional vector space with ordered basis $(\mathbf{e}\_0,\mathbf{e}\_1,\mathbf{e}\_2,\mathbf{e}\_3)$, and let $[X\_0,X\_1,X\_2,X\_3]$ be the corresponding homogeneous coordinates on $\mathbb{P}V$. Let $[T\_0,T\_1]$ be homogeneous coordinates on $\mathbb{P}^1$. For every integ... | 2 | https://mathoverflow.net/users/13265 | 159000 | 83,865 |
https://mathoverflow.net/questions/159001 | 10 | Let $X$ be a variety. Let $D^b(Coh(X))$ be the derived category of bounded complexes of coherent sheaves on $X$, and $D^b\_{coh}(X)$ be the derived category of bounded complexes of sheaves of $\mathcal{O}\_X$-modules with coherent sheaves as cohomologies. Similarly, we have $D^b\_{qc}(X)$ and $D^b(Qco(X))$ by replacing... | https://mathoverflow.net/users/29730 | Are $D^b_{coh}(X)$ and $D^b(Coh(X))$ derived equivalent? | For X noetherian this is still true. (Proposition 3.5 in Daniel Huybrechts' book)
| 11 | https://mathoverflow.net/users/46690 | 159006 | 83,867 |
https://mathoverflow.net/questions/158945 | 25 | This is one of those things I never expected to be hard until I tried to prove it. Why is the right permutohedron order (a.k.a. weak Bruhat order, a.k.a. weak order -- not to be confused with the strong Bruhat order) on the symmetric group $S\_n$ a lattice?
**Details:**
Let $n$ be a nonnegative integer. Consider th... | https://mathoverflow.net/users/2530 | Why is the right permutohedron order (aka weak order) on $S_n$ a lattice? | In my opinion, the most elegant proof is by [Björner, Edelman and Ziegler](http://www.ams.org/mathscinet-getitem?mr=1036875) ([PDF file](http://page.mi.fu-berlin.de/gmziegler/ftp/archiv/007pos-regions.pdf)). They prove the following generalization: Let $\mathcal{H}$ be a finite set of hyperplanes in $\mathbb{R}^n$. Let... | 17 | https://mathoverflow.net/users/297 | 159008 | 83,868 |
https://mathoverflow.net/questions/159007 | 3 | Let us place ourselves in a category $\mathcal C$ with finite coproducts $X\amalg Y$, even cocomplete if necessary. It is well known that the morphism set $\mathcal C(X,Y)$ carries an abelian group structure natural in $Y$ if and only if there are maps
$$\mu\colon X\longrightarrow X\amalg X, \qquad \iota\colon X\lon... | https://mathoverflow.net/users/12166 | Cogroup objects are to groups what --- are to $k$-modules | Just do as you are doing: treat the maps $\alpha^\ast: X \to X$ as unary co-operations and dualize the diagrams you would use for a $k$-module object in a category with finite products.
So for a $k$-module object ($k$ a fixed commutative ring) you would adjoin to the abelian group operations and axioms (for an inter... | 9 | https://mathoverflow.net/users/2926 | 159010 | 83,869 |
https://mathoverflow.net/questions/158331 | 5 | When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi\_1M$ is relatively hyperbolic relative to the parabolic groups $\pi\_1\partial M$, so one can consider $\partial\Gamma$ which is defined as the ideal boundary of the coned off Cayley-G... | https://mathoverflow.net/users/39082 | Boundaries of relatively hyperbolic groups | As pointed out in the comments, the boundary of the
coned-off graph won't be a sphere in this case. In fact, it won't
even be compact.
Bowditch's paper on relatively hyperbolic groups (recently published
in
[*IJAC*](http://www.worldscientific.com/doi/abs/10.1142/S0218196712500166))
gives some details about the relat... | 7 | https://mathoverflow.net/users/46072 | 159019 | 83,874 |
https://mathoverflow.net/questions/159002 | 4 | The theory of generalized stochastic processes was introduced independently in the 50's by Ito\* and Gel'fand in a short paper. The latter then developed his theory more extensively in the fourth tome of his work on Generalized functions\*\*.
I am looking for the initial short paper of Gel'fand, in Russian, with the ... | https://mathoverflow.net/users/39261 | Initial paper of Gel'fand on Generalized Random Processes | A translation of it appears on pp. 529-533 of Gel'fand's *Collected Papers*, Volume III, Springer 1989, which should be available in a library closer to you...
| 6 | https://mathoverflow.net/users/19276 | 159023 | 83,876 |
https://mathoverflow.net/questions/159014 | 3 | This is a simple question, but I can't find explicit discussion in literatures that I can find. Real/imaginary Killing spinor equation
\begin{equation}
\nabla\_\mu \psi = \lambda \gamma\_\mu \psi
\end{equation}
where $\lambda$ is constant is a well known subject. **But I wonder why the case where $\lambda$ is non-co... | https://mathoverflow.net/users/15884 | Killing constant in Killing spinor equation | The freedom to vary $\lambda$ is severely constrained: For a nontrivial solution of the Killing spinor equation you need either a real and constant $\lambda$ or a function $\lambda$ with purely imaginary values [[Lichnerowicz](http://link.springer.com/article/10.1007/BF00401162) (1987)].
The generalization of the Kil... | 2 | https://mathoverflow.net/users/11260 | 159025 | 83,878 |
https://mathoverflow.net/questions/159027 | 1 | I'd like to know if there is an analytical solution of the following Ordinary Differential Equation :
$\displaystyle{\dfrac{d}{dt}x\_i(t) = D\_i + \sum\_{j=1}^{n}L\_{ij}x\_j(t) + \sum\_{j=1}^{n}\sum\_{k=1}^{n}C\_{ijk}x\_j(t)x\_k(t)}$
where $\forall i,j,k=1,...,n$ we have $x\_i(t) \in M\_{T\times 1}(\mathbb{R})$ and... | https://mathoverflow.net/users/41233 | Solution of an Ordinary Differential Equation | Everything depends on the meaning of the word "analytical".
In the common usage there is a word "analytic", which means "can be expanded in a convergent power series". Then there is a general theorem: if you have
a differential equation $y'=F(y)$, where $y$ is a function with values in $R^n$, and $F$ is an analytic f... | 2 | https://mathoverflow.net/users/25510 | 159036 | 83,884 |
https://mathoverflow.net/questions/159015 | 2 | Fenchel-Nielsen coordinates give a coordinatization of Teichmuller space for compact conformal surfaces admitting a pants decomposition. But not all compact conformal surfaces (possibly with boundary, which we assume totally geodesic) of genus less than 2 admit such a decomposition, such as the cylinder. Is there an ex... | https://mathoverflow.net/users/799 | Coordinates for Teichmuller space for compact conformal surfaces | As Lee points out, the only nontrivial examples are the cylinder and the torus. These admit Euclidean metric, which can be normalized to have area one, and for a cylinder we can assume that the boundary components are the same length and parallel. (in other words, metrically the cylinder has the form $a S^1 \times [0, ... | 2 | https://mathoverflow.net/users/11142 | 159054 | 83,894 |
https://mathoverflow.net/questions/159058 | 3 | The moduli space of sheaves over a smooth variety is in general not separated. That is, there exists a flat family of coherent sheaves over a punctured disk which extends to a flat family of coherent sheaves over a disk in several ways. Could anyone give me easy examples?
| https://mathoverflow.net/users/50973 | Non-separatedness of moduli space of sheaves | There is a vector bundle $E$ on $\mathbb{P}^1 \times \mathbb{A}^1$ whose restriction to $\mathbb{P}^1 \times (\mathbb{A}^1 - \{ 0\})$ is isomorphic to $\mathcal{O}^2$, but whose restriction to $\mathbb{P}^1 \times 0$ is $\mathcal{O}(-1)\oplus \mathcal{O}(1)$.
One can take $E$ to be the cokernel of the map
$$ (x, y, t... | 8 | https://mathoverflow.net/users/3847 | 159059 | 83,895 |
https://mathoverflow.net/questions/133050 | 16 | **Context**: First recall some results:
* Actions of finite groups on the hyperfinite type $II\_{1}$ factor $R$ ([Jones 1980](https://www.ams.org/books/memo/0237/)).
* A Galois correspondence for depth 2 irreducible subfactors ([Izumi-Longo-Popa 1998](https://www.sciencedirect.com/science/article/pii/S002212369793228... | https://mathoverflow.net/users/34538 | The cyclic subfactors theory: a quantum arithmetic? | The cyclic subfactors theory defines a too large class$^{\star}$ of subfactors for being an *good* quantum arithmetic (i.e. quantum generalization of the **natural numbers** theory).
Nowadays, my better attempt for this is the **natural subfactors** theory (see the *optional part* [here](https://mathoverflow.net/que... | 1 | https://mathoverflow.net/users/34538 | 159068 | 83,898 |
https://mathoverflow.net/questions/159071 | 8 | Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, such that there exists an isomorphism $\psi\colon X\_L\to Y\_L$ defined over $L$ (but not over $K$ in general).
The map ... | https://mathoverflow.net/users/23758 | Forms of algebraic varieties | CW answer, copied from the comments of Timo Keller and abx:
See J.P. Serre's “Galois Cohomology”, Chapter III, 1.3, Proposition 5.
| 6 | https://mathoverflow.net/users/21815 | 159075 | 83,900 |
https://mathoverflow.net/questions/158879 | 2 | Define $$Q = \bigcup\_{t \in (0,T)}\Gamma \times \{t\}$$ where $\Gamma$ is a compact (without boundary) hypersurface. Assume whatever smoothness is required.
Define $L^2(Q) := L^2(0,T;L^2(\Gamma))$ and $$H^1(Q) := \{ u \in L^2(0,T;H^1(\Gamma)) : u\_t \in L^2(0,T;L^2(\Gamma)) \}.$$
Given a function $u \in H^1(Q)$ wi... | https://mathoverflow.net/users/47571 | Are smooth functions dense in the space $\{u \in H^1(Q) \text{ with } \Delta_\Gamma u \in L^2(Q)\}$? | In my first comment I wrote: "I think the question is off-topic here as not research level." Since there have been no other comments in that direction, and no (visible) downvotes, maybe the question then is on-topic anyway.
