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https://mathoverflow.net/questions/203446 | 6 | I want to know if there is a nice characterization of when a formula is preserved under taking reduced factors.
We say that a formula $\phi$ is closed under taking reduced factors if whenever $I$ is a set, $Z$ is a filter on $I$ and $\prod\_{i\in I}A\_{i}/Z\models\phi([(a\_{i,1})\_{i\in I}],...,[(a\_{i,n})\_{i\in I}]... | https://mathoverflow.net/users/22277 | When is a formula preserved under taking factors in a reduced product or the stalk in a Boolean product? | (Only a sketchy partial answer.)
The family of *factorable formulas*, is the smallest set $F$ of formulas
containing every atomic formula that is closed under conjunction,
existential and universal quantification, and the following rule:
>
> if $\alpha(\vec x),\beta(\vec x, \vec y), \gamma(\vec x, \vec y)\in
> F$... | 4 | https://mathoverflow.net/users/66044 | 203802 | 98,139 |
https://mathoverflow.net/questions/203748 | 2 | I am wondering if there is a standard notation and name for the following. Let $\lambda$ be a partition $\lambda\_1\geq \lambda\_2\geq\cdots\geq \lambda\_r\geq 1$ of $n$ into $r$ parts. Then we can form a partition $\mu$ of $r$ by keeping track of how many times each integer occurs in $\lambda$. For example the partiti... | https://mathoverflow.net/users/15934 | A question of terminology regarding integer partitions | Let $\mu=(\mu\_1,\dots,\mu\_m)$ and $\gamma\_i$ be the number of parts equal $i$ in $\mu$. Then
$$\sum\_{i=1}^r \gamma\_i = m\quad\text{and}\quad\sum\_{i=1}^r i\cdot\gamma\_i = r.$$
Then $C\_{n,\mu}$ equals $\frac{1}{\gamma\_1!\cdots \gamma\_r!}$ times the number of solutions to
$$(\star)\qquad \mu\_1\cdot y\_1 + \do... | 2 | https://mathoverflow.net/users/7076 | 203813 | 98,142 |
https://mathoverflow.net/questions/203807 | 7 | I took this result from Minkowski's book on [Geometry of numbers](https://archive.org/details/geometriederzahl00minkrich):
>
> Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and $\frac{y}{z}$, that have the same denominator and at the s... | https://mathoverflow.net/users/1358 | approximate two different real numbers to order $\frac{1}{z^{3/2}}$ | With the constant $1$, this is Minkowski's higher dimensional extension of Dirichlet's approximation theorem:
*If $\alpha\_1, \ldots,\alpha\_n$ are real numbers, then there are rationals $p\_i/q$ with $|\alpha\_i - p\_i/q| < q^{-(1+1/n)}$. If one at least among the $\alpha\_i$ is irrational, there are infinitely many... | 8 | https://mathoverflow.net/users/26522 | 203816 | 98,143 |
https://mathoverflow.net/questions/203764 | 15 | This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho\_{E,\ell}$ associated to a non-CM elliptic curve $E$ defined over a number field $K$:
1. For each prime $\ell$, the index $I(E,\ell)$ of the image of $\rho\_{... | https://mathoverflow.net/users/37644 | What are the strongest conjectured uniform versions of Serre's Open Image Theorem? | I've seen Conjecture 1 and Conjecture 1' stated in the literature in many places. I don't believe I have seen Conjecture 1'' so stated.
I'd also like to point out that (EDIT: a weaker version of) Conjecture 2'' is true. In particular, if $E$ is a non-CM elliptic curve defined over a number field $K$ and $\ell$ is a p... | 11 | https://mathoverflow.net/users/48142 | 203837 | 98,151 |
https://mathoverflow.net/questions/203722 | 5 | I initially asked this question over at [StackOverflow](https://stackoverflow.com/questions/29804095/ordered-lattice-point-enumeration) as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community.
**Setup:** Let $e\_{i}$ be an orthogonal (but not o... | https://mathoverflow.net/users/22763 | Ordered lattice point enumeration | I am going to try to answer the version of the Observation which seems to imply that the original "orthogonal" basis condition really is meant to mean "positive multiples of the standard basis vectors". So the problem is, given positive real numbers $x\_1$, $x\_2$, ..., $x\_j$, produce the first $M$ elements of the set... | 1 | https://mathoverflow.net/users/1907 | 203842 | 98,153 |
https://mathoverflow.net/questions/203846 | 2 | I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known?
---
The only thing I know is this may be trivial thing : Fix a group element $g \in S\_n$. Let $R(g)$ be the matrix of $g... | https://mathoverflow.net/users/36554 | Is anything known about the eigenspectrum of the regular representation of the permutation group? | For the eigenvalues of matrices in arbitrary irreducible representations of $S\_n$, see the paper by John Stembridge at <http://msp.org/pjm/1989/140-2/pjm-v140-n2-p06-s.pdf>.
| 4 | https://mathoverflow.net/users/2807 | 203849 | 98,157 |
https://mathoverflow.net/questions/203852 | 8 | I am seeking a formalism to define the average height of
the rational points on a curve. This is straightforward
if the number of points is finite, but (to me) not straightforward
when the rational points are dense along the curve.
I will stick to $\mathbb{R}^2$ but all generalizes
to $\mathbb{R}^d$.
The *height* of ... | https://mathoverflow.net/users/6094 | Average height of rational points on a curve | First, may I change your notation a bit? Usually one uses $H(p/q)=\max\{|p|,|q|\}$ for the (multiplicative) height of a rational number, and $h(p/q)=\log H(p/q)$ is the logarithmic height. So I'll use that notation.
One natural way to study the distribution of the infinitely many rational points on a curve is to use ... | 14 | https://mathoverflow.net/users/11926 | 203855 | 98,159 |
https://mathoverflow.net/questions/203864 | 4 | The question is about Proposition 3.8.1 in Laumon and Moret-Bailly book on algebraic stacks.
Let $S$ be a scheme and let $F: \mathscr{X} \rightarrow \mathscr{Y}$ be a morphism of $S$-stacks (for the etale topology, say). Let $F': \mathscr{X}' \rightarrow \mathscr{Y}'$ be the base change of $F$ along an epimorphism o... | https://mathoverflow.net/users/70964 | Descending a monomorphism of stacks | The proof given in Laumon and Moret-Bailly is clear and does not seem to use your claim.
Denote by $G=\Delta\_F :\mathcal X \to \mathcal X \times\_{\mathcal Y} \mathcal X$ the diagonal morphism. The main points used are
A. $F$ is representable iff $\Delta\_F$ is a mono (not really used, but good to know)
B. $F$ ... | 4 | https://mathoverflow.net/users/11682 | 203868 | 98,162 |
https://mathoverflow.net/questions/203801 | 3 | Let $f:X \to Y$ be a flat morphism of projective noetherian integral schemes. Is there any known condition on a morphism $Z \to Y$ under which the resulting fiber product $X \times\_Y Z$ is still integral?
| https://mathoverflow.net/users/58203 | Which base change preserves integrality of schemes | If $Z$ is integral, $Z \to Y$ is dominant, and the field of fractions extension $k(Y) \to k(Z)$ is purely transcendental, you're OK.
First, note that the map from the fiber product to $Z$ is still flat, hence torsion free, so if there are zero divisors on any fiber they are on the generic fiber. So we can replace the... | 4 | https://mathoverflow.net/users/18060 | 203885 | 98,167 |
https://mathoverflow.net/questions/203836 | 39 | I saw a very remarkable asymptotic formula (or a conjecture?) for the volume of of the unitary group $ U(n)$ which is the following:
$$\log[\mathrm{Volume}(U(n))] \sim\_{n\rightarrow \infty} \frac{n^2}{2} \log(n) + \sqrt{2\pi}\log (n) -\sum\_{g\geq 2}n^{2-2g} \chi(\mathcal{M}\_{g}) $$
where the volume is calculated w... | https://mathoverflow.net/users/61328 | Volume of the unitary group | As indicated by Igor Rivin, the volume of the unitary group is given by
$vol(U(N))=(2\pi)^{(N^2+N)/2}/\prod\_{k=1}^{N-1} k!$.
The denominator is the Barnes G-function, which is well-known :
<http://en.wikipedia.org/wiki/Barnes_G-function>
and in particular has a known Stirling-like asymptotic expansion for larg... | 46 | https://mathoverflow.net/users/25309 | 203887 | 98,168 |
https://mathoverflow.net/questions/203893 | 4 | Let X be a compact Hausdorff topological group and let m be the Haar measure on X. Can we find a meager set in X whose complement is m-null? I can do it when X is separable but I don't know if there could be a non separable counterexample. The only non separable examples I know are products of separable groups so this ... | https://mathoverflow.net/users/70986 | Meager set of full measure | I switch to call $G$ your group because $X$ is a weird notation. The answer is: yes iff $G$ is infinite.
I assume you have a proof for $G$ infinite separable.
Next you can deal with the general case thanks to the following lemma: *if $G$ is an infinite compact group, then it has an infinite separable quotient.*
I... | 6 | https://mathoverflow.net/users/14094 | 203894 | 98,169 |
https://mathoverflow.net/questions/203427 | 2 | Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$,
$$|u|\_{W^{m,p}([0,1])}:=\|u^{(m)}\|\_{L^p([0,1])}$$
for all $u\in W^{m,p}$ the Sobolev seminorm where $u^{(m)}$ is the $m$-th der... | https://mathoverflow.net/users/35593 | Interpolation Operator Bounded in Sobolev Norm | I prove here that the upper bound for your inequality is true for $m=1$. However, this approach should help you prove what you want in general. (EDIT: This is true in general after checking with Charles Fefferman--see edit at bottom.)
