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https://mathoverflow.net/questions/245525 | 11 | In many References such as *D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds* chapter 9, and *[Differential Geometric Structures
By Walter A. Poor](https://books.google.com/books?id=6atzBgAAQBAJ&q=horizontal#v=snippet&q=horizontal&f=false)* Page 54; the horizontal and vertical lift(space) of a vector... | https://mathoverflow.net/users/90655 | Geometric interpretation of horizontal and vertical lift of vector field | I find the following viewpoint helpful to translate between the different incarnation of a connection.
To every vector bundle $\pi: E \to M$ (in your case $E = TM$) we have an associated exact sequence of vector bundles (sometimes called the Atiyah sequence, at least in the principal bundle case):
$$ 0 \to V E \to TE... | 19 | https://mathoverflow.net/users/17047 | 245576 | 112,298 |
https://mathoverflow.net/questions/211810 | 0 | I want to find out the correspondences for the following two formulas or whether they are already derivable in the modal logic $KD4.2$, i.e. whether the formulas are valid in serial, transitive and directed frames.
1:
$(\lozenge (\lozenge p\wedge\Box(p\rightarrow q)) \wedge \Box(\lozenge p\rightarrow\lozenge(p\wedge ... | https://mathoverflow.net/users/37385 | A question on two modal formulas | I do have some of my own code that I was able to modify to investigate this.
Short answer
------------
* (1) is not a theorem of $\mathbf{KD4.2}$ since it is invalid on the frame
$F=(W,R)$ where $W=\{A,B,C\}$ and $R=\{(A,A),(B,A),(B,B),(B,C),(C,A)\}$,
and $F$ is serial, transitive and convergent.
* (2) is a theorem... | 2 | https://mathoverflow.net/users/37336 | 245583 | 112,300 |
https://mathoverflow.net/questions/245581 | 3 |
>
> Let $x,y \in \mathbb{Z}$ satisfying $3y^2 = 4x^3 - 1$. Does it follow
> that $x = 1$ and $y = \pm 1$?
>
>
>
Wolfram Alpha says that the answer is positive, but I am not so satisfied with an answer by a computer program since it is (most of the time) not accompanied by a proof, and even if it is, such a proo... | https://mathoverflow.net/users/38889 | Are there integer solutions to $3y^2 = 4x^3-1$ other than $(1,1)$ and $(1,-1)$? | The projective form of your curve is $3y^{2} z = 4x^{3} - z^{3}$. This has three obvious points: $(1 : 1 : 1)$, $(1 : -1 : 1)$, and $(0 : 1 : 0)$.
Your curve is isomorphic over $\mathbb{Q}$ to the Fermat cubic, $x^{3} + y^{3} = z^{3}$. This latter curve has only three rational points on it: $(1 : -1 : 0)$, $(1 : 0 : ... | 17 | https://mathoverflow.net/users/48142 | 245586 | 112,301 |
https://mathoverflow.net/questions/245594 | 7 | Let me first give the definition of a local coefficient system (see also [2, p. 257] and [3, p. 35]):
Let $X$ be a topological space. A local coefficient system is a functor from the category $\Pi\_1(X)$ (= the fundamental groupoid) to the category AbGrp of abelian groups. Such a functor assigns to each $x \in X$ an ... | https://mathoverflow.net/users/95408 | (Non-trivial) local coefficient system which is not a bundle of groups | Strip away the group structure and you get the simpler question: Does every functor from the fundamental groupoid of $X$ to Set correspond to a covering space (a bundle of sets)? As a special case this includes the question, does every subgroup of the fundamental group of a path-connected space come from a connected co... | 14 | https://mathoverflow.net/users/6666 | 245596 | 112,305 |
https://mathoverflow.net/questions/245598 | 3 | Let $X\_1,\dotsc, X\_n$ be $n$ i.i.d. random variables where $X\_1 \in [a,b]$. Similarly, let $Y\_1,\dotsc,Y\_m$ be $m$ i.i.d. random variables where $Y\_1 \in [c,d]$. Furthermore, $X\_i$ and $Y\_j$ are independent for all $i \in \{1,\dotsc,n\}$ and $j \in \{1,\dotsc,m\}$. Intuition tells me that for any $\delta \in (0... | https://mathoverflow.net/users/84393 | Hoeffding's inequality for sums of pairs of random variables | This inequality follows from Theorem 2 in Hoeffding's 1963 paper, and in fact Hoeffding's result yields a better bound. Indeed, Hoeffding's inequality can be written as
\begin{equation}
P(\sum Z\_i<t)\ge1-\exp\Big(-\frac{2t^2}{\sum(B\_i-A\_i)^2}\Big), \tag{1}
\end{equation}
where $t$ is a nonnegative real number, the... | 2 | https://mathoverflow.net/users/36721 | 245604 | 112,309 |
https://mathoverflow.net/questions/245437 | 7 | Is there an example of two different complex manifolds that have the same de Rham cohomology and Dolbeault cohomology but different Bott-Chern/Aeppli cohomology?
| https://mathoverflow.net/users/30172 | Is the Bott-Chern/Aeppli cohomology determined by the de Rham and Dolbeault cohomologies? | I learnt all of the following from section $3.2$ of Angella's *Cohomological Aspects in Complex Non-Kähler Geometry*.
---
Let $R$ be a commutative ring with identity. The *three-dimensional Heisenberg group* over $R$ is
$$\mathbb{H}(3, R) = \left\{\begin{bmatrix} 1 & z^1 & z^3\\ 0 & 1 & z^2\\ 0 & 0 & 1\end{bmat... | 10 | https://mathoverflow.net/users/21564 | 245608 | 112,311 |
https://mathoverflow.net/questions/217804 | 11 | Suppose $X$ is a complex manifold.
If $X$ is Kähler, the cohomology groups decompose into subgroups represented by $(p,q)$-forms.
If $X$ is not Kähler, I think the decomposition may not hold?
Is there an example where we have a nonzero class be represented by both a $(p,q)$-form and a $(p',q')$-form with $(p, ... | https://mathoverflow.net/users/nan | Can a class be represented by both a $(p,q)$-form and a $(p',q')$-form? | I think you want an example of a compact complex manifold $X$ and differential forms $\gamma \in \mathcal{E}^{p,q}(X)$ and $\gamma'\in \mathcal{E}^{p',q'}(X)$ with $(p',q') \neq (p, q)$ such that $[\gamma] = [\gamma']$ in de Rham cohomology.
Let $X$ be a compact complex three-dimensional manifold with a non-closed ho... | 10 | https://mathoverflow.net/users/21564 | 245625 | 112,314 |
https://mathoverflow.net/questions/244945 | 3 | If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many implications of the Birch and Swinnerton-Dyer conjecture.
I want to ask, and excuse me if this is "stupid": Is it enough to show the equality of the ranks for a certain... | https://mathoverflow.net/users/70751 | Selmer and free rank of Elliptic Curves | You first statement is correct, both ranks are expected to be equal. In particular we have:
$$\mathrm{rank}\,\,\mathrm{Sel}\_p(E/K)=\mathrm{rank}(E/K)+\mathrm{rank}\,\,Ш(E/K)[p^\infty]$$
So if either $Ш$ or its $p$-primary part are finite, then the equality holds. And conversely, the equality *for any prime* implie... | 6 | https://mathoverflow.net/users/43108 | 245636 | 112,317 |
https://mathoverflow.net/questions/245585 | 5 | An algebra $A$ is said to be tame if the isomorphism classes of indecomposable $A$-modules in each dimension occur in a finite number of 1-parameter families. $A$ is said to be of finite representation type if the number of distinct isomorphism classes of indecomposable $A$-modules is finite. Thus if an algebra is tame... | https://mathoverflow.net/users/95742 | Classification of indecomposable modules in tame hereditary algebras | In the Dynkin case, [Gabriel's theorem](https://en.wikipedia.org/wiki/Gabriel%27s_theorem) states that the indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram. You can read about it for instance in chapter VII of
*Ibrahim Assem, Daniel Sim... | 4 | https://mathoverflow.net/users/18756 | 245639 | 112,318 |
https://mathoverflow.net/questions/244621 | 12 | Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits analytic continuation to the whole complex plane.
It is known for $1$-dimensional and induced representations, plus a few ot... | https://mathoverflow.net/users/nan | Artin conjecture on L-functions | This is the status as far as I know. For dimension $\leq 2$ it is up to date. For higher dimensional representations I'm sure it is very incomplete, so feel free to edit or comment.
**Dimension 1.** Known by "Artin-Hecke".
**Dimension 2.** Only open case is even $A\_5$ representations. References for the known case... | 8 | https://mathoverflow.net/users/43108 | 245642 | 112,319 |
https://mathoverflow.net/questions/245646 | 5 | Consider the symmetric power series
$$f = \prod\_{i \in I}\left(1+x\_i+x\_i^2+x\_i^4+x\_i^8 + x\_i^{16} +\ldots \right)$$
in variables $(x\_i)\_{i \in I}$ over $\mathbb F\_2$. Fix some degree $r$, smaller than the number of variables, and denote the degree $r$ part of $f$ by $f^{(r)}$.
From looking at small-dimensio... | https://mathoverflow.net/users/14233 | Symmetric power series over $\mathbb{F}_2$ | Let $I=\{i\_0\} \cup J$ and $g(x)=1+x+x^2+x^4+\dots$.
We show that adding the relation $\sum\_{i\in I}x\_i=0$, i.e. $x\_{i\_0}=\sum\_{i\ne i\_0}x\_i$ leads to an even degree power series:
$$ f=g(x\_{i\_0})\prod\_{j\in J} g(x\_j) =\left(1+\sum\_{i\in J} (g(x\_i)-1)\right)\prod\_{j\in J} g(x\_j)\\ = \prod\_{j\in J} ... | 5 | https://mathoverflow.net/users/95545 | 245653 | 112,320 |
https://mathoverflow.net/questions/245616 | 1 | A group $G$ acts freely on a manifold $M$, then $H^\*\_G(M)=H^\*(M/G)$.
>
> Why is $H^\*\_G(M)$ a torsion $H^\*\_G$-module, where $H^\*\_G=H^\*\_G(pt)=H^\*(BG)$?
>
>
> If $G=T=(S^1)^{n+1}$ is a torus then $H^\*\_G=H^\*\_T=\mathbb{Q}[t\_0,...,t\_n]$. Why does $t\_i$ act on $H^\*\_G(M)$ by multiplication by $0$?
> ... | https://mathoverflow.net/users/83786 | Elementary question: Intuition for equivariant cohomology | I'm guessing that, unstated, $M,G$ are finite-dimensional and $G$ is connected Lie.
1. Then $H^\*(M/G)$ vanishes for $\* \gg 0$, but $H^\*\_G$ is positively graded, so $H^\*(M/G)$ must be a torsion module. (Non-example: $M$ is the unit sphere in Hilbert space, $G=U(1)$.)
2. $H^{\*>0}(T/T)=0$, and each $\deg t\_i = 2$... | 1 | https://mathoverflow.net/users/391 | 245658 | 112,323 |
https://mathoverflow.net/questions/245664 | 7 | In many articles (for example in articles given by M.Marcoli) there is statement that there is the following conjecture
>
> Residues of Feynman integrals in scalar field theories are always periods of mixed Tate motives
>
>
>
I have three questions related to this conjecture
1) What is the recent knowledge o... | https://mathoverflow.net/users/75934 | Conjecture of relation between residues of Feynman integrals and mixed Tate motives | 1) Counterexamples were found in the paper Brown, Francis; Schnetz, Oliver:
"A $K3$ in $\phi^4$". Duke Math. J. 161 (2012), no. 10, 1817–1862. It is now the general feeling that most $\phi^4$-Feynman integrals are not mixed Tate.