I have read the OP's questions and comments also in MSE, and there seems to be a similar, but ... | 2 | https://mathoverflow.net/users/12643 | 159076 | 83,901 |
https://mathoverflow.net/questions/159099 | 6 | Suppose $A$ is a finite set and $\Sigma=A\cup A^{-1}$. Let $L\subseteq \Sigma^{\ast}$ be a regular language on the alphabet $\Sigma$. Is there a common name for the group $G$ presented as:
$$G=\langle A: L\rangle ?$$
Is there a reference text or article on this type of groups?
| https://mathoverflow.net/users/44949 | What is the name of this type of groups? | Finitely presented group?
In a regular expression defining $L$, any starred term must represent the identity in $G$. For example if $uv\*w$ defines a set of relators, then $v = v^2 = u^{-1}w^{-1}$. So we can remove any starred term from the regular expression and replace it by the term itself: i.e. replace $uv\*w$ by... | 10 | https://mathoverflow.net/users/35840 | 159100 | 83,908 |
https://mathoverflow.net/questions/158662 | 5 | Let $G$ be a finite group and $H$ a subgroup.
The **normal core** of $H$ in $G$ is $core\_G(H) := \bigcap\_{g \in G}g^{-1}Hg$
**Definition**: $K$ is a **normal intermediate subgroup** of the inclusion $(H \subset G)$ if $H \subset K \subset G$, and $$\forall g \in G \text{ , } KgH=HgK$$ (This definition is motivate... | https://mathoverflow.net/users/34538 | Normal intermediate subgroup and normal core | No, $(D\_{10} \subset A\_6)$ gives a counterexample.
It has exactly two non-trivial intermediate subgroups $K$ and $L$, each isomorphic to $A\_5$ (see [here](http://homepages.ulb.ac.be/~tconnor/atlaslat/alt6.pdf)) and normal intermediate subgroups, thanks to a SAGE-GAP computation (see [here](http://groupprops.subwi... | 3 | https://mathoverflow.net/users/34538 | 159108 | 83,911 |
https://mathoverflow.net/questions/158979 | 18 | This problem is related to my study of the Burau representation of the braid group $B\_3$: I was trying to show that certain "congruence subgroups" are of infinite index.
There is an approach that boils it down to the following question:
>
> Let $\xi$ be a primitive root of unity of degree $2n>12$ (just in case).... | https://mathoverflow.net/users/44953 | An algebraic number is not a root of unity? | Assume that the roots $\zeta$ and $\xi^2/\zeta$ of your equation are roots of unity. Then by Vieta Theorem $\zeta+\xi^2/\zeta=-(\xi^2-\xi+1)$; equivalently sum of 5 roots of unity $\zeta+\xi^2/\zeta+\xi^2-\xi+1$ vanishes. All such relations are classified in Theorem 6 of [paper by Conway and Jones](http://matwbn.icm.ed... | 15 | https://mathoverflow.net/users/4158 | 159120 | 83,916 |
https://mathoverflow.net/questions/159117 | 3 | Let $X$ be a projective, normal variety over complex field with canonical singularities. Suppose $|D|$ is a basepoint free linear system, then is it true that the generic elements in $|D|$ are irreducible?
Besides, I noticed that something might related to "free linear system" (see Mori, Kollár "Birational geometry o... | https://mathoverflow.net/users/29730 | Irreducible divisor in a basepoint free linear system | No, the image of the associated morphism, $$\phi\_{|D|}:X\to \mathbb{P}^n,$$ might be a curve. If the image has dimension $\geq 2$, then the general member is irreducible. Look up "Bertini theorems".
| 7 | https://mathoverflow.net/users/13265 | 159123 | 83,917 |
https://mathoverflow.net/questions/159126 | 28 | The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X\_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a polynomial for the curve. Are the coefficients of the map calculable?
Unless this has changed lately, the proofs rely on som... | https://mathoverflow.net/users/38783 | Is the Modularity Theorem (currently) effective? | Yes. Once you know $E$ is modular, a dominant map $\varphi: X\_0(N) \rightarrow E$ can be computed effectively. That's because one can effectively compute (a bound on) $\deg\varphi$.
Of course you need to compute a model for $X\_0(N)$ for the question to make sense, but we know how to do that and to write $q$-expansi... | 30 | https://mathoverflow.net/users/14830 | 159128 | 83,919 |
https://mathoverflow.net/questions/159113 | 7 | Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l\_i$ for $i \in \{1,2,3\}$. For any two distinct boundary components, is the length of the shortest geodesic connecting them already determined? If so, is there a simple formula for these lengths as a function of the $l\_i$?
| https://mathoverflow.net/users/799 | Distances between boundaries in a hyperbolic pants | Yes. Cutting along the shortest geodesics $\alpha, \beta, \gamma$ between the pants boundaries produces two congruent right angled hexagons. They are congruent because a right angled hexagon is determined by a triple of sides (in this case, $\alpha, \beta, \gamma).$ This means that each of the hexagons has sides $\alph... | 5 | https://mathoverflow.net/users/11142 | 159129 | 83,920 |
https://mathoverflow.net/questions/159057 | 6 | Let $S$ be the spectrum of a DVR, $X/S$ a flat finite type affine scheme. Fix a closed immersion $X\subseteq \mathbb{A}^n\_S$ and consider the closure $\bar X$ of $X$ in $\mathbb{P}^n\_S$. Then $\bar{X}$ is flat over $S$, as $X$ is dense in it. However, the boundary $Y = \bar X - X$ does not have to be flat over $S$ (s... | https://mathoverflow.net/users/3847 | Does there exist a relative compactification with flat boundary? | I believe the answer is no. All of the following will be relative to an algebraically closed field $k$. You may assume $k$ has characteristic $0$, but I think that is unnecessary (use the work of Stefan Schröer).
Let $g',g''>2$ be positive, distinct integers. Let $g$ be $g'+g''$. Let $\overline{M}\_g$ denote the Deli... | 6 | https://mathoverflow.net/users/13265 | 159133 | 83,923 |
https://mathoverflow.net/questions/159138 | 11 | For a fixed natural number $n$ is it possible to obtain an asymptotic for the number of solutions $(a,x,b,y)\in\mathbb{N^4}$ to $ax+by=n$ or equivalently an asymptotic for $\sum\_{k=1}^{n-1}d(k)d(n-k)$?
Using some heuristics I got that possibly:
$$\sum\_{k=1}^{n-1}d(k)d(n-k)\sim \frac{3}{\pi^2}\sigma(n)\ln(n)^2$$
... | https://mathoverflow.net/users/38626 | Asymptotic behavior of $\{\text{#}(a,x,b,y)\in \mathbb{N^4}| \text{ }ax+by=n\}$ for large $n$ | This is a binary additive divisor problem (also known as the shifted convolution problem) and related questions are to understand asymptotics for $\sum\_{n\le x} d(n) d(n+k)$, or $\sum\_{n\le x} \lambda\_f(n) \lambda\_f(n+k)$ where $\lambda\_f$ denotes the Fourier coefficients of a cusp form. For your particular proble... | 13 | https://mathoverflow.net/users/38624 | 159141 | 83,925 |
https://mathoverflow.net/questions/159142 | 9 | The Hamming weight $w(n)$ is the number of 1s in $n$ when written in binary. Is there some effective bound on Fibonacci numbers $F\_n$ with $w(F\_n)\le x$ for a given $x$?
Clearly only $F\_0=0$ has weight 0, and it's not hard to show (e.g. by Carmichael's theorem) that only $F\_1=F\_2=1, F\_3=2, F\_6=8$ have Hamming ... | https://mathoverflow.net/users/6043 | Hamming weight of Fibonacci numbers | The case $x=2$ is still tractable. If $F\_n = 2^e + 2^f$ with $e<f$
then $e < 5$, else $F\_n \equiv 0 \bmod 2^5$, which happens **iff**
$n \equiv 0 \bmod 24$, and then $7 \mid 21 = F\_8 \mid F\_{24} \mid F\_n$,
which is impossible because $2^e + 2^f$ is never a multiple of $7$.
So we have only a few candidates for $e$,... | 13 | https://mathoverflow.net/users/14830 | 159144 | 83,926 |
https://mathoverflow.net/questions/159150 | 4 | Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l\_1,l\_2,l\_3$. Cap off boundary 1 with a conformal disk. The result is a conformal cylinder, which has a unique flat representative of the form $S^1 \times [0,h]$, where we choose the boundaries to have some fixed length $2 \pi$.
What ... | https://mathoverflow.net/users/799 | What's the height of the capped hyperbolic pants? | EDIT - Maxime has answer the question in the comments. The desired $h$ is not a function of the lengths $\ell\_1, \ell\_2, \ell\_3$, but rather lies in an open interval. The minimum (resp. maximum) of $h$ is achieved by sewing the first circle shut so it lies along a concentric (resp. radial) arc of the resulting annul... | 1 | https://mathoverflow.net/users/1650 | 159155 | 83,929 |
https://mathoverflow.net/questions/159173 | 11 | I already posted this question at MSE [here](https://math.stackexchange.com/questions/685767/minimize-norm-of-a-polynomial-on-a-circle), but as it received no significative feedback for a while I cross-post it here.
I also noticed a related question [here](https://mathoverflow.net/questions/35317/minimizing-the-modul... | https://mathoverflow.net/users/10341 | Minimize norm of a polynomial around a circle (count the solutions) | The conjecture is correct.
Let me begin with a simple proof that $|M|\leq 2n$.
The curve $|P(z)|^2=\mu^2$ is a real algebraic curve of degree $2n$.
