parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/218197 | 6 | This question is motivated by trying to establish a converse to Theorem 7.8 of our [paper](http://arxiv.org/abs/1508.05446).
I have a finite poset $P$ with the following properties:
1. $P$ has binary meets (and hence a least element).
2. $P$ is ranked (or graded). This means every maximal chain from the least eleme... | https://mathoverflow.net/users/15934 | Is this algebra isomorphic to an incidence algebra? | Here is a much better exposition. Let $X$ be any connected compact manifold and let $\Delta$ be a simplicial complex realizing $X$. Let $P$ be the face lattice $\Delta$, with added minimal and maximal elements $\hat{0}$ and $\hat{1}$. (I suspect we can replace "simplicial complex" with "regular CW complex", but I don't... | 1 | https://mathoverflow.net/users/297 | 218631 | 102,659 |
https://mathoverflow.net/questions/218583 | 2 | By Lang's theorem, a complete valued field which is the fraction field of a discrete valuation ring with an algebraically closed residue field is quasi-algebraically closed (or $C\_1$).
How much is known about the converse?
Is there a criterion/almost exhaustive list of complete valued fields, which is quasi-algebra... | https://mathoverflow.net/users/43198 | On quasi-algebraically closed fields | If you have a dvr whose residue field is not algebraically closed, then there is a norm form in the residue field, so a form of degree $n$ in $n$ variables with no non trivial zero. Lift this form to a form $f$ over the ring. Now let $\pi$ be an element of value one and consider the form $f(x\_1,...,x\_n)+\pi x\_{n+1}^... | 6 | https://mathoverflow.net/users/2290 | 218633 | 102,661 |
https://mathoverflow.net/questions/218520 | 5 | Let $(V,\langle\,\cdot\,,\,\cdot\,\rangle)$ be an $(n+1)$-dimensional real vector space, equipped with a nondegenerate symmetric bilinear form of indefinite signature, and denote by $\nu(v):=\langle v,v\rangle$ the corresponding quadratic form.
As a homogeneous polynomial, $\nu$ cuts out the **projective** ``light c... | https://mathoverflow.net/users/22606 | Can the conformal structure on the projective light-cone detect hyperplane sections? | For your new question, the answer is at least in part **yes**. I'll outline the proof here for the case your projective hyperplane is originally not a null hyperplane.
First let us work in the ambient $\mathbb{R}^{n+1}$. Let $\Pi$ be a non-null hyperplane through the origin, this is our projective hyperplane. Let $\... | 5 | https://mathoverflow.net/users/3948 | 218636 | 102,663 |
https://mathoverflow.net/questions/218353 | 4 | By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet Energy
$$E(f)=\int\_{\mathcal{M}}\left|\nabla f\right|^2\:d\lambda\_{\mathcal{M}}.$$
A critical point of this energy func... | https://mathoverflow.net/users/41187 | Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces | I think it is a very good exercise to show that a conformal map between Riemann surfaces (equipped with metrics in the conformal classes) are harmonic.
The converse direction is more interesting, and does depend on the assumption that the domain is a compact surface of genus $0.$ In general, one can consider for map... | 5 | https://mathoverflow.net/users/4572 | 218645 | 102,666 |
https://mathoverflow.net/questions/218648 | 19 | Recently I was talking to my friend and I have mentioned to him that it was proven that CH is not provably (over ZFC) equivalent to any statement in second-order arithmetic. However, today I found out that the result I was thinking about is the one mentioned in [this](https://math.stackexchange.com/q/340891) answer, wh... | https://mathoverflow.net/users/30186 | Can we find CH in the analytical hierarchy? | The collapsing forcing by countable partial functions from $\omega\_1$ to $2^\omega$ is $\omega\_1$-closed, hence it preserves $H\_{\omega\_1}$, and a fortiori the truth of all formulas in the analytical hierarchy; it also makes CH hold. Thus CH is not equivalent to any statement in the analytical hierarchy (assuming Z... | 24 | https://mathoverflow.net/users/12705 | 218649 | 102,669 |
https://mathoverflow.net/questions/218640 | 0 | Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z\_t = \int\_0^t f(s)\,dB\_s.$$
* How do I see that $Z\_t$ is normally distributed?
* What is the mean and variance?
I need to know these results for something I am doing with analysis, but unfortunately I do not know any statistics.
... | https://mathoverflow.net/users/80430 | $\int_0^t f(s)\,dB_s$ normally distributed, mean and variance | $Z\_t$ is the $L^2$ limit of Riemann sums like
$$Z\_t^{(n)} = \sum\_{i=1}^n f(it/n) (B\_{it/n} - B\_{(i-1)t/n}).$$
This is a sum of independent normal random variables with mean 0 and variance $\frac{t}{n}|f(it/n)|^2$, so $Z\_t^{(n)}$ is normally distributed with mean 0 and variance $\frac{t}{n} \sum\_{i=1}^n |f(it/n)|... | 4 | https://mathoverflow.net/users/4832 | 218653 | 102,671 |
https://mathoverflow.net/questions/218644 | 11 | Since 2008 we have the following remarkable correspondence:
>
> Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1
> newforms
>
>
>
*note: all Galois representations in this question are ment to be continuous complex linear representations.*
That this is a bijection is the consequence of three... | https://mathoverflow.net/users/43108 | Converse to Modularity I: weight 2 newforms | This should be a comment, but is getting too long.
First, unless you are using the strange convention that Galois representations have by definition complex coefficients, odd, irreducible 2-dimensional Galois representations do not correspond to weight 1 newforms. Only a very small subset of the former set correspond... | 11 | https://mathoverflow.net/users/2284 | 218656 | 102,673 |
https://mathoverflow.net/questions/218655 | 11 | Given a (compact if needed) real smooth surface $V(f)$ defined by $f\in \mathbb{R}[X,Y,Z]$, in particular it is oriented. Is there a formula which gives the Euler character of $V(f)$ ?
Thanks.
| https://mathoverflow.net/users/64541 | Euler Characteristic of Real Algebraic Surfaces | A “formula” is a lot to ask for, but there are *algorithms* based on Morse theory.
E.g. §5 of [Basu (1999)](http://www.ams.org/mathscinet-getitem?mr=1692627), or §3 of [Fortuna-Gianni-Luminati (2004)](http://www.ams.org/mathscinet-getitem?mr=2169368).
| 7 | https://mathoverflow.net/users/19276 | 218657 | 102,674 |
https://mathoverflow.net/questions/218674 | 5 | This conjecture is a generalization of [A theorem for cubic-A generalization of Carnot theorem, in MSE question](https://math.stackexchange.com/questions/1440544/a-theorem-for-cubic-a-generalization-of-carnot-theorem). I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result.
Let $A... | https://mathoverflow.net/users/76698 | A conjecture associated with n-gon cut curve of degree m | I believe you meant to type "curve of degree m". Let's prove the "only if" direction first. If you can prove it for triangles then you can triangulate your $n$-gon and multiply together the expressions for each triangle. So let's assume $n=3$.
Now, pick coordinates $(t\_0,t\_1,t\_2)$ on $\mathbb P^2$ so that the line... | 6 | https://mathoverflow.net/users/2384 | 218675 | 102,678 |
https://mathoverflow.net/questions/218505 | 5 | "Perhaps the most important parts of the Ornstein theory are criteria for determining whether or not a shift or flow is Bernoulli (a Bernoulli shift, $B\_{ct}$ , or $B\_{t}^{\infty}$) because it allows us to prove that certain concrete systems are Bernoulli."[Quote from scholarpedia][1] [1]: <http://www.scholarpedia.or... | https://mathoverflow.net/users/78164 | Importance of Ornstein's isomorphism theorem | You might want to start with this article by Ornstein:
An Application of Ergodic Theory to Probability Theory, Donald S. Ornstein,
The Annals of Probability, Vol. 1, No. 1 (Feb., 1973), pp. 43-58
| 1 | https://mathoverflow.net/users/80449 | 218681 | 102,679 |
https://mathoverflow.net/questions/218325 | 2 | Assume that $H\_t$ is a progressively measurable process such that with probability one $|H\_t| \le k$ for all $t$. Let$$Z\_t = \int\_0^t H\_s\,dB\_s.$$How do I see that for all $s < t$, $\lambda \in \mathbb{R}$, that$$\textbf{E}[\text{exp}\{\lambda(Z\_t - Z\_s)\}] \le \text{exp}\left\{{{k^2\lambda^2}\over2}(t - s)\rig... | https://mathoverflow.net/users/nan | Bound on expectation, not a really simple process, circumvent using Itō's lemma? | You can prove it going back to the definition of the stochastic integral. If $H$ is constant on the interval $[s,t)$ then conditionally to $\mathcal F\_s$ (the underlying filtration at time $s$), $Z\_t-Z\_s$ is a centerer gaussian with variance $H\_s^2 (t-s)$, and hence we have $$\textbf{E}[\text{exp}\{\lambda(Z\_t - Z... | 2 | https://mathoverflow.net/users/10265 | 218688 | 102,681 |
https://mathoverflow.net/questions/218676 | -1 | Let $d$ be an integer and $\sum\_{k=0}^dP\_k(z)y^{(k)}(z)=0$ be a differential equation over $\mathbb C$, where the $P\_k$ are polynomials of degree $\le d$. Consider (if it exists) an entire solution $f$ solution of this equation. Can one assert that $\limsup\_{r\to+\infty}\frac{\ln(\ln|f|\_r)}{\ln r}\le d$, with $|f|... | https://mathoverflow.net/users/33128 | growth of an entire solution of a differential equation | The counterexample is $y'+z^dy=0$, where $P\_d=\ldots=P\_2=0$. The order of solutions is $d+1$.
There is a simple method to determine the orders of entire solutions.
It is called the Newton polygon.
Plot the points with coordinates $(k,\deg P\_k-k)$, $k=0,\ldots,d$ in the plane.
Newton's polygon is the smallest concave... | 4 | https://mathoverflow.net/users/25510 | 218695 | 102,683 |
https://mathoverflow.net/questions/218696 | 8 | For $n \in \mathbb{Z}\_{\geq 0}$, let $[n]\_q := (1-q^n)/(1-q) = (1+q+...+q^{n-1})$ as is customary, with $[0]\_q=0$.
Let $R$ be the subring of $\mathbb{Q}(q)[x]$ consisting of all $f$ such that $f([n]\_q) \in \mathbb{Z}[q]$ for all $n \in \mathbb{Z}\_{\geq 0}$.
Define $f\_0(x) = 1$ and $f\_k(x) = f\_{k-1}(x)\cdot ... | https://mathoverflow.net/users/25028 | q-Integer-valued polynomials | (Below is the proof that module $R$ is generated by $f\_i(x)$, without calculation of structure constants.)