2) In 2008, Schnetz has compiled a list of Feynman integrals in [Quantum periods: A cens... | 16 | https://mathoverflow.net/users/89948 | 245666 | 112,328 |
https://mathoverflow.net/questions/245617 | 2 |
>
> Let $\pi: X\to Y$ be an Iitaka fibration of projective varieties
> $X,Y$, then is there always the following decomposition
>
>
> $$K\_Y+\frac{1}{m!}\pi\_\*\mathcal O\_X(m!K\_{X/Y})=P+N$$
>
>
> where $P$ is semiample and $N$ is effective and
>
>
> $$H^0(Y,\mathcal O\_X(maP))\cong H^0(X,\mathcal O(ma(K\_X+... | https://mathoverflow.net/users/nan | A Decomposition for Iitaka fibration | If $X$ is smooth (projective over the complex numbers), then $R(K\_X)$ is finitely generated by BCHM. We may thus assume that $R(kK\_X)$ is generated in degree 1 for some $k>0$. Passing to a log resolution of $|kK\_X|$ we may assume that $|kK\_X|=M+F$ where $F$ is the fixed divisor and $M$ is base point free and so $M$... | 3 | https://mathoverflow.net/users/19369 | 245682 | 112,332 |
https://mathoverflow.net/questions/245676 | 5 | Grothendieck (EGA I 0.7 & 1.10) defined a category of (topologically Noetherian) "formal rings" and a corresponding global category of formal schemes. Roughly, a formal ring is a topological commutative ring $R$ whose topology is $I$-adic for $I\subset R$ an open ideal. Its formal Spec is $Spf(R) : = Spec(R\_{red})$ fo... | https://mathoverflow.net/users/7108 | A derived category of formal sheaves | For a start, you have an approach to ind-coherent sheaves and its derived category in
Duality and Flat Base Change on Formal Schemes
*Contemp. Math.* **244** (1999), pp. 3-90.
[On line version, with some corrections incorporated.](http://www.math.purdue.edu/~lipman/papers/formal-duality.pdf)
The theory gets inter... | 8 | https://mathoverflow.net/users/6348 | 245683 | 112,333 |
https://mathoverflow.net/questions/245686 | 2 | Let $q\geq 2$ be an integer and $\alpha \in R$ such that $(q-1)\alpha \in R \setminus Z.$ For every positve integer $n$ there exists a unique sequence $(a\_j(n)\_{j\geq 0},$ $a\_j(n) \in \{0,1,...,q-1\}$ such that $$n=\sum\_{k=0}^{\infty} a\_k(n)q^{k}.$$ Define the function sum of digits in the base $q$ by
$$S\_q(n):=... | https://mathoverflow.net/users/76102 | On an open problem of Gelfond | Michael Drmota and Christian Mauduit [EDIT: and Joel Rivat], The sum-of-digits function of polynomial sequences, J. Lond. Math. Soc. (2) 84 (2011), no. 1, 81–102, MR2819691 (2012f:11193) Theorem 1:
Let $d\ge2$ be an integer, $q\ge q\_0(d)$ be a sufficiently large prime number, and $P$ a polynomial of degree $d$ with... | 7 | https://mathoverflow.net/users/3684 | 245687 | 112,334 |
https://mathoverflow.net/questions/245657 | 4 | Did not get an answer from the Stack Exchange.
Let $MonCat$ and $Cat$ denote the 2-categories of monoidal categories with strict monoidal functors and small categories, respectively.
There is a forgetful functor $Forget:MonCat\rightarrow Cat$ Does this preserve (filtered) colimits? I am thinking of 2-colimits in Mo... | https://mathoverflow.net/users/84563 | Forgetful Functor $MonCat\rightarrow Cat$ preserves filtered colimits? | Yes. Intuitively, this is because all the operations of a monoidal category are "finitary".
One way to prove this formally is that the 2-monad for monoidal categories can be given a presentation in the category of finitary 2-monads, hence it is finitary (which is equivaent to its forgetful functor preserving filtered... | 9 | https://mathoverflow.net/users/49 | 245693 | 112,335 |
https://mathoverflow.net/questions/245667 | 4 | I am having trouble with the following matrix equation:
$(K + MU)(K + MU) = U $
$K$, $M$, and $U$ are all square matrices, the values of $K$ and $M$ are known (but they don't have a particularly simple form, e.g. they are not diagonal). I would like to find a solution for $U$.
Does anyone know how this can be don... | https://mathoverflow.net/users/95787 | Specific quadratic matrix equation | Set $Y=K+MU$. You have $Y^2=U$, so $K+MY^2=K+MU=Y$, which gives an equation in $Y$ only:
$$
K-Y+MY^2 = 0
$$
This is a widely studied equation; see for instance Higham and Kim, <http://www.maths.manchester.ac.uk/~higham/narep/narep347.pdf>. In particular, there are several solutions, whose eigenvalues are $n$ out of the... | 9 | https://mathoverflow.net/users/1898 | 245699 | 112,337 |
https://mathoverflow.net/questions/245672 | 4 | Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) for the following: The set of points in $H$ corresponding to a Cohen-Macaulay subscheme is an open subset.
Does someone... | https://mathoverflow.net/users/36563 | Being Cohen-Macaulay open in Hilbert scheme? | I am just posting the comment as an answer. One reference is EGA $\textrm{IV}\_2$, Section 6.11, pp. 158-163.
**Edit.** Hans points out that in EGA only the absolute version of the results are proved, whereas he is asking about the relative version. However, the same proofs as in that section prove the relative resul... | 6 | https://mathoverflow.net/users/13265 | 245709 | 112,339 |
https://mathoverflow.net/questions/245706 | 4 | Let $X/k$ be a surface (over some field), smooth except for an isolated (closed) point $x$. One may look at the punctured local ring
$X:=\mathrm{Spec}(\mathcal{O}\_{X,x}) - x$.
Are there non-trivial vector bundles on $X$? If $X$ is smooth at the puncture as well, every vector bundle on $X$ must be trivial, thanks t... | https://mathoverflow.net/users/95819 | Vector bundles on punctured disc around isolated surface singularity | I am just posting my comment as an answer. For a Noetherian local ring $\mathcal{O}\_{X,x}$ with maximal ideal $\mathfrak{m}\_{X,x}$, the $\mathcal{O}\_{X,x}$-module $\mathcal{O}\_{X,x}/\mathfrak{m}\_{X,x}$ has a finite free resolution if and only if the local ring is regular. Thus, for every surjection of $\mathcal{O}... | 4 | https://mathoverflow.net/users/13265 | 245710 | 112,340 |
https://mathoverflow.net/questions/245030 | 11 | For a connected $n$-manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a point $p\in M$ we define $L\_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\}$. Of course $L\_{p}$ is a Lie subalgebra of $\chi^{\infty}(M)$ whose codimension is equal to $n$.
>
> Is it true that every codi... | https://mathoverflow.net/users/36688 | Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n = \dim M$ | Correction:
===========
The argument I gave initially is wrong. I treated $\mathfrak X(M)'$ like the space of differential forms. Only operations on $\mathfrak X(M)$ go over to the dual as (negative) adjoint operations, so $\mathcal L\_X$ makes sense but $i\_X$ and $d$ do not. Since it created some interest I leave t... | 3 | https://mathoverflow.net/users/26935 | 245713 | 112,341 |
https://mathoverflow.net/questions/245690 | 3 | Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish].
Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume that $\mathcal{S}$ is an element of the $\sigma$-algebra on the space of probability measures on $(X,\Sigma)$ generated b... | https://mathoverflow.net/users/15570 | "Strongly mutually singular" families of measures, and the set of ergodic measures | The set of all $f$-invariant Borel probability measures on $\mathbb{R}$ is $A$-strongly mutually singular. To see this let $L(\mu)$ be the set of all points $x$ such that $\frac{1}{n}\sum\_{k=0}^{n-1}\phi(f^k(x)) \to \int \phi\,d\mu$ for all compactly supported continuous $\phi \colon \mathbb{R}\to \mathbb{R}$. Note th... | 4 | https://mathoverflow.net/users/1840 | 245722 | 112,344 |
https://mathoverflow.net/questions/245712 | 1 | Let $M$ be a smooth Riemannian manifold of dimension $d$. I wish to choose in a measurable way a map $C\_x:T\_xM\rightarrow \mathbb{R}^d$ s.t.
$$\forall u,v\in T\_xM: \langle C\_xu,C\_xv\rangle=\langle u,v\rangle\_x'$$
where $\langle\cdot , \cdot\rangle\_x'$ is some inner-product on the tangent space, which is defined ... | https://mathoverflow.net/users/70853 | A measurable choice of inner-product preserving linear maps between two vector spaces | Use a selection theorem to choose measurably a unit vector in $V\_x$. Consider now a new measurable collection of vector spaces $V'\_x$ given as the orthogonal complements of the chosen vectors. Iterate $d$-times. By now you got a measurable choice of orthogonal basis.
Fix an orthonormal basis in $W$. Use these bases t... | 1 | https://mathoverflow.net/users/89334 | 245729 | 112,349 |
https://mathoverflow.net/questions/245733 | 3 | Let $E \to X$ be a Lie algebroid over the manifold $X$. Let $x\_1,...x\_n$ be local coordinates on $X$ and $e\_1,...e\_m$ be the basis of local sections of $E$. In terms of these coordinate functions Lie bracket and the anchor map $\rho$ are described like this:
$$
[e\_i,e\_j]\_E = \sum\limits\_k c\_{ijk}e\_k
$$
$$
\rh... | https://mathoverflow.net/users/88385 | Poisson structure on the dual Lie algebroid | Depending on you sign convention, this goes as follows. First you denote the bundle projection by $pr\colon E^\* \longrightarrow X$. For a section $s \in \Gamma^\infty(E)$ you have a linear function $J(s) \in C^\infty(E^\*)$ defined by pointwise evaluation, i.e. $J(s)(\alpha\_p) = \alpha(s(p))$ where $\alpha\_p \in E^\... | 2 | https://mathoverflow.net/users/12482 | 245737 | 112,351 |
https://mathoverflow.net/questions/245714 | 4 | Let $F(z)=\displaystyle \sum\_{k=0}^\infty a\_kz^k,\;|z|<R $ and $F(R)=\displaystyle \sum\_{k=0}^\infty a\_kR^k$ (the series converges).
Assume that $F(\alpha\_j)=0,\;j=1,2,\dots ,m$, where all $|\alpha\_j|<R$, Then $$F(z)=(z-\alpha\_1)\dots (z-\alpha\_m)\cdot \displaystyle \sum\_{k=0}^\infty b\_kz^k,\;|z|<R. $$
This... | https://mathoverflow.net/users/78726 | Convergence of a series | We can normalise $R=1\ $. By induction the question boils down to this: Let
$$
F(z)=(z-a)\sum\_{n=0}^\infty b\_nz^n=-ab\_0+\sum\_{n=1}^\infty (b\_{n-1}-ab\_n)z^n.
$$
If this series converges for $z=1$, we have to show that $\sum\_{n=0}^\infty b\_n$ converges.