At the points of $M$ this curve must be tangent to the unit circle. The unit circle is
a curve of degree $2$, therefore by Bezout theorem these two curves have at most... | 14 | https://mathoverflow.net/users/25510 | 159176 | 83,935 |
https://mathoverflow.net/questions/129955 | 25 | I asked the [same question on math.se](https://math.stackexchange.com/questions/334962/int-01-mathrmd-u-1-cdots-int-01-mathrmd-u-n-frac-delta1-u-1-cdots) but got no answer there. Since it pertains to my current research, I decided to ask here:
Let $n\in 2\mathbb{N}$ be an even number. I want to evaluate
$$I\_n
:=
\in... | https://mathoverflow.net/users/21850 | Evaluation of an $n$-dimensional integral | We can think of $I\_{n}$ as being a classical partition function for $n$ beads on a circle which cannot pass through each other, with logarithmic interaction potential between each bead and its next-to-nearest neighbors on either side. For $I\_{2n}$ the beads fall into two ``colors" which do not have logarithmic intera... | 23 | https://mathoverflow.net/users/44331 | 159180 | 83,936 |
https://mathoverflow.net/questions/159060 | 5 | I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V\_0(x) = 1-a^2$
$V\_1(x) = 1-a^2 - x$
$V\_{k \geq 2}(x) = (1+a^2 - x)V\_{k-1}(x) - a^2V\_{k-2}(x)$
I know that $a \in (0,1)$.
Another formulation of the problem could be to find the eigenvalues of the tridia... | https://mathoverflow.net/users/47640 | Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix | For $k\in\mathbb{N}$ and $0<a<1$ your polynomial $V\_k$ has indeed $k$ distinct real roots in the interval
$\big( (1-a)^2, (1+a)^2 \big)$ ; these can be written
$$x\_j:=1+a^2-2a\cos(t\_j)$$
where $ t\_1<\dots < t\_k $ are the $k=\lceil k/2\rceil+\lfloor k/2\rfloor$ solutions in the interval $(0,\pi)$ to either
$$\cos... | 5 | https://mathoverflow.net/users/6101 | 159184 | 83,938 |
https://mathoverflow.net/questions/159185 | 4 | Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets restrict the canonical n plane bundle over $G(n,2n)$ to $\Lambda(n)$. We denote this restricted bundle by $E$.
**Questio... | https://mathoverflow.net/users/36688 | Canonical n plane bundle over Lagrangian Grassmanian | It is not trivial. Its characteristic classes were worked out by Dmitrii Fuchs. In particular, the Maslov class is one of its nontrivial characteristic classes, and was the subject of a famous paper of Vladimir Arnol'd.
| 4 | https://mathoverflow.net/users/13268 | 159190 | 83,941 |
https://mathoverflow.net/questions/159198 | 6 | Vaught's Conjecture is a dual form of Continuum Hypothesis in model theory. It asserts that for each complete consistent theory $T$ in a countable language if $I(T,\aleph\_{0})>\aleph\_{0}$ then $I(T,\aleph\_0)=2^{\aleph\_{0}}$.
The best known result here is proved by Morley.
**For each complete theory $T$ in a cou... | https://mathoverflow.net/users/nan | Morley Phenomena for Special Families of Reals | The classical example is ${\mathbf\Pi}^1\_1$ sets of reals.
We say that $X\subset \mathbb R$ is ${\mathbf\Pi}^1\_1$ if the complement of the projection
of a Borel set in ${\mathbb R}^n$. Any such set of cardinality greater than $\aleph\_1$ has cardinality continuum.
| 7 | https://mathoverflow.net/users/5849 | 159200 | 83,945 |
https://mathoverflow.net/questions/159169 | 4 | Let $T$ be a stable $L$-theory with elimination of imaginaries. We work in the monster model $\mathfrak C$ of $T$. Let $A$ be a small (infinite) set of the monster, $\phi(x,y)$ be a $L(A)$-formula and $a\_i:i\in \omega$ be a Morley sequence over $A$ such that $\mathfrak C\models \phi(a\_0,a\_1)$.
Can we find a Morley... | https://mathoverflow.net/users/47687 | Replace Morley sequence over some set by one over a finite set, s.t. they both satiesfy a certain formula | Your question comes down to asking whether given a formula $\phi(x,b)$ which does not fork over $A$, there is a finite subset of $A$, $A\_0$, such that $\phi(x,b)$ does not fork over $A\_0$.
The answer is yes. The reason depends on your preferred definition of forking, of course. Using the definability of types defi... | 3 | https://mathoverflow.net/users/2126 | 159206 | 83,948 |
https://mathoverflow.net/questions/158639 | 3 | I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go directly to the question.
Fix a field $k$, of characteristic $0$ and consider the ring of power series in $n+1$ variables... | https://mathoverflow.net/users/11475 | Tor dimension in polynomial rings over Artin rings | If $B/I$ is regular local, then $I$ is a prime ideal in $B$. Notice that $x^m \not \in I$ (if $x \in I$, then the equations defining $I$ would imply $y\_1, \ldots, y\_n \in I$, so then $I$ would be the maximal ideal, which it is not, being $n$-generated). Thus $x^m$ is regular on $B/I$ (and on $B$), so taking a finite ... | 1 | https://mathoverflow.net/users/44201 | 159213 | 83,950 |
https://mathoverflow.net/questions/159214 | -1 | First version: Let $A$ and $B$ be (complex) Hankel matrix. Is it true that $\det (A+B)\neq 0$ if $\det A=0$ and $\det B\neq0$? **No**.
Reformulating: For which $B$ is it true that $\det (A+B)\neq 0$ if $\det A=0$?
| https://mathoverflow.net/users/47711 | Determinant of a sum of two Hankel matrices | No, let
$$
B=\left(\begin{matrix}
0 & 1\\
1 & 1\\
\end{matrix}\right)
$$
$$
A=\left(\begin{matrix}
1 & 0\\
0 & 0\\
\end{matrix}\right)
$$
| 1 | https://mathoverflow.net/users/4600 | 159215 | 83,951 |
https://mathoverflow.net/questions/159210 | 2 | I'm trying to prove the following:
Let $G$ be a group with finite symmetric generating set $S$ and let $\Gamma(G,S)$ be the corresponding Cayley graph. Let $X\_1, X\_2,\cdots$ be a simple random walk starting from a fixed vertex $x\in\Gamma(G,S)$. For each $n$ let $\mu\_n$ denote the law of $X\_n$. Fix an integer $k>... | https://mathoverflow.net/users/47709 | Proving that a closed walk of some odd length k exists on a Cayley graph | There is a theorem of Foguel (MR0400332) according to which for any probability measure $\mu$ on **any** group (not necessarily countable) and any $k>0$ the limit of the (monotone) sequence $\|\mu^{\*n}-\mu^{\*(n+k)}\|$ is either 0 or 2. In the case of of countable groups it implies that if the supports of $\mu^{\*N}$ ... | 3 | https://mathoverflow.net/users/8588 | 159218 | 83,953 |
https://mathoverflow.net/questions/159220 | 6 | Let $G$ be a finite non-abelian simple group and $t$ is equal to the number of involutions of $G$. We know that $t<|G|/3$ or $3t+1 \leq |G|$. Is this the best upper bound for the number of involutions of $G$? and if not what is it (if there is any)?
| https://mathoverflow.net/users/47713 | The best upper bound for the number of involutions in a finite non-abelian simple group | If $n$ denotes the number of involutions of the non-Abelian simple group $G,$ then we have $n + 1 = \sum\_{\chi} \nu(\chi)\chi(1),$ where $\chi$ runs over the irreducible characters of $G$ and $\nu(\chi) \in \{0,1,-1\}$ denotes the Frobenius-Schur indicator of $\chi.$ Hence, by Cauchy-Schwarz, we have $n < \sqrt{k(G)} ... | 18 | https://mathoverflow.net/users/14450 | 159225 | 83,955 |
https://mathoverflow.net/questions/159227 | 2 | Suppose $G$ is a finite linear group, and I have a $G$-torsor $Z \to X$. Suppose also I have a morphism $f : Y \to X$ with some properties $P$. What should these properties $P$ be in order to make the fiber product $Y \times\_X Z \to Y $ (with the natural projection) a $G$-torsor over $Y$?
| https://mathoverflow.net/users/47386 | Fiber products and torsors | I think P should be empty. The group $G$ acts on the fibre product by acting on $Z$, pulling back a trivialising étale cover for $Z \to X$ gives a trivial étale cover for $Y \times\_X Z \to Y$.
No?
| 6 | https://mathoverflow.net/users/46690 | 159231 | 83,957 |
https://mathoverflow.net/questions/159229 | 5 | Which are the finite groups $G$ such that the element orders of $G$ form an arithmetic progression? Several remarks:
1. $S\_3$, $A\_4$ and any $p$-group of exponent $p$ satisfy this property.
2. If $G$ satisfies this property and $p\_1<p\_2<...<p\_k$ are the prime divisors of $n=\mid G\mid$, then $p\_2=2p\_1-1$, and ... | https://mathoverflow.net/users/17565 | Finite groups for which the element orders form an arithmetic progression | No, your impression is not correct. -- A counterexample is the group ${\rm S}\_5$, whose elements have orders $1, 2, 3, 4, 5$ and $6$.
As to groups with *spectrum* (i.e. set of element orders) equal to $\{1, \dots, n\}$: the maximum here is 8, and for $n = 8$ the only such group is an extension of ${\rm PSL}(3,4)$ by... | 10 | https://mathoverflow.net/users/28104 | 159232 | 83,958 |
https://mathoverflow.net/questions/159234 | 5 | By **algebraic theory** I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe universal algebraists call these abstract clones?)
A **strict terminal object** in a category is a terminal object $1... | https://mathoverflow.net/users/11640 | What do algebraic theories with strictly terminal trivial models look like? | I believe these would be the algebraic theories where there is a finite set of identities of closed terms that together imply all identities. (If there are no closed terms, just add a fresh constant symbol with no new axioms.) This is the case for rings, where the single identity $0 = 1$ implies that all ring identitie... | 4 | https://mathoverflow.net/users/2000 | 159238 | 83,960 |
https://mathoverflow.net/questions/159230 | 4 | Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My question is: can we choose the frame such that $a$ falls in a bounded set in $\mathfrak{g}$ for any connection $A$?
| https://mathoverflow.net/users/47719 | connections on principal bundles over $S^1$ | Yes. Let $g\in G$ be the holonomy of the connection $A$; since $G$ is compact and connected, one can choose a frame (i.e., trivialization of $P$) such that $A = a\,\mathrm{d}\theta$ where $g = \exp(2\pi a)$. Since $G$ is assumed compact, there is a compact set $K\subset\frak{g}$ such that $\exp(2\pi K) = G$. Choose $a$... | 6 | https://mathoverflow.net/users/13972 | 159239 | 83,961 |
https://mathoverflow.net/questions/159237 | 5 | I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment !
So here is the situation. Let $k'\subset k$ be a finite field extension. Take affine algebraic group schemes defined over $k'$ together with a centr... | https://mathoverflow.net/users/47722 | rationality question while dealing with an isogeny | The answer is negative (even under all of the given hypotheses); this expresses a standard difficulty in the arithmetic aspects of connected semisimple groups over local function fields in contrast with $p$-adic fields.
As S.Carnahan notes, it is equivalent to ask that the subgroup $\Gamma\_{\rm{down}}$ of $G(k')$ co... | 7 | https://mathoverflow.net/users/43107 | 159241 | 83,963 |
https://mathoverflow.net/questions/159083 | 5 | Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$.
We shall say: $D$ is *connected* if its support is connected. $D$ is *numerically connected* if for any non-trivial effective decomposition $D\sim D\_1+D\_2$ we have $D\_1\cdot D\_2\geq1$. Notice that these two notions are not equivalent (if $D$ is... | https://mathoverflow.net/users/40038 | Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces? | Interesting question. I think the answer is yes, let me try to prove it.
As you noticed, the ideal sheaf sequence shows that $h^1(D)=0$ is equivalent to the fact that $H^0({\mathcal O}\_D)$ is 1-dimensional generated by the constant function $1$.
Considering, for every effective decomposition $D=A+B$, the exact se... | 5 | https://mathoverflow.net/users/46104 | 159243 | 83,964 |
https://mathoverflow.net/questions/158776 | 20 | Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ .
>
> Is there a global real analytic vector field $X$ on $M$ such that $D(f)=X.f$ for all $f\in C^{\omega}(M)$?
>
>
>
The motivati... | https://mathoverflow.net/users/36688 | Derivation on real analytic manifolds | $\def\RR{\mathbb{R}}\def\CC{\mathbb{C}}\def\NN{\mathbb{N}}$I'm going to try once more. As my previous flawed answers should have made obvious, the real analytic category is not my home, so read with caution.
Conveniently, the two big results I need are stated as Corollary 5.43 and 5.44 [here](http://www.mathematik.un... | 10 | https://mathoverflow.net/users/297 | 159246 | 83,965 |
https://mathoverflow.net/questions/158900 | 7 | I am currently sketching a paper in the general area of symbolic dynamics in which I would like to be able to use the following fact:
>
>
> >
> > Proposition (proposed): there exists a shift-invariant measure $\mu$ on $\{0,1\}^{\mathbb{Z}}$ such that $\mu$ is weak-mixing and has zero entropy with respect to the s... | https://mathoverflow.net/users/1840 | A weak-mixing, zero entropy measure on the 2-shift which gives equal weight to both symbols | So with help from Ian and Anthony it looks like my comment above can be turned into an answer. To summarize:
Let S be the substitution on two symbols given by $0\to001, 1\to11100$. Let $x\in\{0,1\}^{\mathbb N}$ be the limit of $S^n(01)$. Let $\Sigma$ be the orbit closure of $x$ under the shift map.
$(\Sigma,\sigma... | 3 | https://mathoverflow.net/users/24586 | 159250 | 83,967 |
https://mathoverflow.net/questions/159261 | 3 | This fact might be either trivial, wrong, or well known.
Let $R$ be a commutative ring. Let $u\_1,\dots,u\_{s-1},u\_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy
$$(m-u\_1) \dots (m-u\_{s-1})=0,$$
$$(M-m)(M-u\_s)=0.$$
**Question.** Is it true that $M$ satisfies:
$$(M-u\_1)\dots (M-u\_{s-1})(M-u\_{s})=0?$... | https://mathoverflow.net/users/16183 | An identity in an arbitrary commutative ring | This is actually trivial. Write $M-u\_k=(M-m)+(m-u\_k)$; this gives $(M-u\_1)\dots (M-u\_{s-1})\in R(M-m)$. Multiplying by $(M-u\_s)$ gives $(M-u\_1)\dots (M-u\_{s})=0$.
| 8 | https://mathoverflow.net/users/40297 | 159266 | 83,972 |
https://mathoverflow.net/questions/159268 | 18 | So I walked into this very innocent-looking combinatorics problem,
and quite soon I ended up with the problem to prove that any doubly stochastic $n \times n$ matrix has a non-zero permanent.
Now clearly, this follows from the Van der Waerden conjecture (which is now a theorem), which give a lower (positive) bound fo... | https://mathoverflow.net/users/1056 | Silly me & Van der Waerden conjecture | It's known (again using Hall's theorem, or using convex analysis) that the doubly stochastic matrices are a convex combination of the permutation matrices (these are the extreme points of the collection of doubly stochastic matrices). Accordingly each doubly stochastic matrix is a finite positive linear combination of ... | 28 | https://mathoverflow.net/users/11054 | 159270 | 83,975 |
https://mathoverflow.net/questions/159272 | 1 | My question is about domains of trivializability of distributions on a smooth compact manifold. Assume that you have a sequence $\{E^k\}\_k$ of $C^r$ distributions of rank $n$ which is $C^0$ close to a fixed $C^r$ distribution $E$ of rank $n$ with respect to a given Riemannian metric.
My question is that for each $x... | https://mathoverflow.net/users/34518 | A question about the "size" of the neighbourhoods in which bundles are trivializable | I may not have understood your question correctly, but assuming that "distribution" stands for a $n$-plane field, any vector bundle over any contractible base is trivial, so you only need to make sure that your neighborhoods are contractible. Speaking about "size", you can introduce a Riemannian metric and take all $U$... | 2 | https://mathoverflow.net/users/44953 | 159273 | 83,976 |
https://mathoverflow.net/questions/158751 | 11 | Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:
>
> The Hodge-De Rham spectral sequence $E^{a,b}\_1=R^af∗Ω^b(X/S)⇒H^{a+b}\_{DR}(X/S)$ degenerates in $E\_1$.
>
>
>
This is also true when $S=Spec(k)$ with $k$ a ... | https://mathoverflow.net/users/4504 | Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring? | First of all, the statement of Deligne-Illusie as you've given it is slightly wrong; there is only degeneration in degree $\leq \dim(X)$ (and thus full degeneration if $\dim(X)\leq p$).
Yukiyoshi Nakkajima [proves an analogous result](http://ac.els-cdn.com/S0022314X97921120/1-s2.0-S0022314X97921120-main.pdf?_tid=980... | 9 | https://mathoverflow.net/users/6950 | 159274 | 83,977 |
https://mathoverflow.net/questions/159279 | 15 | I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's *Algebraic Stacks and Moduli of Vector Bundles*.
Explicitly, let $G$ be an affine smooth group $S$-scheme with right action $\rho:X\times G\to X$ on a noetherian $S$-scheme $X$. The **quotient stack** $[X/G]$ is the p... | https://mathoverflow.net/users/33377 | Understanding the definition of the quotient stack $[X/G]$ | The functor $[X/G]$ is not representable whenever there is non-trivial isotropy of the action of $G$ on $X$.
Let us consider the most extreme case: when $X = \bullet$ is a point (the terminal object) and $G$ is any non-trivial group. In such a case, $Hom(-,\bullet/G)$ is a singleton, as $\bullet / G = \bullet$, which... | 11 | https://mathoverflow.net/users/1703 | 159280 | 83,978 |
https://mathoverflow.net/questions/159297 | 2 | Let $(M,\omega)$ be a manifold endowed with symplectic form. Then the product manifold $M\times M$ with symplectic form $\omega\times -\omega$ is symplectic, and the diagonal submanifold $\Delta\equiv\{(x,x)|x\in M\}$ is Lagrangian. Does the product manifold in fact admit a Lagrangian fibration over $\Delta$?
| https://mathoverflow.net/users/26762 | Does $(M\times M, \omega\times -\omega)$ admit a Lagrangian fibration? | No (independent of the full meaning of the question) in general it does not.
Any manifold which admits a Lagrangian fibration to a half dimensional base, also admits a Lagrangian foliation, and hence admits a global continuous choice of Lagrangian at each point.
If this were the case then it means that the classif... | 7 | https://mathoverflow.net/users/4500 | 159302 | 83,986 |
https://mathoverflow.net/questions/159257 | 7 | Let $X\_k$ be a family of spectra equipped with maps $f\_k: X\_k \to X\_{k+1}$. If $Y$ is a compact object (such as a sphere), then I can compute homotopy classes of maps from $Y$ into the homotopy colimit $X\_\infty := hocolim \; X\_k$ of the sequence by
$$[Y, X\_\infty] = \lim\_{k \to \infty} [Y, X\_k]$$
Now if I... | https://mathoverflow.net/users/4649 | Homotopy of localisations of colimits | There are two natural finiteness conditions that you might impose on $Y$. The stronger one says that the Morava $E$-theory of $Y$ is finite in each degree, or equivalently that $Y$ is a retract of $L\_{K(n)}Y\_0$ for some finite spectrum $Y\_0$ with $E(n-1)\_\*Y\_0=0$. If this holds, then $[Y,L\_{K(n)}X\_\infty]$ is th... | 6 | https://mathoverflow.net/users/10366 | 159306 | 83,988 |
https://mathoverflow.net/questions/159310 | 19 | Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base?
In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X \rightarrow E$ denote the inclusion given by the zero section. Then for $V$ a vector bundle over $E$, we have the restriction... | https://mathoverflow.net/users/12830 | Vector bundles on vector bundles | This is definitely false in the algebraic or holomorphic setting, even in dimension 1. There is a well-known example (see [this post](https://mathoverflow.net/questions/159058/non-separatedness-of-moduli-space-of-sheaves)) of a rank 2 vector bundle $E$ on $\mathbb{C}\times \mathbb{P}^1$ such that $E\_{|\{t\}\times \mat... | 17 | https://mathoverflow.net/users/40297 | 159314 | 83,990 |
https://mathoverflow.net/questions/159318 | 3 | Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I\_2[g]$
$$
f(x)=\int\_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,
$$
which solves $-\Delta f=g$ (up to positive normalizing constants, which I shall ignore), and assume that $g\in L^q$ for all $q\in[1,2d/(d+2)]$. By the [Hardy-Littlewood-Sobolev](http://en.wiki... | https://mathoverflow.net/users/33741 | Is this integration by parts legitimate? | This "integration by parts" is true in larger generality:
Let $\mu$ be a measure,
$$u(x)=\int\frac{1}{|x-y|^{n-2}}d\mu(y).$$
Then
$$\int u(x)d\mu=\int\frac{1}{|x-y|^{n-2}}d\mu(x)d\mu(y)$$
is called the energy of $\mu$. If this is finite, the measure is said of
finite energy. Gradient of the potential of such measure ... | 3 | https://mathoverflow.net/users/25510 | 159322 | 83,991 |
https://mathoverflow.net/questions/159289 | 3 | Let $M$ be a complete Riemannian manifold, endowed with its exponential map $\exp: TM \longrightarrow M$. For any $C^k$- function $u$, we get the Pullback
$$ \exp^\* u = u \circ \exp$$
which is in $C^k(TM)$, as $\exp$ is smooth.