Polynomial $f(x)$ of degree $n$ may be interpolated in points $[0],[1],\dots,[n]$ (I omit index $q$): $f(x)=\sum\_{i=0}^n f([i])\prod\_{j\ne i} \frac{x-[j]}{[j]-[i]}$. For any summand leading term $f([i])\prod\... | 7 | https://mathoverflow.net/users/4312 | 218698 | 102,685 |
https://mathoverflow.net/questions/218705 | 7 | I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at <http://ncatlab.org/nlab/show/category+of+simplices>. **Is this subcategory a full subcategory?** I *think* it is, because I *think* that I can... | https://mathoverflow.net/users/25477 | For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$? | As far as I can tell you are correct. In fact, any map in the category of simplices coming out of a nondegenerate simplex must be injective. If $a:\Delta^n\to X$ is a nondegenerate simplex and $f:\Delta^n\to\Delta^m$ is a map to some other simplex $b:\Delta^m\to X$, factor $f=hg$ as a surjection followed by an injectio... | 6 | https://mathoverflow.net/users/75 | 218711 | 102,688 |
https://mathoverflow.net/questions/218703 | 2 | Let $n\geq 2$ and $\mathbb{P}^n(\mathbf{C})$ be the complexe projective space of dimension $n$. Let $H\subseteq \mathbb{P}^n(\mathbf{C})$ be a hypersurface of degree $d$ where the coordinates in $\mathbb{P}^n(\mathbf{C})$ are chosen to be $x=[x\_1,\ldots,x\_{n+1}]$. Since $H$ is a hypersurface of degree $d$, it is defi... | https://mathoverflow.net/users/11765 | Biregular maps between hypersurfaces of the same degree | Since $\phi$ is biregular, it induces an isomorphism $\mathrm{Pic}(H')\stackrel{\sim}{\rightarrow }\mathrm{Pic}(H)$, which maps $\mathcal{O}\_{H'}(1)$ to $\mathcal{O}\_H(e)$. If $e>1$ this implies that $\mathcal{O}\_{H'}(1)$ is divisible in $\mathrm{Pic}(H')$, which is impossible if your hypersurfaces have reasonable s... | 2 | https://mathoverflow.net/users/40297 | 218715 | 102,692 |
https://mathoverflow.net/questions/218692 | 2 | Let $n$ be an even positive integer and $K\_{2n}$ be the complete graph on $2n$ vertices. There are $\dfrac{1}{2}{{2n}\choose n}={{2n-1}\choose n}$ subgraphs of $K\_{2n}$ which is isomorphic to $K\_{n,n}$ and we use $E\_1,E\_2,\dots,E\_{{2n-1}\choose n}$ to denote the edge sets of the ${2n-1}\choose n$ subgraphs respec... | https://mathoverflow.net/users/58096 | The existence of a specific partition of the edge set of $K_{2n}$ | **No,** this is not possible. Towards a contradiction, suppose such a partition $S\_1, \dots, S\_{2n-1}$ of $E(K\_{2n})$ exists. I first claim that each $S\_i$ must be a matching. If not, then some vertex $v$ has degree at least 2 in say $S\_1$. But now if we sort the $S\_i$ according to the degree of $v$ in $S\_i$ and... | 4 | https://mathoverflow.net/users/2233 | 218721 | 102,694 |
https://mathoverflow.net/questions/218609 | 6 | Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with the set $\mathbb Z$ of integers?
We recall that a function $f:X\to\mathbb R$ is of the *first Baire class* if it is ... | https://mathoverflow.net/users/61536 | On continuous perturbations of functions of the first Baire class on the Cantor set | After some thinking I realized that the answer to this question is negative. A counterexample can be constructed by a standard diagonal method of killing all possible candidatures.
We shall construct a function $f:X\to\mathbb R$ of the first Baire class on the Cantor cube $X=\{0,1\}^\omega$ such that for any continuo... | 6 | https://mathoverflow.net/users/61536 | 218726 | 102,697 |
https://mathoverflow.net/questions/218730 | 7 | I asked this here <https://math.stackexchange.com/questions/1441408/orbit-under-lie-group-and-projective-variety> but did not get any answers, so I am asking here. I apologize if this is not appropriate for this site.
On <https://en.wikipedia.org/wiki/Generalized_flag_variety#Highest_weight_orbits_and_homogeneous_pr... | https://mathoverflow.net/users/80175 | Flag varieties and orbit of highest weight vector | This is part of the Borel-Weil theorem, more precisely it's Theorem 3 in Serre's 1954 [exposition](http://www.numdam.org/item?id=SB_1951-1954__2__447_0). As he notes there, the algebraicity you ask about is a consequence of [Chow's theorem](http://www.ams.org/mathscinet-getitem?mr=33093). Also, the orbit of the highest... | 9 | https://mathoverflow.net/users/19276 | 218731 | 102,699 |
https://mathoverflow.net/questions/218733 | 4 | I apologize that this question is a bit vague, however that is partially the point.
In subsystems of second order arithmetic, one considers $\omega$-models, these are models of $\mathsf{RCA}\_0$ whose first order part is $\omega = \mathbb{N}$ and whose second order part is a subset of $2^\omega$ closed under Turing r... | https://mathoverflow.net/users/12978 | Analogy of $\omega$-models in constructive mathematics | It is not clear to me exactly what you are asking, but here are some pointers. Let us stay within the realm of realizability theory.
There is a way to extend any given model by an "oracle" or by a "non-computable" object, see Section 1.7 of Jaap van Ooosten's book "Realizability: An Introduction to its Categorical Si... | 6 | https://mathoverflow.net/users/1176 | 218735 | 102,701 |
https://mathoverflow.net/questions/218439 | 2 | Let $(Y,\left\|\cdot\right\|\_Y)$ be a Banach space and $A:D(A)\subset Y \to Y$ a closed operator. Studying dynamical systems of the form
\begin{equation}
u'=Au
\end{equation}
quickly leads to the notion of one-parameter semigroups:
$\textbf{Definition}$: A one-parameter semigroup $G$ is a collection $\left\{S(t):Y\t... | https://mathoverflow.net/users/33927 | "Generalisation" of one-parameter semigroups | In fact, even a snake does not have to look beyond its nose to find a nice example (I post it here such that the original question does not get to long, discouraging potential readers).
Take $X$ as the complex-valued sequences with the usual vector-space structure and with norm
\begin{equation}
\left\|x\right\|\_X^2=|x... | 0 | https://mathoverflow.net/users/33927 | 218739 | 102,702 |
https://mathoverflow.net/questions/218027 | 10 | I am looking for the definition of a parahoric group scheme in the sense of Bruhat and Tits? I couldn't find a reference for that? at least a "clear" reference!
thanks
| https://mathoverflow.net/users/75343 | Parahoric Group Scheme | Well, I won't give a definition, but I can make some comments which may (?) be somewhat helpful.
The parahoric group schemes associated to a reductive group $G$ over a local field $F$ are certain smooth, affine group schemes $\mathcal{P}$ over the ring of integers $R$ of $F$ with the property that their generic fiber... | 7 | https://mathoverflow.net/users/4653 | 218743 | 102,704 |
https://mathoverflow.net/questions/153105 | 4 | Let
\begin{align\*}
p(z) & = \ \prod\_{j = 1}^{n} (z - z\_j) \, ; \ |z\_j| \ = \ 1 \\
& = \ \prod\_{j = 1}^{l+1} (z - z\_j)^{M\_j}
\end{align\*}
be a non-constant complex polynomial with $l+1$ distinct zeros of multiplicity $M\_j$, which all lie on the unit circle. Then there are $l$ not necessarily distinct zeros $z... | https://mathoverflow.net/users/37855 | The Poisson-kernel in the plane and polynomials | Here is an easy algebraic proof of your identity. I present it in the case of a simple zero, a general case requires some obvious modifications.
So, let $z\_1$ be a simple zero of $p$. Your equation (1) is a special case of more general equation:
$$\sum\_{l=1}^{n-1}\frac{|z\_1|^2-|z\_l'|^2}{|z\_1-z\_l'|^2}= n-1 + 2... | 2 | https://mathoverflow.net/users/29557 | 218755 | 102,708 |
https://mathoverflow.net/questions/218759 | 11 | A special case of Cheboratev's density theorem states that, for $K/\mathbb{Q}$ a Galois number field of degree $n$, then the rational primes that split completely in $K$ have density $1/n$.
Is there an elementary proof of this fact?
Actually I'm asking this only to apply it to $K=\mathbb{Q}(\zeta\_n)$ and get that ... | https://mathoverflow.net/users/43195 | Elementary proof of a special case of Chebotarev's density theorem | There does exist an elementary proof, and is given in many books on algebraic number theory. I think it is in Lang's book. I reproduce the proof below.
Consider $ \zeta \_K(s)=\prod \_{\mathfrak p}(1-\frac{1}{(N\mathfrak p)^{s}})^{-1}$ where $\mathfrak p$ runs over all prime ideals in the ring of integers in $K$. The... | 20 | https://mathoverflow.net/users/23291 | 218763 | 102,711 |
https://mathoverflow.net/questions/218765 | 4 | Does there exist a function $\tau(\varepsilon)=\tau(\varepsilon,n,K,\mu)$ such that $\lim\_{\varepsilon\to +0}\tau(\varepsilon)=0$ and for any $n$-dimensional complete Riemannian manifold $M^n$ with sectional curvature $\geq K$ and with the injectivity radius $\geq \mu>0$ the following property is satisfied: if in a ge... | https://mathoverflow.net/users/16183 | A property of geodesic triangles in manifolds with lower bounds on curvature and injectivity radius | Yes it is true and the statement follows from Toponogov's comparison.
Assume that triangle $[xyz]$ is small and $\measuredangle [y^x\_z]>\pi-\varepsilon$.
Extend the side $[zy]$ behind $y$ and marke the point $v$ on the extension such that $|y-v|=\tfrac\mu2$.
Note that $\measuredangle [y^x\_v]<\varepsilon$.
By Topono... | 3 | https://mathoverflow.net/users/1441 | 218781 | 102,718 |
https://mathoverflow.net/questions/218766 | 11 | **QUESTION. Given a Riemannian metric on the sphere $S^n$ with positive sectional survature. Can it be isometrically imbedded into $\mathbb{R}^{n+1}$ (of any class of regularity) as a boundary of a convex set?**
This question was motivated by the following three well known facts on isometric imbeddings (please correc... | https://mathoverflow.net/users/16183 | Isometric imbedding of a sphere with positively curved metric | Igor is essentially right. If $n > 2$ and the embedding is convex, then the second fundamental form exists almost everywhere. Using the Gauss equations per Robert's answer, it extends uniquely and smoothly to everywhere. The rest is straightforward. I'm omitting it because I'm typing this on an iPhone.
