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https://mathoverflow.net/questions/129485 | 0 | I was reading <http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials/Bernstein-Polynomials.html>, the section on the derivative, which gives the derivative for a Bezier curve as:
$B\_{k,n}(t) \frac{d}{dt} = n \left ( B\_{k-1,n-1}(t) - B\_{k,n-1}(t)\right )$
which it then shows the derivation steps f... | https://mathoverflow.net/users/31558 | Deriving the rule for direct differentiating Bezier functions | The minus sign should be in the second line, due to the chain rule:
$B\_{k,n}(t) \frac{d}{dt} = {n \choose k} t^k (1-t)^{n-k} \frac{d}{dt}$
... = ${n \choose k} \cdot f(t) g(t) \frac{d}{dt} $
... = ${n \choose k} \cdot \left ( f'(t) g(t) + f(t)g'(t) \right )$
Due to the chain rule applying to $g(t)$, its deriva... | 0 | https://mathoverflow.net/users/31558 | 129730 | 72,066 |
https://mathoverflow.net/questions/129538 | 2 | Edited after @J. Martel's comment: Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$). We know that if $X\_i$ represent the vector fields on $S^2$ giving the rotation about the $x\_i$-axis, then the Laplacian on $S^2$ is given by $X^2\_1 + X^2\_2 + X^2\_3$. My question is:
Can we have a similar express... | https://mathoverflow.net/users/33655 | Laplacian on coset spaces | I'll give an answer in a few parts:
First, for the $n$-sphere $S^n\subset\mathbb{R}^{n+1}$: The obvious thing to do is to consider the vector fields
$$
X\_{ij} = x\_i\frac{\partial }{\partial x\_j} - x\_j\frac{\partial }{\partial x\_i}\ ,
\qquad 0\le i < j\le n.
$$
Then one easily computes that the operator
$$
L = \s... | 1 | https://mathoverflow.net/users/13972 | 129747 | 72,074 |
https://mathoverflow.net/questions/129745 | 0 | Let F(x) be a formula belonging to the language of first order ZF in which x is the one and only
variable that occurs free and let N(x) be the negation of F(x). Are any examples known of an F(x)
and an N(x) such that not just one but each one of the two sentences "The set (x:F(X)) exists" and
"The set (x:N(x)) exists"-... | https://mathoverflow.net/users/4423 | A question about "paradoxical" sentences in the language of ZF set theory. | Here are two examples:
1. Let $F(x)$ be the statement "There exists $y$ such that $x$ is the power set of $y$", or formally, $$F(x)=\exists y\forall u(u\subseteq y\leftrightarrow u\in x).$$
Since every set has a (unique) power set, we have that the class of power sets is not a set; but also the class of those which... | 4 | https://mathoverflow.net/users/7206 | 129749 | 72,076 |
https://mathoverflow.net/questions/129752 | 9 | A version of the "hairy ball" theorem, due probably to Chern, says that the Euler-characteristic of a closed (i.e. compact without boundary) manifold $M$ can be computed as follows. Choose any vector field $\vec v \in \Gamma(\mathrm T M)$. If $p$ is a zero of $\vec v$, then the matrix of first derivatives at $p$ makes ... | https://mathoverflow.net/users/78 | Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field? | Yes, if M is closed and connected and χ(M)=0, then M admits a nowhere-vanishing vector field.
Start with generic vector field, it has zero of index $\pm 1$.
Two zeros of opposite sign can kill each other (maybe it is called Whitney trick?).
So you get a field with zeros of the same sign.
The result follows since th... | 17 | https://mathoverflow.net/users/1441 | 129757 | 72,080 |
https://mathoverflow.net/questions/129755 | 3 | Using a simple java application, I have noticed that for $x > 25$:
$$\psi\left(\frac{x}{5}\right) \ge \psi\left(\frac{x}{3}\right) - \psi\left(\frac{x}{4}\right)$$
where:
$$\psi\left(x\right) = \sum\_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$
and
$$\vartheta\left(x\right) = \sum\_{p \le x} \log p$$
... | https://mathoverflow.net/users/15915 | A question about the second Chebyshev function $\psi(x) = \sum_{m=1}^{\infty}\vartheta(\sqrt[m]{x})$ | The inequality is simpler than PNT but also I would not consider it as straightforward; what seems to be needed are results slightly weaker than those of Chebyshev.
Suppose we can bound $c\_1 y < \psi(y) < c\_2 y$. Then for your inequality it would suffice to have
$$
c\_1/5 \ge c\_2/3 - c\_1/4
$$
which translates... | 5 | https://mathoverflow.net/users/nan | 129761 | 72,082 |
https://mathoverflow.net/questions/129758 | 10 |
>
> Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with oriented fibers)?