If $u$ is only in $L^p\_{\mathrm{loc}}$, however, the pullback is harder to define: For e... | https://mathoverflow.net/users/16702 | Pullback of $L^p$ functions via exponential map | Let us denote the geodesic flow for time $t$ by $\Theta^t$,
this is the map $TM\to TM$
and let $\sigma$ be the projection $TM\to M$.
Then $\exp=\sigma\circ \Theta^1$.
Note that $\Theta^1$ preserves volume on $TM$.
Threfore the $\Theta^1$-pullback of $L^p\_{loc}$-function on $TM$ is $L^p\_{loc}$.
Hence your statement... | 3 | https://mathoverflow.net/users/1441 | 159325 | 83,992 |
https://mathoverflow.net/questions/159340 | 6 | Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega\_1, \ldots , \omega\_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma\_{W}$ generated by $W$ is formed by bi-infinite concatenations of words of $W$. My first question is, does every renewal system is intrinsically er... | https://mathoverflow.net/users/10518 | Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture | There are some examples related to your third question in "Renewal Systems, Sharp-Eyed Snakes, and Shifts of Finite Type" by Johnson and Madden, Amer. Math. Monthly 109 (2002), 258-272. A long time ago Goldberger, Smorodinsky, and I showed that for every possible entropy of a shift of finite type (or, what amounts to t... | 6 | https://mathoverflow.net/users/8112 | 159345 | 83,997 |
https://mathoverflow.net/questions/159331 | 9 | Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote the $\infty$-category of small stable $\infty$-categories and exact functors by $Cat^{\rm ex}\_{\infty}$. Suppose I hav... | https://mathoverflow.net/users/3995 | coends of stable infinity categories | There is some discussion of coends in $\infty$-categories in [my PhD thesis](http://arxiv.org/abs/1011.3243) (Sheffield, 2010). I think that, given that the target category $\mathrm{Cat}\_\infty^\mathrm{ex}$ is cocomplete, this gives you one way to do it.
I have another way, which is arguably rather less fussy, but i... | 8 | https://mathoverflow.net/users/14901 | 159360 | 84,000 |
https://mathoverflow.net/questions/159354 | 4 | Let $N = \{ (z\_0,z\_1,z\_2) \in S^5 \mid z\_0^3+z\_1^3+z\_2^3 = 0 \}$, where we consider $S^5\subset\mathbb{C}^3$.
The circle $U(1)$ acts on $N$ by
$$e^{i\theta} \cdot (z\_0,z\_1,z\_2) = (e^{i\theta}z\_0,e^{i\theta}z\_1,e^{i\theta}z\_2) ,$$ and the quotient $E = N/U(1)$ is an elliptic curve in $S^5/U(1) = \mathbb{C}P^... | https://mathoverflow.net/users/19956 | Chern class of Hopf fibration over elliptic curve | The circle bundle you get is the restriction to $E$ of the unit circle bundle of the tautological line bundle $\mathcal{O}(-1)$ over $\mathbb{CP}^2$. We will compute the Chern class of this line bundle. It is dual to $\mathcal{O}(1)|\_E$, the restriction of the hyperplane bundle to $E$. Since a hyperplane section of $E... | 4 | https://mathoverflow.net/users/24525 | 159367 | 84,004 |
https://mathoverflow.net/questions/159368 | 0 | Given a number $x \geq 3$, let $b(x) \in \{0,1\}$ be the second most significant digit (bit) of its binary representation, and $t(x)\in \{1,2\}$ the most significant digit of its ternary representation.
In other words: $x = 2^n + b(x) 2^{n-1} + ... = t(x) 3^m + ...$ for some $n,m \geq 1$
Let $A$ be the set of $K \g... | https://mathoverflow.net/users/35419 | Arithmetic progression and most significant digits in different bases | The problem is a little harder than it seems at first glance.
Pick m large, say m > 8. There are 3k=3^m numbers with first
ternary digit 1 and m other ternary digits. Suppose 0<= a < 3^m is
smallest such that 3k +a= 3\*2^l. Then 2^(l+2)=4k +4a/3, so there
will be either at least k/2 numbers before 3k+a or after 4k+ 4a/... | 2 | https://mathoverflow.net/users/35626 | 159377 | 84,005 |
https://mathoverflow.net/questions/159222 | 12 | Let $X$ be an integer-valued random variable and let $X\_n$ be the sum of $n$ independent realizations of $X$. I would like to understand the behavior of $X\_n/n$ for large $n$ in some cases where $X$ has no expected value and therefore the central limit theorem (along with Chebyshev's Theorem, etc.) does not apply.
... | https://mathoverflow.net/users/10503 | Mean of i.i.d Random Variables With No Expected Value | We can attack this problem by using Fourier transforms (i.e. characteristic functions). I'll consider
the example in the problem where $X$ is a random variable taking the value $0$ with probability $1/3$, and $4^{k}$ and
$-4^{k}$ with probability $1/4^{k+1}$ (for $k=0$, $1$, $\ldots$). I'll show that the probabilit... | 13 | https://mathoverflow.net/users/38624 | 159379 | 84,007 |
https://mathoverflow.net/questions/74603 | 3 | In writing a paper I need to refer to the following type of objects, does anybody know if it has an established name?
**What we are given**
We have $M$ a smooth manifold and $\Sigma \subset M$ a codimension 1, smooth compact submanifold with boundary $\partial\Sigma$ and interior $\hat{\Sigma}$.
**What I am inte... | https://mathoverflow.net/users/3948 | Terminology for a type of smooth foliation | (Just to get this off the unanswered list)
Deane's suggestion of "lens-shaped domain" and "ravioli-shaped domain" are actually quite descriptive. And they are sufficiently close in their original definitions to what I have in mind that I don't feel uncomfortable abusing them. :-)
| 1 | https://mathoverflow.net/users/3948 | 159383 | 84,010 |
https://mathoverflow.net/questions/98580 | 1 | Let $f\_k$ be a sequence of measurable functions on $\mathbb{R}^k$ where $k > 1$. (Let us be generous and also assume that $f\_k$ is locally integrable.) Does anyone know what the phrase
>
> uniform equicontinuity of the indefinite integrals $\int f\_k(x) \mathrm{d}x$
>
>
>
means?
(For that matter, what is... | https://mathoverflow.net/users/3948 | Uniform equicontinuity of a family of indefinite integrals | (Copying [fedja's comment](https://mathoverflow.net/questions/98580/uniform-equicontinuity-of-a-family-of-indefinite-integrals#comment252944_98580) to get this off the unanswered list.)
>
> That's what I would think. Just check that it works with this interpretation (it certainly does in the passage you quoted but ... | 1 | https://mathoverflow.net/users/3948 | 159384 | 84,011 |
https://mathoverflow.net/questions/156639 | 7 | Given $N$ pairs of distinct real numbers $t\_i, t'\_i \in [0,1]$, $i = 1,\ldots,N$, we ask if there is a function $f(x) = \cos(2\pi mx+\alpha) + \gamma\cdot \cos(2\pi nx+\beta)$, with $m, n \in \mathbb{N}$, $\alpha, \beta, \gamma \in \mathbb{R}$, so that for all $i$: $f(t\_i) > f(t'\_i)$?
If yes, this proves that all... | https://mathoverflow.net/users/14973 | Are Fourier series of length 2 'asymmetric enough' to generate all crossing patterns? - A reformulation of the Fourier-(1,1,2) knot question | If all $t\_i$, $t'\_i$ are rational numbers with common denominator
$d$, then it is enough to examine frequencies $m, n \le d$.
After David's answer I decided that it could be worthwhile to
look for counterexamples for small $d$ and used an optimization routine to
minimize the sum over $\max(0,f(t'\_i)-f(t\_i))$ for ea... | 3 | https://mathoverflow.net/users/14973 | 159385 | 84,012 |
https://mathoverflow.net/questions/159392 | 5 | The idea of the question is whether it is ever possible that $L$ is so nice in the sense that $\{L\_\alpha\}$ does not incorrectly "guess" a bigger inaccessible than $L$ really has, as long as $L\_\alpha \models ZFC + V = L$.
More precisely, is it ever possible that $L \models "\kappa$ is the largest inaccessible" a... | https://mathoverflow.net/users/43354 | Property of $L$ Relating to Reflection | The answer to the "More precisely ..." question is that yes, it may be the case that $L$ has a single inaccessible cardinal $\kappa$ and that for no larger $\beta >\kappa$ is there any $\alpha$ with $L\_\alpha\models$``$\beta$ is inaccessible.'' For example if $\kappa$ is the first inaccessible in $L$ (if such exists) ... | 7 | https://mathoverflow.net/users/6942 | 159395 | 84,015 |
https://mathoverflow.net/questions/159348 | 7 | I'm trying understand a computation on page 371 of Besse's book on [Einstein Manifolds](http://rads.stackoverflow.com/amzn/click/3540741208).
I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form relative to the direct sum decomposition:
$$R=\begin{pmatrix}A&B\\C&D\... | https://mathoverflow.net/users/39997 | The trace of a wedge product of matrices | The confusion is caused by using $R$ to denote two different things.