The $C^1$ but ... | 5 | https://mathoverflow.net/users/613 | 218782 | 102,719 |
https://mathoverflow.net/questions/218741 | 6 | A sequence $(x\_{n})\_{n}$ in a Banach space $X$ is said to be unconditionally $p$-summable if $$\sup\_{x^{\*}\in B\_{X^{\*}}}\Bigl(\sum\_{n=m}^{\infty}\lvert\langle x^{\*},x\_{n}\rangle\rvert^{p}\Bigr)^{1/p}\rightarrow 0\qquad(m\rightarrow \infty).$$ We say that an operator $T:X\rightarrow Y$ is unconditionally $p$-su... | https://mathoverflow.net/users/41619 | Is $T^{**}$ unconditionally $p$-summing whenever $T$ is unconditionally $p$-summing? | The answer is no. Bourgain and Delbaen constructed a Banach space $X$ that has the Schur property and $X^{\*\*}$ is isomorphically universal for separable Banach space (it is even isomorphic to the the second dual of $C[0,1]$). Every operator from $\ell\_p$, $1<p<\infty$, and from $c\_0$ into $X$ is thus compact, so th... | 3 | https://mathoverflow.net/users/2554 | 218790 | 102,723 |
https://mathoverflow.net/questions/196132 | 6 | Given a map of spaces $f:X\to BGL\_1(R)$ for $R$ an $E\_\infty$-ring spectrum (of course this can be done more generally) one can produce a Thom spectrum $Mf$ by a number of methods. Let's denote such a datum by $(X,f)$ and let $(X,\ast)$ denote the datum $X\to \ast\to BGL\_1(R)$, whose associated Thom spectrum is $R\w... | https://mathoverflow.net/users/11546 | Is the Thom diagonal co-$E_\infty$? | The answer to this is yes, it is co-$E\_\infty$. Let $\iota\colon BGL\_1(R)\to Mod\_R$ be the inclusion. Since colimit is left adjoint to the strong monoidal diagonal functor, it's oplax monoidal. Note that the constant functor $\kappa\_R\colon X\to Mod\_R$ is the monoidal unit for the pointwise monoidal structure in $... | 4 | https://mathoverflow.net/users/11546 | 218791 | 102,724 |
https://mathoverflow.net/questions/218488 | 4 | Let $X=\mathbb{C}^n$ be affine n-space (with the Zariski topology), $\mathcal{D}$ its sheaf of differential operators. Let $D$ be the $n$th Weyl algebra, $M$ a right $D$-module, and $N$ a left $D$-module. Write $\tilde M$ for the $\mathcal{D}$-module associated to $M$, and similarly for $N$. Is the $D$-module tensor pr... | https://mathoverflow.net/users/36720 | Global sections of D-module tensor product | As discussed in comments, the claim holds in the derived world; here is a counterexample to the naive statement. As I was writing it, I realized that I looked for a counterexample in the classical topology, but the same idea can be used to produce an easier counterexample that works in both Zariski and classical topolo... | 5 | https://mathoverflow.net/users/2653 | 218794 | 102,726 |
https://mathoverflow.net/questions/218803 | 9 | Let $M$ by an compact, connected $n$-dimensional manifold without boundary.
Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, \mathbb{H}P^m$?
| https://mathoverflow.net/users/65800 | Examples of Stiefel-Whitney classes of manifolds | It goes back to Wu in the 1950's that if one can compute the mod 2 cohomology of a manifold, with its Steenrod operations, then one can explicitly compute its Stiefel-Whitney classes,
via the Wu formula. See for example the Theorem on page 188 of ``A concise course on algebraic topology'' (no originality claimed, just ... | 12 | https://mathoverflow.net/users/14447 | 218805 | 102,729 |
https://mathoverflow.net/questions/218809 | 3 | Let $A = F\_2$ be the free group on two generators (not sure if this is important.)
Suppose you have a semidirect product $A\rtimes C$ coming from some homomorphism $\beta: C\rightarrow \text{Aut}(A)$.
Let $G$ be a finite group, and let $f\_A: A\twoheadrightarrow G$ be a surjection such that
$$f\_A(^c\cdot) := f\_A... | https://mathoverflow.net/users/15242 | extending homomorphisms to semidirect products | No. For a counterexample, let $G = \langle i,j \rangle$ be the quaternion group of order $8$, let $f\_A$ map the generating set $\{a,b\}$ of $F\_2$ to $\{i,j\}$, and let $C = \langle c \rangle$, where $c$ is the automorphism that fixes $a$ and inverts $b$.
We can choose $g\_c$ to be $i$, so the hypothesis is satisfi... | 5 | https://mathoverflow.net/users/68305 | 218815 | 102,732 |
https://mathoverflow.net/questions/218774 | 3 | I am having a trouble in understanding the Example 4.7 (pages 65-66), the genus two fibrations with $p\_g=0$, $q=1$, and $K^2 = -3$, in "Surfaces fibrées en courbes de genre deux", Lecture Notes in Mathematics, 1137, by Xiao Gang(<http://link.springer.com/book/10.1007%2FBFb0075351>). My French is not good and the Googl... | https://mathoverflow.net/users/80501 | On surfaces with $p_g=0$, $q=1$, and $K^2=-3$ | Xiao Gang is taking a configuration of six lines in the plane with 4 triple points $x,z\_1,z\_2,z\_3$ and three double points $y\_1,y\_2,y\_3$. He considers a general quartic through the seven points which is double on the $y\_i$ (the first part of the discussion is the proof that such a quartic is irreducible).
Then... | 3 | https://mathoverflow.net/users/46104 | 218820 | 102,737 |
https://mathoverflow.net/questions/218182 | 3 | It is well-known that $\{e^{i n t}\}\_{n\in\mathbb Z}$ is an orthonormal basis for $L^2(-\pi,\pi)$. A theorem by Kadec (Kadec $1/4$ theorem) studies the perturbed exponential system:
>
> If $\{\lambda\_n\}$ is a sequence of real numbers for which
> $$|\lambda\_n-n|\leqq L<\frac{1}{4}, \ \ n=0, \pm 1, \pm 2, \dots$... | https://mathoverflow.net/users/69931 | Transform Riesz basis $\{e^{i \lambda_n t}\}_{n\in\mathbb Z}$ to an orthogonal basis | In the conditions of Kadec's result , you can write $A=(I-S)^{-1}=\sum\_{m=0}^\infty S^m,$ where $S(f)(x)=\sum\_{n=-\infty}^\infty \hat f(n) (e^{inx}-e^{i\lambda\_n x})$ verifies $\Vert S\Vert<1.$ Here $\{ \hat f (n)\} $ are the Fourier coefficients of $f$.
Then $A(e^{i\lambda\_n x})= e^{in x}$ and so $A$ orthogonal... | 1 | https://mathoverflow.net/users/nan | 218821 | 102,738 |
https://mathoverflow.net/questions/218690 | 2 | I want to know literature about maximal inequalities for dependent random variables i.e. upper bound for $P(\max\_{n\ge k\ge 1}\sum\_{i=1}^{k}X\_i > \delta)$ where $X\_i$ are dependent random variables. I am aware of martingale cases, want to know about other scenarios. Thanks.
| https://mathoverflow.net/users/77923 | maximal inequalities for dependent random variables | If you are interested in non-asymptotic bounds, the following references can be useful (of course, the list is far from being complete).
* The martingale case is addressed in Nagaev, S. V. [On probability and moment inequalities for supermartingales and martingales.](http://link.springer.com/article/10.1023%2FA%3A102... | 3 | https://mathoverflow.net/users/17118 | 218831 | 102,743 |
https://mathoverflow.net/questions/218040 | 12 | Let k be a field of characteristic $0$.
>
>
> >
> > Let $X$ be a variety over k which is isomorphic to a smooth cubic threefold over $\bar{k}$. Then is $X$ isomorphic to a smooth cubic threefold over $k$?
> >
> >
> >
>
>
>
For motivation, let's consider some other cases.
* For cubic curves, the analogo... | https://mathoverflow.net/users/5101 | Twists of cubic threefolds | As pointed out by Noam Elkies, my comment above can be turned into a positive answer to the question.
In the exact sequence
>
> $\DeclareMathOperator{\Pic}{Pic}\DeclareMathOperator{\Br}{Br}0 \to \Pic X \to (\Pic \bar{X})^{G\_k} \to \Br k$,
>
>
>
the arrow $\delta \colon (\Pic \bar{X})^{G\_k} \to \Br k$ sends... | 7 | https://mathoverflow.net/users/3753 | 218833 | 102,745 |
https://mathoverflow.net/questions/117754 | 33 | Notation: identify an element of $\{-1,1\}^n$ with the set $S \subseteq \{1, \ldots, n\}$ on which it takes the value $-1$.
The following is an asymptotic question. "Close to one" means "more than $r\_n$" and "away from $\frac{n}{2}$" means "outside the interval $[\frac{n}{2} - k\_n\sqrt{n}, \frac{n}{2} + k\_n\sqrt{n... | https://mathoverflow.net/users/23141 | Fourier transform on the discrete cube | In the language of [Marcus-Spielman-Srivastava](http://arxiv.org/abs/1306.3969), Corollary 1.5, identify $l^2(S)$ with ${\mathbb C}^d$ where $d = |S|$ and let the vectors $u\_i \in {\mathbb C}^d$ be the orthogonal projections into $l^2(S)$ of the Fourier transforms of the standard basis vectors of $l^2(\{-1,1\}^n)$. By... | 7 | https://mathoverflow.net/users/23141 | 218843 | 102,746 |
https://mathoverflow.net/questions/218797 | 5 | Can you suggest books or lecture notes (for beginners) covering basic material about flag varieties and Grassmannians (of reductive groups), with emphasis on the usual flag variety, i.e. flag variety of $GL(n, \mathbb{C})$. Topics like: classical Plücker embedding and its generalizations for other reductive groups. Lin... | https://mathoverflow.net/users/21491 | Notes on flag varieties and Grassmannians for beginners | The book "Flag varieties" by V. Lakshmibai and N. Gonciulea (Hermann 2001) covers several of these topics.