What you're trying to do is similar to recent work by Charles Fefferman, Arie Israe... | 1 | https://mathoverflow.net/users/68222 | 203901 | 98,170 |
https://mathoverflow.net/questions/203902 | 1 |
>
> Let $A\_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such that $1<tuv|N$. For any such triple, define
> \begin{equation}
> E\_2^{\psi,\varphi,t}(\tau)=\begin{cases}
> E\_2^{\p... | https://mathoverflow.net/users/66686 | Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong? | The first part of Theorem 4.6.2 in the book says that $\left\{E\_2^{\psi,\varphi,t}:(\psi,\varphi,t)\in A\_{N,2}\right\}$ represents a basis of the Eisenstein space of weight $2$ with respect to $\Gamma\_1(N)$. As $E\_2^{\psi,\varphi,t}$ as a modular form for $\Gamma\_0(N)$ has nebentypus $\psi\phi$ regarded as a modul... | 4 | https://mathoverflow.net/users/11919 | 203909 | 98,173 |
https://mathoverflow.net/questions/203913 | 7 | As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is so easy to count the number of integers $x < p\#$ where $\gcd(x(x+2),p\#)=1.$
When I do a google search on *reduced r... | https://mathoverflow.net/users/15915 | Are reduced residue systems relative primorials an active area of research? If not, why not? | The Chinese remainder theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of the finite fields ${\bf Z}/q {\bf Z}$, where $q$ ranges over the primes up to $p$. As such, many "global" or "multiplicative" questions about the residue classes modulo a primorial quickl... | 13 | https://mathoverflow.net/users/766 | 203928 | 98,179 |
https://mathoverflow.net/questions/203892 | 2 | It is well known fact that a (f.g.) group is hyperbolic if and only if it admits a (finite) Dehn presentation.
My question concerns something I'm struggling with since the first time I read the proof of this Theorem (in Bridson, Haefliger, Metric spaces of non-positive curvature). Are there a lot of known examples of... | https://mathoverflow.net/users/70809 | Explicit examples of Dehn presentations of hyperbolic groups | I'll turn YCor's comment (above) into an answer. Surface groups are the simplest (and historically first) non-free example. Let $S\_g$ be the surface of genus $g$. Then the fundamental group has presentation
>
> $\pi\_1(S\_g) \cong \langle a\_1, a\_2, \ldots, a\_{2g-1}, a\_{2g} \mid a\_1 a\_2 \ldots a\_{2g} A\_1 A\... | 3 | https://mathoverflow.net/users/1650 | 203933 | 98,180 |
https://mathoverflow.net/questions/203936 | 2 | Let $X$ be a quasi projective variety over $\mathbb{C}$. By the tangent cone of $X$ at a point $p \in X$, I mean the subvariety of the tangent space of $X$ at $p$ as it is defined in Harris' "Algebraic Geometry: A first course" (Lecture 20). In particular, the tangent cone is a *reduced* subscheme.
Now let $X$ be loc... | https://mathoverflow.net/users/36563 | Tangent cone of a complete intersection | If you use the *wrong* definition of tangent cone, then certainly there are counterexamples. For instance, for the origin $p=(0,0,0)$ in $\mathbb{A}^3$, consider the curve $$X=\text{Zero}(\ s(t+u) + f(s,t,u),\ tu + g(s,t,u)\ ),$$ where $f$ and $g$ are sufficiently general polynomials of high degree. The tangent cone is... | 4 | https://mathoverflow.net/users/13265 | 203937 | 98,182 |
https://mathoverflow.net/questions/203938 | 5 | Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times\_{\rho}\Delta$ its spinor bundle, where $\rho\colon Spin(d)\to\Delta$ is the corresponding spinor representation. My question is:
When a reduction on $SP... | https://mathoverflow.net/users/66688 | Frame-bundle reduction from spinor-bundle reduction | Well, in one sense, this is always true. If $Q\subset SP(M)$ is a principal right $H$-bundle, where $H\subset\mathrm{Spin}(n)$ is a subgroup, then $\delta(Q)\subset F(M)$ is a principal right $\pi(H)$-bundle, where $\pi(H)\subset\mathrm{SO}(n)$ is the image subgroup. (Here, I am using $\delta:SP(M)\to F(M)$ to denote t... | 5 | https://mathoverflow.net/users/13972 | 203941 | 98,183 |
https://mathoverflow.net/questions/203929 | 14 | Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that is being operated on is of exponential size, so it cannot be run efficiently.
However, let us make the following observ... | https://mathoverflow.net/users/7089 | Can Shor's Algorithm be modified to run efficiently on a classical computer? | If this kind of simple trick involving just the Fourier transform, and not taking advantage of special properties of multiplication modulo $n$, worked, then it would provide a fast classical algorithm not just for factoring but for the more general abelian hidden subgroup problem - e.g. identifying the period of an arb... | 17 | https://mathoverflow.net/users/18060 | 203951 | 98,186 |
https://mathoverflow.net/questions/203848 | 2 | Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$
I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and $F(0)=a+b$.
I was wondering what the conditions on a program and constraints should be, so that the problem is continuous in it... | https://mathoverflow.net/users/42371 | When is a convex program continuous in its constraint vectors? | So I think you're missing one key fact. When you do this minimization problem you should find that the points satisfying the minimization problem are $$x^\* = \frac{z}{2} + \frac{1}{2} \log(\frac{a}{b})$$ $$y^\* = \frac{z}{2} - \frac{1}{2} \log(\frac{a}{b})$$ Now, you require that $x^\* \geq 0$ and $y^\* \geq 0$ and so... | 2 | https://mathoverflow.net/users/49404 | 203952 | 98,187 |
https://mathoverflow.net/questions/203948 | 11 | The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see [wikipedia](http://en.wikipedia.org/wiki/Topological_K-theory)).There is the Bott spectrum associated to this generalized cohomology theory.Using that spectrum,we could surely produce a generalized ***homology**... | https://mathoverflow.net/users/67140 | A survey for various $K$-homology theories and their relationship | I would say that there are really only two definitions of K-homology commonly used in the literature (apart from the naive definition via the Bott spectrum): "analytic K-homology" and "geometric K-homology". KK theory is a bivariant theory which includes topological K-theory as the special case $KK(\mathbb{C},C(X))$ an... | 14 | https://mathoverflow.net/users/4362 | 203953 | 98,188 |
https://mathoverflow.net/questions/203955 | 1 | Is an arbitrary Brownian path a viscosity solution of every differential equation?
My intuition is that a path of Brownian motion is so ill-behaved that it not only does not have derivatives anywhere but it also does not have (local) subdifferentials and superdifferentials. Equivalently, there exists no positive mea... | https://mathoverflow.net/users/71015 | Is an arbitrary Brownian-motion path a viscosity solution of every differential equation? | Yes, I think you're missing something in the definition.
Quoting from Wikipedia's definition,
>
> An equation $ H(x,u,Du,D^2 u) = 0 $ in a domain $ \Omega $ is defined to be ''degenerate elliptic'' if [...]
>
>
>
In particular,
>
> Any first order equation is degenerate elliptic.
>
>
>
So let's consid... | 1 | https://mathoverflow.net/users/4600 | 203960 | 98,189 |
https://mathoverflow.net/questions/203970 | 15 | Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v\_1)+\dots +F(v\_n)$ over an orthonormal basis is independent of the basis. Does it follow that $F$ is a constant function?
This is clearly false for $n=2$. I am wondering if it is true for ... | https://mathoverflow.net/users/6666 | A combinatorial question about orthonormal bases | Given a vector $u$ and an orthonormal basis $x\_1,\ldots,x\_n$, we have $||u||^2 = \left<u,x\_1\right>^2 + \cdots + \left<u,x\_n\right>^2$. But that means that, if you choose a nonzero $u$, then the function $F(x) = \left<u,x\right>^2$ gives a counterexample.
| 29 | https://mathoverflow.net/users/14901 | 203974 | 98,195 |
https://mathoverflow.net/questions/203966 | 10 | Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The *vertex enumeration problem* asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be performed in $\mathcal{O}(|V|^{\lfloor d/2 \rfloor})$, cf. [[1]](http://www.mathematik.tu-darmstadt.de/~pfetsch/Publications... | https://mathoverflow.net/users/56325 | Computionally efficient vertex enumeration for (convex) polytopes | cddlib is rather old; a much more efficient implementation of the double description method is in PPL (Parma Polyhedra Library). One frontend to PPL can be found in Sagemath: <http://www.sagemath.org/doc/reference/geometry/sage/geometry/polyhedron/constructor.html>
PPL will perform computations exactly.
Apart from th... | 10 | https://mathoverflow.net/users/11100 | 203978 | 98,197 |
https://mathoverflow.net/questions/203947 | 10 | Is it true that the following two statements are equiconsistent?
(1) $2^\mu>\mu^+$ for some strong limit singular cardinal $\mu$
(2) $cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular cardinal $\mu$
Thanks.
| https://mathoverflow.net/users/71011 | Are the failure of SCH and "$cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular" equiconsistent? | As it is stated by Yair Hayut, in the comment above, statement $(1)$ implies statement $(2)$. Let's show that statement $(2)$ does not imply statement $(1)$.
Let $u(\kappa, \lambda)=cf(P\_\kappa(\lambda), \subseteq),$ so that $cf([\mu]^{cf(\mu)}, \subseteq)=u(cf(\mu)^+, \mu).$
As it is stated in [Large cardinals a... | 5 | https://mathoverflow.net/users/11115 | 203994 | 98,205 |
https://mathoverflow.net/questions/203995 | 2 | Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.
Let $f,g\in \mathbb{C}[x,y]$.
Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$.
Given $h\in \mathbb{C}[x,y]$, how to determine whether $h\in (f,g)$ or not?