Let $S\_N=\sum\_{n=1}^Nb\_{n-1}$ and let $T\_N=S\_N-aS\_{... | 7 | https://mathoverflow.net/users/nan | 245740 | 112,352 |
https://mathoverflow.net/questions/245738 | 10 | I've been taking the ideas expressed in ["Functors are Type Refinement Systems"](http://noamz.org/papers/funts.pdf) seriously lately and it's lead me to a form of the Grothendieck construction I've never seen before.
The idea in that paper is to think of a programming language $T$ as a category of terms/substitutions... | https://mathoverflow.net/users/82445 | Profunctorial Grothendieck Construction? | First of all, the phrase "Grothendieck construction" generally refers to the inverse construction, starting with a functor $T^{\mathrm{op}}\to \mathrm{Cat}$ and constructing a fibration $D\to T$.
Now, if you have an arbitrary functor $D\to T$, what you get is actually a [normal lax functor](http://ncatlab.org/nlab/sh... | 11 | https://mathoverflow.net/users/49 | 245743 | 112,353 |
https://mathoverflow.net/questions/245744 | 3 | Let $f=\sum\_{n\ge 1} a(n)q^n\in M\_{k+\frac{1}{2}}(\Gamma\_0(4N),\chi)$ be a modular form of half-integral wieght.
Can someone prove or disprove that:
$$X\ll \dfrac{\left(\sum\_{n\le X}a(n)\right)^2}{\underset{n\le X}{\sum} a(n)^2}$$
Thanks !
| https://mathoverflow.net/users/95750 | Estimate the ratio $\dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\underset{n\le X}{\sum} a(n)^2}$ | If your $f$ is a Hecke-cusp form then the estimate is certainly false, and the ratio is $o(X)$, since the coefficients $a(n)$ oscillate and there is a lot of cancellations in $\sum a(n)$. If $f$ is something like an Eisenstein series then it's probably true, because the coefficients are positive and mildly behaved. You... | 6 | https://mathoverflow.net/users/95842 | 245745 | 112,354 |
https://mathoverflow.net/questions/245741 | 0 | Polymath8b project allowed, building on Zhang's 2013 breakthrough, to prove that there are infinitely prime gaps of size less or equal to 600. Under the generalized Elliott-Halberstam conjecture, one can reach the upper bound 6.
My question is: in early August 2016, what is the narrowest interval $I=[a,b]$ such that... | https://mathoverflow.net/users/13625 | What is the narrowest interval I=[a,b] such that there are infinitely prime gaps of size in I? | You should find [this wiki page useful.](http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes) The current unconditional record is 246. Assuming Elliot Halberstam, the current record is 12, and assuming generalized Elliot Halberstam, the current record is 6.
[Here is the polymath paper.](h... | 5 | https://mathoverflow.net/users/50426 | 245748 | 112,355 |
https://mathoverflow.net/questions/245747 | 2 | This seems like it should be easy, but unfortunately I don't see how to do it.
Let $X$ be a variety; I'm happy to assume that $X$ is quasiprojective. If $L\_1$ and $L\_2$ are two non-isomorphic line bundles on $X$, then can we find a curve $C$ in $X$ such that $L\_1$ and $L\_2$ restrict to non-isomorphic bundles on ... | https://mathoverflow.net/users/84144 | Non-isomorphic line bundles detected by sub-curves? | Here is a proof for $\dim X=2$, projective and smooth. We may replace $L\_1, L\_2$ by $L\_1\otimes L\_2^{-1}=L$ and thus suffices to prove that if $L$ is not trivial, it is not trivial restricted to some curve. Take $H$ a large hypersurface section. Then $H^1(L-H)$ can be assumed to be zero and so if $L\_{|H}$ is trivi... | 5 | https://mathoverflow.net/users/9502 | 245754 | 112,356 |
https://mathoverflow.net/questions/245755 | 8 | Let $A(m,n)$ denote the [Eulerian numbers](https://en.wikipedia.org/wiki/Eulerian_number).
I'm looking for a simple combinatorial proof of the following fact.
>
> **Fact.** If $p$ is prime and $0\le k < p-1$, then $A(p-1,k) \equiv 1 \pmod{p}$.
>
>
>
The closest thing I'm aware of is an argument of S. Tanimoto,... | https://mathoverflow.net/users/3106 | Combinatorial proof of fact about Eulerian numbers? | We identify the permutations of $1,2,\dots,p-1$ and the cyclic permutations $c=(c\_0,\dots,c\_{p-1})$ of $0,\dots,p-1$: if $c\_k=0$, $c$ corresponds to $\pi(c):=(c\_{k+1},c\_{k+2},\dots,c\_{k-1})$. There are $p-1$ cyclic permutations which are arithmetic progressions $(0,a,2a,\dots,(p-1)a)$, and other cyclic permutatio... | 6 | https://mathoverflow.net/users/4312 | 245761 | 112,359 |
https://mathoverflow.net/questions/246778 | 1 | I have asked this question few days ago in MathStackExchange but I got only one response which gave a partial answer to my question, so I decided to ask it here.
I am reading Kulkarni's "Proper action and Pseudo-Riemannian space forms" article.
His work describes the action on the general space $S^{p,q}$, but I be... | https://mathoverflow.net/users/95174 | orthogonal transformations of one sheeted hyperboloid $S^{1,1}$ | This is from Magnus, *Noneuclidean Tesselations and Their Groups,* pages 123-124. In turn, this part is quoting fairly directly from [Fricke and Klein (1897), the first volume on automorphic forms, the volume on group theory](https://books.google.com/books?id=H5kLAAAAYAAJ&printsec=frontcover&dq=Vorlesungen%20%C3%BCber%... | 1 | https://mathoverflow.net/users/3324 | 246780 | 112,367 |
https://mathoverflow.net/questions/245750 | 0 | Question edited after the answer of Sándor Kovács:
>
> Let $f:X\to B$ be a holomorphic fibre space of smooth projective
> varieties which $f$ is relatively semi-ample and take $\mu$ as $m$-th root of holomorphic section of
> direct image of relative line bundle $f\_\*(K\_{X/B}^{\otimes m})$ then
> why $\mu$ mus... | https://mathoverflow.net/users/nan | $m$-th root of holomorphic section of direct image of relative line bundle | If I understand the question correctly, then here is a likely answer.
But before getting there, let me say that this is a very poorly formed question. If you are asking for help, then put at least as much effort into writing your question as the people who respond put into their answer.
---
For any line bundle $... | 6 | https://mathoverflow.net/users/10076 | 246782 | 112,369 |
https://mathoverflow.net/questions/194700 | 9 | Let $R=\bigoplus\_{i \geq 0} R\_i$ be a Cohen-Macaulay graded ring ($R\_0$ is a field and $R$ is generated by $R\_1$) of dimension $d$ with canonical module $\omega\_R$, and $M$ a graded Cohen-Macaulay $R$-module of dimension $t$. Assume that we know the Hilbert series, Hilbert polynomial and all Betti numbers of $M$. ... | https://mathoverflow.net/users/36563 | Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$) | It is equal to the multiplicity of $M$. In fact, you do not need graded or even Cohen-Macaulayness of $M$. Let $N= \textrm{Ext}^{d-t}(M,\omega\_R)$. Let $S(M) := \{P \in \textrm{Supp}(M), \dim R/P = t\}$. Then we have the so-called associativity formula:
$e(M) = \sum\_{P \in S(M)} \textrm{length}\_{R\_P}(M\_P)e(R/P)$... | 4 | https://mathoverflow.net/users/2083 | 246794 | 112,371 |
https://mathoverflow.net/questions/246792 | -1 | Let $G=(V,E)$ be a connected simple undirected graph and let $k>0$ be an integer such that
1. $\delta(G) \geq k$ (that is every vertex has at least $k$ neighbours), and
2. $K\_{k+1}$ is not a minor of $G$.
**Question:** In terms of $k$, how many vertices does a graph satisfying 1. and 2. above to contain at least? ... | https://mathoverflow.net/users/8628 | Lower bound for number of vertices in graph with certain forbidden minor | There is such a graph with $k+2$ vertices for all $k \geq 4$. To see this, first assume that $k$ is even. Let $G$ be $K\_{k+2}$ minus the edges of a perfect matching. Note that every vertex of $G$ has degree $k$, but $G$ does not contain a $K\_{k+1}$-minor. For $k$ odd, just take the even example and add an apex vertex... | 1 | https://mathoverflow.net/users/2233 | 246818 | 112,381 |
https://mathoverflow.net/questions/246820 | 1 | The most primitive formulation of the [Stone-Weiestrass theorem](https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem) states that any continuous functions, $f(x)$, defined on $[0, 1]$ can be uniformly approximated by a polynomial, $p(x)$, to an arbitrary precision. Basically, for any given function, we can ... | https://mathoverflow.net/users/76501 | The Stone-Weiestrass convergence for polynomials in different bases | Regarding the density of the span of monomials in the algebra of continuous functions with the uniform norm, there is the [Müntz–Szász theorem](https://en.wikipedia.org/wiki/M%C3%BCntz%E2%80%93Sz%C3%A1sz_theorem). One simple version says that a necessary and sufficient condition for the monomials $x^n, n \in S\subset \... | 9 | https://mathoverflow.net/users/14493 | 246824 | 112,383 |
https://mathoverflow.net/questions/245348 | 5 | Given $\tau$ in the upper half plane, define the normalized real-analytic Eisenstein series by
$$
E(\tau, s) = \frac{1}{2} \sum\_{(m,n)}' \frac{y^s}{|m\tau + n|^{2s}}
$$
It is initially defined for $\text{Re} (s) > 1$, but then analytically continued to the whole plane, except for a simple pole at $s=1$.
What are so... | https://mathoverflow.net/users/401 | Special values of real analytic Eisenstein series | Combining paul garret's comments (see also his wonderful notes, [Standard compact periods for Eisenstein series](http://www.math.umn.edu/~garrett/m/v/eis_std_periods.pdf)) with the class number formula, the functional equation of the Dedekind zeta function and the fact that $\zeta(0)=-1/2$, we get:
$$E(\tau,0)=\frac{... | 6 | https://mathoverflow.net/users/43108 | 246830 | 112,386 |
https://mathoverflow.net/questions/246787 | 2 | Let $X$ be a Gorenstein (not necessarily smooth) projective $\mathbb{C}$-scheme and $S$ another $k$-scheme. Let $I$ be an injective sheaf on $X$. Denote by $p:X \times\_k S \to X$ the natural projection map. Is there any known condition, under which $p^\*I$ is an injective sheaf on $X \times\_k S$?
If there is no ge... | https://mathoverflow.net/users/43198 | When is the pullback of an injective sheaf injective? | Let's say that $X={\rm Spec\,} k$ for a field $k$. Then it is certainly Gorenstein and $k$ is an injective sheaf on $X$. For any $S$ and $p$ as defined in the question, $p^\*k\simeq \mathscr O\_S$. If this is injective, then the injective dimension of $S$ is $0$, and hence $\dim S=0$.
I am pretty sure that this can ... | 4 | https://mathoverflow.net/users/10076 | 246836 | 112,388 |
https://mathoverflow.net/questions/246827 | 9 | Let $A$ and $B$ be two positive definite $n \times n$ matrices. It is, of course, not true that $AB+BA$ is necessarily positive definite.