Michel Brion wrote lecture notes "Lectures on the geometry of flag varieties" which appeared in "Topics in Cohomological Studies of Algebraic Varieties", Impanga Lecture Notes, Ed. Piotr Pragacz, Birkhäuser Tren... | 1 | https://mathoverflow.net/users/12221 | 218845 | 102,748 |
https://mathoverflow.net/questions/218824 | 0 | From [SMOOTH NUMBERS: COMPUTATIONAL NUMBER THEORY AND BEYOND Andrew Granville pp.13-14](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.5030&rep=rep1&type=pdf):
>
> **2j. Lenstra’s polynomial time test as to whether an integer that is conjecturally prime, is rigorously squarefree.**
> If $n > 32$ ... (... | https://mathoverflow.net/users/12481 | What is wrong with this counterexample to primality test assuming GRH? | This is a revised version of my original answer. I fixed some inconsistencies and made the text more readable.
The correct statement is given by the deterministic Miller-Rabin test coupled with an estimate of Bach under GRH. Let us follow Sections 10.2 and 10.5 of [Shoup: A computational introduction to number theory... | 14 | https://mathoverflow.net/users/11919 | 218853 | 102,752 |
https://mathoverflow.net/questions/218839 | 4 | I propose a conjecture variant of [Cayley-Bacharach's theorem](https://en.wikipedia.org/wiki/Cayley%E2%80%93Bacharach_theorem).
I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know some more information for the conjecture:
**Conjectu... | https://mathoverflow.net/users/76698 | A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem | I do not understand how your $d$ is related to the degrees of $C\_1$ and $C\_2$. Anyway, many variants and generalizations of the classical Cayley-Bacharach for cubics are known. One of them is the statement below, whose proof can be found in the beautiful paper by Eisenbud, Green and Harris
*[Cayley-Bacharach Theor... | 4 | https://mathoverflow.net/users/7460 | 218858 | 102,754 |
https://mathoverflow.net/questions/218660 | 16 | Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.)
In, e.g., Fulton's *Intersection Theory* chapter 15, and Soule's *Lectures on Arakelov Geometry* chapter 2, there are brief expositions on the associated graded $\mathbb{Z}$-algebra... | https://mathoverflow.net/users/58688 | What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring? | $\def\ZZ{\mathbb{Z}}\def\QQ{\mathbb{Q}}\def\cO{\mathcal{O}}\def\FF{\mathbb{F}}$Does this help?
$K(X) \otimes \mathbb{Q}$ is a graded ring in the sense that it is isomorphic (by the Chern character map) to $A(X) \otimes \mathbb{Q}$, which is graded. I would perhaps prefer to say that it is "gradeable", since the grading... | 9 | https://mathoverflow.net/users/297 | 218861 | 102,755 |
https://mathoverflow.net/questions/218778 | 4 | Let $(X,\Sigma)$ be a standard measurable space, and let $\,\preceq\,$ be a total order on $X$ with the property that $\,\{(x,y) \in X \times X: x \preceq y\} \in \Sigma \otimes \Sigma$.
Let $A \subset X$ be a set with the property that for all $x,z \in A$ and $y \in X$ with $x \preceq y \preceq z$, we have $y \in A$... | https://mathoverflow.net/users/15570 | Is every convex subset of a Borel-linearly ordered space measurable? | I believe the answer is yes. (Please check this answer carefully, as this is rather outside my field. There may well be a much easier solution.)
In the paper ["Borel Orderings" by Harrington, Marker and Shelah](http://www.ams.org/journals/tran/1988-310-01/S0002-9947-1988-0965754-3/) (Trans. AMS 310 (1988), 293-302, [... | 2 | https://mathoverflow.net/users/4832 | 218864 | 102,756 |
https://mathoverflow.net/questions/218867 | 3 | Let $l=\min\{s\in \mathbb{N}|0\in s\cdot (\mathbb{F}\_{p^n}^\times)^k\}$. Is any information known about this number already as a function of $k$? Any reference would be greatly appreciated!
| https://mathoverflow.net/users/69078 | Minimal expression of 0 as a sum of kth powers in a finite field | Write $q=p^n$. You can replace $k$ by $gcd(k,q-1)$ and assume $k | (q-1)$. Now, trivially, $l=2$ if and only if $(-1)^{(q-1)/k} = 1$. If $l >2$, then $l=3$ if the Fermat curve has a point which, from the Weil bound, happens if $k < q^{1/4}$, approximately. For larger $k$ you may want to use techniques from additive com... | 6 | https://mathoverflow.net/users/2290 | 218871 | 102,759 |
https://mathoverflow.net/questions/218874 | 10 | I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these.
If I understand correctly, mathematicians tend to be quite happy working with ZF+DC, but other forms of choice that are not implied by DC can be more controversial.
[Therefore it seems natural th... | https://mathoverflow.net/users/15570 | Some "axiom of choice" and "dependent choice" issues | To elaborate a bit on Ashutosh's comment, [my answer to this question](https://mathoverflow.net/a/32722/1946) shows that even countable AC, which is strictly weaker than DC, suffices to show that most sets of reals are not Borel. Thus, the answer to question (1) is negative. The argument there shows also that the answe... | 4 | https://mathoverflow.net/users/1946 | 218882 | 102,762 |
https://mathoverflow.net/questions/218610 | 11 | This is a short question:
Is it just unproven folklore (yet), or is it definitively known that $E\_n$-operads are not formal, if the characteristic of the underlying field is not equal to zero?
| https://mathoverflow.net/users/21965 | Are $E_n$-operads not formal in characteristic not equal to zero? | Ok, Sean. I'll write in terms of homology operations. Let $\mathcal C$ be any $\Sigma$-free operad,
in spaces or in chain complexes, makes no real difference to the answer. For definiteness, take
chain complexes over a field $K$; $\Sigma$-free means that $\mathcal C(j)$ is a free $K[\Sigma\_j]$-module
for each $j$. Fo... | 9 | https://mathoverflow.net/users/14447 | 218886 | 102,765 |
https://mathoverflow.net/questions/218837 | 4 | I'm reading the following article by Newman
<http://scitation.aip.org/content/aip/journal/jmp/4/7/10.1063/1.1704018>
about the generalization of the Schwarzschild metric. My question is the following: after the author obtained the tetrad (resp. the metric) on page 5 of the article, he then proceeds to write down th... | https://mathoverflow.net/users/51137 | Obtaining Killing fields from the tetrad | My answer will be a bit more charitable than Willie's. There is an algorithm (sort of) to compute the dimension of the solution space of the Killing equation. Whether it is simple or not, you can decide for yourself.
The Killing equation is an example of an (overdetermined) equation of *finite type*. This means that ... | 5 | https://mathoverflow.net/users/2622 | 218889 | 102,766 |
https://mathoverflow.net/questions/218887 | 9 | This interesting question resulted from a query of Mushfeq: In ZFC, can we find a non null set of pairwise Turing incomparable reals?
| https://mathoverflow.net/users/2689 | Non null Turing antichain | This is a $ZFC$-theorem. Actually for any $\Sigma^1\_1$-locally countable partial ordering, there is a non-null antichain.
The proof is not quite simple. See my paper
<http://ims.nju.edu.cn/~yuliang/lcpfinal>
I feel that it might be helpful to give more details.
The idea is as following:
By Harrison's theore... | 12 | https://mathoverflow.net/users/14340 | 218893 | 102,768 |
https://mathoverflow.net/questions/218856 | 10 | Let $\theta\_1,\dots,\theta\_n$ be algebraic numbers of degree $\leq D$ and Weil height $\leq H$ such that $1,\theta\_1,\dots,\theta\_n$ are $\mathbb Q$-linearly independent. An effective version of Kronecker's approximation theorem would then assert that, given $\epsilon > 0$ and $\alpha \in \mathbb R^n$, there exist ... | https://mathoverflow.net/users/36982 | An effective version of Kronecker's approximation theorem and its variations | The numbers $\theta\_i$ here are real of course. Also there is only one question really; upon replacing $\boldsymbol{\alpha}$ with $(\boldsymbol{\alpha} - t \boldsymbol{\theta})/s$ the second problem reduces to the case that $s = 1$ and $t = 0$.
That being said, there is nothing inherently ineffective in neither of t... | 9 | https://mathoverflow.net/users/26522 | 218895 | 102,769 |
https://mathoverflow.net/questions/218811 | 20 | Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by
\begin{equation}
s\_\lambda(x\_1,\ldots, x\_n) = a\_{\lambda + \delta}(x\_1, \ldots, x\_n) / a\_\delta (x\_1, \ldots, x\_n),
\end{equation}
where $\delta = (n-1,n-2,\ldots, 0)$ and $a\_\lambda = ... | https://mathoverflow.net/users/4923 | Bounding Schur symmetric polynomials on the unit circle | In the special case mentioned in the problem, I'll show the bound
$$
|e\_{n/2}(x\_1,\ldots, x\_n)| \le 2^{n/2}.
$$
Let
$$
F(z) = \prod\_{j=1}^{n} (1+zx\_j) = C \prod\_{j=1}^{n} (z + \overline{x\_j}),
$$
where $C= \prod\_{j} x\_j$ has magnitude $1$. (The roots of $F$ are $-\overline{x\_j}$.)
Now by Cauchy's th... | 14 | https://mathoverflow.net/users/38624 | 218900 | 102,771 |
https://mathoverflow.net/questions/218903 | 1 | Consider $K$ vectors $x\_1,\dots,x\_K$ in $\mathbb{R}^N$. Define the $K\times K$ matrix $A$ whose $(i,j)$ entry is given as $$A\_{ij}=\exp(-\frac{||x\_i-x\_j||^2}{2})$$ Is this matrix Positive Semi-Definite?
| https://mathoverflow.net/users/27249 | Is this matrix positive semi-definite? | Yes, it is. We write $A\_{ij}=B\_{ij}C\_{ij}$, where
$$B\_{ij}=e^{-\|x\_i\|^2/2}e^{-\|x\_j\|^2/2}; \quad C\_{ij}=e^{(x\_i,x\_j)}.$$
By Schur's theorem on elementwise product of positive definite matrices, it is enough to show that $B\_{ij}$ and $C\_{ij}$ are positive semidefinite. Positive definiteness of $B\_{ij}$ is... | 6 | https://mathoverflow.net/users/29557 | 218906 | 102,773 |
https://mathoverflow.net/questions/218913 | 0 | For any set $X$ we let $[X]^2 = \big\{\{x,y\}: x\neq y \in X\big\}$.
Let $G=(V,E)$ be a simple, undirected graph. Its open neighborhood hypergraph $\mathcal{H}(G)$ has the same vertex set $V$ with a hyperedge for the open neighborhood of every vertex $v \in V$. (The open neighborhood of $v\in V$ is the set $N\_v = \{... | https://mathoverflow.net/users/8628 | Different graphs with the same open neighborhood hypergraph | The answer to the first question is positive. Consider two graphs on eight vertices each consisting of two disjoint 4-cycles: the first one's cycles are $abcd$ and $efgh$, the second's ones are $afch$ and $ebgd$.