I have tried some examples by the onli... | https://mathoverflow.net/users/41075 | ideals of polynomial ring with complex number coefficients | You should use "**GrΓΆbner basis**", (Groebner) . see the book by "Cox D., Little J., O'Shea D.": named "*Ideals, Varieties, and Algorithms*", for example. In page.82 they have:
Corollary.2. Let $G = \{g\_1, \cdots , g\_t\}$ be a *Groebner basis* for an ideal $I \subset k[x\_1, \cdots , x\_n]$ and let $f \in k[x\_1, ... | 5 | https://mathoverflow.net/users/47763 | 204003 | 98,209 |
https://mathoverflow.net/questions/203979 | 5 | Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b\_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a strictly full triangulated subcategory of $D^b\_{coh}(X)$. Then following [Orlov 2003](http://arxiv.org/pdf/math/0302304v... | https://mathoverflow.net/users/24965 | Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left? | The exact sequence of triangulated categories
$$ Perf(X)\to D^b\_{coh}(X)\to D\_{sg}(X) $$
may be lifted to an exact sequence of stable $\infty$-categories or dg-categories in the sense of [BGT](http://arxiv.org/pdf/1001.2282.pdf): choose an enhancement of $D^b\_{coh}(X)$, take the induced enhancement on the subcatego... | 8 | https://mathoverflow.net/users/2503 | 204012 | 98,214 |
https://mathoverflow.net/questions/204009 | 1 | Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A\_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the completition of $C^\infty(X, E)$ with the norm
$$
\lVert f \rVert\_{W^{k,p}} = \int\_X |f|^p + |\nabla\_{A\_0}f|^p + \cdots ... | https://mathoverflow.net/users/39229 | Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles | The rank of $E, F$ essentially doesn't enter the discussion, since given a basis of $e\_i$ and a basis $f\_j$ of $E$ and $F$, you have a basis $e\_i\otimes f\_j$ of $E\otimes F$ and representing your connection against this basis you are down to dealing with scalar functions.
Suppose first that you can choose bases ... | 2 | https://mathoverflow.net/users/3948 | 204022 | 98,216 |
https://mathoverflow.net/questions/203999 | 3 | Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional Hilbert spaces.
Let $B\_1, \ldots, B\_k \in B(\mathcal{H}).$
Define $L: B(\mathcal{K})^k \rightarrow B(\mathcal{H}\otimes \mathcal{K})$ via the formula
$$L(X\_1,\ldots, X\_k) = \sum B\_i \otimes X\_i.$$
For each $\varepsilon > 0,$ does there exist a finite r... | https://mathoverflow.net/users/32470 | Approximating the norm of an operator-valued linear function with operator inputs via a matrix-valued linear function | No, there does not always exist such a finite rank projection.
Indeed, this implies that the linear space spanned by $B\_1,\dots,B\_k$ is **exact** as an operator space, and there are non-exact operator spaces. Actually one can check that the converse holds: if $B\_1,\dots,B\_n$ span an exact operator space, then suc... | 5 | https://mathoverflow.net/users/10265 | 204023 | 98,217 |
https://mathoverflow.net/questions/204019 | 6 | Is there any references concerning the computation of the fundamental groups and Hodge numbers of Fano schemes of lines in a smooth hypersurface in $\mathbb{P}^n$?
| https://mathoverflow.net/users/43423 | The topology of Fano schemes of lines | As Daniel Loughran points out, my claim regarding vanishing of the fundamental group is wrong. In all of the following, $F$ is the Fano scheme parameterizing lines on a degree $d$ hypersurface $X$ in $\mathbb{P}^n$. I am assuming that $F$ is smooth of the expected dimension $2n-d-3$. Here is what I can prove.
(1) Whe... | 9 | https://mathoverflow.net/users/13265 | 204041 | 98,224 |
https://mathoverflow.net/questions/204040 | 10 | Let us denote the Riesz potential in $\mathbb R^d$ by
$$
I\_\alpha (f)(x) := c\_{d, \alpha} \int\_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
By the classical Hardy-Littlewood-Sobolev theorem on fractional integration we have for $1 < p <d/\alpha$ that
$$
\|I\_\alpha(f)\|\_{L^q} \le C\_{d, \alpha, p} \|f\|... | https://mathoverflow.net/users/46298 | Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$? | Let $a \in \mathbb{R}^d$ and $r > 0$. We have
$$
\frac{1}{\vert B\_r \vert^2}
\int\_{B\_r} \int\_{B\_r} \vert I\_\alpha (f) (x) - I\_\alpha (f) (y) \vert\,\mathrm{d}x\,\mathrm{d}y
\le \frac{c\_{d, \alpha}}{\vert B\_r \vert^2} \int\_{\mathbb{R^d}} \int\_{B\_r} \int\_{B\_r} \vert f (z)\vert \,\Big\vert \frac{1}{\vert z ... | 5 | https://mathoverflow.net/users/42047 | 204045 | 98,226 |
https://mathoverflow.net/questions/204052 | 1 | Is there a topological space $X$ with a nonzero sheaf $\mathcal{F}$ of abelian groups such that $H^i(X,\mathcal{F})=0$ for all $i=0,1,2...$?
| https://mathoverflow.net/users/nan | Is there a nonzero sheaf with all cohomologies vanish? | What about the "Moebius" local system over $S^1$ with fibers $\mathbb{Q}$ and monodromy $-1$? (It has $H^0=H^1$ by Poincare duality with coefficients).
| 4 | https://mathoverflow.net/users/7108 | 204054 | 98,230 |
https://mathoverflow.net/questions/204027 | 13 | Assume we have a homeomoprhism $\phi:M\rightarrow M$, where $M$ is a topological manifold which admits at least one smooth structure.
Is it always possible to construct a smooth structure on $M$ w.r.t to it $\phi$ will be a diffeomorphism?
Of course when there is little freedom in defining the smooth structure the ... | https://mathoverflow.net/users/46290 | When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure? | Let me first answer your last question in the negative: there exist homeomorphisms $f:M \rightarrow M$ of smoothable manifolds $M$ such that neither $f$ nor $f^{-1}$ are smooth with respect to any smooth structure on $M$. In fact, we can take $M = S^3$. Bing has constructed homeomorphisms $f:S^3 \rightarrow S^3$ such t... | 18 | https://mathoverflow.net/users/317 | 204059 | 98,233 |
https://mathoverflow.net/questions/203969 | 1 | I'm trying to read this paper of Bogomolov and Tschinkel <http://arxiv.org/pdf/math/9902092.pdf> about potential density of rational points on elliptic K3 Surfaces.
I got quite stuck in Corollary 3.27 and Proposition 3.24. I do not understand when they say the following(line 3 Corollary 3.27):
"Dividing (the cocyc... | https://mathoverflow.net/users/71024 | Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory | Question 1: $S'$ is an elliptic surface without a section such that it is Jacobian is $S$. The surfaces with this property are parametrized by a certain cohomology group and the cocycle the authors refer to, lives in this group. More details about non-Jacobian elliptic fibrations can be found in e.g., the final chapter... | 1 | https://mathoverflow.net/users/8621 | 204067 | 98,234 |
https://mathoverflow.net/questions/204064 | 14 | It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when the base is the same, but not for when the power is the same like here.
For example, does $-\sqrt 2$ satisfy this? I w... | https://mathoverflow.net/users/17086 | For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$? | Robert Israel's guess that this is true for $t$ an algebraic irrational does actually follow from Schanuel's conjecture.
Apply Schanuel's conjecture to $\log p$ and $t \log p$ for $n$ different primes $p$. These are linearly independent unless $t$ is the ratio of the logarithms of two integers, which it isn't because... | 19 | https://mathoverflow.net/users/18060 | 204077 | 98,237 |
https://mathoverflow.net/questions/22007 | 20 | While attending a very nice talk on the geometric group theory of fundamental groups of Kahler manifolds by Pierre Py last weekend, I realized that I don't know the answer to the following question. Let $X$ be a smooth projective variety over $\mathbb{C}$. Is the [word problem](http://en.wikipedia.org/wiki/Word_problem... | https://mathoverflow.net/users/317 | The word problem for fundamental groups of smooth projective varieties | I was going through my old questions and realized that this one did not have a good answer (as far as I know, the reference that Ben Wieland gave in his answer does not work). I've since learned that it is a well-known open question. However, I thought I'd point out the recent paper
Kapovich, Michael,
Dirichlet funda... | 7 | https://mathoverflow.net/users/317 | 204083 | 98,240 |
https://mathoverflow.net/questions/204084 | -1 | "Let (u\_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u\_j\rightharpoonup u$ in $W^{1,p}\_0(\Omega)$ and $|\nabla u\_j|\rightharpoonup d\mu,$ $|u\_j|^{p^\*}\rightharpoonup d\nu$ weakly\* in the sense of measures."
| https://mathoverflow.net/users/49045 | Question about measure lemma? | Assuming $1 < p < \infty$, $W^{1,p}(\Omega)$ is reflexive, so bounded sets are weakly precompact by Alaoglu's theorem (the weak-\* and weak topologies coincide). Thus $u\_j$ has a subsequence converging weakly to some $u \in W^{1,p}(\Omega)$. Now $W^{1,p}\_0(\Omega)$ is convex and strongly closed in $W^{1,p}(\Omega)$, ... | 2 | https://mathoverflow.net/users/4832 | 204088 | 98,242 |
https://mathoverflow.net/questions/204092 | -1 | By coincidence i stumbled over this page
<http://www.fields.utoronto.ca/programs/scientific/11-12/exceptional/abstracts.html>
, which was installed for a workshop on algebraic groups in 2012.
In the upper left you will see several Dynkin diagramms of Root Systems.
Just under $F\_4$ there is one named $H\_3$, anot... | https://mathoverflow.net/users/51251 | Algebraic Groups of Type H_3 and H_4 | It's actually $I\_n$, not $L\_2$ (the print is very small). These diagrams classify finite Coxeter groups ($I\_n$ is the family of dihedral groups and $H\_3$ and $H\_4$ are automorphism groups of certain exceptional polytopes). The three that you mention aren't Weyl groups, so don't come from an algebraic group like th... | 4 | https://mathoverflow.net/users/321 | 204094 | 98,244 |
https://mathoverflow.net/questions/204031 | 3 | Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L\_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a subset $P$ of $n$ points in $\mathbb{F}^2$, then we call $L\_m$ an even cover of $P$ if every line in $L\_m$ contains a... | https://mathoverflow.net/users/71042 | An upper bound on the number of sets of parallel lines covering points in a finite plane? | If a set $S$ is evenly covered by lines in $n$ slopes, then $n \le |S|-1$ because through every point $p$, there are at most $|S|-1$ lines connecting $p$ to other points in the set, and any other slope of line would include a line intersecting $S$ in just $p$.
Here are some examples achieving that bound: Hyperovals i... | 3 | https://mathoverflow.net/users/2954 | 204103 | 98,245 |
https://mathoverflow.net/questions/204106 | 6 | I would like to know what the definition of a short proof is.