Consider, though, the results of the following numerical experiment. I generated $A$ by letting its eigenvalues be random in $[0,1]$, and selecting its eigenvectors by generating ... | https://mathoverflow.net/users/96890 | For positive definite $A,B$ why does $AB+BA$ tend to be positive definite? | $\text{tr}(AB+BA) = 2 \operatorname{tr}(A^{1/2} B A^{1/2}) > 0$, so that may produce some bias toward positive eigenvalues. In particular if you generate your "random" matrices in such a way that the eigenvalues of $AB+BA$ will tend to be concentrated very close together, this may produce the results you observed.
Bu... | 16 | https://mathoverflow.net/users/13650 | 246837 | 112,389 |
https://mathoverflow.net/questions/246795 | 9 | Let $X\subset \mathbb P^n$ be a surface (possibly singular) and $\omega\_X$ be its dualizing sheaf.
Let $G$ be a finite group acting on $X$ (possibly with fixed points).
We know how to calculate the dualizing sheaf for surfaces in projective space
(ref <https://mathoverflow.net/q/125724>). I want to know what will hap... | https://mathoverflow.net/users/20282 | Dualizing sheaf after a finite group action | If $f:X\to Y$ is a finite morphism, and assuming that both $X$ and $Y$ admit dualizing sheaves, then by duality (see [Hartshorne, Ex.III.6.10]) you have the first map of the following:
$$
\eta: f\_\*\omega\_X\to \mathscr Hom \_Y(f\_\*\mathscr O\_X,\omega\_Y)\to \omega\_Y,
$$
where the second map is induced by the natur... | 9 | https://mathoverflow.net/users/10076 | 246843 | 112,390 |
https://mathoverflow.net/questions/245773 | 0 | Let $(x\_{n})\_{n}$ be a normalized basic sequence in $X=L\_{p}$, with $1<p<2$.
Does there exist a subsequence $(x\_{k\_{n}})\_{n}$ of $(x\_{n})\_{n}$ and a weakly null sequence $(x^{\*}\_{n})\_{n}$ in $X^{\*}$ such that $(x\_{k\_{n}})\_{n}$ and $(x^{\*}\_{n})\_{n}$ are biorthogonal?
This question may be obvious or... | https://mathoverflow.net/users/41619 | Basic sequences in $ L_{p}$ | I think it is worthwhile to point out that you do not need to pass to a subsequence of $(x\_n)$. To see that, let $(y\_n^\*)$ be any Hahn-Banach extensions to $X^\*$ of the functionals biorthogonal to $(x\_n)$ and observe that all weak$^\*$ cluster points of $(y\_n^\*)$ in $X^\*$ are in $(x\_n)^\perp$. By the separabil... | 2 | https://mathoverflow.net/users/2554 | 246846 | 112,392 |
https://mathoverflow.net/questions/246838 | 5 | I'm reading a few papers on reflective factorization systems and I've just noticed they're all mentioning a procedure which seems very similar to the small object argument.
First of all, some background. My first encounter with the small object argument was [Garner's paper](http://arxiv.org/abs/0712.0724) about which... | https://mathoverflow.net/users/69037 | Finite well-completeness and the small object argument? | This isn't a full answer because it isn't completely precise, but I would say that the relationship is between "predicative" and "impredicative" constructions of universal objects.
Suppose $P$ is a poset and $f:P\to P$ is monotone and inflationary, i.e. $x\le y \Rightarrow f(x)\le f(y)$ and $x\le f(x)$. And say we ha... | 3 | https://mathoverflow.net/users/49 | 246852 | 112,396 |
https://mathoverflow.net/questions/245143 | 5 | Let $q$ be a prime power and $\mathbb{F}\_q$ the field of cardinality $q$. Let $A = \mathbb{F}\_q[T]$ and let $A\_+ \subset A$ be the monic polynomials. Choose any ordering $<$ of $A\_+$ and let $k$ be a positive integer. Set
$$e(k) = \sum\_{\begin{matrix} a\_1, a\_2, \ldots, a\_k \in A\_+ \\ a\_1<a\_2<\cdots<a\_k \end... | https://mathoverflow.net/users/297 | Elementary symmetric functions of reciprocals of monic polynomials in function fields | Many of the various types of function field valued multiple zeta values (MZV's) were first defined by Dinesh Thakur in his book "Function Field Arithmetic" from 2004 (see section 5.10). He considers several possible definitions, for example
$$
\zeta\_l(s\_1, \ldots, s\_k) = \sum\_{\substack{a\_1, \ldots, a\_k \in A\_+ ... | 3 | https://mathoverflow.net/users/7263 | 246853 | 112,397 |
https://mathoverflow.net/questions/246870 | 2 | I understand that via the [Borel density theorem](http://www3.nd.edu/~andyp/notes/BorelDensity.pdf) given a finite dimensional (polynomial) representation of the simple non-compact Lie groups $SL\_n \mathbb R$ or $Sp\_n \mathbb R$, I get an irreducible representation when I restrict to $SL\_n \mathbb Z$ or $Sp\_n \math... | https://mathoverflow.net/users/41840 | Irreducible representations of $SL_n \mathbb Z$ | (This is too long for a comment.) This follows from the fact that $\text{SL}\_n\mathbb{Z}$ is Zariski dense in $\text{SL}\_n\mathbb{R}$. Let $\phi:V\to W$ be a linear isomorphism such that for every $g\in \text{SL}\_n\mathbb{Z}$, the "conjugate" $\phi^g = g^{-1}\cdot\phi(g\cdot -)$ equals $\phi$. The subset of $\text{S... | 8 | https://mathoverflow.net/users/13265 | 246871 | 112,403 |
https://mathoverflow.net/questions/246881 | 9 | I am try to understand the concept: an algebra in a category. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. $A$ is an algebra in $\mathcal{C}$ means the multiplication $m: A \otimes A \to A$ is a morphism in $\mathcal{C}$?
Let $H$ be a bialgebra and $V$ a Yetter-Drinfeld module over $H$. Let $Y... | https://mathoverflow.net/users/11877 | Algebra in a category | What you are talking about is the notion of monoid in a monoidal category. To show $A$ is a monoid ('algebra'), you need to construct a multiplication map $\mu: A \times A \to A$, that is associative, where $\times$ is the monoidal product for your monoidal category. In an example like a tensor algebra, you already hav... | 10 | https://mathoverflow.net/users/82938 | 246882 | 112,406 |
https://mathoverflow.net/questions/246876 | 8 | Let $F\_n$, $n\geq 0$, be the sequence of Fibonacci numbers, where $F\_0=F\_1=1$ and $F\_{n+1}=F\_n+F\_{n-1}$ for $n\geq 1$. A number is squarefree if it is is not divisible by the square of a prime number.
**Question**: Are there infinitely many squarefree Fibonacci numbers?
| https://mathoverflow.net/users/81443 | Squarefree Fibonacci Numbers | I assume the traditional definition with $F\_0=0$ and $F\_1=1$.
Most likely there are infinitely many squarefree Fibonacci numbers. A simple way to construct them is to consider a subsequence $F\_p$ for prime $p$. Notice that if $q^2\mid F\_p$ for some prime $q$, then $q$ must be a [Wall-Sun-Sun prime](https://en.wi... | 6 | https://mathoverflow.net/users/7076 | 246883 | 112,407 |
https://mathoverflow.net/questions/246884 | 1 | Feel free to restrict the function space to a Hilbert space or to a RKHS. Given a probability distribution on it when can we define a ``covariance operator" for it and when would it also have a well-defined notion of eigenfunctions for it?
| https://mathoverflow.net/users/89451 | About covariance operators for probability distributions on a function space | Let $X$ be a random vector taking values in a
separable Hilbert space $H$ such that $E\|X\|^2<\infty$ and $E X=\mu$. Then the corresponding covariance operator $R\colon H\to H$ is defined by the formula
\begin{equation\*}
Rx:=E\langle x,X-\mu\rangle (X-\mu)=E\overline{\langle X-\mu,x\rangle}(X-\mu)
\end{equation\*} ... | 2 | https://mathoverflow.net/users/36721 | 246887 | 112,409 |
https://mathoverflow.net/questions/246868 | 1 | In the context of control theory,
The algebraic design tradeoff by Freudenberg and Looze is a constraint that relates the sensitivity function $\sigma$, and the complementary sensitivity function $\tau : \sigma + \tau = 1$.
However, does this constraint exist in a feedback system designed using an observer and stat... | https://mathoverflow.net/users/96923 | Algebraic design tradeoff by Freudenberg and Looze and State Feedback with Observer | Of course, why would be interested otherwise :-)?
The constraints on sensitivity and complementary sensitivity function are formulated in the frequency domain. When you are talking about feedback systems designed using an observer and state feedback then you are talking about objects defined in the time domain. So th... | 0 | https://mathoverflow.net/users/85570 | 246895 | 112,412 |
https://mathoverflow.net/questions/246890 | 2 | We have the following identity (see Bateman, H. (1953). Higher Transcendental Functions [Volumes I], p. 25.)
$$(\*)\quad \Gamma(\mu)\, \zeta(\mu,\nu) = \int\_{0}^{1} x^{\nu-1} \,(1-x)^{-1} \Bigr(\log 1/x\Big)^{\mu-1} \, dx; \quad \Re e (\mu)>1,\Re e (\nu)>0,$$
where $\Gamma(\mu)$ is the Gamma function and $\zeta(\mu,\... | https://mathoverflow.net/users/84558 | How to compute the following integral $I_{\alpha,\beta}$ | Assume first $\beta>1$ so that the integral converges and let
$$f(x)=x^{\alpha}(1-x)^{-1}(-\log x)^{\beta}.$$
Then
$$0=\int\_{0}^{1}df\\=\alpha\int\_{0}^{1}x^{\alpha-1}(1-x)^{-1}(-\log x)^{\beta}dx
+ I\_{\alpha,\beta}-\beta\int\_{0}^{1}x^{\alpha-1}(1-x)^{-1}(-\log x)^{\beta-1}dx,$$
where the last two integrals can be e... | 5 | https://mathoverflow.net/users/89429 | 246902 | 112,415 |
https://mathoverflow.net/questions/245632 | 3 | Roth's Theorem states that any subset $A$ of $\{1, \dots, n\}$ with no solution to the equation $$x + y = 2z,\, (x, y, z) \in A^3,\, x \neq y$$ has size $o(n)$. Similar results hold when dealing with the same question in, for instance, $\mathbb{Z}/n\mathbb{Z}$.
I was wondering if similar questions have been studied i... | https://mathoverflow.net/users/46573 | (Extremal) arithmetic combinatorics in non-abelian groups | For the sake of getting this question off the unanswered stack, let me turn some of the comments into a question.
1. Noam Elkies' comment: if one considers arbitrary subsets of $S\_n$, then one can find equations for which there are very large subsets of $S\_n$ containing no solutions. For instance $x\_1x\_2x\_3=1$ h... | 4 | https://mathoverflow.net/users/801 | 246903 | 112,416 |
https://mathoverflow.net/questions/246823 | 7 | *Prenote: I have asked this question first on [math stackexhange](https://math.stackexchange.com/questions/1882475/gauss-theorem-for-null-boundaries), but a user suggested that mathoverflow might be a better place for this question. Upon thinking about it I have agreed with him and copy-pasted the question here.*
*No... | https://mathoverflow.net/users/85500 | Gauss' theorem for null boundaries | Gauss' Theorem has nothing to do with the (pseudo-)metric. Is just a consequence of Stokes' theorem.