The answer for the second question is also positive, even if we consider $\mathcal H(G)$ to be a multi-hy... | 4 | https://mathoverflow.net/users/17581 | 218916 | 102,777 |
https://mathoverflow.net/questions/218914 | 5 | Is this statement true for all positive integers $n\in\mathbb{N}$?
>
>
> >
> > For all $\varepsilon >0$ there are prime numbers $p,q$ such that $|\frac{p}{q} - n| < \varepsilon$.
> >
> >
> >
>
>
>
| https://mathoverflow.net/users/8628 | Approximating integers with prime quotients | This is an addendum to my comment above, but too long for another comment. I think it's safe to say that the following (more general) result: "For every $\alpha \in \mathbf R$ and $\varepsilon \in \mathbf R^+$ there exist (rational) primes $p,q \in \bf Z$ such that $|\alpha−p/q|<\varepsilon$" is well-known, and here is... | 12 | https://mathoverflow.net/users/16537 | 218917 | 102,778 |
https://mathoverflow.net/questions/218918 | 6 | The question is in the title : are there spaces X such that the adjoint of the identity on the loop space $\Omega X$, i.e. $\Sigma\Omega X \to X$, is a homotopy equivalence ?
| https://mathoverflow.net/users/80559 | Can the standard map $\Sigma \Omega X \to X$ be a homotopy equivalence? | If $X$ is such a space (a CW complex, say), then it must be a suspension $\Sigma Y$ (as it is the suspension of $y :=\Omega X$). The James splitting gives
$$\Sigma \Omega \Sigma Y \simeq \bigvee\_{n=1}^\infty \Sigma Y^{\wedge n},$$
the wedge of suspensions of smash products, so under your assumption $\Sigma \Omega \Sig... | 15 | https://mathoverflow.net/users/318 | 218922 | 102,779 |
https://mathoverflow.net/questions/218925 | 1 | Consider the following function $f: \omega\to \{0,1\}$:
* Set $f(n) = 1$ if for all $k\in \omega$ there are prime numbers $p,q > k$ such that $n = p-q$, and
* set $f(n) = 0$ otherwise.
(Trivially, if $n$ is odd, we have $f(n) = 0$. Moreover, the question whether $f(2) = 1$ is the subject of the [twin prime conjectu... | https://mathoverflow.net/users/8628 | Computability of prime difference function | I once heard Harvey Friedman suggest that the set of prime-differences, that is, the set of all natural numbers $n$ for which there are primes $p,q$ with $p-q=n$, as a possible candidate for all we knew for an intermediate Turing degree — a noncomputable set between $0$ and $0'$ — that was *natural*, not specifically c... | 2 | https://mathoverflow.net/users/1946 | 218926 | 102,781 |
https://mathoverflow.net/questions/218921 | 2 | Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
>
> **Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The following functionals,
> $$L(u,v)= \Big ( \Delta u - \frac{u}{v} \Delta v \Big)^2,$$
> $$R(u,v)=|\Delta u|^2 - \Delt... | https://mathoverflow.net/users/76453 | interpret of Picone inequality for non-regular functions | $\def\sp{\kern.4mm}$I see no problem in interpreting
$$\int\_{\Omega} \frac{\Delta ^2 v}{v} u^2 \, \mathrm{d}x\ .$$
It is just the integral of the function $\Omega\owns x\mapsto(v(x))^{-1}(u(x))^2$ w.r.t. the positive (Radon) measure $\mu\,$. In the proof of Theorem 3.9, the authors use Fatou's lemma for suitably chose... | 2 | https://mathoverflow.net/users/12643 | 218957 | 102,790 |
https://mathoverflow.net/questions/218961 | 2 | Famously, Rosser introduced a provability predicate $\pi[A]$ that holds iff $\exists x(xP[A]\wedge\forall y(y\le x\to\lnot yP[\lnot A]))$.
Supposing $PA$ is consistent, what are the adequacy conditions for $\pi$ as compared with the Hilbert-Bernays-Löb derivability conditions for P?
In particular, do we have $\vdas... | https://mathoverflow.net/users/37385 | What are the adequacy conditions for Rosser Provability? | If we assume that PA is consistent, then the two provability predicates are equivalent, since Rosser provability implies regular provability; and conversely, if there is a proof of $A$ in PA, then by Con(PA) there will be no proof of $\neg A$ at all, let alone one with a shorter proof, and so $A$ is Rosser provable. Th... | 1 | https://mathoverflow.net/users/1946 | 218967 | 102,794 |
https://mathoverflow.net/questions/218947 | 5 | I'd like to know if there exists (and, in this case, where I can find it) some computer program/programming language/any kind of software that can find explicitly the conetypes of a hyperbolic group on which I am working (a presentation of which is known).
| https://mathoverflow.net/users/80571 | Computations with conetypes of hyperbolic groups | My [KBMAG](http://homepages.warwick.ac.uk/~mareg/download/kbmag2/) package can compute a finite state automaton that accepts the language of geodesic words in a hyperbolic group, and I think the states of that automaton correspond exactly to the conetypes.
It is not particularly easy to use. If you write down the pre... | 9 | https://mathoverflow.net/users/35840 | 218979 | 102,800 |
https://mathoverflow.net/questions/218970 | 1 | Let $F(d\_1,d\_2,\ldots,d\_k;n )$ be the variety of all flags $\mathbb A^{d\_1} \subset\mathbb A^{d\_2}\subset\ldots\subset \mathbb A^{d\_k}\subset \mathbb A^{n}$. This variety has the natural maps to Grassmann varieties:
$$
f\_i:F(d\_1,d\_2,\ldots,d\_k; n) \to F(d\_i;n),
$$
and pull-backs $f\_i^\*\mathcal O(1)$ from G... | https://mathoverflow.net/users/19436 | Canonical class of partial flag variety | If $G$ is simply connected (so, $SL\_n$ in your less general question), then every line bundle $\mathcal L$ is uniquely of the form $G \times^B \mathbb C\_\lambda$ where $\mathbb C\_\lambda$ is the $T$-irrep with weight $\lambda$. From there, you can figure out $\lambda$ from $\mathcal L$ by looking at the $T$-weight o... | 4 | https://mathoverflow.net/users/391 | 218980 | 102,801 |
https://mathoverflow.net/questions/218942 | 13 | Let $X$ be a domain in the Riemann sphere $\widehat{\mathbb{C}}$. We say that $X$ is a *circle domain* if every connected component of its boundary is either a circle or a point.
It was conjectured by Koebe in 1909 that circle domains represent all planar domains up to conformal equivalence. This was proved by Koebe ... | https://mathoverflow.net/users/1162 | How bad can a circle domain get? | Let $K$ be the boundary of your circle domain $\Omega$.
Let us suppose that every point of $K$ is accumulated on by a sequence of (pairwise different) circle components. Such an example is easy to construct (see e.g. the already existing answer to your Question 2, or simply add the circles inductively - see below fo... | 6 | https://mathoverflow.net/users/3651 | 218985 | 102,802 |
https://mathoverflow.net/questions/218995 | 3 | Let $G\_1$ and $G\_2$ be two finitely generated groups which are quasi-isometric in the sense of geometric group theory.
Are their rational cohomology rings $H^{\ast}(G\_i; \mathbb Q)$ necessarily isomorphic?
| https://mathoverflow.net/users/14233 | Do quasi-isometric groups have the same rational cohomology? | No. For example, $F\_2$ is quasi-isometric to $F\_3$ because the latter is finite index in the former, but their rational $H^1$s differ.
| 5 | https://mathoverflow.net/users/290 | 218999 | 102,808 |
https://mathoverflow.net/questions/218866 | 6 | If one has a normal Lie group inclusion $H\to G$, with quotient $G/H$, and a $G$-manifold $X$, one can take the quotient $X/H$. Then the $G$-action on $X/H$ factors thru a $G/H$-action, so one can take the quotient again to obtain $(X/H)/(G/H)$. It is known classically that $(X/H)/(G/H)\cong X/G$ (see, e.g. Bourbaki's ... | https://mathoverflow.net/users/11546 | Iterated Homotopy Quotient | Here's a sketch. The sequence of maps to look at is actually
$$BH \to BG \to B(G/H).$$
Recall that an object $X$ with a $G$-action in an $\infty$-category $C$ is the same thing as a functor $BG \to C$. I claim that the left Kan extension of this to a functor $B(G/H) \to C$ has the effect of quotienting by $H$ and t... | 7 | https://mathoverflow.net/users/290 | 219001 | 102,810 |
https://mathoverflow.net/questions/219003 | 4 | We know that from prime number theorem that the number of primes below $n$ is approximately $$\frac{n}{\log\_en}.$$
**$\star$** Given $n,m$, what is the largest list of pairwise coprime numbers that one can come up above $n$ and below $m$? Will it be asymptotically same as number of primes or something different? In ... | https://mathoverflow.net/users/nan | Greatest number of coprime numbers between two numbers | For a given $n$, let $P\subset\{2,3\dots,n\}$ be the set of primes up to $n$, and let $S\subset\{2,3\dots,n\}$ be any subset with pairwise coprime elements. Consider the function $f:S\to P$ that assigns to any $s\in S$ its smallest prime factor $p\in P$. It is clear that $f$ is an injection, whence $|S|\leq|P|$.