In Lance Fortnowβs article β[The Status of the P Versus NP Problem](http://cacm.acm.org/magazines/2009/9/38904-the-status-of-the-p-versus-np-problem/fulltext)β, Communications of the ACM, Vol. 52 No. 9, he says,
>
> If a formula ΞΈ is not a tautology, w... | https://mathoverflow.net/users/71074 | What defines a "short proof"? | The statement you quoted is somewhat sloppy, since there is no precise notion of a short proof for a single formula. There is, however, a notion of short proofs for a class $C$ of formulas, when the class contains formulas of arbitrarily high length. One says that $C$ admits short proofs if there is a polynomial $p(x)$... | 22 | https://mathoverflow.net/users/6794 | 204108 | 98,247 |
https://mathoverflow.net/questions/204096 | 0 | Does the wave equation $u\_{tt} - \Delta u = 0$ have any backward uniqueness results that are similar to the ones for the heat equation (see for example Theorem 11 page 64 in Evans)? If not, are there any counterexamples?
| https://mathoverflow.net/users/71070 | Backward Uniqueness for the wave equation | Suppose $u$ solves
\begin{cases}
(\partial\_t^2-\Delta)u = 0 & \text{on } U\\
u(T,x) = 0 \\
\partial\_t u(T,x) = 0\\
u(t,x) = 0 & \text{on } \partial U.
\end{cases}
Since the energy
$$
E(t) = \int\_U |\nabla u|^2 + |\partial\_t u|^2\,dx.
$$
is constant, $E(t) = E(T) = 0$, it follows that $u \equiv 0$. From linearity o... | 1 | https://mathoverflow.net/users/30178 | 204111 | 98,248 |
https://mathoverflow.net/questions/204104 | 0 | I give [here](https://mathoverflow.net/questions/132973/would-the-following-conjectures-imply-lim-inf-n-to-inftyp-nk-p-n-ok-lo) a heuristics that suggests that the quantity $\displaystyle{G\_{k}:=\liminf\_{n\to\infty}p\_{n+k}-p\_{n}}$ should be approximately equal to $k(1+H\_{k})$, where $H\_{k}$ is the $k$-th harmonic... | https://mathoverflow.net/users/13625 | Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured? | The [Hardy-Littlewood prime tuples conjecture](http://mathworld.wolfram.com/k-TupleConjecture.html) implies that $G\_k$ equals the smallest diameter of an admissible $(k+1)$-tuple. In particular, it implies that $G\_{10}=36$ (cf. [here](http://math.mit.edu/~primegaps/)).
Your conjecture implies for $k>3$ that
$$ G\_k... | 6 | https://mathoverflow.net/users/11919 | 204113 | 98,250 |
https://mathoverflow.net/questions/204095 | 4 | Let $\varphi$ be the Euler totient function, and let us define the function $f(z)$ by the series
$$
f(z) := \sum\_{n=1}^{\infty} \varphi(n) z^n
$$
Since $0\le \varphi(n)\le n$, I believe this gives a well-defined function in some region of the complex plane. Where could I find a discussion on the analytic behavior of ... | https://mathoverflow.net/users/59322 | Residue for the generating function of the Euler totient function | Note that $f(z)=\sum\_{n=1}^\infty \phi(n) z^n$ can be written as a Lambert series,
$$f(z) = \sum a\_n\frac{z^n}{1-z^n},$$ where $a\_n = \sum\_{d|n}\phi(d) \mu(n/d).$ It is clear that the Lambert series converges inside the unit disk, and $f(z)$ has a natural boundary on the unit circle (since it will have a pole at ev... | 7 | https://mathoverflow.net/users/11142 | 204115 | 98,252 |
https://mathoverflow.net/questions/204110 | 1 | Let $\mathsf{A}$ be an Abelian category (perhaps vector spaces or modules over your favorite ring), and let $\mathsf{A}(x,y)$ denote the set of morphisms in $\mathsf{A}$ from an object $x$ to another object $y$.
>
> Does there exist a non-trivial partial order on the morphism-sets of $\mathsf{A}$ which behaves well... | https://mathoverflow.net/users/18263 | Poset-enrichment of abelian categories | Suppose that $V, W$ are vector spaces and $f, g : V \to W$ are two parallel morphisms such that $f \le g$, for some preorder $\le$ satisfying your conditions.
>
> **Claim:** $f$ is a scalar multiple of $g$.
>
>
>
*Proof.* The first condition implies that if $v : 1 \to V$ is any vector in $V$ (here $1$ denotes... | 6 | https://mathoverflow.net/users/290 | 204117 | 98,253 |
https://mathoverflow.net/questions/204098 | 2 | (Please see a few paragraphs below by what I mean by βcolored node graph isomorphismβ.)
Some basic definitions for completeness:
Given two graphs $G\_1=(V\_1, E\_1)$ and $G\_2=(V\_2, E\_2)$ the graph isomorphism problem (GI) asks whether there exists a one-to-one mapping $\sigma: V\_1 \rightarrow V\_2$ such that $(... | https://mathoverflow.net/users/71071 | Linear algebra formulation for colored node graph isomorphism | It is equivalent (up to polynomials). Given a coloured graph $G\_1$, add a new node (a *colour vertex*) for each colour and add edges from it to all vertices with that colour. Next add one more new node (the *supernode*) that is adjacent to all the colour vertices. Finally, attach to the supernode a clique that is so b... | 5 | https://mathoverflow.net/users/9025 | 204121 | 98,257 |
https://mathoverflow.net/questions/200607 | 4 | [The nLab page on partitions of unity](http://ncatlab.org/nlab/show/partition+of+unity) mentions the application of partitions of unity as a way to construct continuous maps to geometric realizations of simplicial spaces. However I often feel uncomfortable with the continuity of maps constructed in the realm of this ex... | https://mathoverflow.net/users/69525 | Continuous maps to fat geometric realizations of simplicial spaces | My first remark concerns Segal's (4.1). The technical details are key. Segal begins by choosing a *locally finite partition of unity* $f\_i$ subordinate to the cover $U\_i$. The definition of this term on the nLab seems to be non-standard, and probably does not make the statement true.
A partition of unity $\{f\_i\}... | 4 | https://mathoverflow.net/users/318 | 204135 | 98,261 |
https://mathoverflow.net/questions/204132 | 1 | Is the category, $\textbf{FHILB}$, of finite dimensional Hilbert spaces and linear maps locally regular, where `locally regular' is defined like this
<http://ncatlab.org/nlab/show/locally+regular+category>
| https://mathoverflow.net/users/45570 | Is $\textbf{FHILB}$ locally regular? | Yes. Since it has a terminal object (namely the zero-dimensional space 0=$\{0\}$), the slice category $\mathbf{FHilb} / 0$ is isomorphic to $\mathbf{FHilb}$ itself. Because all slices of a locally regular category are regular, local regularity is equivalent to regularity.
And in fact, $\mathbf{FHilb}$ is regular. It ... | 3 | https://mathoverflow.net/users/10368 | 204136 | 98,262 |
https://mathoverflow.net/questions/204128 | 3 | It [turns out](https://mathoverflow.net/questions/36085/minimal-hausdorff) that not every Hausdorff topology is contained in a minimal Hausdorff topology. Let's put this question on its head: is every non-$T\_2$ topology contained in a topology that is maximal with respect to the property of not being $T\_2$?
| https://mathoverflow.net/users/8628 | Minimal Hausdorffness reversed | Yes.
Let $\tau$ be a non-Hausdorff topology on a set $X$.
Suppose first that $\tau$ is not $T\_1$, so there are two points $a,b\in X$ such that every open neighbourhood of $a$ contains $b$. Then $\tau$ can be refined to the topology consisting of all subsets of $X$ except for those containing $a$ but not $b$, whic... | 8 | https://mathoverflow.net/users/22989 | 204142 | 98,264 |
https://mathoverflow.net/questions/204099 | 2 | I'm reading through Atiyah's paper that classifies vector bundles over an elliptic curve, and I'm a little confused about one of his proofs.
Lemma 15(i) states that if $E \in \mathcal{E}(r,d)$ is a vector bundle of rank $r$ and degree $d\geq0$ over $X,$ then $s:=h^0(X,E)=d$ if $d>0$ and $s=0$ or $1$ if $d=0.$
For... | https://mathoverflow.net/users/52914 | Atiyah's vector bundles over an elliptic curve | I'm pretty sure that by $L\_i>1$ he means that the line bundle has sections. Now this either means that $L\_i=1$ or $\deg L\_i\geq 1$. On an elliptic curve, any line bundle (divisor) of positive degree is non-special so for any of these $L\_i$ with $\deg L\_i\geq 1$, we must have $H^1(L\_i)=0$. Also in general on a cur... | 5 | https://mathoverflow.net/users/13139 | 204153 | 98,270 |
https://mathoverflow.net/questions/204168 | 9 | It is not very hard to see that for each prime power $q$ and natural numbers $n,h$, we have an embedding
$$\iota \colon \mathrm{GL}(n,q^h) \hookrightarrow \mathrm{GL}(nh, q),$$
obtained by choosing a basis for the finite field $\mathbb{F}\_{q^h}$ over the base field $\mathbb{F}\_q$.
It is not very hard either to see th... | https://mathoverflow.net/users/12858 | Embedding $\mathrm{PGL}(n,q^h)$ in $\mathrm{PGL}(nh,q)$ | Yes, ${\rm PGL}(2,q^2)$ is a subgroup of ${\rm PGL}(4,q)$, but I would guess that that is an exception, and in general there is no such embedding.
${\rm GL}(4,q)$ contains the subgroup that I denote by ${\rm CO}^-(4,q)$, which is the conformal orthogonal group of minus-type (and equal to the normalizer in ${\rm GL}(4... | 8 | https://mathoverflow.net/users/35840 | 204173 | 98,274 |
https://mathoverflow.net/questions/186723 | 20 | More precisely:
>
> Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
> are integral curves of genus $g$. Must the fppf relative Picard functor
> $\operatorname{\bf Pic}\_{X/S}$ be representable by a scheme?