Stokes's theorem says that, for any $n-1$ form $\omega$,
$$ \int\_M d\omega = \int\_{\partial M} \omega. $$
Now fix any smooth measure $\mu$ (i.e. given by a smooth non-vanishing top dimensional form, or a density... | 4 | https://mathoverflow.net/users/13915 | 246906 | 112,417 |
https://mathoverflow.net/questions/246905 | 0 | Suppose that $P(x) = a\_m x^m + \dots + a\_0$ and $Q(x) = b\_n x^n + \dots + b\_0$ are two polynomials, with $m > n > 1$ and $a\_m > b\_n > 0$. Suppose that $P$ has $m$ distinct real roots $y\_1<\dots<y\_m$ and $Q$ has $n$ distinct real roots $z\_1<\dots<z\_n$.
Is the following claim true:
$P(x) - Q(x)$ is strictly ... | https://mathoverflow.net/users/49831 | Comparing tails of polynomial functions | Try $P(x) = 2 x (x+1)(x+1/3)$ and $Q(x) = x(x+1)$. Note that $P'(0) - Q'(0) = -1/3$.
| 2 | https://mathoverflow.net/users/13650 | 246909 | 112,418 |
https://mathoverflow.net/questions/246913 | 1 | Let $\mathcal{A}$ be a central hyperplane arrangement in a (finite dimensional) real vector space $V$. Assume for each hyperplane $H\in\mathcal{A}$ that we're given a labelling $H^+$, $H^-$ of the connected components of $V\setminus H$.
1. Given a subset $\mathcal{B}\subseteq \mathcal{A}$, is it possible for the set ... | https://mathoverflow.net/users/36720 | Chambers of central hyperplane arrangements | 1. Yes: let $V=\mathbb{R}$, let $\mathcal{B}$ be empty, and let $\mathcal{A}$ be two copies of the origin, with $H^+$ being the positive numbers once, and the negative numbers once (this is a pretty degenerate example, but you can also get 3 hyperplanes in $\mathbb{R}^2$ giving you a line, etc).
2. Yes: the problem wit... | 4 | https://mathoverflow.net/users/66 | 246915 | 112,420 |
https://mathoverflow.net/questions/245643 | 4 | Let $X$ be a compact Riemann surface, i.e. compact smooth complex analytic (hence automatically algebraic) curve. Let $A\subset X$ be a finite subset, and $X\_0:=X\backslash A$.
Let $Y\_0$ be a smooth complex analytic curve (necessarily non-compact) with a holomorphic map $f\_0\colon Y\_0\to X\_0$ which is a finite ... | https://mathoverflow.net/users/16183 | Finite covers of punctured Riemann surfaces | The positive answer to the above question (even in a more general form) in explicitly contained in Theorem 8.4 in the book "Lectures on Riemann surfaces" by Otto Forster (1981).
| 2 | https://mathoverflow.net/users/16183 | 246928 | 112,424 |
https://mathoverflow.net/questions/245623 | 5 | There are all sorts of curios in low-dimensional Lie groups and Lie algebras, many of them due to the presence of the quaternions. There is, I have recently learned, an isomorphism $SO(6,2) \simeq SO(4,\mathbb{H})$ (this *isn't* listed on Wikipedia, for instance). I'm curious to know
>
> is there is any correspond... | https://mathoverflow.net/users/4177 | Exceptional isomorphism with Spin(6,2)? | To put the problem to a rest, I add my comment as an answer which is, in a nutshell, $Spin(6,2)\cong Spin(4,\mathbb H)$. The existence of this isomorphism follows from the isomorphism of the Satake diagrams and simple connectedness.
The tricky thing is the definition of a spin group over the quaternions which is expl... | 7 | https://mathoverflow.net/users/89948 | 246933 | 112,427 |
https://mathoverflow.net/questions/246918 | 0 | Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ a convex function. Since convex functions are locally Lipschitz, they are differentiable almost everywhere. Let $\delta f(x)$ be the set of subgradients to $f$ at $x$. Suppose that $f$ is not differentiable at $x\_0$ and we are interested in a certain subgradient at this poin... | https://mathoverflow.net/users/58218 | For a convex function, can subgradients be formed from finite convex combinations of gradients? | The answer is "no".
Consider the function
$$f(x,y)=\sqrt{x^2+y^2}+|y|$$
Note that $v\_0=(1,0)$ is a subgradient at $(0,0)$.
The gradient of $f$ is defined if $y\ne0$ and at all these points its first coordinate is strictly less than 1.
Hence the statement follows.
| 2 | https://mathoverflow.net/users/1441 | 246940 | 112,428 |
https://mathoverflow.net/questions/246944 | 0 | I'm reading the probabilistic book write by çinlar, but I don't understand the Kernel theory, in details:
$ (E,\mathcal{E}),(F,\mathcal{F})$ are two measurable space
$$K:E \times \mathcal{F} \rightarrow R\_{+}$$
with these proprieties: they are measurable on $E$ fixed $B\in\mathcal{F}$, and they are a measure on $F$ fi... | https://mathoverflow.net/users/nan | Theory of integration of Kernel in çinlar probability and stochastic | I've resolved! since the kernel is a measure fixed $x$ I can integrate the function $f$ respect to this measure (notation $\nu(dx)$ means integrate respet to measure $\nu$ and variables $x$)!
| 1 | https://mathoverflow.net/users/nan | 246945 | 112,429 |
https://mathoverflow.net/questions/244940 | 8 | Assume that $M$ is an arbitrary manifold.
>
> Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?
>
>
>
If not, what is a counter example?
In particular what is the answer to this question for $M=\mathbb{R}^{2}$ or $M=S^{2}$?
The que... | https://mathoverflow.net/users/36688 | The minimum codimension of Lie subalgebra of $\chi^{\infty}(M)$ | Let $L$ be a sub-algebra of $\mathrm{Vect}(M)$. I think one might be able to prove $\mathrm{codim}\ L \geq \dim M$ by using a recent result of [Hurtado](http://arxiv.org/abs/1307.4447). Here is a sketch of the proposed proof, every step of which is difficult:
(1) Let $G \subset \mathrm{Diff}(M)$ be the group generate... | 2 | https://mathoverflow.net/users/297 | 246952 | 112,431 |
https://mathoverflow.net/questions/246935 | 3 | By (r, s, t)-identity I mean any sort of such identity:
$$
(x\_1^2+\ldots + x\_r^2)(y\_1^2+\ldots +y\_s^2)=(z\_1(x,y)^2+\ldots + z\_t(x,y)^2),
$$
where $z\_i(x,y)$ is a polynomial for every $i$.
See this for some further reading: <https://en.wikipedia.org/wiki/Hurwitz_problem>
I can't find in the web any non-stan... | https://mathoverflow.net/users/11072 | On Hurwitz Square (r, s, t)-Identities examples | From pages 137-138 of [Rajwade](http://oskicat.berkeley.edu/record=b14952300~S1), "Notes on Chapter 10"
the Hurwitz-Radon theorem gives
$$ (2,2,2), \; (4,4,4), \; (8,8,8), \; (9,16,16), \; (10,32,32), \ldots $$
K. Y. Lam found $(10,10,16)$ in 1966.
In 1975, Adem found
$$ (3,5,7), \; (10,10,16), \; (12,12,28), \;... | 3 | https://mathoverflow.net/users/3324 | 246956 | 112,432 |
https://mathoverflow.net/questions/246948 | 7 | Fix a field $k$ and suppose $\mathcal{C}$ and $\mathcal{D}$ are $k$-linear additive categories and are enriched over the category $\mathcal{V}$ of finite-dimensional $k$-vector spaces. So we have copower functors
$$
\mathcal{C} \times \mathcal{V} \to \mathcal{C} \quad \text{and} \quad \mathcal{D} \times \mathcal{V} \t... | https://mathoverflow.net/users/29738 | Do copowers commute with k-linear functors? | Yes, this is true (up to natural isomorphism). The simplest way to see this is just to write down what the copowers are explicitly. Let us take a skeleton of $\mathcal{V}$ consisting of all vector spaces of the form $k^n$, and write $\otimes$ for copowers. Then on objects, $A\otimes k^n$ is just a direct sum of $n$ cop... | 5 | https://mathoverflow.net/users/75 | 246958 | 112,433 |
https://mathoverflow.net/questions/246969 | 1 | Recall that the James $p$-space $J\_{p}(1<p<\infty)$ is the (real) Banach space of all sequences $(a\_{n})\_{n}$ of real numbers such that $\lim\_{n\rightarrow \infty}a\_{n}=0$ and
$$\|(a\_{n})\_{n}\|\_{pv}=\sup\{(\sum\_{j=1}^{m}|a\_{i\_{j-1}}-a\_{i\_{j}}|^{p})^{\frac{1}{p}}:1\leq i\_{0}<i\_{1}<\cdots<i\_{m}, m\in \ma... | https://mathoverflow.net/users/41619 | Non-weakly compact operators on the James $p$-space $J_{p}(1<p<\infty)$ | The answer to both questions is no.
Let $I:J\_p\to c\_0$ denote the formal identity defined by
\begin{equation\*}Ie\_n=f\_n,\;\;\;n\in\mathbb{N},\end{equation\*}
where $(e\_n)\_{n=1}^\infty$ is the canonical basis for $J\_p$ and $(f\_n)\_{n=1}^\infty$ is the canonical basis for $c\_0$. It is well-known that $x\_n=\su... | 2 | https://mathoverflow.net/users/73784 | 246975 | 112,437 |
https://mathoverflow.net/questions/246965 | 2 | Let $\mu$ be the $n$-dimensional Lebesgue measure and $\lambda$ be a complex Borel measure on $\mathbb{R}^n$.
Let $S$ be the set of points $x\in \mathbb{R}^n$ where $\lim\_{r\to 0} \frac{\lambda (B(x,r))}{\mu (B(x,r))}$ exists in $\mathbb{C}$.
Then, is $S$ a Borel set? Moreover, is $\lambda (S)=0$?
| https://mathoverflow.net/users/83098 | Is the domain of symmetric derivative borel set? | Yes, $S$ is Borel. Assume, to be specific, that $B(x,r)$ denotes the open ball of radius $r$ centered at $x$.
Lemma 1. The function $(0,\infty)\ni r\mapsto\ell(r):=\lambda (B(x,r))$ is left-continuous, for each $x\in \mathbb{R}^n$.
Proof. By the Hahn decomposition theorem, $\lambda$ is a linear combination (possi... | 1 | https://mathoverflow.net/users/36721 | 246978 | 112,439 |
https://mathoverflow.net/questions/246904 | 2 | Let $\Omega\subset \mathbb{R}^N$ and $H\_0^1(\Omega)$ the standard Sobolev space. Assume that $1<q<p<2^\star$ and $$\mathcal{S}=\{u\in H\_0^1(\Omega):\ \|u\|=1\}.$$
Define $C\_q,C\_p$ by $$C\_q=\inf\_{u\in \mathcal{S}}\frac{1}{\|u\|\_q},$$
and $$C\_p=\inf\_{u\in \mathcal{S}}\frac{1}{\|u\|\_p}.$$
Once $q<p<2^\star... | https://mathoverflow.net/users/53175 | Maximizing $\|u\|_q\|u\|_p$ over the unitary sphere in the Sobolev space $H_0^1(\Omega)$ | Any maximizer $u\in\{ H^1\_0(\Omega): \|\nabla u\|\_2=1 \}$ of $\|u\|\_p$ is a nonconstant, nonnegative function solving $-\Delta u = \lambda u^{p-1}$, with $\lambda=\lambda\_p>0$. So if $u$ maximizes both $\|u\|\_p$ and $\|u\|\_q$, then $\lambda\_p u^p=\lambda\_q u^q$ a.e., which is only possible if $p=q$.
| 1 | https://mathoverflow.net/users/6101 | 246991 | 112,441 |
https://mathoverflow.net/questions/246993 | 0 | Let $R$ be a ring. Take the polynomial ring over $R$
$$R[x\_1,\dots, x\_n]$$
nonzerodivisors $f,g\in R[x\_1,\dots, x\_n]$ such that $f$ is a polynomial in the first $k$ indeterminates, $g$ a polynomial in the last $n-k$, $0\le k\le n$.