In ... | 7 | https://mathoverflow.net/users/11919 | 219006 | 102,811 |
https://mathoverflow.net/questions/219022 | 2 | I am perturbing a dynamical system with a perturbation that looks like $$\vec u e^{-i k x} e^{\sigma(p;k)t}$$ where $\sigma$ is a function of the parameters of the dynamical system and the wavenumber, coming out from the diffusion part of the system. Hopf bifurcation will be when for a $k=0$ I will have a non zero $\Im... | https://mathoverflow.net/users/80604 | Difference between Hopf-Turing bifurcation and traveling-waves bifurcation in reaction-diffusion systems | You want a codimension-two point in parameter space where a time-independent, spatially periodic (Turing) mode (wave vector $\vec{q}\_T$) and a spatially homogeneous, time-periodic (Hopf) mode (frequency $\omega\_H$) bifurcate simultaneously. These modes may have a different vector $\vec{u}\_T$ and $\vec{u}\_H$, and a ... | 1 | https://mathoverflow.net/users/11260 | 219023 | 102,816 |
https://mathoverflow.net/questions/163630 | 1 | The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by:
$$\log(\sum\_{i=0}^n |z\_i|^2)).$$
What is the analogous formula for a Kaehler potential in a weighted projective space $\mathbb{C} P(m\_0, \ldots, m\_n)$ with weights $m\_0, \ldots, m\_n \in \mathbb{Z... | https://mathoverflow.net/users/43696 | Kaehler form on weighted projective space | Weighted projective spaces are the simplest projective toric varieties and you can find the Kahler potential of singular toric varieties [here](http://msp.org/pjm/2008/238-1/pjm-v238-n1-p03-p.pdf)
| 0 | https://mathoverflow.net/users/nan | 219025 | 102,818 |
https://mathoverflow.net/questions/219021 | 1 | The paper [1] introduced the category $\mathcal{O}$ for rational Cherednik algebras $H\_{t,c}(W)$. This construction is tailored for the $t=1$ case (equivalently, the $t\neq 0$ case). The general setup for the structural theory introduced is that of an algebra with a triangular decomposition $A=\overline{B}\otimes H\ot... | https://mathoverflow.net/users/33854 | Graded category O for for rational Cherednik algebras, but at t=0 | This is extremely false: for generic $\mathbf{c}$, the algebra $H\_{0,\mathbf{c}}(W)$ is Morita equivalent to the functions on Calogero-Moser space, a finite dimensional smooth affine variety, and category $\mathcal O$ is the subcategory of coherent sheaves which are supported set-theoretically on a subvariety isomorph... | 2 | https://mathoverflow.net/users/66 | 219028 | 102,819 |
https://mathoverflow.net/questions/219029 | 2 | Suppose $X$ is a surface, can it have infinitely many $(-1)$ curves?(If they are disjoint, we can see this since Neron Severi group has finite rank, but how to deal with the case when they are not disjoint?)
Suppose $X$ is a projective variety, is it true that the codimension $1$ subvarieties with negative top self-... | https://mathoverflow.net/users/nan | Negative self-intersection and finiteness | Let $C\_1$ and $C\_2$ be two general cubic plane curves. Let $S$ be the surface obtained by blwoing-up the nine base points. Then $S$ is a rational elliptic surface. Every section of the elliptic fibration $S\to \mathbb{P}^1$ is a $-1$ curve. After choosing one section as the zero-section you find that the sections for... | 5 | https://mathoverflow.net/users/8621 | 219030 | 102,820 |
https://mathoverflow.net/questions/219009 | 3 | Is there some standard technique or approach to determine when a (irreducible) subvariety of a rationally connected variety is again rationally connected? Any reference/text dealing with this kind of question will be most welcome.
The example that I have in mind is the following: Let $P$ be the Hilbert polynomial of ... | https://mathoverflow.net/users/58203 | Standard techniques on rationally connected varieties | Let $[x,y,z,w]$ be homogeneous coordinates on $\mathbb{P}^3$ so that $\Gamma\_\*(\mathcal{O}\_{\mathbb{P}^3})$ equals $k[x,y,z,w]$. Let $p$ be the point $[0,0,0,1]$ in these coordinates, whose associated homogeneous ideal $\Gamma\_\*(\mathcal{I}\_{p/\mathbb{P}^3})$ is $\langle x,y,z \rangle$. Let $A\subset \mathbb{P}^3... | 4 | https://mathoverflow.net/users/13265 | 219032 | 102,822 |
https://mathoverflow.net/questions/110177 | 7 | **Question:**Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} f(t,x)=\frac{f(t,x)^2-f(t,0)^2}{x}, x>0,$$
$$\frac{\partial}{\partial t} f(t,0)=2f(t,0) \frac{\partial f}{\partial x}(t,0).$$
I want to k... | https://mathoverflow.net/users/27404 | Uniqueness for a non-local differential equation | The answer is yes: f=g. I wrote up a paper with a more general result [here](http://arxiv.org/abs/1509.06631). The idea is the following. If $f$ were assumed to be of Laplace transform type
$$
f(t,x) = \frac{1}{x} \int\_0^\infty e^{-w/x} A(t,w)\: dw,
$$
then, by taking the inverse Laplace transform, one finds that $A(t... | 3 | https://mathoverflow.net/users/27404 | 219033 | 102,823 |
https://mathoverflow.net/questions/219045 | 3 | Let $X$ be a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. Consider the exact sequence $$0 \rightarrow \text{Pic}(X) \rightarrow \text{Cl}(X) \rightarrow \bigoplus\_{x \in X^{\text{sing}}} \text{Cl}(\mathcal{O}\_{X,x})$$ and the injections $\text{Cl}(\m... | https://mathoverflow.net/users/80607 | Jaffe's exact sequence | The cokernel of the right arrow is in general $H^2(X, \, \mathcal{O}\_X^\*)$. In many cases, for instance when the strict henselization of every $\mathcal{O}\_{X, \, x}$ is a factorial ring, one can conclude that this group is a torsion group.
See [this](https://mathoverflow.net/questions/137155/exact-sequence-for-di... | 1 | https://mathoverflow.net/users/7460 | 219047 | 102,827 |
https://mathoverflow.net/questions/219002 | 4 | If $M$ is a smooth manifold and $\mu$ is a $1$-density thereon then we may define a Borel measure (on Borel sets $A$) on $M$ as:
\begin{equation}
\nu(A) = \int\_M I\_A \mu.
\end{equation}
My question is does the converse hold also? Is not, then when would it.
That is, if $\nu$ is a measure on $M$ then when does t... | https://mathoverflow.net/users/36886 | Riemannian Measures, Densities and Radon–Nikodym Theorem | This follows from the usual Radon-Nikodym theorem. Observe that, given two metrics $g\_0,g\_1$ on $M$ with volume densities $dV\_{g\_0}$, $dV\_{g\_1}$ then there exists a *positive* smooth function $\rho\_{10}: M\to (0,\infty)$ such that
$$ dV\_{g\_1}= \rho\_{10} dV\_{g\_0}. $$
Hence, if $B\subset M$ is a Borel sub... | 3 | https://mathoverflow.net/users/20302 | 219051 | 102,829 |
https://mathoverflow.net/questions/218981 | 5 | Given positive integers $k$, $m$, $n$, with $m,n >> k$, suppose we have
$n$ boxes each containing $k$ randomly (uniformly) selected positive integers $x$ satisfying $1 \leq x \leq m$ (duplicates in the box are permitted).
I begin selecting distinct positive integers $y$ such that $1 \leq y \leq m$ until one integer f... | https://mathoverflow.net/users/48949 | Integers in Boxes Problem | It sounds like the following paper should be relevant.
W. Fernandez de la Vega, V. Th. Paschos, and R. Saad,
Average case analysis of a greedy algorithm for the minimum hitting set problem, *LATIN '92* (São Paulo, 1992), 130–138,
Lecture Notes in Comput. Sci., 583, Springer, Berlin, 1992.
Unfortunately it's behind... | 1 | https://mathoverflow.net/users/3106 | 219058 | 102,830 |
https://mathoverflow.net/questions/218851 | 6 | I recently noticed that I could mostly prove a special case of the following statement. I think it's true in general, though perhaps only for nice spaces.
>
> Let $X$ be a topological space, and suppose $X=X\_0\cup X\_1$, where $X\_0$, $X\_1$, and $Y:=X\_0\cap X\_1$ are all nonempty and path-connected. Then
> $$
>... | https://mathoverflow.net/users/300 | Seifert--van Kampen for the loop space dga | Sketch of proof. I will use the following ingredients
0) the category of spaces will be the category of simplicial sets. All the computation are in the derived sense.
1) use the Quillen adjunction $$F: sSet^{\otimes}\longleftrightarrow sMod\_{k}^{\otimes}: U$$
between the category of monoids in simplicial sets an... | 5 | https://mathoverflow.net/users/21369 | 219062 | 102,833 |
https://mathoverflow.net/questions/219056 | 12 | (This is a re-post of [1](https://math.stackexchange.com/questions/1446061/non-realizability-of-mathbbq-as-a-cohomology-group))
In the paper "On the realizability of singular cohomology groups" by Kan and Whitehead, it is shown that there is no space $X$ and integer $n\geq 1$ such that $H^{n−1}(X)=0$ and $H^n(X)=\mat... | https://mathoverflow.net/users/49520 | Non-realizability of $\mathbb{Q}$ as a cohomology group | This may depend on your axioms, see
S. Shelah "The consistency of Ext(G,Z)=Q", Israel J. Math. 39 (1981), no. 1-2, 74–82.
There it is shown that it is consistent with the generalised continuum hypothesis that there exists a group $G$ having $Ext(G, \mathbb{Z})=\mathbb{Q}$. Then a Moore space $M(G,n-1)$ has the req... | 15 | https://mathoverflow.net/users/318 | 219070 | 102,836 |
https://mathoverflow.net/questions/219071 | 6 | **Initially [posted](https://math.stackexchange.com/questions/1445367/universal-covering-and-double-cover-functors) on MSE**
Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}\_\*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to know whether there... | https://mathoverflow.net/users/47757 | Universal covering and double cover functors | I am supposing that you want to take all continuous maps between CW-complexes in $\mathsf{CW}$, and I am also supposing for simplicity that you want the purported universal cover functor $U : \mathsf{CW} \to \mathsf{CW}$ to be continuous, and that you want it to be augmented $\eta: U \Rightarrow Id\_{\mathsf{CW}}$ so t... | 5 | https://mathoverflow.net/users/318 | 219072 | 102,837 |
https://mathoverflow.net/questions/219013 | 9 | I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field.
According to Mazza--Voevodsky--Weibel "Lectures on motivic cohomology", these groups have the following *covariant* functoriality properties (in addition to their... | https://mathoverflow.net/users/2481 | Motivic cohomology and pushforward maps | The answer is **yes, they are compatible**. This is consequence of the machinery developped for the Riemann-Roch theorem.
Recall that motivic cohomology, $l$-adic cohomology, and absolute Hodge cohomology have *additive* Chern classes (that is to say, $c\_1(L\otimes L')=c\_1(L)+c\_1(L')$) and that the realization ma... | 5 | https://mathoverflow.net/users/12204 | 219081 | 102,838 |
https://mathoverflow.net/questions/218994 | 10 | Fourier analysis is useful for analysis in the frequency domain. SVD on the other hand is useful for analysis of data, and expressing noise in the data. I have a problem that needs extensive data analysis, it is in the area medicine. This could be generalized to other problems.
The problem is that of gene expression... | https://mathoverflow.net/users/58718 | SVD vs Fourier analysis for data. | When you say SVD, you probably mean something like the [Karhunen–Loève decomposition](https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem), or maybe just a corresponding [Arnoldi process](https://en.wikipedia.org/wiki/Arnoldi_iteration). (I also regard [Prony's method](https://en.wikipedia.org/wiki/Prony'... | 3 | https://mathoverflow.net/users/20781 | 219082 | 102,839 |
https://mathoverflow.net/questions/219040 | 10 | Here's a precise question. Does Wiles' proof of FLT run just fine in the set theory that logicians would perhaps call "Zermelo + choice" -- i.e. drop the axiom schema of replacement but assume the axiom of choice to make analysis work sensibly? [Wiles' proof needs some cyclic base change, which involves a lot of functi... | https://mathoverflow.net/users/43076 | Is the axiom schema of replacement used in algebraic number theory (or more generally outside logic) | At eric's request, I am expanding my comment into an answer.