>
>
>
If $g \ne 1$, then some integer power of $\omega\_{X/S}$
sho... | https://mathoverflow.net/users/2757 | Does every relative curve have a Picard scheme? | The answer is yes: $\mathbf{Pic}\_{X/S}$ is representable by a scheme. I will argue that this follows from the SGA 3 result mentioned by user27920 and from Theorem 2 (c) in section 6.6 of *Neron models* (which itself is based on a nonflat descent result due to Raynaud).
**Preliminary reductions:** Since the $g \neq 1... | 10 | https://mathoverflow.net/users/5498 | 204182 | 98,277 |
https://mathoverflow.net/questions/204187 | 3 | Assume that we are working in ZF set theory without the Axiom of Choice. If S is an infinite set, let $S(f)$ denote the set of all finite subsets of $S$, let $S(I)$ denote the set of all infinite subsets of $S$ and let $\operatorname{Card}(S)$ denote the cardinal number of $S$.
Even though we can prove Cantor's theor... | https://mathoverflow.net/users/4423 | A question about Cantor's Power Set theorem without the Axiom of Choice | For the first equality, the answer is true.
It is quite easy to construct examples where the set of finite subsets is strictly larger. For example if $X$ is an infinite Dedekind-finite set which is the countable union of finite sets (e.g. Russell socks sets), then the set $X(f)$ is not Dedekind-finite anymore, since ... | 5 | https://mathoverflow.net/users/7206 | 204191 | 98,281 |
https://mathoverflow.net/questions/204209 | 3 | Let $X$ be a measurable space whose $\sigma$-algebra is generated by a family $\mathcal{G}=\bigcup\_n \mathcal{G}\_n$ of subsets of $X$, where $(\mathcal{G}\_n)$ is a sequence of $\sigma$-algebras on $X$ of increasing fineness. I am not sure if the "$=\bigcup\dots$" is helpful, but I included it just in case.
Further... | https://mathoverflow.net/users/12713 | Quotient sigma-algebra generated by quotient-measurable generating sets | No, it isn't.
Let $X = \omega = \{0,1,2,\dots\}$. Let $\mathcal{G}\_n = \sigma(\{\{0\}, \{1\}, \dots, \{n\}\})$ be the $\sigma$-field in which subsets of $\{0,1,\dots, n\}$, and their complements, are measurable. If $\mathcal{G} = \bigcup\_n \mathcal{G}\_n$, then $\sigma(\mathcal{G}) = 2^\omega$, but all sets in $\ma... | 4 | https://mathoverflow.net/users/4832 | 204219 | 98,289 |
https://mathoverflow.net/questions/204167 | 35 | This is a follow-up to Dan Ramras' answer of [this question](https://mathoverflow.net/questions/47702/why-the-w-in-cgwh-compactly-generated-weakly-hausdorff-spaces).
The following correction can be found in the errata to [The Geometry of Iterated Loop space](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.17... | https://mathoverflow.net/users/32022 | Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"? | I'm not quite certain what Peter May had in mind 40 years ago,
but probably he had in mind the fact that pushouts are a lot better
behaved in CGWH than in CGH. Specifically, CGWH is closed
under pushouts, one leg of which is the inclusion of a closed
subspace. CGH does not have such nice behavior, and pushouts
like tha... | 81 | https://mathoverflow.net/users/14447 | 204221 | 98,290 |
https://mathoverflow.net/questions/204185 | 2 | Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. I would like to know when this induces a homotopy equivalence
$$B(\text{colim}\, D)\stackrel{\sim}{\rightarrow}\text{ho... | https://mathoverflow.net/users/27870 | Classifying space of a colimit of topological categories | Espen, I would disagree with your description of the classifying
space functor. Your question starts with a diagram in Cat(Top). The standard classifying space functor is the composite of the nerve
functor $N$ from there to simplicial spaces and geometric realization. Here $N$ is defined in what should be an obvious w... | 5 | https://mathoverflow.net/users/14447 | 204225 | 98,293 |
https://mathoverflow.net/questions/92951 | 5 | Let $k$ be a valued field.
Is there a special term for a commutative (Banach) $k$-algebra $A$ such that for any maximal ideal $m$ we have $A/m=k$?
Is there an easy to check criterion that would imply this property?
| https://mathoverflow.net/users/6027 | When quotient of a $k$-algebra by any maximal ideal is $k$? | I don't think that there is a standard term for what you are looking for, but I would be inclined to call such an algebra "compact". My reason is the following. Suppose that $X$ is a manifold (resp. locally compact Hausdorff space). Then the maximal spectrum of the ring $C^\infty(X)$ of smooth real-valued functions (re... | 3 | https://mathoverflow.net/users/78 | 204229 | 98,295 |
https://mathoverflow.net/questions/204204 | 5 | It seems to me that there are (at least) two notions of exact sequences in a category:
1) Let $\mathcal{C}$ be a pointed category with kernels and images. Then we call a complex (i.e. the composite of two adjacent maps is zero) $$\dotsc \to A\_0 \to A\_1 \to A\_2 \to \dotsc$$ *exact* if the canonical map $\mathrm{im}... | https://mathoverflow.net/users/2841 | Exact sequences of pointed sets - two definitions | The basic reason for the appearance of 1) in the long exact sequence in homotopy is that it is exactly the kind of exactness you get if you apply $\pi\_0$ to a fiber sequence $F \to E \to B$ of pointed spaces. You simply don't get 2) because $F$ cannot see what happens outside of the connected component of the basepoin... | 3 | https://mathoverflow.net/users/290 | 204234 | 98,297 |
https://mathoverflow.net/questions/204240 | 3 | A hyperconnected space is a topological space such that every two non-empty open sets have non-empty intersection. Let's call a space $(X,\cal{T})$ maximally hyperconnected if it is hyperconnected and for every topology $\cal{T'}$ with $\cal{T'} \supseteq \cal{T}$ and $\cal{T}\neq\cal{T'}$ the space $(X,\cal{T'})$ is n... | https://mathoverflow.net/users/nan | Extending hyperconnected spaces | There is a positive answer involving ultrafilters. Let $(X,\mathcal{T})$ be hyperconnected. Then note that $\mathcal{F}:=\{V\subseteq X: V\supseteq U \text{ for some non-empty } U\in\mathcal{T}\}$ is a filter. So by Zorn's Lemma, $\cal{F}$ is contained in an ultrafilter $\cal{U}$.
**Claim 1**: $(X,(\mathcal{U}\cup\{\... | 2 | https://mathoverflow.net/users/8628 | 204242 | 98,302 |
https://mathoverflow.net/questions/204251 | 0 | Let $\mathbf{r}:(a,b)\times (0,1)\to\mathbb{R}^2\subseteq\mathbb{R}^3$ be a **injective** application, given by:
$$\mathbf{r}(u,v)=A(u)+v\cdot (B(u)-A(u)), \forall\ (u,v)\in (a,b)\times (0,1)$$
where $A,B:(a,b)\to\mathbb{R}^2$ are two functions of class $C^1((a,b))$.
Is it true that $\dfrac{\partial\mathbf{r}}{\p... | https://mathoverflow.net/users/61629 | Area of a plane surface that gives a lot of theoretical problems | You don't need the map $r$ to be differentiable and the jacobian to be non-zero to compute the area of the image r((a,b)Γ(0,1)). It suffices that $r$ is injective Lipschitz. This guarantees that the Jacobian is defined almost everywhere and the usual formula holds. So even if r((a,b)Γ(0,1)) is not regular, you are ok. ... | 3 | https://mathoverflow.net/users/6129 | 204252 | 98,304 |
https://mathoverflow.net/questions/204250 | 1 | Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function.
For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U\_y(b,a)$ be the time evolution operator (solution operator) to the SchrΓΆdinger equation $i\psi'(x)=H(x,y)\psi(x)$.
More precisely, the operator is def... | https://mathoverflow.net/users/55893 | Derivative of a time evolution operator w.r.t. a parameter | Come to think of it, though, since you are in the ODE world, isn't what you wrote a direct consequence of variation of parameters?
I write it in a slightly easier (for me) notations.
**Proposition**: Let $S(t\_1,t\_2)$ be the solution operator for $\phi'(t) = A(t) \phi(t)$ with initial data prescribed at $t\_2$, a... | 2 | https://mathoverflow.net/users/3948 | 204266 | 98,310 |
https://mathoverflow.net/questions/204255 | 3 | Let $k$ be a field and $(A,m)$ be the completion of the local ring of a smooth point of a $k$-variety. Let $x\_1,x\_2\in m\backslash m^2$ be regular elements. I am interested in knowing if one can find a $k$-linear automorphism of $A$ which takes the ideal $(x\_1)$ to $(x\_2)$.
If $k$ is perfect, it is easy to see t... | https://mathoverflow.net/users/45347 | Automorphisms of complete local rings | Here is a counterexample. Fix $a\in k\smallsetminus k^p$ and take the completion of the affine plane at the point $(0,a^{1/p})$. In other words, $A$ is the completion of the local ring $k[u,v]\_{(u,v^p-a)}$.
Now take $x\_1=u$ and $x\_2=v^p-a$. Then $A/(x\_1)$ is the completion of $k[v]$ at $(v^p-a)$, while $A/(x\_2)\... | 4 | https://mathoverflow.net/users/7666 | 204273 | 98,313 |
https://mathoverflow.net/questions/204188 | 27 | Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this does not look providing stable regular answer.
(update)
Many thanks for comments.
My motivation for the question is ... | https://mathoverflow.net/users/2702 | Combinatorics of K(Z,2)? | Here is my model <https://arxiv.org/abs/1908.04029> called in the paper $\pmb SC$. The homotopy issue only mentioned and postponed to more general later writings but it is exactly what was mentioned by @AndrΓ©Henriques β factor of symmetric cross-simplicial group $\pmb S$ (which is contractible) by free right acton of C... | 2 | https://mathoverflow.net/users/2702 | 204277 | 98,314 |
https://mathoverflow.net/questions/204268 | 4 | Let $A$ be a C\*-algebra such that $A \otimes\_{\min} A$ is nuclear.