In Section 13.6, Besse introduces $R$ as a $6$-by-$6$ matrix of (scalar) curvature coefficients, which is the matrix of the linear transformation $\Lambda^2\to\Lambda^2$ induced by the curvature operator with respect to a basis adapted to the stand... | 5 | https://mathoverflow.net/users/13972 | 159404 | 84,020 |
https://mathoverflow.net/questions/159355 | 5 | In a paper by J.-L. Krivine, [*Modèles de ZF+AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue*](http://gallica.bnf.fr/ark:/12148/bpt6k4802973/f547.image) [C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A549–A552, [MR0253894](http://www.ams.org/mathscinet-getitem?mr=253894)],... | https://mathoverflow.net/users/38200 | A model of Krivine | 1. Yes, that is correct. Krivine's observation is that by collapsing the continuum (or any larger cardinal) to $\aleph\_0$ using finite conditions then any set of reals definable from ground model parameters (e.g. ordinals) is Lebesgue measurable in the extension.
2. No, not at all countable sequences of ordinals are i... | 5 | https://mathoverflow.net/users/2000 | 159407 | 84,022 |
https://mathoverflow.net/questions/159393 | 12 | In this question $\lfloor a\rfloor$ means the greatest integer not exceeding $a$.
Using van der Corput's inequalities one is able to show that $\log(n!)\alpha$ is equidistributed on the unit circle ($\mathbb{R}\setminus\mathbb{Z}$) for all $\alpha \in \mathbb{R}^{+}$.
I am however having trouble trying to show th... | https://mathoverflow.net/users/47797 | Is $\lfloor \log(n!)\rfloor \alpha$ equidistributed on the unit circle? | Yes. This follows from Theorem 3.2 of my [paper](http://www.math.uvic.ca/faculty/aquas/papers/paper18.pdf) with Michael Boshernitzan, Gregori Kolesnik and Máté Wierdl, 'Ergodic Averaging Sequences'.
| 15 | https://mathoverflow.net/users/11054 | 159410 | 84,023 |
https://mathoverflow.net/questions/159323 | 5 | I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here:
Given a regular tiling of the hyperbolic plane is there a criterion to say it is not a Cayley graph?
By regular tiling, I mean that the graph is vertex-transitive an... | https://mathoverflow.net/users/18974 | Which hyperbolic tilings are Cayley graphs? | Actually the requirement that your graph is vertex transitive is *very* strong. Any such tiling is "close" to the Cayley graph of a hyperbolic surface group, i.e. the fundamental group of a closed surface, which is not a sphere, torus, projective plane, or Klein bottle. Unfortunately the best description I can give you... | 5 | https://mathoverflow.net/users/38698 | 159413 | 84,024 |
https://mathoverflow.net/questions/159411 | 4 | Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f\_\*([S])$ is zero in $H\_2(M)$, is there a compact oriented 3-manifold $W$ with boundary $S$ and a map $F : W \to M$ that extends $f$?
| https://mathoverflow.net/users/23193 | 0-homologous surface bounds | Yes, there is. This follows from the definition of the (singular oriented) bordism group $\Omega\_2(M)$ and the fact that the natural map $\Omega\_2(M)\to H\_2(M;\mathbb{Z})$ is an isomorphism. In fact the conclusion holds for much more general spaces $M$ (such as CW-complexes).
In more detail, $\Omega\_2(M)$ denotes... | 7 | https://mathoverflow.net/users/8103 | 159414 | 84,025 |
https://mathoverflow.net/questions/159399 | 3 | For various reasons I became interested in bounds on the norm of commutators of operators. For instance, if $B(H)$ is the algebra of bounded operators on a Hilbert space, one may ask for a lower bound on $||[A,B]||$ in terms of properties of $A$ and $B$. (It is clear that one can not have simple norm estimates since fo... | https://mathoverflow.net/users/15488 | Lower bounds for norms of commutators | This doesn't address the full generality of your question but it might at least suggest places to hunt in the literature.
Thinking of $B$ as fixed, you're asking for a lower bound on the norm of $\operatorname{ad}\_B(A)$ as $A$ varies in some subspace. If we take $A$ to range over all of $B(H)$ then MathSciNet direct... | 4 | https://mathoverflow.net/users/763 | 159416 | 84,026 |
https://mathoverflow.net/questions/159277 | -2 | Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$.
The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ through the action on the coefficients of the equations defining $X$.
So, if for example $X$ is defined in $\mathbb{P}^... | https://mathoverflow.net/users/47739 | Order of the zero of a meromorphic function under the action of $Gal(\mathbb{C},\mathbb{Q})$ | This question amounts to tracing the ideal-theoretic interpretations of all of the objects in question. You have a coordinate ring $S$ for projective space, and a homogeneous ideal $I$ defining $X$. Complex conjugation takes $I$ to $I^\sigma$ (i.e., the coefficients of all functions are conjugated), and the homogeneous... | 1 | https://mathoverflow.net/users/121 | 159417 | 84,027 |
https://mathoverflow.net/questions/156727 | 3 | This question arose while reading <http://prd.aps.org/abstract/PRD/v13/i6/p1592_1> (Duality transformations of Abelian and non-Abelian gauge fields, by Stanley Deser and Claudio Teitelboim). It is well known that the source-free Maxwell equations $dF=0,\;d\;^\star F=0$ are invariant under duality transformations $$\mat... | https://mathoverflow.net/users/32389 | Electromagnetic duality symmetry | If one considers the reduced phase space of the electromagnetic field,
i.e., the space of the fields and their conjugate momenta modulo gauge transformations (which is an infinite dimensional symplectic space), then the electric-magnetic duality can be obtained from the Hamiltonian version of the Noether's theorem, giv... | 3 | https://mathoverflow.net/users/1059 | 159421 | 84,028 |
https://mathoverflow.net/questions/159419 | 2 | Calculating the [domination number](https://en.wikipedia.org/wiki/Dominating_set) is an NP-Hard problem. Does it remain NP-Hard if we restrict it to non-bipartite graphs?
| https://mathoverflow.net/users/22051 | Is the domination number NP for non-bipartite graphs? | The general case of the domination number problem reduces to the non-bipartite case. Given a (possibly bipartite graph) $G$, create new graph $G'$ consisting of an copy of $G$ and a disjoint copy of $K\_3$. The domination number of the new (non-bipartite) graph $G'$ is exactly one more than that of the original graph $... | 9 | https://mathoverflow.net/users/2000 | 159424 | 84,029 |
https://mathoverflow.net/questions/159446 | 2 | The ordinary Thom isomorphism says $H^{\*+n}(E,E\_{0}) \simeq H^{\*}(X)$, where $E$ is a vector bundle over $X$ and $E\_{0}$ is $E$ minus the zero section. Now assume that $S$ is a non vanishing section for the vector bundle $E$. In each fiber $E\_{x}$ we remove two points $0\_{x}$ and $S(x)$. Then we put $E\_{0,1}$for... | https://mathoverflow.net/users/36688 | A generalized Thom Isomorphism | You get two copies of $H^\*(X)$. In general ($k$ pairwise disjoint sections), by excision it's just like disjoint union of $k$ copies of the original bundle, hence $k$ copies of $H^\*(X)$. (An extra observation is that, for one section, the result does not depend on its choice, as any section is homotopic to $0$.)
Th... | 4 | https://mathoverflow.net/users/44953 | 159448 | 84,037 |
https://mathoverflow.net/questions/159435 | 4 | Let $H$ be some Polish group and $X$ some standard Borel space. Assume that $H$ acts measurably on $X$, i.e. $(h,x) \mapsto hx$ is Borel. Let $H\_0 \subset H$ and $A \subset X$ be some Borel sets. Is it true that the set of all $x \in X$ such that $H\_0 x \subset A$ is Borel?
The following theorem of descriptive set ... | https://mathoverflow.net/users/46931 | Measurability of $\{ x \in X ; H_0 x \subset A \}$ | No, it is not true.
It suffices to show that the complement is not necessarily Borel.
Let $B=X\setminus A$, which is a general Borel set. We have
$$
\{x:H\_0x\not\subseteq A\} = \{x:H\_0x\cap B\ne\varnothing\} = \{x: (\exists h\in H\_0)(hx\in B)\} = \{x: x\in H\_0^{-1}B\} = H\_0^{-1}B
$$
Let $G=H\_0^{-1}$ which is ag... | 4 | https://mathoverflow.net/users/4600 | 159453 | 84,038 |
https://mathoverflow.net/questions/159433 | 6 | I'm sorry for asking such a specific question, but i have trouble understanding one detail in the proof of Belyi's theorem in the book "Graphs on surfaces and their applications" by Lando and Zvonkin" and i don't know how to figure it out.
I'm referring to page 149 where they want to prove the implication "existence ... | https://mathoverflow.net/users/47807 | The proof of Belyi theorem by Lando and Zvonkin | The idea is that if you fix $X$, $f$, $\sigma$, and one such isomorphism $u\_0$, then for any other isomorphism $u:(X,f)\to (X^{\sigma},f^{\sigma})$ it follows that the composition $u^{-1}\circ u\_0$ is an automorphism of $(X,f)$. I don't know how Lando and Zvonkin define things, but anyway, an automorphism of $(X,f)$ ... | 1 | https://mathoverflow.net/users/30412 | 159461 | 84,041 |
https://mathoverflow.net/questions/159452 | 8 | Let $T$ be an Aronszajn-tree, $C\subset \omega\_1$ a club set and $f:\bigcup\limits\_{\alpha\in C}T\_\alpha\longrightarrow \mathbb Q$ a strictly increasing function (where $T\_\alpha$ is the $\alpha$-level of $T$). Is $T$ special (i.e. there exists such an $f$ defined on the whole $T$)?
I suspect that this is true an... | https://mathoverflow.net/users/41274 | Does being special on a club imply being special? | This is a classic result that I learned by reading *The Souslin Problem* by Devlin and Johnsbråten (Lecture Notes in Mathematics 405).