Suppose both $R[x\_1,\dots, x\_n]/(f)$ and $R[x\_1,\dots, x\_n]/(g)$ are flat ... | https://mathoverflow.net/users/nan | Tor independence | With the flatness hypothesis, it looks both statements are true.
The first being true implies the morphism
$$\text{Spec}(R[x\_1,\dots, x\_n]/(f,g)) \to \text{Spec}(R)$$
is a flat lci. But then the ideal $(f,g)\subset R[x\_1,\dots, x\_n]$ has to be regular, hence, calling $A := R[x\_1,\dots, x\_n]$, $B := A/(f)$, ... | 3 | https://mathoverflow.net/users/nan | 246996 | 112,443 |
https://mathoverflow.net/questions/247006 | 2 | Is there a proof that $BO(k)$ is not of the homotopy type of a finite dimensional complex?
The Grassmannian $BO(k) := \{ k\text{-dim subspaces of } \mathbb{R}^\infty \}$ classifies the $k$-dimensional vectorbundles on a $CW$ complex $X$ as:
$Vect^k(X) \cong [X,BO(k)]$.
$BO(k)$ can be constructed as the direct limit... | https://mathoverflow.net/users/91925 | Infinite Grassmannian does not have the homotopy type of a finite-dimensional complex | We have $H^\*(BO(k); \mathbb{Z}\_2) \cong \mathbb{Z}\_2[w\_1, \dots, w\_k]$ where $\deg w\_i = i$. In particular, $H^n(BO(k); \mathbb{Z}\_2) \neq 0$ for every $n$ as $w\_1^n$ is a non-zero element. Therefore $BO(k)$ cannot be homotopy equivalent to a finite-dimensional CW complex.
That is, the degrees of the usual ch... | 14 | https://mathoverflow.net/users/21564 | 247007 | 112,446 |
https://mathoverflow.net/questions/241814 | 2 | Let $( \mathbb{R}^n, \| \cdot \|\_P)$ be the $n$-dimensional Euclidean space equipped with $\ell\_p$-norm $\| \cdot \|\_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and define
\begin{align}
A^{\epsilon} = \{ y \in \mathbb{R}^n \colon \exists x \in A~\text{such that}~\| x -y \| \_{p} \leq... | https://mathoverflow.net/users/81633 | Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$ | Let's pursue Jochen's idea. We assume $A \ne \emptyset.$
Let
$$ \varphi(t) = \begin{cases}
e^{-\frac{1}{t}} &\text{if $ t>0$}\\
0 &\text{otherwise.}
\end{cases}$$
This function is $\mathcal C^{\infty}$, and $0 < \varphi(t)$ iff $0<t.$
Define $\rho$ as
$$\rho(x) := k \varphi(1- \|x\|\_2^2)$$
where $k$ is ... | 1 | https://mathoverflow.net/users/47322 | 247015 | 112,449 |
https://mathoverflow.net/questions/247009 | -1 | I took a quick glance on a survey paper about superzeta functions where one considers a pair $\rho\leftrightarrow 1-\rho$ of non trivial zeroes of the Riemann zeta function. The assumption of RH, i.e $\rho=1/2+it\_{k}$ yields $\rho(1-\rho)=1/4+t\_{k}^{2}$. Hence my question: can the eigenvalues considered in Selberg ei... | https://mathoverflow.net/users/13625 | Is Selberg's eigenvalue conjecture related to RH? | Selberg's 1/4 conjecture can be phrased like this: For a congruence subgroup all non-trivial zeros of the Selberg zeta function either come from resonances, which means that they lie at $\mathrm{Re}(s) < 1/2\ $ or they come from eigenvalues and lie at $\mathrm{Re}(s)= 1/2\ $.
| 3 | https://mathoverflow.net/users/nan | 247020 | 112,450 |
https://mathoverflow.net/questions/247013 | 1 | Consider a real sequence $(x\_k)$ for $k=0,1,2,\dots,N$ as $x\_0=1$ and for $k>0$
$$ x\_k=x\_{k-1}+\frac{\gamma}{N}x\_{k-1}^2,\qquad (\gamma>0).$$
I wonder to show that the sequence is bounded as $N\to\infty$. I appreciate any idea for proving that.
Hint 1: Numerical experiments suggest me that if and only if $\gamma... | https://mathoverflow.net/users/75491 | boundedness of a nonlinear recursive sequence | You can view this difference equation as the Euler method for the IVP $y'=\gamma y^2$, $y(0)=1$, on the interval $0\le t\le 1$, using a grid of width $1/N$ and setting $x\_k=y(k/N)$.
By solving the ODE, we find that $y$ blows up at $t=1/\gamma$. We want to know if $x\_N=y(1)$ stays bounded, and it now follows that th... | 1 | https://mathoverflow.net/users/48839 | 247028 | 112,455 |
https://mathoverflow.net/questions/247034 | 5 | Let $d$ be a positive, non-square integer, and define $c\_d$ to be the smallest positive number with the following property: for all pairs of co-prime integers $(p,q)$ with $q > 0$, the inequality
$$\displaystyle \left \lvert \frac{p}{q} - \sqrt{d} \right \rvert > \frac{c\_d}{q^2}$$
holds. The existence of such a n... | https://mathoverflow.net/users/10898 | Constant related to continued fraction of quadratic irrationals | Edit: I answered a different question than the one asked. The question asks for the least $c\_d = q^2|\sqrt{d}-p/q|$. I evaluated the liminf.
If $d = a^2+b$ with $1\le b \le 2a$ then the simple continued fraction for $\sqrt{d}$ is preperiodic, and the period ends with $2a$, which is the largest coefficient, with $a$ ... | 5 | https://mathoverflow.net/users/2954 | 247039 | 112,459 |
https://mathoverflow.net/questions/246990 | 1 | In the [lecture notes](https://docs.google.com/viewer?url=http://www.msri.org/workshops/595/schedules/15557/documents/1433/assets/16929), on page 24, there is an example of drawing a quiver for a pseudoline arragement. What is the rule to draw a quiver for a pseudoline arragement? I don't know how to put the directions... | https://mathoverflow.net/users/11877 | How to draw a quiver for a pseudoline arragement? | Recall how the quiver and its mutations encode the clusters the their mutations. The vertices of the quiver correspond to cluster variables. When we mutate at a vertex $x$ the arrows change as prescribed while all vertices except $x$ stay the same. The vertex $x$ is replaced by $x'$ and we have the relation
$$xx' = M\_... | 2 | https://mathoverflow.net/users/51668 | 247043 | 112,460 |
https://mathoverflow.net/questions/247044 | 5 | I was reading the section about sites in the Stacks Project.
Small sites are studied only in few cases, but I don't get why.
As some users say, it's because of this.
If $f: X\to Y$ is a morphism of schemes, it's known it's not generally true that the small fppf inverse image functor $f^{-1}$ is exact, as opposed to ... | https://mathoverflow.net/users/nan | Small fppf/syntomic/smooth sites? | Good point. Meanwhile I'll say a few things, which I plan to make into an answer at some point, though I suggest you wait for a bunch of examples to come from users all over.
To fix ideas, we consider the category of all smooth $S$-schemes with arbotrary $S$-morphisms between them, denoted $\mathcal{C}$, and endow it... | 5 | https://mathoverflow.net/users/nan | 247045 | 112,461 |
https://mathoverflow.net/questions/247037 | 2 | I'm reading some notes on hodge theory by Charles Siegel which makes a claim on page 16 relating the space of deformations of a smooth projective hypersurface $X$ with the jacobian ideal. More specifically, let
$$
Proj(\mathbb{C}[x\_1,\ldots, x\_n]/(f)) = Proj(S\_\bullet) = X
$$
then
$$
H^1(X,T\_X) = \frac{S\_d}{\text{... | https://mathoverflow.net/users/78824 | Where can I find a proof of identity of $H^1(X,T_X)$ and a quotient by the jacobian? | This is just a standard application of Griffiths's Residue theory.
For a complete treatment you may consult Claire Voisin's book no.2 on Hodge Theory, but let me just give you the idea.
In general, under some mild hypotheses, the whole Jacobian ring $R=S/Jac(f)$ describes the (embedded) deformations of the affine co... | 3 | https://mathoverflow.net/users/52811 | 247049 | 112,462 |
https://mathoverflow.net/questions/168526 | 25 | It can be shown (see [Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?](https://mathoverflow.net/questions/167954/is-every-paracompact-hausdorff-locally-contractible-space-homotopy-equivalent/168523#168523)) that if $X$ is a locally contractible paracompact Hausdorff spac... | https://mathoverflow.net/users/51164 | Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex? | **ETA** The answer is **yes** in general. Replace 2 below with a reference to HTT, Prop. 7.1.5.8.
Since this has been open for a while, let me give a partial answer which hopefully is already interesting: I believe that the ∞-topos of sheaves on any **locally finite** (equivalently, locally compact) CW complex $X$ is... | 16 | https://mathoverflow.net/users/20233 | 247061 | 112,463 |
https://mathoverflow.net/questions/247058 | 6 | Fix $n\in \mathbb N$ and a partition $\lambda$ with at most $n-1$ parts (of length at most $n-1$). Let $V$ be the irreducible $GL\_n \mathbb R$-representation with highest weight $\lambda$ and $D$ the determinant representation.
Is it possible that
$$ Res\_{GL\_n \mathbb Z} V \cong Res\_{GL\_n \mathbb Z} (V \otimes D... | https://mathoverflow.net/users/41840 | Can these two irreducible $GL_n \mathbb Z$-representations be isomorphic? | First of all, if $V$ is not irreducible, this can happen. Every matrix in $GL\_n(\mathbb{Z})$ has determinant $\pm 1$ so $D^{\otimes 2}|\_{GL\_n(\mathbb{Z})}$ is trivial. Therefore, $(1 \oplus D) \cong (1 \oplus D) \otimes D$ on $GL\_n(\mathbb{Z})$.
However, with the stated hypothesis that $V$ is the irrep of $GL\_n$... | 6 | https://mathoverflow.net/users/297 | 247063 | 112,464 |
https://mathoverflow.net/questions/247062 | 4 | It is well known that any knot diagram can be unknotted by a sequence of crossing changes (i.e., changing an overcrossing with an undercrossing or vice versa) and of Reidemeister moves. More precisely, one can first perform a certain number of crossing exchanges to modify the given knot diagram into a diagram represent... | https://mathoverflow.net/users/8320 | Unknotting knot diagrams by Reidemeister moves and crossing changes | ***Edit: This answer has been edited to correct a mistake graciously pointed out by Ian Agol in the comments below.***
The answer to you question is yes. It is possible to take a sequence of crossing changes and Reidemeister moves to unknot a diagram in a non-decreasing manner.