Does it follow that $A$ is nuclear?
| https://mathoverflow.net/users/22052 | Does nuclearity pass to un-tensoring? | I have in mind the following argument, which I am a little suspicious of since it seems dangerously simple, but here goes: since $A\otimes\_{min}A$ is nuclear, there is a net of cp maps $\pi\_\lambda:A\otimes\_{min}A\to A\otimes\_{min}A$ factoring through matrix algebras $M\_{n\_\lambda}(\mathbb C)$ and converging poin... | 5 | https://mathoverflow.net/users/13360 | 204280 | 98,315 |
https://mathoverflow.net/questions/204278 | 7 | I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex associated and, more importantly, the differential graded Lie algebra structure that apparently appears in this problem.
... | https://mathoverflow.net/users/66688 | References for the moduli space of complex structures | Concerning the deformation theory of complex manifolds, there are of course the seminal papers of Kodaira-Spencer. There are also some more recent notes of Manetti, *Lectures on deformations of complex manifolds*, which are available on arxiv and could be of interest for you.
The general principle relating deformatio... | 8 | https://mathoverflow.net/users/36625 | 204288 | 98,320 |
https://mathoverflow.net/questions/204126 | 2 | thank you for spending time on the following question.
In [1] Khovanov and Rozansky categorifized $sl\_n$ version of HOMFLY Polynomial, in page 11, they mention that what they defined in [1] is equivalence to what Khovanov "original" defined in [2] and give a short argument:
(I) For a closed link diagram $\Gamma$ w... | https://mathoverflow.net/users/25054 | A basic question of Khovanov-Rozansky Homology | <http://www.worldscientific.com/doi/abs/10.1142/S0218216514500576>
In the following paper, the author give a explicit isomorphism.
Mark C. Hughes, A note on KhovanovβRozansky sl2-homology and ordinary Khovanov homology, J. Knot Theory Ramifications, 23, 1450057 (2014)
| 2 | https://mathoverflow.net/users/25054 | 204296 | 98,325 |
https://mathoverflow.net/questions/204292 | 3 | Let $\rho: \pi\_1(S,s\_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V\_{s\_0} = V$.
Let $H$ be the Zariski closure of the image of $\rho$ in $GL(V)$. We say that $\rho$ is big if the identity component $H^0$ of $H$ acts irreducibly on $V$.... | https://mathoverflow.net/users/71143 | What are the easiest examples of irreducible, but not big, monodromy representations | I have seen "big monodromy" used before, in some papers of Katz I think, with a somewhat different meaning (basically that $H^0$ should as big as possible). But I'll use your definition, since that's what you seem to be interested in. One can certainly construct many non big irreducible representations as follows: Star... | 3 | https://mathoverflow.net/users/4144 | 204299 | 98,326 |
https://mathoverflow.net/questions/204274 | 13 | I have heard that one can do algebraic geometry internal to symmetric monoidal categories. Topological quantum field theories also exist internal to symmetric monoidal categories, and the usual definition is recovered by inserting the category of vector spaces.
I wonder whether this can, or has, been done for noncomm... | https://mathoverflow.net/users/13767 | Is there something like "Noncommutative geometry internal to a category"? | I'll leave open how this might connect to "internal noncommutative geometry", but you can say something about internal $\*$-algebras.
You can formulate $\*$-algebras, anti-linear involution and all, internal to [*dagger* monoidal categories](https://en.wikipedia.org/wiki/Dagger_symmetric_monoidal_category), such as t... | 6 | https://mathoverflow.net/users/10368 | 204312 | 98,331 |
https://mathoverflow.net/questions/204325 | 2 | I am relatively new to complexity and computability theory. I just came across the concept of [Permanent](http://en.wikipedia.org/wiki/Permanent) of a matrix and read that it is NP hard problem to compute the permanent of 0-1 matrix.
Of course it struck me with surprise as it would have to anyone new that it is NP har... | https://mathoverflow.net/users/71158 | NP Hardness proof for permanent of 0-1 matrix | [Les Valiant's original paper](http://www.math.washington.edu/~billey/colombia/references/valiant.permanent.1979pdf.pdf) is beautifully written.
**EDIT** A simpler proof, with a nice explanation (see Section 3) is given by [Ben-Dor and Halevy](http://people.csail.mit.edu/shaih/pubs/01perm.pdf)
| 7 | https://mathoverflow.net/users/11142 | 204326 | 98,334 |
https://mathoverflow.net/questions/203863 | 14 | I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x\_j, w^j \} \_{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it, and why should it work?
*To those who need some background:*
The goal is to approximate an integral by a discret... | https://mathoverflow.net/users/42864 | Computing Gauss Legendre quadrature for large $N$ | There are asymptotic methods that essentially give you $N$ nodes and weights in $O(N)$ time if the precision is assumed to be fixed (e.g. at double precision).
See Nicholas Hale and Alex Townsend, "Fast and Accurate Computation of Gauss-Legendre and Gauss-Jacobi Quadrature Nodes and Weights", SIAM J. Sci. Comput., 35... | 6 | https://mathoverflow.net/users/4854 | 204350 | 98,339 |
https://mathoverflow.net/questions/204378 | 5 | Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space and $M$ is a finite dim real smooth manifold?
| https://mathoverflow.net/users/41654 | constant rank theorem for banach spaces | Yes, there is. The (Constant) Rank Theorem for Banach spaces is Theorem 2.5.15 of the book of R. Abraham, J.E. Marsden and T. Ratiu, *Manifolds, Tensor Analysis and Applications* (3rd. edition, Springer-Verlag, 2001). There is a demand that the image of $DF[u\_0]$ and the kernel of $DF[u\_0]$ are closed direct summands... | 6 | https://mathoverflow.net/users/11211 | 204381 | 98,352 |
https://mathoverflow.net/questions/203769 | 0 | Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space.
Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action defined in the usual way).
A common goal in working with such things is to find canonical decompositions for classes of t... | https://mathoverflow.net/users/14835 | Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds? | No. (See Ian Agol's comment.)
| 0 | https://mathoverflow.net/users/14835 | 204386 | 98,354 |
https://mathoverflow.net/questions/204387 | 0 | Consider the Hilbert space $H = L^2(\mathbb{R})$, and a bounded operator $A \in B(H)$ which satisfies:
$$
\forall f \in H, \quad Af \text{ is trace class and } Tr(Af) < C \| f \|\_{H},
$$
where $f$ is seen as the multiplicative operator by the function $f$. Can we deduce
$$
\forall f \in H, \quad fA \text{ is trace cla... | https://mathoverflow.net/users/71190 | On the equality Tr(Af) = Tr(fA) | The answer to your original question is NO. Here is an counterexample:
For simplicity, assume everything is real-valued and the Hilbert space is over $\mathbb{R}$, let $$ \varphi \in L^2(\mathbb{R}) \setminus L^\infty(\mathbb{R}) \text{ and } \varphi\chi\_{[0,1]}\in L^\infty([0,1]);$$
$$\psi \in L^2([0,1]) \cap L^\i... | 3 | https://mathoverflow.net/users/17506 | 204394 | 98,356 |
https://mathoverflow.net/questions/204393 | 2 | Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset of $X$.
Let $\mathcal{L}$ be a line bundle on $U$. Could we always extend $\mathcal{L}$ to a line bundle on $X$, i.e. ... | https://mathoverflow.net/users/24965 | Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve? | The answer is 'yes'. One way to argue this is to first find a Cartier divisor $D$ on $U$ whose associated line bundle is $\mathcal{L}$ (the existence of such a divisor is ensured, for instance, by [EGA IV$\_4$, 21.3.4 a)]), extend $D$ to a Cartier divisor $\widetilde{D}$ on the whole $X$ (e.g., by applying [EGA IV$\_4$... | 5 | https://mathoverflow.net/users/5498 | 204395 | 98,357 |
https://mathoverflow.net/questions/204334 | 7 | Typical level sets of smooth real-valued functions are manifolds, so they cannot be fractals. If we coarse grain a bit though, sometimes we get space-filling behavior, eg. every point could be within some epsilon of the level set. Obviously one can find such a level set for any epsilon in any compact region, but for sm... | https://mathoverflow.net/users/71164 | how wiggly is a generic level set? | You can have any epsilon and any $\Lambda$: there is no estimate you ask.
Say in dimension $1$, take a polynomial $p(x)$ whose zeros are $n\epsilon,\; |n|<1/\epsilon$. They are within epsilon of any point of the unit ball. Now
multiply it on any entire function $g$ of exponential type $\delta$, which decreases on the r... | 4 | https://mathoverflow.net/users/25510 | 204397 | 98,358 |
https://mathoverflow.net/questions/204335 | 5 | In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}\_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then in all generic extension, one continues to have $A = p[U]$, where $A$ as defined by the $\mathbf{\Pi}\_2^1$.
Is this... | https://mathoverflow.net/users/43354 | Universally Baire Tree Representation of Projective Sets | If we assume the existence of large cardinals (as in the latest version of the question) then such trees for $\Pi^1\_3$ formulas exist. First let's get a local version, meaning a tree for a given $\Pi^1\_3$ formula that works for all posets below a certain cardinality. At the end I will mention how to get trees that wo... | 5 | https://mathoverflow.net/users/1682 | 204398 | 98,359 |
https://mathoverflow.net/questions/204358 | 11 | Suppose $X$ is a complete algebraic variety of dimension $n$. Must there exist an affine covering with $n+1$ pieces?
(For a projective variety in $\mathbf{P}^m$, we can always project it to some subspace $\mathbf{P}^n$ with $n$ equals the dimension of X, by a composition of projection from points. Since projection i... | https://mathoverflow.net/users/nan | Can every algebraic variety of dimension $n$ be covered by $n+1$ affine opens? | No. Example 4.9 of [Roth and Vakil](http://arxiv.org/abs/math/0406384) shows that, for any $m$, there is a singular, integral complete $3$-fold which cannot be covered by $m$ open affines. The authors mention as an open problem whether there is a smooth example. If you don't require varieties to be integral, Example 4.... | 14 | https://mathoverflow.net/users/297 | 204410 | 98,363 |
https://mathoverflow.net/questions/204368 | 6 | Suppose that $S$ is smooth and that $U\subset S$ is a dense open subscheme. Let $X$ be a scheme (not necessarily smooth) and let $f:X\to U$ be a finite flat morphism. I would like to know whether this finite flat family can be extended over $S$, and whether such an extension is unique. More precisely:
1. Does there a... | https://mathoverflow.net/users/10273 | Existence and uniqueness of extensions of a finite flat map | $\def\cO{\mathcal{O}}\def\cE{\mathcal{E}}$I will show that the answer to (1) is yes if and only if the vector bundle $f\_{\ast} \cO\_X$ on $U$ extends to a vector bundle on $S$. Moreover, if $S \setminus U$ is codimension $\geq 2$ in $S$, I will show that the extension is unique as well.