>
>
>
For example, if $M \to N$ is a fiber bundle of compact manifolds with fiber $F$, then their Euler characteristics ... | https://mathoverflow.net/users/78 | When does an even-dimensional manifold fiber over an odd-dimensional manifold? | Being the total space of a fiber bundle is not invariant under homotopy equivalences, so I doubt there is a criterion of the type you state (eg in terms of homological invariants).
However, one can say something a little weaker. If a manifold $M^k$ fibers over a manifold $N^{\ell}$, then the fibers form a codimension... | 10 | https://mathoverflow.net/users/317 | 129774 | 72,090 |
https://mathoverflow.net/questions/129741 | 5 | I encountered an inequality when reading a paper. Can someone help to show how to prove it?
Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of matrix }A\}$. For matrices $S$ and $T$ with positive spectral radii, and two arbitrary real positive numbers $a$ and $b$, ... | https://mathoverflow.net/users/32660 | spectral radius monotonicity | Not true in general, as noted by @SergeiIvanov, but true for (element-wise) nonnegative matrices.
Note that if $\rho(S) < b$, then $b(bI-S)^{-1}=(I-\frac{S}{b})^{-1}=\sum\_{i=0}^\infty \frac{S^i}{b^i}$. In particular, thanks to this expansion, if $\rho(S)< a < b$, then $b(bI-S)^{-1}< a(aI-S)^{-1}$ in the componentwis... | 9 | https://mathoverflow.net/users/1898 | 129789 | 72,099 |
https://mathoverflow.net/questions/129762 | 56 | I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about Algebraic $K$-theory mention that we don't know $K(\mathbb{Z})$ and that somehow $K(\mathbb{Z})$ is worth computing becaus... | https://mathoverflow.net/users/11540 | What arithmetic information is contained in the algebraic K-theory of the integers | $\newcommand\Z{\mathbf{Z}}$
$\newcommand\Q{\mathbf{Q}}$
I'm a number theorist who already thinks of the algebraic $K$-theory of $\Z$ as part of number theory anyway, but let me make some general remarks.
**A narrow answer**: Since (following work of Voevodsky, Rost, and many others) the $K$-groups of $\Z$ may be id... | 68 | https://mathoverflow.net/users/nan | 129791 | 72,100 |
https://mathoverflow.net/questions/129792 | 4 | Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply
$$\sum\_{j=0}^k\frac{(-1/2)\_{k-j}(m+1)\_j(-m)\_j}{(1/2)\_j(k-j)!j!}
=\frac{(-m-\frac{1}{2})\_k(m+\frac{1}{2})\_k}{(\frac{1}{2})\_kk!}$$
where $a\_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer's symbol.
| https://mathoverflow.net/users/10583 | Hypergeometric identities | Treating both sides as coefficients of $x^k$ in a power series with variable $x$, the left-hand side turns into a power-series product, one factor representing $\sqrt{1-x}$. We end up with a special case of the known hypergeometric identity
$$(1-x)^{a+b-c} {}\_2F\_1(a,b;c;x) = {}\_2F\_1(c-a,c-b;c;x)$$
with $a = m+1$, $... | 10 | https://mathoverflow.net/users/43871 | 129795 | 72,103 |
https://mathoverflow.net/questions/129218 | 11 | My question is simple:
Given an algebraic group $G$ acting on a variety $X$ algebraically. If the orbits are of finite number then they form what is called an algebraic stratification of $X$.
Now my question is: **Is this (the stratification by $G$-orbits) also a Whitney Stratification?**
I think showing this by ... | https://mathoverflow.net/users/32972 | Algebraic Stratifications of $G$-varieties | So Ulrich and Geordie were right, Tom Braden was the right person to ask and here is what he told me:
The answer is yes, in the case above $X$ is Whitney stratified.
The argument goes roughly as follows. In the paper [K] the following is shown. Let $X$ be an algebraic stratified variety and $ S$ a stratum. The set ... | 4 | https://mathoverflow.net/users/32972 | 129806 | 72,109 |
https://mathoverflow.net/questions/129697 | 4 | I'm have difficultly nailing down the direction of some implications. For $2 \leq q < \infty$, there are (at least) two ways to say that a Banach space $B$ has "cotype $q$".
* $B$ has **cotype** q.