Here is an algorithm to accomplish t... | 6 | https://mathoverflow.net/users/27453 | 247066 | 112,465 |
https://mathoverflow.net/questions/227130 | 0 | Let $\mathcal M\_g$ be the moduli space of curves of genus $g$. If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow
$X^{reg}$ with a complete Kahler metric which has a type of singularities normal to each component of $D$; in local coordinates, if $D = (z\_1,...,z\_k)$, the Weil-Peterss... | https://mathoverflow.net/users/nan | Weil-Petersson metric is quasi isometric with which model? | Ken-Ichi Yoshikawa, found an asymptotic formula for Weil-Petersson metric on moduli space of Calabi-Yau varieties as follows
$$\omega\_{WP}=\left\{\frac{\ell }{|s|^2(\log |s|)^2}+O\left(\frac{1 }{|s|^2(\log |s|)^3}\right)\right\}\sqrt[]{-1}ds\wedge d\bar s$$
| 1 | https://mathoverflow.net/users/nan | 247077 | 112,467 |
https://mathoverflow.net/questions/247041 | 6 | Let $X$ and $Y$ be graphs and consider the Kronecker product: $Z = X \otimes Y$. Is it true that if $X$ excludes an $M$-minor, $Z$ excludes an $M \otimes Y$ minor?
I am particularly interested in the case where $Y$ is just an edge, and $Z$ is just the bipartite double cover of $X$.
| https://mathoverflow.net/users/96990 | Graph minors, and Kronecker product | The diamond cubic is a subgraph of the Kronecker product of three infinite paths, and $K\times K\times K$ patches of the diamond cubic are subgraphs of the Kronecker product of three length-$K$ paths. But there are no forbidden minors for diamond cubics (one way to see this is that they have treewidth $\Omega(K^2)$ whe... | 3 | https://mathoverflow.net/users/440 | 247080 | 112,468 |
https://mathoverflow.net/questions/247067 | 8 | I am an undergrad. I have taken courses in algebraic number theory and have a basic idea about $p$-adic numbers. I have also read a little bit of infinite Galois theory. But I have no idea about modular forms and automorphic forms and Galois cohomology.
I want to study about Galois Representations. All the books I ha... | https://mathoverflow.net/users/92281 | Reference book for Galois Representations | Galois representations have to come from somewhere.
If you are not interested in learning about modular forms and automorphic forms at this point, the other best source of representations are elliptic curves. You can find a gentle introduction in Silverman's book "The Arithmetic of Elliptic Curves", particularly the ... | 15 | https://mathoverflow.net/users/43108 | 247081 | 112,469 |
https://mathoverflow.net/questions/247072 | 6 | I am looking for a finite group $G$ and an irreducible projective representation $\rho: G \to PGL(\mathbb C^n)$ such that for *any* abelian subgroup $A\subset G$, the restricted representation $\rho|\_A$ is reducible.
(EDIT: Moreover $\rho$ should be a `genuine' projective representation, in the sense that it represe... | https://mathoverflow.net/users/50893 | (ir)reducibility of projective representation when restricted to abelian subgroup | The group ${\rm SL}(2,7)$ has a faithful complex irreducible representation of degree $4$. Viewing this as a projective representation of $G = {\rm PSL}(2,7)$ gives an example where every Abelian subgroup of $G$ acts reducibly, but $G$ acts irreducibly. This is because the only non-cyclic Abelian subgroups of $G$ are K... | 7 | https://mathoverflow.net/users/14450 | 247082 | 112,470 |
https://mathoverflow.net/questions/240419 | 5 | Super Grassmannians are introduced by Manin, see [for example](https://www.researchgate.net/publication/227218661_On_the_rigidity_of_super-Grassmannians). We have [Plucker relation](https://www.academia.edu/5178360/PLUCKER_EMBDDING_OF_GRASSMANNIAN_AND_A_TOY_EXAMPLE_OF_CLUSTER_ALGEBRA) for Grassmannian.
Are there som... | https://mathoverflow.net/users/11877 | Do we have super Plucker relations for a super Grassmannian? | In [The quantum chiral Minkowski and conformal superspaces](https://projecteuclid.org/euclid.atmp/1337951930) by Cervantes, Fioresi, and Lledó the super Grassmannian of $(2|0)$ planes in $\mathbb{C}^{4|1}$ is considered. In Equation (4.9) some "super Plücker relations" are given for this particular super Grassmannian. ... | 3 | https://mathoverflow.net/users/51668 | 247084 | 112,471 |
https://mathoverflow.net/questions/247075 | 3 | I'm trying to understand better conic bundles on quartic del Pezzo surfaces *over non-algebraically closed fields*.
Let $k$ be a field. A conic bundle surface is a smooth projective surface $S$ over $k$ equipped with a dominant morphism $S \to C$ to some smooth curve $C$, whose fibres are isomorphic to plane conics.
... | https://mathoverflow.net/users/5101 | Conic bundles on quartic del Pezzo surfaces | Let $S\_0 \subset P^3$ be a quadric surface (defined over $k$) with no 0-cycles of odd degree. Let $S\_1 \subset P^3$ be another quadric surface (also defined over $k$), such that the intersection $E := S\_0 \cap S\_1$ is smooth and in the pencil generated by $S\_0$ and $S\_1$ there are no degenerate quadrics defined o... | 5 | https://mathoverflow.net/users/4428 | 247091 | 112,473 |
https://mathoverflow.net/questions/247088 | 1 | Suppose that $r$ is a homogeneous linear recurrence sequence of order $>1$ with nonnegative integer coefficients, not all $0$, and nonnegative initial values, not all $0$. Suppose that $S$ and $T$ are finite sets of numbers in $r$. Let $S'$ be the product of numbers in $S$, and let $T'$ be the product of numbers in $T$... | https://mathoverflow.net/users/61426 | Distinct products of terms from a linear recurrence sequence | Consider the recurrence $a\_{n+2} = a\_{n+1} + 2 a\_n$, $a\_0 = 1$, $a\_1 = 2$, whose solution is $a\_n = 2^n$. Then $\prod\_{j \in J} a\_j = 2^{\sum J}$, so there are infinitely many counterexamples to your conjecture.
EDIT: Similarly for $a\_{n+2} = c a\_{n+1} + d a\_n$, $a\_0 = 1$, $a\_1 = t$, where $t^2 = c t + d... | 3 | https://mathoverflow.net/users/13650 | 247093 | 112,475 |
https://mathoverflow.net/questions/247090 | 9 | A strong version of the loop theorem implies that if an essential closed curve on the boundary of a 3-manifold $M$ is nullhomotopic, then realizing its image in $\partial M$ as a 4-valent graph, you can draw a cycle with no edge repeats that is homotopic on $\partial M$ to the boundary of an essential embedded disk in ... | https://mathoverflow.net/users/74169 | A strong annulus theorem for 3-manifolds | I think this might follow from JSJ theory. Assume that $M$ is irreducible with incompressible boundary. Then any essential annulus is homotopic into an $I$-bundle region or a Seifert-fibered region of the JSJ decomposition. In the Seifert case, the region meets the boundary in annuli, in which case the boundaries of th... | 6 | https://mathoverflow.net/users/1345 | 247097 | 112,478 |
https://mathoverflow.net/questions/247101 | 4 | What methods do we know about proving-disproving existence of rational points on surfaces of general type?
I was recently asked. My gut answer was- 'nothing'.
| https://mathoverflow.net/users/nan | Surfaces of general type | Usually nothing, as you guessed $-$ though you might get lucky:
i) There might be a local obstruction (e.g. no rational points on the
twisted Fermat sextic surfaces $x^6+y^6+z^6+t^6=0$ and $x^6+2y^6+4z^6=8t^6$).
ii) the surface, say $S$, may map to a curve with finitely many rational points.
iii) $S$ may be conta... | 7 | https://mathoverflow.net/users/14830 | 247103 | 112,480 |
https://mathoverflow.net/questions/247118 | 25 | Is there a Hausdorff topological space $X$ such that for any continuous map $f: X\longrightarrow \mathbb{R}$ and any $x\in \mathbb{R}$, the set $f^{-1}(x)$ is either empty or infinite?
| https://mathoverflow.net/users/86088 | A rare property of Hausdorff spaces | Yes, there is such a space. Let $X=2^{\omega\_1}$ be the space of
binary sequences of length $\omega\_1$, in the order topology
generated by the lexical order. So $X$ consists of the branches
through the tree $2^{<\omega\_1}$, with the left-to-right order on
branches. This is an order topology of a linear order and hen... | 35 | https://mathoverflow.net/users/1946 | 247120 | 112,484 |
https://mathoverflow.net/questions/247065 | 14 | It is well known among historians of Fermat that, while his technique of [adequality](https://en.wikipedia.org/wiki/Adequality) prepared the ground for the general framework later developed by Leibniz and Newton, Fermat himself gave very little in the way of explanation of his technique exploiting a symbol $E$ that app... | https://mathoverflow.net/users/28128 | Fermat's opponents | Maybe the following article <http://arxiv.org/abs/1306.5973> (Is mathematical history written by the victors?) and references therein will be useful :-).
Fermat's life and work is carefully investigated in the book "The Mathematical Career of Pierre de Fermat, 1601-1665" by Michael Sean Mahoney:
<http://press.princet... | 16 | https://mathoverflow.net/users/32389 | 247125 | 112,485 |
https://mathoverflow.net/questions/247135 | 1 | Suppose $X, U \in \mathbb{R}^{n \times r}$, $n>r$, where $U$ is a fixed matrix and $X$ is a variable, and both are of full column ranks. Let $\mathfrak{R} = \{ \Psi \in \mathbb{R}^{r \times r}: \Psi \Psi^\top = \Psi^\top \Psi = I\_r \}$ be the set of rotation matrices in dimension $r$. Also suppose there exists constan... | https://mathoverflow.net/users/97042 | lower bound on the norm of (correlated) matrix multiplication | No, there is no lower bound. Take for $\epsilon\neq 0$ arbitrarily small:
$$ X=\left( \begin{matrix} 1 & 0 \\ 0 & \epsilon \\ 0 & 0\end{matrix} \right) \ \ \mbox{and} \ \ U=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix} \right) .$$
However, if you add a condition on $X^T X$ having a uniformly bounded inverse... | 2 | https://mathoverflow.net/users/95413 | 247144 | 112,489 |
https://mathoverflow.net/questions/247119 | 5 | If $n$ is composite, then $\phi(n) < n-1$: hence, there is at least one number $d$ which does not divide $\phi(n)$ but divides$(n-1)$. We shall call $d$ the totient divisor of $n$. *The purist will say that totient non-divisor is a more appropriate name but for the sake of simplicity we shall stay with totient divisor*... | https://mathoverflow.net/users/23388 | Congruences for the non-divisors of Euler's $\phi(n)$ | I can prove the first congruence, so $\tau(4k+3) \equiv 0 \mod 2$.
Let $\sigma\_0(n)$ be the number of divisiors of $n$. Note that if $k \mid \varphi(n)$ and $k \mid n-1$ then $k \mid \gcd(\varphi(n),n-1)$. So $\tau(n)$ is the number of divisiors of $n-1$ that are not a divisor of $\gcd(\varphi(n),n-1)$. In other wo... | 4 | https://mathoverflow.net/users/74951 | 247152 | 112,490 |
https://mathoverflow.net/questions/246809 | 6 | Let $q$ be a power of a prime $p$. Deligne's paper "*Variétés abéliennes ordinaires sur un corps fini*" seems to describe an equivalence of categories between
1. ordinary abelian varieties over a finite field $\mathbf F\_q$,
2. complex abelian varieties equipped with an endomorphism $\pi$ which is a $q$-Weil number.