First, recall that another way to think of special trees is that they are the $\omega\_1$-trees that admit a cover by countably many antichains. Suppose $C = \{\gamma\_\alpha:\alpha\l... | 6 | https://mathoverflow.net/users/2000 | 159475 | 84,047 |
https://mathoverflow.net/questions/158769 | 41 | To three elements $a\_1$, $a\_2$, $a\_3$ in the finite field $\mathbb F\_q$
of $q$ elements we associate the number $N(a\_1,a\_2,a\_3)$
of elements $a\_0\in \mathbb F\_q$ such that the polynomial
$x^4+a\_3x^3+a\_2x^2+a\_1x+a\_0$ splits into four distinct linear factors over
$\mathbb F\_q$. The number
$$\sum\_{(a\_1,a\... | https://mathoverflow.net/users/4556 | A curious identity related to finite fields | For prime $q \geq 5$ write the count as
$$
\frac1{1152} q (q-1) (q^3 - 21q^2 + 171 q - c\_q)
$$
where
$$
c\_q = 483 + 36 \left(\frac{-1}{q}\right) + 64 \left(\frac{-3}{q}\right)
+ \delta\_q.
$$
Then for $(\frac{-2}{q}) = -1$ **Ronald Bacher**'s calculations
indicate $\delta\_q=0$. If $(\frac{-2}{q}) = +1$ then
$q$ can ... | 25 | https://mathoverflow.net/users/14830 | 159480 | 84,050 |
https://mathoverflow.net/questions/159464 | 11 | Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C\_{\cdot}(A)$ where
$$
C\_n(A):=A^{\otimes n+1}
$$
and the differntial $b: C\_n(A)\rightarrow C\_{n-1}(A)$ is defined to be
$$
b(a\_0\otimes\ldots\otimes a\_n):=\sum\... | https://mathoverflow.net/users/24965 | Is there any relation between the simplicial $S^1$ and the Hochschild homology of a noncommutative algebras | The article "[Hochschild and cyclic homology via functor homology](http://www.math.uni-hamburg.de/home/richter/hochcyc.pdf)," by Pirashvili and Richter, gives precisely such a description.
In this paper they describe a category of "noncommutative sets," isomorphic to one described earlier by Fiedorowicz and Loday. Th... | 8 | https://mathoverflow.net/users/360 | 159482 | 84,051 |
https://mathoverflow.net/questions/159459 | 0 | Let $(X, d)$ be complete metric space, $q \in [0, 1)$ be a real number, and $f$ be a map that satisfies $$d(f(x), f(y)) \leq q \cdot d(x, y)$$ for all $x, y \in X$. Then, Banach fixed-point theorem says that there is a unique fixed-point of $f$ in $X$.
I am interested in figuring out what happens if we relax the cond... | https://mathoverflow.net/users/24274 | A variation of the Banach fixed-point theorem | If you really mean $m=m(x,y)$ and $n=n(x,y)$ (as per your comments), your assumption is equivalent to say that every pair of orbits have minimum distance $0$: for any $x\in X$ and $y\in X$
$$\inf\_{ n\in\mathbb{N}\atop m\in\mathbb{N}}d\big((f^n(x),f^m(y)\big)=0\, .$$
A map $f$ satisfying this condition may be discont... | 5 | https://mathoverflow.net/users/6101 | 159492 | 84,055 |
https://mathoverflow.net/questions/159497 | 0 | I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for small $\varepsilon$. I'm also interested in $f\in L^1\log L^1$ or the limit case $f\in L^1$ but not in general measure da... | https://mathoverflow.net/users/33741 | uniqueness for Poisson equation in R^d with mildly regular data | If $v$ is a temperate distribution such that $\Delta v=0$ then the Fourier transform of $v$ must be supported at 0 hence it is a finite combination of derivatives of the Dirac delta, which in the end means that $v$ is a polynomial. Thus if I'm not mistaken the condition $u\in L^p$ for some finite $p$ should guarantee u... | 2 | https://mathoverflow.net/users/7294 | 159500 | 84,056 |
https://mathoverflow.net/questions/159463 | 9 | Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1\_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which particularly requires boundedness for compactness of the solution operator of the corresponding elliptic problem.
**Is... | https://mathoverflow.net/users/nan | Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$ | If $u\_j$ is such a basis, from $(\nabla u\_j,\nabla u\_k)=0$ for all $j\not=k$ it follows $(\Delta u\_j,u\_k)=0$ which means $\Delta u\_j$ is a multiple of $u\_j$, and unfortunately $\Delta$ has no eigenvalues in $L^2$.
| 8 | https://mathoverflow.net/users/7294 | 159501 | 84,057 |
https://mathoverflow.net/questions/159447 | 2 | $$X\_{solution}=\arg\min\_X \|XAX-B|\_F \quad\mathrm{subject\ to}$$
X is square and diagonal
A is square and positive semi-definite
B is square and positive semi-definite
Any pointers or relevant search terms?
| https://mathoverflow.net/users/47818 | Matrix equation XAX=B where the solution must be diagonal | @Robert, we can assume that $A$ is diagonal and $X$ is symmetric, that I shall do in the sequel. We suppose that the matrices are real. You seek the argmin of the function $f:X\in Sym\rightarrow trace(UU^T)$ where $U=XAX-B$. Then $D\_Xf=0$ iff, for every symmetric matrix $H$,
$trace((AX^2AX+XAX^2A-AXB-BXA)H)=0$. There... | 1 | https://mathoverflow.net/users/9091 | 159505 | 84,060 |
https://mathoverflow.net/questions/159517 | 0 | Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int \_{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space,
$$A(\mathbb T):= \{f\in L^{1}(\mathbb T): \hat{f}\in \ell^{1}(\mathbb Z), \ \text {that is,} \ \sum\_{n\in \mathbb Z} |\hat{f}(n)| < \infty \}.$... | https://mathoverflow.net/users/33018 | Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ? | What's your motivation?
The Fourier transform is a \*-homomorphism for the natural adjoint on $\ell^1(\mathbb Z)$ and so if $f$ is the transform of $a\in\ell^1(\mathbb Z)$ then $|f|^2f$ is the transform of $a^\*aa$. Similarly let $g$ and $b$ be related. As $\|a\|,\|b\| \leq M$ we perform a simply triangle-inequality ... | 3 | https://mathoverflow.net/users/406 | 159524 | 84,069 |
https://mathoverflow.net/questions/159536 | 12 | Let $R$ be a local Noetherian ring.
What is the *geometric interpretation* of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric interpretation?
| https://mathoverflow.net/users/47763 | geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings | *Regular* rings have clear and famous geometric interpretation: They correspond to *non-singular* (affine) varieties.
A (local) *complete intersection* rings can be thought of as a local ring at a point, such that at this point the variety can be defined by minimal number of relations, i.e. a regular sequence. In mo... | 16 | https://mathoverflow.net/users/37808 | 159546 | 84,075 |
https://mathoverflow.net/questions/159544 | 2 | Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the [going-up theorem](http://en.wikipedia.org/wiki/Going_up_and_going_down#Going-up) hold between $R$ and $S$?
| https://mathoverflow.net/users/47763 | Does the going-up theorem hold between flat algebras? | For a counterexample, take
$$R={\mathbb Z}\qquad S=R[x]\qquad P=(1+2x)\subset S$$.
Then $P\cap R=(0)\subset (2)$, so if going-up holds, then there is a prime $Q$ in $S$ containing $(1+2x)$ and such that $Q\cap R=(2)$. But then $Q$ contains $2$, so $Q$ contains $(2,1+2x)=S$, contradiction.
| 6 | https://mathoverflow.net/users/10503 | 159548 | 84,076 |
https://mathoverflow.net/questions/159402 | 2 | Let $G$ be a compact connected Lie group and $O\_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^\*$ and $\mathfrak{g}^\*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let $P$ be an invariant polarization of the coadjoint orbit $O\_{\mu}$ and $D=P\cap\bar P\cap TO\_μ$ (here $\bar P$ i.e. com... | https://mathoverflow.net/users/nan | The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$ | The leaves of $D$ are points and the leaf space $O\_\mu/D$ is $O\_\mu$ itself.
Indeed, a $G$-invariant polarization $P$ (involutive lagrangian subbundle of $(TO\_\mu){}^\mathbf C$) is determined by the preimage $\mathfrak p\subset\mathfrak g^\mathbf C$ of its value $P\_\mu\subset (T\_\mu O\_\mu){}^\mathbf C=(\mathfr... | 0 | https://mathoverflow.net/users/19276 | 159564 | 84,080 |
https://mathoverflow.net/questions/159547 | 5 | In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter II, section 4, p.93), that is one has
$$\otimes: D^{-}(X) \times D^{-}(X) \to D(X), $$ where $X$ is a variety, and $D^{-}(X), D(X)$ are derived category of **bounded above** complex of $\ma... | https://mathoverflow.net/users/29730 | Why is the derived tensor product only defined for bounded above derived categories? | I would like to add my 2 euro ¢. In a nutshell: your information is **outdated**. "Residues and Duality" by Hartshorne was the first available text on derived categories of coherent sheaves and Grothedieck duality but it was published in 1966. In 1988, Spaltenstein showed in ("[Resolutions of unbounded complexes](http:... | 18 | https://mathoverflow.net/users/6348 | 159572 | 84,084 |
https://mathoverflow.net/questions/159449 | 15 | It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative setting. So my question is:
>
> Why can't we define the category of "affine non-commutative schemes" by simply putting it t... | https://mathoverflow.net/users/32741 | Why is "naive" definition of non-commutative spectrum bad? | I have also wondered about this question and recently came across some papers that seem to answer it.
First of all, the paper
* Manuel L. Reyes, *Obstructing extensions of the functor Spec to noncommutative rings*, Israel J. Math. 192 (2012), no. 2, 667–698, [arXiv](http://arxiv.org/abs/1101.2239).
shows that the... | 19 | https://mathoverflow.net/users/2503 | 159573 | 84,085 |
https://mathoverflow.net/questions/159534 | 6 | Does the following 'alternating' Euler product, with $p\_n$ the $n$-th prime number, converge for $\Re(s)>0$ ?
$$\displaystyle \prod\_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p\_{n}^{s}}} \right)^{(-1)^n}$$
Based on numerical evidence, I dare to conjecture that this is indeed the case (note that I could not find a... | https://mathoverflow.net/users/12489 | Does this 'alternating' Euler product converge for all $\Re(s) > 0$? | Taking logarithms, we arrive at the sum
$$-\sum\_{n=1}^{\infty} (-1)^n \log(1-p\_n^{-s}).$$ For $Re(s)>1/k$, we can approximate the logarithm with its taylor series to $2k+1$ terms with an error of $o(p\_n^{-2})$ which converges absolutely. Thus, it suffices to show that for each $s>0$, the sum
$$\sum\_{n=1}^{\in... | 11 | https://mathoverflow.net/users/4181 | 159580 | 84,089 |
https://mathoverflow.net/questions/159559 | 11 | Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in Goldman-Millson (1988).