If we set aside specific theorems in algebraic number theory for the moment, the question of whether Replacement is used in "ordinary mathematics" has come up on MO before, e.g.,
[here](https://mathoverflow.net/questions/121406/where-in-ordinary-math-do-we-... | 7 | https://mathoverflow.net/users/3106 | 219086 | 102,842 |
https://mathoverflow.net/questions/219069 | 4 | We propose a (probably not new) definition. Let $\varphi\_e$ be an effective enumeration of the partial computable functions.
A total function $f$ is promptly non-computable (PNC) [or promptly non-recursive, PNR] if there exists a computable function $h$ such that, for all $\varphi\_e$, there is some $m\le h(e)$ with... | https://mathoverflow.net/users/38989 | Is DNC/DNR stronger than "prompt" non-computability? | The graph of the course-of-values variant
$$\{(x,(f(0),\dots,f(x))): x\in \mathbb N\}$$
of such a function would be [effectively immune](https://en.wikipedia.org/wiki/Simple_set). Namely, if we enumerate a subset of this graph then there is an associated partial recursive function. Therefore it is equivalent to DNR.
| 7 | https://mathoverflow.net/users/4600 | 219090 | 102,844 |
https://mathoverflow.net/questions/202338 | 7 | Let $R$ be a noetherian domain and let $\mathcal{O}$ be an $R$-algebra that is finitely generated and projective as an $R$-module. The set of invertible fractional ideals of $\mathcal{O}$ is a group under multiplication. Is this group always abelian?
| https://mathoverflow.net/users/4433 | Noncommutative group of invertible ideals of a ring | No; it is not abelian in general.
Let $R$ be a discrete valuation ring with maximal ideal generated by an element $\pi$, let $k := R / \pi R$ be the residue field of $R$ and let $\mathcal{O}$ be the inverse image in $M\_2(R)$ of the scalar matrices in $M\_2(k)$. Note that $\mathcal{O}$ is free of rank $4$ as an $R$-m... | 7 | https://mathoverflow.net/users/6827 | 219093 | 102,846 |
https://mathoverflow.net/questions/219092 | 9 | Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown.
Alternatively, has this been worked out for the various hyperreal fields, or real-closed fields in general?
| https://mathoverflow.net/users/24611 | Pontryagin dual of the surreal numbers? | For any infinite cardinal $\kappa$, let $S\_\kappa$ be the surreal numbers of rank $<\kappa$, considered as a group under addition and topologized with the order topology (if you want to consider *all* the surreal numbers, suppose $\kappa$ is inaccessible). Suppose now that $\kappa$ has uncountable cofinality and $f:S\... | 9 | https://mathoverflow.net/users/75 | 219102 | 102,849 |
https://mathoverflow.net/questions/212009 | 3 | Is there anything known about the (mock)modular properties, if any, of the following theta series,
$\sum\_{n\in {\mathbb Z}^r\_+} e^{2\pi i \langle b, n\rangle} q^{\frac12 \langle n,n\rangle}$,
where $\langle x, y\rangle$ might be degenerate. For example,
$\sum\_{\lambda,\mu,\nu=0}^\infty q^{\lambda+\mu+\nu+\lamb... | https://mathoverflow.net/users/31476 | Modular property of indefinite degenerate theta series | For indefinite theta series of higher signature than (1,1) (which as Jeff mentions above are studied in Zwegers' thesis), in general there will be non-holomorphic completions which are modular. These will generically be "higher depth" mock modular forms, which were considered by Zagier and Zwegers and by Raum, and are ... | 5 | https://mathoverflow.net/users/7998 | 219105 | 102,851 |
https://mathoverflow.net/questions/219096 | 7 | It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.
My question is: If $S$ is a connected scheme such that has every prime as a residue characteristic, then can there exist an elliptic curve over $S$?
| https://mathoverflow.net/users/15242 | Do there exist elliptic curves over schemes which have all primes as residue characteristics? | Yes, there can. Choose any elliptic curve $E$ over $\mathbb{Q}$ with potential good reduction (for instance, a curve with potential CM) and pass to a number field $K$ over which the reduction is everywhere good. Then $E\_K$ extends to an elliptic curve over the ring of integers $\mathcal{O}\_K$, and the latter has all ... | 15 | https://mathoverflow.net/users/5498 | 219107 | 102,852 |
https://mathoverflow.net/questions/218954 | 24 | SnapPea (<http://www.math.uic.edu/~t3m/SnapPy/>) is a program with extensive facilities for doing various kinds of calculations with hyperbolic 3-manifolds. The official documentation assumes that the reader is intimately familiar with all the relevant mathematical background. Does there exist any other document which ... | https://mathoverflow.net/users/10366 | SnapPea for the uninitiated | First as pointed out in the comments, the documentation of the SnapPea kernel (now maintained as SnapPy) is extensive. It contains theorems and proofs as well as a fairly thorough treatment many of the functions of SnapPea/SnapPy.
Also, in addition to Thurston's notes, one might also consider Section E.6 of Benedetti... | 8 | https://mathoverflow.net/users/27453 | 219110 | 102,853 |
https://mathoverflow.net/questions/219109 | 38 | Let $(M,g)$ be a compact manifold without boundary.
>
> **Question:** For which $(M,g)$ are the eigenvalues of the Laplace operator on functions *explicitly* known?
>
>
>
An important example is the $n$-sphere with its standard metric. To find eigenvalues, we embed $S^n$ inside $\mathbb{R}^{n+1}-\{0\}$ in the ... | https://mathoverflow.net/users/41626 | Explicit eigenvalues of the Laplacian | [Besse (1978, p.202)](http://ams.org/mathscinet-getitem?mr=496885) has the spectra of compact rank 1 symmetric spaces (CROSSes). In addition to $\mathrm S^n$ due apparently to Heine ([1863](https://zbmath.org/?q=an:062.1614cj), [§19](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002151928&physid=phys139); [1878](h... | 47 | https://mathoverflow.net/users/19276 | 219112 | 102,855 |
https://mathoverflow.net/questions/218637 | 0 | Let $\Lambda(x,n)$ be the the number of totatives of $x$ which are less than or equal to $n$, and $\Phi(x)$ be Euler's totient function.
For now assume $x>n$.
>
> Is there a general formula for $\Lambda(x,n)$? Furthermore, has the result stated below been documented elsewhere?
>
>
>
Let $l = gcd(x,n)$, $x'=... | https://mathoverflow.net/users/41928 | Results regarding the relative-totient function | As I understand the claim $\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, it is false for some $n$ and $x$ with $n$ close to $x$. Let us take $x$ to be $P\_4=210$, the fourth primorial. Let us take $n$ close to and less than $210$, say $208$. (I use Gerry's observation that $n'/x' = n/x$.)
The claim says that the number ... | 2 | https://mathoverflow.net/users/3206 | 219119 | 102,858 |
https://mathoverflow.net/questions/219118 | 28 | I'm curious what languages contribute the largest fraction of published research mathematics. That is, for a given language the percent of new research being published in that language. I'm especially curious to see how you come up with such numbers.
From some googling I managed to get a very crude estimate of total ... | https://mathoverflow.net/users/72302 | Amount of math research published in other languages? | This question has no definite answer if time frame is not specified. The situation in 20s century changed very quickly. In the first half of the century, German and French dominated.
(More German than French). In the second half, it is clearly English, and one does not need any research to see this. But English dominat... | 23 | https://mathoverflow.net/users/25510 | 219129 | 102,861 |
https://mathoverflow.net/questions/218993 | 16 | I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that
$$\frac{1}{2} \log \left| B\_{2 n}\right|\approx \vartheta (2n)\ ,\quad n \in \mathbb{N}$$
where $B\_n$ is the $n$th Bernoulli number, and $\vartheta,$ the Riemann-Siegel theta function. Mo... | https://mathoverflow.net/users/45057 | Connection between Bernoulli numbers and Riemann-Siegel theta function? |
---
**Summary: Your approximation is very good, but not quite perfect.** The absolute error does not go to $0$ as $n\to\infty$. But it is astonishingly close: the error is approximately $1.3\times 10^{-8}+o(1)$.
---
**Detailed answer:**
we can easily check your proposed approximation for $\vartheta(2n)$, sinc... | 16 | https://mathoverflow.net/users/78525 | 219141 | 102,865 |
https://mathoverflow.net/questions/219136 | 2 | Let $X$ be an $n$-dimensional Alexandrov space with curvature at least -1. Assume that at every point it has an $(n,\delta)$-strainer of length $\mu$, where $\delta$ and $\mu$ are independent of a point.
**Does there exist $\sigma=\sigma(\delta, \mu)$ such that for any geodesic triangle of diameter less than $\sigma... | https://mathoverflow.net/users/16183 | A property of geodesic triangles in Alexandrov spaces | First note that for any direction $\xi$ at $x$ there is a geodesic $[xz]$ of length $\tfrac{\mu}{10^n}$ which runs in the $\varepsilon$-close direction to $\xi$.
(This part follows from volume comparison. The space is locally-bi-Lipschitz-Euclidean, therefore voulme of small balls are almost as in the Euclidean spac... | 1 | https://mathoverflow.net/users/1441 | 219146 | 102,867 |
https://mathoverflow.net/questions/219150 | 1 | My question in the most simple form:
Let $\mathfrak{g}=\mathfrak{g}\_1\oplus \mathfrak{g}\_2$ be a direct sum of simple finite-dimensional Lie algebras over $\mathbb{C}$ and let $M$ be a finite-dimensional simple $\mathfrak{g}$-module (it is known that $M$ is some $V(\lambda)$ highest weight module).
Since $\mathfr... | https://mathoverflow.net/users/80651 | An irreducible Lie algebra module decomposition over a subalgebra | A highest weight of $\mathfrak g$ is a pair of highest weights $\lambda=(\lambda\_1,\lambda\_2)$ for the two factors. The highest weight module is just the tensor product $V(\lambda)=V(\lambda\_1)\otimes V(\lambda\_2)$. So when you restrict, that just means you forget that $V(\lambda\_2)$ has an action, and it just bec... | 5 | https://mathoverflow.net/users/66 | 219155 | 102,871 |
https://mathoverflow.net/questions/219140 | 11 | Does every vector bundle on a Stein space have a finite local trivialisation?
Definitions:
* *Stein space* means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of Kiehl.