* $B$ is isomorphic to a $q$-**uniformly convex Banach space**, i.e. a uniformly convex Banach space with a "power type"... | https://mathoverflow.net/users/12978 | Martingale-cotype vs cotype on super-reflexive spaces | I'll attempt to answer this with what I've found:
---
(1) The answer to my main question is that it is not true. In this [document of Pisier's](http://students.mimuw.edu.pl/~tt249057/other/figel_fulltext.pdf), he states
>
> It is possible to find a uniformly convex space $B$ for which the index of type $p(B)$... | 3 | https://mathoverflow.net/users/12978 | 129809 | 72,112 |
https://mathoverflow.net/questions/129811 | 7 | Let $X\_0$ be a smooth variety (for simplicity I'm willing to assume that X is a curve) over a finite field $k$, $X$ its geometric base change, and $\mathcal{F}$ an $l$-adic etale sheaf on $X$ with $\mathcal{F}(m)$ its Tate-twist.
>
> What is the relation between the weights of Frobenius on $H^r(X,\mathcal{F})$ an... | https://mathoverflow.net/users/448 | Frobenius weights on etale cohomology and purity | You remember correctly, that the effect of Tate twisting the sheaf $\mathcal F$ is just to multiply the Frobenius eigenvalues by a factor of $q$. This result is not surprising because the Tate twist $\mathcal F(m)$ has no relation to the Serre twist on a projective variety, they are just denoted the same way.
The an... | 8 | https://mathoverflow.net/users/1310 | 129813 | 72,113 |
https://mathoverflow.net/questions/129547 | 9 | The paper I'm referring to can be found [here](https://arxiv.org/abs/math/0206094), and is absolutely seminal work in area of homotopy theory for operads. It came out in 2003. I remember someone once told me there was a mistake somewhere in this paper, but I can't remember where.
>
> Is there some error in this pap... | https://mathoverflow.net/users/11540 | Are there any errors in Axiomatic Homotopy Theory for Operads? | The only errors that I am aware of in this paper are the following:
* A $G$ missing in the statement of Lemma 5.10 (pointed out by the authors after Lemma 2.5.3 in [1])
* A small mistake in the proof of Proposition 5.1 (also pointed out by the authors and fixed in the Appendix of [2])
[1] I. Moerdijk and C. Berger,... | 10 | https://mathoverflow.net/users/33739 | 129817 | 72,114 |
https://mathoverflow.net/questions/129804 | 17 | Let $\mathcal{C}$ be the field of constructible numbers, that is, the complex numbers constructible by compass and straightedge. It can be shown that it consists of all the numbers obtainable by adding square roots iteratively to the rational numbers, and it's easy to see that the extension $\mathcal{C}/\mathbb{Q}$ is ... | https://mathoverflow.net/users/4619 | Galois group of constructible numbers | It is always a good idea to look at the local question first. What is the maximal pro-$p$ extension of $\mathbf{Q}\_p$ ? There is a vast literature on the subject, starting with Demushkin whose results are exposed in an old Bourbaki talk by Serre on the *Structure de certains pro-p-groupes (d'après Demuškin),* (<http:/... | 17 | https://mathoverflow.net/users/2821 | 129828 | 72,118 |
https://mathoverflow.net/questions/129508 | 3 | Hello. I have some question on Cartan decomposition of unitary group, especially $U(2)$.
I am interested in local situation, that is p-adic or archimedian.
Let $F$ be a local field and $E$ be its quadratic extension. Let $V$ be a hyperbolic hermitian vector space of dimension $2$ and we consider $U(V)$.
Then by C... | https://mathoverflow.net/users/29334 | On the Cartan decomposition of unitary group | *Theorem*: Let $G$ be a reductive algebraic group over a local field $F$, let $K$ be any maximal compact subgroup of $G(F)$, and let $Z = Z(G)$. Then $K \cap Z(F)$ is the unique maximal compact subgroup of $Z(F)$.
*Proof*: Let $W$ be the maximal compact subgroup of $Z(F)$. Then the natural multiplication map $K \time... | 4 | https://mathoverflow.net/users/2481 | 129829 | 72,119 |
https://mathoverflow.net/questions/129823 | 6 | It is customary to define the class of [partial recursive functions](https://en.wikipedia.org/wiki/Mu-recursive_function) by taking the set of [primitive recursive functions](https://en.wikipedia.org/wiki/Primitive_recursive) $PR$ and taking closure over unbound search operation.