... | https://mathoverflow.net/users/48499 | Ordinary abelian varieties over a finite field | Because the abelian variety is ordinary, $\pi$ has the property that it's $p$-adic valuation at every place is either $0$ or $1$ (this follows easily from condition (IV) on the first page of Deligne's paper.) That means no prime $p'$of $\mathbb Q(\pi)$ lying over $p$ is equal to its complex conjugate, because if the $p... | 4 | https://mathoverflow.net/users/18060 | 247158 | 112,492 |
https://mathoverflow.net/questions/247159 | 2 | Let $V \subseteq \{0,1\}^n$, $\log|V| = k$. Consider
$V\_r:= \bigcup\_{x \in V} V\_r(x)$, where $V\_r(x)$ is a Hamming full-ball of radius $r$ and center $x$.
What is a lower bound for the cardinality of $V\_r$? Is the cardinality of $V\_r$ the smallest when $V$ is a Hamming full-ball (as in Harper's theorem)?
| https://mathoverflow.net/users/31356 | Union of Hamming balls | This is Lemma 2.2 in Ahslwede-Katona
<http://www.sciencedirect.com/science/article/pii/0012365X77900176>
| 3 | https://mathoverflow.net/users/3637 | 247163 | 112,494 |
https://mathoverflow.net/questions/247189 | 3 | Let $X$ be a scheme with finitely many irreducible components $V\_1,\dots,V\_r\subset X$. Its normalization $X^\nu\to X$ is the morphism obtained as follows: $$X^\nu=\coprod\_iV\_i^\nu\to X\_{red}\to X.$$ Here $V\_i^\nu$ is the normalization of $V\_i$. They are the irreducible components of $X^\nu$.
However, there is... | https://mathoverflow.net/users/30827 | Irreducible components of the seminormalization | I think your question is equivalent to this one:
>
> Is it true that the irreducible components of a seminormal scheme are themselves seminormal?
>
>
>
A curve is seminormal if it is locally analytically isomorphic to the coordinate axis in an affine space. Any subset of that has the same property (for a smal... | 2 | https://mathoverflow.net/users/10076 | 247192 | 112,500 |
https://mathoverflow.net/questions/247194 | 0 | We say that a function $f:\mathbb{R}\to\mathbb{R}$ has the *intermediate value property* (ivp) if for $a<b$ in $\mathbb{R}$ we have $$f([a,b]) \supseteq [\min\{f(a),f(b)\}, \max\{f(a), f(b)\}].$$
The intermediate value theorem states that continous functions have the ivp. Is there a non-continuous function $f:\mathbb{R... | https://mathoverflow.net/users/8628 | Intermediate value property and continuity | What kind of conditions do you prefer?
Say, local bounded variation is enough: if $f$ is discontinuous at a point $a$, there exist two numbers $A<B$ such that $f$ takes the values less than $A$ and more than $B$ in any (punctured) neighborhood of $a$. Clearly variation of $f$ in any such neighborhood is infinite.
... | 2 | https://mathoverflow.net/users/4312 | 247196 | 112,501 |
https://mathoverflow.net/questions/247079 | 8 | This is a question I've [implicitly asked on AoPS five years ago](http://artofproblemsolving.com/community/c6h427146p2418487), and has not been answered. I apologize for its possible simplicity, as I am not a graph theorist.
>
> Let $n$ and $k$ be two positive integers. Let $G$ be a graph with vertex set $V$. Assum... | https://mathoverflow.net/users/2530 | Turan-like bound for $k$-partite graphs | It is not true in general that $G$ under your conditions contains a $k$-clique. For example, if $k = 4$ consider the following construction by Gouping Jin (see the proof of Theorem 3.1 in [1]):
Let $V\_i = A\_i \cup B\_i$ for $i \in \{1,2,3,4\}$, where $\left|A\_i\right| = \lfloor n/3 \rfloor$ and $\left|B\_i\right| ... | 3 | https://mathoverflow.net/users/69775 | 247211 | 112,508 |
https://mathoverflow.net/questions/247213 | 3 | Let $\{X\_t\}\_{t\ge1}$ and $\{Y\_t\}\_{t\ge1}$ be two iid sequences of random variables that have full support. That is, if $A\subseteq\mathbb{R}$ has positive Lebesgue measure, then $P(X\in A) >0$ and $P(Y\in A)>0$. For such a sequence we almost surely have for any $M>0$ that $|X\_t|<M$ for infinitely many $t$. That ... | https://mathoverflow.net/users/52978 | Liminf of the maximum of two iid sequences | take a k so that $P(|X\_i | < k, | Y\_i | < k ) > \epsilon > 0$. They must exist because any k for $P(|X| < k) > \frac 3 4 $ and same for Y works, Let $A\_i = \{ |X\_i | < k, | Y\_i | < k \}$ By borel cantelli $A\_i$ happens infinitely often, and so the liminf is < k
sorry, had misinterpreted dependence structure. In... | 1 | https://mathoverflow.net/users/nan | 247216 | 112,511 |
https://mathoverflow.net/questions/242875 | 2 | For an elliptic curve $E$ over $\mathbb{Q}$, it is well-known that the torsion points on $E$ are integral points.
Then, is it possible that there exists an example whose all of non-torsion rational points (or all of points of a subgroup of $E\_{free}(\mathbb{Q}$)) are integral points (of course, with respect to affi... | https://mathoverflow.net/users/85711 | An example of "all non-torsion rational points on an elliptic curve are integral points''? | It’s not possible for a much simpler reason than Siegel’s Theorem. Let $p$ be a prime of good reduction, and $\tilde X$ the reduction mod $p$ of your putative integral nontorsion point. Since $\tilde X$ is a torsion point mod $p$, say $[n](\tilde X)=\mathbb O$, the neutral point. Then $[n](X))$ is in the $p$-neighborho... | 7 | https://mathoverflow.net/users/11417 | 247223 | 112,513 |
https://mathoverflow.net/questions/247199 | 2 | Let $\{x\_i\mid i\in \mathbb{Z}\}$ be a partition of $\mathbb{R}$ with equal distance $h>0$, and a given function $f\in L^2(\mathbb{R})$. I approximate $f$ by $P\_hf$, the $L^2$ projection of $f$ on piecewise constant function defined as
$$P\_hf(x)=\sum\_j a\_j 1\_{(x\_j,x\_{j+1}]}(x),\; x\in \mathbb{R},\quad a\_j=\fra... | https://mathoverflow.net/users/91196 | Approximation rate of $L^2$ function by piecewise constant functions | Additional smoothness will typically not improve the rate of convergence. If you have a well behaved smooth function with a derivative $f'$ that doesn't vary wildly on the intervals $I\_j=(x\_j,x\_j+h)$, then $|f(x)-a|\gtrsim |f'(x\_j)|h$ on a substantial portion of $I\_j$, so this interval makes a contribution $\gtrsi... | 5 | https://mathoverflow.net/users/48839 | 247241 | 112,520 |
https://mathoverflow.net/questions/247248 | 2 | Motivated by the concept of [diagonally dominated matrices](https://en.wikipedia.org/wiki/Diagonally_dominant_matrix#cite_note-3) we consider the space $S$ of all complex $n\times n$ matrices with $|a\_{ii}|>\sum\_{j\neq i} |a\_{ij}|$, for every $i$. [Every element of $S$ is invertible](https://mathoverflow.net/questio... | https://mathoverflow.net/users/36688 | A certain subset of general linear group | We can construct a homotopy retract of this to the set of diagonal matrices with nonzero diagonal entries by scaling nondiagonal entries to 0, and from there to a torus by scaling the diagonal entries to norm 1. Therefore, it is homotopy equivalent to $(S^1)^n$, and is connected.
For real matrices, the retract above ... | 7 | https://mathoverflow.net/users/44191 | 247249 | 112,522 |
https://mathoverflow.net/questions/247255 | 0 | Let $p\ge 11$ be a prime number, $n \ge 5$ be an odd positive divisor of $p-1$ and $s \in \mathbb Z\_p$ such that $ord\_p(s) = n$.
Is it true that the geometric progression $\{s^k\}\_{k \in \mathbb Z\_n}$ intersects some of the classes $\overline{p-n}, \;\; \overline{p-n+1}, \;\; \dots, \;\; \overline{p-1} \pmod p$?
... | https://mathoverflow.net/users/89000 | Geometric progression modulo p | I don't see a reason why this should hold.
Counterexample: $p=31$, $n=5$, $s=16$.
The powers of $s$ give 16, 8, 4, 2, 1 modulo 31.
| 2 | https://mathoverflow.net/users/7076 | 247260 | 112,524 |
https://mathoverflow.net/questions/247257 | 3 | All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural morphism $\pi\colon \Bbb A^n \to X=\operatorname{Spec}(R^G)$ is an almost geometric quotient. (**EDIT:** See [this quest... | https://mathoverflow.net/users/36720 | When is an almost geometric quotient flat? | Let $G$ be a reductive group acting linearly on affine space $\mathbb A^n$ and let $\pi:\mathbb A^n\to X$ be the categorical quotient. Then the following conditions are equivalent:
1. $\pi$ is flat.
2. $\mathcal O(\mathbb A^n)$ is a free $\mathcal O(X)$-module.
3. The morphism $\pi$ is equidimensional and $X$ is smoo... | 7 | https://mathoverflow.net/users/89948 | 247262 | 112,525 |
https://mathoverflow.net/questions/247143 | 7 | There are a number of famous results to the effect that "countable structures" of a certain type have a universal, "homogeneous" structure of the same type into which they all embed with various nice properties. For example, Urysohn's metric space, Rado's graph and Higman's finitely-presented group all follow this patt... | https://mathoverflow.net/users/4336 | "Universal embedding structures" in a general setting? | The following is essentially a copy-paste from a stackexchange answer I gave [here](https://math.stackexchange.com/questions/1084450/universal-object/1085906#1085906), so I'll make it community wiki.
[Trevor Irwin's thesis](http://gradworks.umi.com/32/97/3297082.html) gives a categorical treatment of [Fraïssé limits]... | 7 | https://mathoverflow.net/users/2362 | 247265 | 112,526 |
https://mathoverflow.net/questions/247266 | 3 | Given a smooth algebraic variety $X$ and a Q-Cartier Q-divisor $D$ which is semiample and big. Assume that $Y=Spec ( \oplus\_{m\geq 0}H^0 (X, mD))$ is Cohen-Macaulay. Do it follows that $H^i (X,mD)=0$ for $0 <i<\dim (X)$?
I know the answer is yes if $D$ is Q-ample and this characterize Cohen-Macaulayness for ring of ... | https://mathoverflow.net/users/37338 | Cohen-Macaulay ring of sections | This is not true in this form.
Think about it this way: For simplicity let us assume that $D$ is already basepoint-free (and to give a counter-example this is certainly enough), so there exists a proper surjective birational morphism $f:X\to Z$ and a very ample line bundle $\mathscr L$ on $Z$ such that $\mathscr N:=\... | 9 | https://mathoverflow.net/users/10076 | 247270 | 112,529 |
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