This condition is obviously n... | 4 | https://mathoverflow.net/users/297 | 204413 | 98,365 |
https://mathoverflow.net/questions/204414 | 7 | Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred from the projective model structure on chain complexes, it follows formally that the projective model structure on $cdga$ i... | https://mathoverflow.net/users/4528 | Direct proof that the model category of cdgas is left proper | The model structure on this category has a set of generating cofibrations: if we write $F$ for the free cdga functor, $k$ for the complex with value $k$ concentrated in degree zero, and $I$ for the mapping cone of the identity $k \to k$, then the maps $F(k[n]) \to F(I[n])$ form a set of generating cofibrations. Weak eq... | 12 | https://mathoverflow.net/users/360 | 204420 | 98,366 |
https://mathoverflow.net/questions/204419 | 4 | Does anyone out there know if Seymour's second neighborhood conjecture is still open? if not, I would appreciate any references.
| https://mathoverflow.net/users/23850 | Seymour's second neighborhood conjecture | As far as I know, it is still open. Here are some of related results (Probably you know all of them):
* Chen-Shen-Yuster proved that for any digraph $D$, there exists a vertex $v$ such that $|N^{++}(v)|\geq\gamma|N^+(v)|$, where $\gamma=0.67815.$. See "Second neighborhood via first neighborhood in digraphs", Ann. Com... | 6 | https://mathoverflow.net/users/30375 | 204427 | 98,367 |
https://mathoverflow.net/questions/204418 | 3 | If $X$ is aspherical, we know that $H^\*(X,k) = \text{Ext}\_R(k,k)$, with $R = k\pi\_1$. For non-aspherical spaces, do we ever have $H^\*(X,k) = \text{Ext}\_R(k,k)$ for some ring $R$? Obviously we need this algebra to be graded-commutative, so perhaps we want $R$ to be a Hopf algebra.
Or, instead of a ring $R$, maybe... | https://mathoverflow.net/users/5279 | When is the cohomology of a space Ext(k,k)? | Yes, for any connected X we can find a $\pi$ such that $H^\*(X;k) \cong Ext\_{{k}\pi}(k,k)$
This is the Kan-Thurston theorem:
Kan, D. M.; Thurston, W. P.
Every connected space has the homology of a K(Ο,1).
Topology 15 (1976), no. 3, 253β258.
combined with the universal coefficient theorem and the isomorphism $H... | 12 | https://mathoverflow.net/users/6574 | 204429 | 98,368 |
https://mathoverflow.net/questions/203220 | 13 | Here is a cohomology theory for a Hopf algebra, which I am sure has appeared elsewhere. I met it in the van Est spectral sequence for Hopf algebras. Apologies for my being stupid here, but it would be really helpful if someone would tell me where it comes from, and where to look for results on it!
Let $H$ be a Hopf a... | https://mathoverflow.net/users/29625 | Identifying a Hopf algebra cohomology theory | Looks to me as if you have not used the product of your Hopf algebra, and it looks to me that you have written an example of a cotorsion product, as defined by Eilenberg and Moore in their paper Homology and fibrations I. Coalgebras, cotensor product and its derived functors. Comm. Math. Hel. 40(1965), 199--236, availa... | 8 | https://mathoverflow.net/users/14447 | 204440 | 98,371 |
https://mathoverflow.net/questions/203186 | 5 | I have been led to believe that there is a result giving a description of the quotient of a Bruhat-Tits building $\Delta(G,k)$, for a semisimple algebraic group $G$ over a non-archimedean local field of positive characteristic $k$, by a non-uniform arithmetic lattice $\Gamma$. I believe it says that such a quotient is ... | https://mathoverflow.net/users/15482 | Arithmetic quotients of Bruhat-Tits buildings for groups over local fields of positive characteristic | I guess it is more convenient to collect some of the relevant literature references in an answer. The disclaimer is that I do not know of any results for general non-uniform arithmetic lattices $\Gamma\leq G(k)$; the results I know of always concern lattices of the form $G(\mathbb{F}\_q[C])$ where $C$ is a smooth affin... | 4 | https://mathoverflow.net/users/50846 | 204448 | 98,374 |
https://mathoverflow.net/questions/204415 | 7 | If $T$ is a nonlinear surjective isometry from Lipschitz-free space $\mathcal{F}(M)$ to $\mathcal{F}(N)$($M,N$ are metric spaces), is $M$ homeomorphic to $N$?
| https://mathoverflow.net/users/41619 | Banach-Stone Theorem in Lipschitz-free spaces | No, this is false --- trivially, because if $M$ is the completion of $N$ then $M$ and $N$ have the same Arens-Eells space. But if you require $M$ and $N$ to both be complete there is a more interesting counterexample.
Let $M$ be three copies of the interval $[0,1]$, joined at the $0$'s, with path metric. (That is, a ... | 9 | https://mathoverflow.net/users/23141 | 204453 | 98,376 |
https://mathoverflow.net/questions/201416 | 14 | Dimension refers to the Krull dimension of a commutative ring.
In the paper "Prime ideals in power series rings" J. Arnold gives an example of such a ring:
Let $k$ be a field and $K=k(t)$ a simple transcendental extension of $k$. Suppose that $V=K+M$ is a discrete valuation ring with maximal ideal $M$. Let $D=k+M... | https://mathoverflow.net/users/69591 | Example of a ring $R$ such that $\dim(R[[X]])<\dim(R[X])$ | The ring $D$ given in the question indeed has $\dim(D[[X]])$=2. This is proved in detail in the paper "Power series rings over Pruefer domains" by J. Arnold.
| 3 | https://mathoverflow.net/users/69591 | 204473 | 98,380 |
https://mathoverflow.net/questions/191742 | 6 | Consider a non-trivial elementary embedding $j:V\_\lambda\to V\_\lambda$ and, for each $A\subset V\_\lambda$, set $j(A)=\bigcup\_{\delta<\lambda}j(A\cap V\_\delta)$.
In *Implications between strong large cardinals,* Annals of Pure and Applied Logic 90 (1997) 79-90, Laver says that $j:(V\_\lambda, \in,A)\to (V\_\lambd... | https://mathoverflow.net/users/41274 | A question on rank-to-rank embeddings | Here is the author of the notes you quoted. Thank you very much for pointing this out, there is indeed a gap in the proof: I truly took for granted that if $f$ is total then $j^+(f)$ is total (more specifically, that if $f$ is a Skolem function, then $j^+(f)$ is a Skolem function, but once one has totality, the rest is... | 7 | https://mathoverflow.net/users/69827 | 204476 | 98,381 |
https://mathoverflow.net/questions/204491 | 2 | Im confuse..I read in an article that in dealing with polynomials, a quadratic equation can have either 2 real roots, 1 equal real root or 2 complex roots...but in dealing with random polynomials only two cases are possible either 2 real roots or 2 complex roots...why is that so? the article also said that using the de... | https://mathoverflow.net/users/71234 | existence of multiplicity of roots | Choose $b$ and $c$ randomly. The value $b^2/(4c)$ is now determined. Note that the probability that $b=0$ is zero, since almost all real numbers are irrational. The measure of rationals in reals is zero!
In any case, this means that the probability that a random $a$ will equal $b^2/(4c)$ is also zero.
In general, i... | 0 | https://mathoverflow.net/users/17773 | 204492 | 98,388 |
https://mathoverflow.net/questions/201637 | 1 | Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= \frac{E(S,\bar{S})}{d \vert S \vert}$. Let $f$ be the characteristic vector of the set $S$. Let $V\_{\geq \lambda}$ and $V\_{< \lambda... | https://mathoverflow.net/users/36554 | About expectation norms on graphs | The decomposition should be taken with respect to the eigenvalues of $A/d$.
---
Discussion of the first inequality. There seems to be a slight mistake here: $V\_{\ge\lambda}$ should be replaced with the orthogonal projection $P\_{\ge \lambda}(G)$ onto this subspace. (Note that in the cited paper, both the stateme... | 1 | https://mathoverflow.net/users/68305 | 204494 | 98,389 |
https://mathoverflow.net/questions/204487 | 2 | Let $X=M(v,w)$ be a Nakajima quiver variety for a quiver $Q$. Can one calculate the second singular cohomology groups $H^2(X,\mathbb Z)$ or $H^2(X,\mathbb C)$ explicitly, and if not, are there some particular cohomology classes in these groups which one can identify, or a class of quivers where one can calculate these ... | https://mathoverflow.net/users/12395 | Second cohomology groups of Nakajima quiver varieties | There's a very natural set of cohomology classes in $H^2(X,\mathbb{Z})$; for each node $i$ in your quiver, there's a tautological bundle $\mathcal{V}\_i$. The Chern classes $c\_1(\mathcal{V}\_i)$ give natural elements of $H^2(X,\mathbb{Z})$. These are usually linearly independent, but not always (for example, sometimes... | 5 | https://mathoverflow.net/users/66 | 204506 | 98,391 |
https://mathoverflow.net/questions/204472 | 9 | We are given a collection of sets $A\_1,\ldots,A\_s$, pairwise different and each of cardinality $k$, and a collection of sets $B\_1,\ldots,B\_s$, pairwise different and each of cardinality $l>k+1$, such that $A\_i\subseteq B\_i$ for all $i=1,\ldots,s$. Can we find elements $a\_i$ from $B\_i\setminus A\_i$ for all $i=1... | https://mathoverflow.net/users/31441 | Extending subsets to supersets in different ways | The answer is no. Here is a list of sets $A\_i$ and $B\_i$ which fails.