* *Vector bundle* means either holomorphic vector bundle or rigid analytic vector bundle (= locally free sheaf... | https://mathoverflow.net/users/62434 | Trivialisation of vector bundles on Stein spaces | In the complex analytic world\*, every vector bundle which is globally generated by a finite set of global sections has a finite trivialising cover:
Let $X$ be a complex space and let $E$ be a locally free (hence coherent) $\mathcal{O}\_X$-module of constant rank $r$, generated by the global sections $s\_1,s\_2,\dots... | 11 | https://mathoverflow.net/users/15782 | 219158 | 102,872 |
https://mathoverflow.net/questions/219148 | 3 | I am confused about the following: can one describe the action of the Weyl group on the cohomology of each fiber of the Grothendieck-Springer resolution? I only need the case of ${\mathfrak sl}\_n$. The Grothendieck-Springer resolution for ${\mathfrak s \mathfrak l}\_n = {\mathfrak s \mathfrak l}(V)$, where $V$ is an $... | https://mathoverflow.net/users/12395 | Is it possible to describe the action of the Weyl group on the cohomology of the fibers of the Grothendieck-Springer resolution? | As a $W$ module, the cohomology of a fiber of the Grothendieck-Springer resolution over $g$ is isomorphic to the cohomology of the fiber of the Springer resolution over its nilpotent part $g\_n$. Note, this is not true as a graded representation: for example for a regular semi-simple element, we get $\# W$ points with ... | 3 | https://mathoverflow.net/users/66 | 219159 | 102,873 |
https://mathoverflow.net/questions/219132 | 20 | Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (<http://cantorsattic.info/Upper_attic>)
Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type B cardinals can coexist), I define:
\*Type A > Type B if smallest Type A cardinal has higher cardinality than small... | https://mathoverflow.net/users/76572 | Ordering of large cardinals by cardinality | The usual relations to consider in the large cardinal hierarchy
are
* **Direct implication:** every A cardinal is also a B cardinal
* **Consistency strength implication:** if ZFC + there is an A cardinal
is consistent, then so is ZFC + there is a B cardinal.
Your concept, however, is focused on the least instance o... | 33 | https://mathoverflow.net/users/1946 | 219165 | 102,877 |
https://mathoverflow.net/questions/219060 | 5 | Suppose $\langle s\_\alpha : \alpha \in \omega\_2 \cap \mathrm{cof}(\omega\_1) \rangle$ is a sequence such that each $s\_\alpha$ is an increasing cofinal map from $\omega\_1$ to $\alpha$. Is it possible that for all $\alpha < \beta$, $\mathrm{ran}(s\_\alpha) \cap \mathrm{ran}(s\_\beta)$ is finite? Note that a pressing-... | https://mathoverflow.net/users/11145 | almost disjoint ladder system on $\omega_2$ | Following the suggestion, here is the solution, from Baumgartner: Almost-disjoint sets, the dense set problem and the partition calculus, Annals of Math. Logic, 10(1976), p. 424, part 6.
Let $\{A\_\xi:\xi<\omega\_2\}$ be sets of size $\aleph\_1$ with pairwise countable intersection.
Set $p\in P$ iff $p$ is a functio... | 7 | https://mathoverflow.net/users/6647 | 219180 | 102,885 |
https://mathoverflow.net/questions/219193 | 22 | Equip $S^6$ with the almost complex structure coming from the cross product on $\mathbb R^7$ (i.e. the product on the pure imaginary octonions). What is known about the psudo-holomorphic curves in this $S^6$?
There are certainly a bunch of holomorphic $S^2$'s coming from the inclusions $\mathbb R^3\to\mathbb R^7$ com... | https://mathoverflow.net/users/35353 | Pseudo-holomorphic curves in the six-sphere | It might be immodest of me to mention my own work, but there is quite a lot known about the pseudo-holomorphic curves in $S^6$. For example, it is known that the pseudoholomorphic rational curves are all algebraic and have area a multiple of $4\pi$ (and, yes, there are many of them). (It's true that the moduli space is... | 29 | https://mathoverflow.net/users/13972 | 219197 | 102,891 |
https://mathoverflow.net/questions/219195 | 2 | This may be an extremely stupid and elementary question, but is there a name for sequences $\{a\_i\}$ such that $\lim\_{n\to\infty} \left( \frac{1}{n}\sum\_{i=1}^{n} a\_i\right)$ exists? This seems to be a separate condition from boundedness or summability.
| https://mathoverflow.net/users/70190 | A term for sequences whose mean is defined? | The standard term is *Cesàro summable*, named after [Ernesto Cesàro](https://en.wikipedia.org/wiki/Ernesto_Ces%C3%A0ro). Note that a convergent sequence is also Cesàro summable (with the same limit), but the converse does not always hold.
**Edit.** I realize that there is some confusion, thanks to the comments of Hu... | 10 | https://mathoverflow.net/users/2233 | 219198 | 102,892 |
https://mathoverflow.net/questions/219196 | 23 | Let $q$ be a number. Let us consider the $q^2-1$-th line of the Pascal triangle (i.e. numbers ${{q^2-1} \choose i}$, $i=0,1,...q^2-1$). We have $q^2$ numbers.
Let us form naively a $q \times q$ matrix from them. Example for $q=2$:
\[\begin{pmatrix} 1 & 3 \\
3 & 1 \end{pmatrix}\]
And let us take its determinant: ... | https://mathoverflow.net/users/80668 | On determinants formed by binomial coefficients | Looking at [Krattenthaler's famous "determinant calculus" survey](http://arxiv.org/abs/math/9902004) (also, suggested by *Per Alexandersson*) we find Theorem 26 in there, which answers a more general question, and should (most likely) yield the claim in the OP.
>
> **Theorem** Let $q$ be a nonnegative integer, and ... | 26 | https://mathoverflow.net/users/8430 | 219209 | 102,898 |
https://mathoverflow.net/questions/219227 | 3 | Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar problem. Let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves on $X$, $Z$ a closed subset of $X$, could we always extend a ma... | https://mathoverflow.net/users/24965 | For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space? | I don't think this works out for soft sheaves, and I think the following is an example: Let $X$ be the real line, and let $Z$ be the origin. Let $\mathcal F$ be the skyscraper sheaf on $X$ with stalk $\mathbb Z$ at the origin. I believe that should be soft (any section on any subset is determined by its germ at the ori... | 4 | https://mathoverflow.net/users/6646 | 219232 | 102,907 |
https://mathoverflow.net/questions/219220 | 6 | Let $(M,g)$ be a Riemannian manifold. Then there is the well-known
Sasaki metric that makes $(TM,\hat{g})$ a Riemannian manifold. In a
similar way, one can construct a Sasaki metric $\bar{g}$ on the
cotangent bundle. My question is, is the map
$g\colon (TM,\hat{g}) \to (T^\*M,\bar{g})$ an isometry?
For completeness, ... | https://mathoverflow.net/users/3928 | Are the Sasaki metrics on tangent and cotangent bundle isomorphic? | The answer is yes, and you can see this by realizing that the vertical (resp. the horizontal) subbundle of $TTM$ is mapped to the vertical (resp. the horizontal) subbundle of $TT^\*M.$ For the vertical part, this follows from the fact that $g\colon TM\to T^\*M$ is a bundle homomorphism, and for the horizontal bundle th... | 7 | https://mathoverflow.net/users/4572 | 219233 | 102,908 |
https://mathoverflow.net/questions/219226 | 13 | For any scheme $X$, let $\operatorname{Br}X$ denote the (Azumaya) Brauer group of $X$, namely the Morita equivalence classes of Azumaya $\mathcal{O}\_{X}$-algebras.
>
> Is the functor $$\operatorname{Br} : \operatorname{Sch}^{\operatorname{op}} \to \operatorname{Ab}$$ sending $X \mapsto \operatorname{Br}X$ a sheaf ... | https://mathoverflow.net/users/15505 | Is the Brauer group functor a Zariski sheaf? | No, $\mathrm{Br}$ is not a Zariski sheaf: it is possible for a non-trivial Azumaya algebra on a variety to become trivial when restricted to a Zariski cover. This can happen even for a normal surface with rational singularities. Some references:
* M. Ojanguren, "A non-trivial locally trivial algebra", *J. Algebra* 29... | 17 | https://mathoverflow.net/users/3753 | 219234 | 102,909 |
https://mathoverflow.net/questions/219260 | 3 | I'm reading the following paper on universality considerations in VLSI circuits
<http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf>
In Theorem 2 On the second page it states there exists constants $c\_{1},c\_{2}$ such that $c\_{1}n^{2}\leq X\ll(n) \leq c\_{2}n^{2}$
In the construction it states $9n^{2}$ ... | https://mathoverflow.net/users/54239 | Construction of planar embedding | Here are some more details. We will prove the following stronger claim.
**Theorem.** Let $G$ be a planar graph with $n$ vertices and maximum degree 4. For every planar embedding $\Gamma$ of $G$, there is a rectilinear embedding $\Gamma'$ of $G$ such that $\Gamma$ and $\Gamma'$ have the same outer face, and $\Gamma'$... | 4 | https://mathoverflow.net/users/2233 | 219269 | 102,918 |
https://mathoverflow.net/questions/219255 | 5 | I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra over a simplicial ring.
2. Let $A$ be a simplicial commutative ring.
Is the category of simplicial modules over $A$ abel... | https://mathoverflow.net/users/78856 | Basic questions about simplicial commutative rings | Yes, there are many references that discuss this. The first is Quillen's *Homotopical Algebra*. Chapter II contains much of what you're asking about, especially II.4 and II.6
For a more modern version, see Schwede-Shipley *Algebras and Modules in Monoidal Model Categories*. Section 5 is all examples and contains prec... | 6 | https://mathoverflow.net/users/11540 | 219271 | 102,919 |
https://mathoverflow.net/questions/219248 | 5 | In Quantum Physics one often has to deal with commutators.
Here I want to denote by $H\_0$ the set of all hermitian matrices with trace equal to zero!
One can easily relate it to $\mathfrak{su}(N)=iH\_0.$
Now, my question is basically if there has been a detailed study of the centralizer in $H\_0$ or $\mathfrak{su... | https://mathoverflow.net/users/80695 | Centralizer of hermitian matrices with zero trace | As you have pointed out, one can work in the Lie algebra ${\frak{su}}(N)$. You want to classify the centralizers of various subspaces of this Lie algebra, and without loss of generality you can assume that the subspace is a Lie subalgebra, ${\frak{m}}$. Its action on $W={\Bbb{C}}^N$ preserves the inner product, hence c... | 2 | https://mathoverflow.net/users/5740 | 219277 | 102,921 |
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