So I wonder what precisely goes wrong in (mixed) characteristic p>2?
| https://mathoverflow.net/users/47856 | Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?) | More precisely, the "Deligne principle" of deformation theory (but actually one could add a lot of other names) is that every deformation problem corresponds to a deformation functor, which in turn is defined by a certain dg Lie algebra controlling the deformations.
More precisely, from any dg Lie algebra $g$ and any c... | 8 | https://mathoverflow.net/users/36625 | 159582 | 84,091 |
https://mathoverflow.net/questions/159585 | 0 | Suppose $A$ and $B$ are operators on a (separable) Hilbert space $H$ and $A \leq B$. Is it true that if $B$ is compact then $A$ is compact too? If not, could you please show a counterexample?
| https://mathoverflow.net/users/33804 | Does positivity preserve compactness? | What you need to assume is that $|A|\leqslant B$.
Assume first that for all $x$ one has $\Vert A x \Vert \leqslant \Vert B x \Vert$ and $B$ is compact. Then choose any net $P\_i$ of finite rank projections which converge weakly to the identity, one has $\Vert A x - A P\_i x\Vert \leqslant \Vert B x - B P\_i x\Vert$ w... | 2 | https://mathoverflow.net/users/22131 | 159591 | 84,095 |
https://mathoverflow.net/questions/159596 | 7 | In $K$ theory, the Borel construction of equivariant cohomology is somehow not the right one. The $G$-equivaraint $K$ theory of a point should be the representation ring of $G$, but $K(BG)$ is this ring completed at the ideal of zero dimensional representations (this is the Atiyah-Segal completion theorem).
I am wond... | https://mathoverflow.net/users/36591 | The Borel construction of equivariant cobordism | No, that map is not an isomorphism.
First, there are a couple of different things that you might mean by $\Omega^\*\_G(\*)$, and if you want the full story then you need to distinguish between them, but there is no standard interpretation under which your map is iso. The basic point is that, just as in the Atiyah-Se... | 11 | https://mathoverflow.net/users/10366 | 159599 | 84,097 |
https://mathoverflow.net/questions/159539 | 1 | Let $X=(X\_1,\dots,X\_d)$ be a random vector, and a.s. $X \in [0,1]^d$. Suppose that for every $a \in \mathbb{R}^d$, we know the probability distribution of the random variable $Y\_a = <a,X>$. My question is that can one uniquely determine the distribution of $X$ (i.e., the joint distribution of $X\_1,\dots,X\_d$)?
I... | https://mathoverflow.net/users/30853 | Determine joint distribution from projections | The answer to the first question is yes. Think of the characteristic function of X, namely
$$
\varphi\_X(a)
= \mathbb{E}[e^{i\langle a,X\rangle}]
= \mathbb{E}[e^{i Y\_a}]
$$
For the second question, I would say the answer is no. If you know $(Y\_{a\_i})\_{i=1,\dots,n}$, it amounts to know $\varphi\_X$ along the dire... | 3 | https://mathoverflow.net/users/8966 | 159605 | 84,099 |
https://mathoverflow.net/questions/159609 | 3 | Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int\_{a}^{b} t^nf(t)dt=0$ for all integer n.
We know that $f\equiv 0$. It's call **Hausdorff theorem**.
* This theorem is wrong on $\mathbb{R^+}$, a counter example is :
$$f(x)=\exp(-x^{\frac{1}{4}})\sin(x^\frac{1}{4})$$
>
> In fact this exercice was poste... | https://mathoverflow.net/users/nan | Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? | The standard reference is N. I. Akhiezer, The Classical moment problem and some related questions of analysis, MR0184042.
There is also a paper:
MR1627806 Simon, Barry The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137 (1998), no. 1, 82–203.
| 5 | https://mathoverflow.net/users/25510 | 159618 | 84,103 |
https://mathoverflow.net/questions/159615 | 5 | Take a prime other than 2,3 or 5 and look at the part of it that repeats in base 10. Is it true that the sum of the digits in the end divided by the period(number of repeated digits id always $\frac{9}{2}$ Can we extend this result to other bases? Why does this happen?
Regards.
This is a crosspost from math.se
| https://mathoverflow.net/users/24478 | Sum of digits of repeating end of reciprocal of prime over period is $\frac{9}{2}$ | Note first that the counterexamples to user4140's observation involve decimal expansions with odd period length. In the case of even period length, what user4140 has observed is a consequence of Midy's theorem, which says the following: let p be a prime not dividing 10, and suppose that the repeating part of the decima... | 11 | https://mathoverflow.net/users/47900 | 159621 | 84,105 |
https://mathoverflow.net/questions/159626 | 4 | Recently I'm reading *Stochastic Equations in Infinite Dimensions*, a result is used many times. It is
>
> *If $E$ is a separable Banach spaces, then there is a sequence $\{ \phi\_n \}$ in its dual $E^{\star}$ such that $$\|x\|=\sup\_n |\phi\_n(x)|$$*
>
>
>
my question is
(1) How to prove it? Or where can I... | https://mathoverflow.net/users/44175 | The norm of a separable Banach space can be determined by countable continuous linear functionals? | For (1): Pick a countable dense set $x\_i\in E$, and for each $x\_i$ in that set, pick (by Hahn–Banach) a functional $\phi\_i$ of norm $1$ such that $\|x\_i\|=\phi\_i(x\_i)$.
For (2): duals of separable Banach spaces also have that property. For example, $L^{\infty}(\mathbb R)$ is the dual of $L^1(\mathbb R)$ and its... | 8 | https://mathoverflow.net/users/5690 | 159629 | 84,107 |
https://mathoverflow.net/questions/159649 | 3 | We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is rationally connected. (This is
established as a corollary of Graber, Harris and Starr’s theorem).
My question is why ... | https://mathoverflow.net/users/31724 | $\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational | First, a $\mathbb{P}^1$-bundle over $\mathbb{P}^2$ is by definition rational, because it is locally trivial. What you want is probably a fibration over $\mathbb{P}^2$ with general fibre rational (but not the generic fibre), and which is not rational. For this, consider conic bundles over $\mathbb{P}^2$. The discriminan... | 6 | https://mathoverflow.net/users/23758 | 159651 | 84,110 |
https://mathoverflow.net/questions/159644 | 5 | Let $C$ be a projective curve (over an algebraically closed field, not necessarily of characteristic zero) which is smooth except for exact one node. Let $\pi:\tilde{C} \to C$ be its normalization. Let $\mathcal{V}$ be a locally free sheaf over $\tilde{C}$. Is it true that the push forward, $\pi\_\*(\mathcal{V})$ is a ... | https://mathoverflow.net/users/46578 | Normalization of a curve and push forward of vector bundles | Quoting answer\_bot: "No and yes for the rank zero vector bundle." In other words, the answer is "no" except for one case: the rank zero vector bundle. That means the answer is already "no" for *every* rank 1 vector bundle. To user46578, I recommend that you contemplate what happens for the trivial rank 1 vector bundle... | 3 | https://mathoverflow.net/users/13265 | 159659 | 84,114 |
https://mathoverflow.net/questions/159668 | 2 | This question is base on my previous [question](https://mathoverflow.net/questions/159547/why-is-the-derived-tensor-product-only-defined-for-bounded-above-derived-categor/159549#comment408017_159549), and I repeat it here:
Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below com... | https://mathoverflow.net/users/29730 | Injective resolution for right derived functor | You are certainly allowed to do this and you will get a **right** derived functor because you are looking at resolutions $G^\bullet \to I^\bullet$ (note how the arrow is pointing to the right!). This works because any two injective resolutions $I^\bullet$ of $G^\bullet$ are homotopy equivalent, so you can use this to c... | 3 | https://mathoverflow.net/users/44862 | 159670 | 84,121 |
https://mathoverflow.net/questions/159669 | 5 | Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega\_1^V$ is actually a limit cardinal.
The original proof is in German, and I've been looking for an English account of the proof. I couldn't find it in all the usual places (usually just a reference). I do... | https://mathoverflow.net/users/7206 | Perfect set property implies $\omega_1$ is a limit cardinal in $L$ | The usual proof (as in Kanamori's book, section 11) is as follows: Work in $\mathsf{ZF}$. Note first, with Bernstein, that if $\omega\_1\le\mathfrak c$, then there is a set of reals without the perfect set property: Either $\omega\_1=\mathfrak c$, so $\mathbb R$ can be well-ordered, and we can build Bernstein sets usin... | 8 | https://mathoverflow.net/users/6085 | 159679 | 84,127 |
https://mathoverflow.net/questions/159329 | 5 | Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$.
(Q1). do we have $L\cdot D\geq0$ ?
If either one has positive self-intersection, the answer is yes by index theorem. Assume $L^2=D^2=0$. If either one is basepoint free then it is a multiple of an ellipt... | https://mathoverflow.net/users/40038 | Positivity question on K3 surfaces | Let me prove that $(L,D)\geq 0$.
Consider the set of all (1,1)-classes which satisfy
$\eta^2\geq 0$. It is a union of two components $P\_+$ and $P\_-$,
intersecting in 0. If $\eta, \eta'$ are in the same component
($P\_+$ or $P\_-$), they intersect non-negatively, by Hodge index
theorem. Let $P\_+$ be a component ... | 3 | https://mathoverflow.net/users/3377 | 159680 | 84,128 |
https://mathoverflow.net/questions/159663 | 12 | A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical characterization of when functors give covering maps on the topological side (i.e. the map is a local homeomorphism)?
| https://mathoverflow.net/users/27923 | Functors and coverings | Coverings of groupoids are described for example in an ancient book <http://www.maths.ed.ac.uk/~aar/papers/gz.pdf>.
The condition on a groupoid functor C' -> C is that for any arrow f in C and an object x in C' which is taken to the source or the target of f by the given functor, there is a unique lifting of f in C'... | 5 | https://mathoverflow.net/users/39004 | 159686 | 84,131 |
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