12, 1234
23, 1235
13, 1236
14, 1245
25, 2356
36, 1346
45, 1456
56, 2456
46, 3456
The failure can be seen by drawing the picture which consists of vertices for each $A\_i$ and for each $A\_i \cup j$ with $j\in B\_i\se... | 4 | https://mathoverflow.net/users/468 | 204512 | 98,394 |
https://mathoverflow.net/questions/204495 | 4 | The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) :
Let $(X,\mathcal{U})$ be a uniform space such that for each member $U$ of some subbase for $\mathcal{U}$ there is a finite cover $A\_{1}$, $A\_{2}$,...,$A\_{n... | https://mathoverflow.net/users/50259 | The subbase theorem for total boundedness | It seems the following.
The answer is positive. Let $\mathcal S$ be a subbase satisfying the condition and $U\in\mathcal U$ be an arbitrary entourage. Then there exists a finite subfamily $\mathcal V=\{V\_1,V\_2\dots, V\_n\}$ of the family $\mathcal S$ such that $\bigcap\mathcal V\subset U$. For each member $V\_i$ o... | 2 | https://mathoverflow.net/users/43954 | 204518 | 98,395 |
https://mathoverflow.net/questions/204516 | 10 | I [asked this in stackexchange](https://math.stackexchange.com/questions/1253347/is-there-a-diffeomorphism-with-only-finite-orbits-but-of-infinite-order), but got no answer, so I am trying here.
Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following properties:
1. **All** its o... | https://mathoverflow.net/users/46290 | is there a diffeomorphism with only finite orbits but of infinite order? | No, there is no such animal. This is due to Montgomery (1938), see my answer to this question: [Nonperiodic points of homeomorphisms of a ball](https://mathoverflow.net/questions/191133/nonperiodic-points-of-homeomorphisms-of-a-ball/191143#191143)
| 11 | https://mathoverflow.net/users/11142 | 204524 | 98,396 |
https://mathoverflow.net/questions/202385 | 3 | I'm reading one of the classical theorems presented in Bowen's lecture notes, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms." I'm trying to figure out a very short line of reasoning that's eluding me in the proof of the following:
>
> Let $f$ be a transitive $C^2$ Anosov diffeomorphism. If $... | https://mathoverflow.net/users/70317 | Differential form equation in Bowen's lecture notes | There is indeed a typo. The correct normalisation should be $h= e^{-u}$ and thus $Jac(f)=h/h\circ f$. Then we get for all $g$
$$
\int g\ h\ dm = \int g\circ f\ h \circ f\ Jac(f)\ dm = \int g\circ f\ h\ dm
$$
See the book of Katok-Hasselblatt, introduction to the modern theory of dynamical systems, theorem 19.2.7 (s... | 1 | https://mathoverflow.net/users/6129 | 204535 | 98,402 |
https://mathoverflow.net/questions/204539 | 4 | Let n be a positive integer not less than 2. Does anyone know of a theorem stating that- for each n- there exists a simple closed curve c(n), which (1) is a subset of n-dimensional Euclidean space E(n) and (2) does not contain n+1 pairwise distinct points all belonging to the same (n-1)-dimensional Hyperplane of E(n)? ... | https://mathoverflow.net/users/4423 | A question about simple closed curves in finite dimensional Euclidean spaces | Extending my comment, I can give an almost complete answer. If $n$ is even, an example is an appropriate normal rational curve, e.g., $t\mapsto(t/p,t^2/p,\ldots,t^n/p)$, where $p(t)$ is any real polynomial of degree $n$ and without real roots. I conjecture that, for $n$ odd, such a curve doesn't exist. Here is a proof ... | 7 | https://mathoverflow.net/users/44953 | 204547 | 98,404 |
https://mathoverflow.net/questions/204526 | 1 | Given a number field $K$, the Dirichlet Unit Theorem tells us about the structure of the unit group $O\_K^\times$. However, the proofs do not seems to give any way to explicitly write out a set of generators.
I wonder if there are special cases in which we could do that. Here are a few that comes up to me: (will edi... | https://mathoverflow.net/users/32631 | When can we write fundamental units explicitly | For cyclotomic fields, there are the cyclotomic units, which are formed in a very simple manner from ratios of differences of roots of unity (or for non-prime power roots of unity, just differences). An important theorem is that the set of cyclotomic units generate a subgroup $\mathcal C$ of finite index in the full gr... | 4 | https://mathoverflow.net/users/11926 | 204551 | 98,407 |
https://mathoverflow.net/questions/204489 | 13 | For a final project in my class, I decided to try to simulate a quantum computer and implement Grover's algorithm. I followed [this excellently written blog post](http://twistedoakstudios.com/blog/Post2644_grovers-quantum-search-algorithm) by Craig Gidney, and was successful in getting Grover's algorithm to work using ... | https://mathoverflow.net/users/71200 | Constructing the oracle for Grover's algorithm | The simplest way to flip the phase of a single amplitude is to use a Z gate with controls on every wire.
```
βββ’ββ
β
βββ’ββ
β
βββ’ββ
β
βββ’ββ
β
βββ’ββ
β
ββZββ
```
The above gate flips the phase of the all-on |111111> state.
Note that it doesn't matter which wire gets the Z gate instead of a control. Also n... | 8 | https://mathoverflow.net/users/40272 | 204553 | 98,408 |
https://mathoverflow.net/questions/204507 | 1 | Let $$U \mapsto U \otimes U^\* \otimes U \otimes U^\*$$ be a unitary representation of the unitary group $U(n)$ acting on the vector space $V$ (where $U^\*$ is the complex conjugate of $U$). We can decompose $V$ into invariant subspaces in which the representation is irreducible. Let $P\_i$'s be projectors onto invaria... | https://mathoverflow.net/users/66471 | Projectors onto the invariant subspaces of a unitary representation $U \otimes U^* \otimes U \otimes U^*$ | One way to do this is to solve the Clebsch-Gordan problem. [This](http://arxiv.org/abs/1009.0437) paper discusses an algorithm for this on the unitary group.
A similar question was asked earlier (see [here](https://mathoverflow.net/questions/198022/decomposing-a-reducible-representation-of-the-unitary-group/198222#19... | 1 | https://mathoverflow.net/users/38947 | 204556 | 98,410 |
https://mathoverflow.net/questions/204546 | 3 | I've read about free cocompletion of categories discussing on the adjunction between Cat and cocompleteCat (Cat: category of small categories, cocompleteCat: category of small cocomplete categories and cocontinuous functors) where the adjunction is about a free cocompletion functor F which sends each category C to PSh(... | https://mathoverflow.net/users/64534 | a (pseudo)adjunction for the functor sending a category C to PSh(C) the category of presheaves | The free cocompletion functor $C \mapsto \widehat{C}$ (which, as Zhen Lin says, does not agree with the presheaf functor when $C$ is not essentially small) should in no reasonable sense have a left adjoint, since it is very far from preserving limits.
It already fails to preserve products: if $C, D$ are two small ca... | 6 | https://mathoverflow.net/users/290 | 204558 | 98,412 |
https://mathoverflow.net/questions/204462 | 1 | Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?
| https://mathoverflow.net/users/56353 | symplectic reduction for pair $(M,D)$ | Let me illustrate this using the simplest example $(\mathbb{C}^2,D)$, where $D$ is the conic defined by $xy+1=0$ with $x,y$ the coordinates on $\mathbb{C}^2$. Consider the Hamiltonian $S^1$-action on $\mathbb{C}^2$ defined by $e^{i\theta}(x,y)=(e^{i\theta}x,e^{-i\theta}y)$, then the moment map is $\mu=\frac{|x|^2-|y|^2... | 1 | https://mathoverflow.net/users/43423 | 204566 | 98,415 |
https://mathoverflow.net/questions/204536 | 0 | What is the cardinality of the set of values corresponding to the first $n$ rationals generated in Cantor's enumeration scheme for proving their countability?
Edit:
following the [suggestion](https://mathoverflow.net/questions/204536/redundancy-of-the-cantor-enumeration-of-the-rationals#comment508412_204536) of Todd,... | https://mathoverflow.net/users/31310 | Redundancy of the Cantor enumeration of the rationals | I'm not sure, but it seems to me that you are asking about an asymptotic formula for the cardinality of the set $\left\{\frac{n}{m} : n + m \leq x\right\}$, as $x \to +\infty$, where $n$ and $m$ are positive integers. If so, the answer is the following
$$\#\left\{\frac{n}{m} : n + m \leq x\right\} = \sum\_{k \leq x} ... | 5 | https://mathoverflow.net/users/nan | 204571 | 98,417 |
https://mathoverflow.net/questions/204577 | -1 | Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in C^{\infty}([a,b])$ such that:
$$\begin{cases} x(t)\sin\theta(t)=y(t)\cos\theta(t),\ \forall\ t\in [a,b].\\ (x(t\_0),y(t\_0))... | https://mathoverflow.net/users/72276 | Uniqueness of a smooth function | You need $(x(t),y(t))$ to be non-zero for all $t$. If the curve $t\mapsto (x(t),y(t))$ spends some time at the origin, you can choose $\theta$ as you wish there.
Assuming that $(x,y)$ is non-zero, your question boils down to representing the curve in polar coordinates. Writing $x=r\cos(\eta)$, $y=r\cos(\eta)$, we ge... | 1 | https://mathoverflow.net/users/6129 | 204578 | 98,420 |
https://mathoverflow.net/questions/201761 | 4 | Thank you for spending time on the following question.
I am trying to make an explicit example of Korevaar-Schoen convergence. The problem I am facing is that I cannot find the limit of the harmonic maps in the following example I hope to consider. The process I am doing mostly follows from [1].
Let $M = S^1\times ... | https://mathoverflow.net/users/25054 | Equivariant Harmonic Maps to R-tree and Korevaar-Schoen Convergence | The Korevaar-Schoen limit of the $u\_k$ function is the following map
\begin{equation}
\mathbb{R}^1\times S^2 \rightarrow \mathbb{R}^1\\
(t,x)\rightarrow (0,0,t)
\end{equation}
The essential part of [1] is that we have the pointwise bound the energy function of a harmonic map.
Lemma[3]: Let $\Omega\subseteq M$ b... | 3 | https://mathoverflow.net/users/25054 | 204593 | 98,423 |
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