Do we need the "whole" set of primiti... | https://mathoverflow.net/users/nan | Smallest base to reach partial recursive functions as a closure of unbound search | $\def\dotm{\mathbin{\scriptstyle\dot{\smash{\textstyle-}}}}$It follows from the MRDP theorem that every partial recursive function $f(\vec x)$ can be written as
$$f(\vec x)\simeq l\bigl(\mu z\,[p(\vec x,l(z),l(r(z)),\dots,l(r^{k-1}(z)),r^k(z))=0]\bigr),$$
where $p(\vec x,y\_0,\dots,y\_k)$ is a polynomial with integer c... | 9 | https://mathoverflow.net/users/12705 | 129835 | 72,120 |
https://mathoverflow.net/questions/129831 | 3 | Let $\mathcal{I}$ be a proper [ideal](http://en.wikipedia.org/wiki/Ideal_%28set_theory%29) on $\omega$. If $\mathcal{I}$ is Borel as a subset of $2^\omega$, does it follow that $\mathcal{I}$ is meager?
**Edit:** What if $\mathcal{I}$ contains all finite subsets of $\omega$?
| https://mathoverflow.net/users/33743 | Borel ideals on $\omega$ are meager? | Let us consider a prime ideal J containing I. Suppose I is non meager. Then J would have non empty interior (modulo meager). Since complementing a set of integers is a meager preserving operation so the dual ultrafilter U of J has the same property. But both J and U are closed under rational translations and they both ... | 2 | https://mathoverflow.net/users/2689 | 129841 | 72,123 |
https://mathoverflow.net/questions/129833 | 9 | It is known that a 2-connected closed smooth 6-manifold is homeomorphic to S^{6}
or connected sum of (S^{3}xS^{3}). My question is whether we have a similar statement for (n-1)-connected closed smooth 2n-manifold (at least when n=4). If not, do we have a clear classification theorem?
| https://mathoverflow.net/users/26222 | Classification of higher dimensional manifolds | There is a series of papers from the mid-1960's by C.T.C. Wall on classification of highly connected smooth manifolds, starting with Wall, C. T. C., Classification of (n−1)-connected 2n-manifolds. Ann. of Math. vol. 75 1962 163–189. I think this paper answers your question.
The Math Review of this paper by Kervaire s... | 10 | https://mathoverflow.net/users/3460 | 129842 | 72,124 |
https://mathoverflow.net/questions/129802 | 0 | We have known from Ehrhart theory that if $P$ is a $d$-dimensional polytope of $\mathbb R^n$ which has integer vertices then the number of integer points in $nP$ is a polynomial of degree $d$. We also know the leading coefficients, the second and the constant coefficients.
I wonder if we have a similar conclusion in th... | https://mathoverflow.net/users/33219 | Counting integer points in a Minkowski sum | Proving this is actually problem 3 on page 164 of Integer Points in Polyhedra by Alexander Barvinok - the number of integer points is a polynomial in $t\_1,...,t\_k$ as long as they are non-negative integers.
Proof omitted at the moment because I'm too rusty to produce one
| 2 | https://mathoverflow.net/users/3669 | 129845 | 72,126 |
https://mathoverflow.net/questions/129820 | 1 | $f(n+1) = f(n) + f(n)^{a}$ where $a \in (0,1)$ and $n \ge 1$ with $f(1) = m$.
If $a=0$, we see $f(n) = m + n - 1$ and if $a=1$, we see $f(n) = 2^{n-1}m$. So the recursion seems to interpolate between linear and exponential forms.
Is there a closed form for $f(n)$ in terms of $n$, $a$ and $m$?
| https://mathoverflow.net/users/10035 | Closed form solution to an iterative equation. | There is no closed form except for the cases $a=0,1$. But you can find the asymptotic behavior.
See, for example Fatou, Sur les equations fonctionnelles, Bull Soc. Math. France, 47 (1919),
section 8 and further. Available here:
<http://archive.numdam.org/ARCHIVE/BSMF/>
BSMF\_1919\_*47*/BSMF\_1919\_*47*\_161\_0/BSMF\_... | 2 | https://mathoverflow.net/users/25510 | 129848 | 72,129 |
https://mathoverflow.net/questions/94250 | 2 | Let us consider $S(M) = \{(f\_0, f\_1) | f\_0, f\_1: M \rightarrow M\}$, where $M$ is a finite set. Each element of $S(M)$ is equivalent to a finite directed
graph with the set of nodes $M$, which has exactly two arrows
from each node, the arrows being labeled $0$ and $1$.
Then, the simplest operations on those graph... | https://mathoverflow.net/users/22795 | Universality of blind graph rewriting | The construction is similar to that of [Schönhage's Storage Modification Machine (SMM) model](http://en.wikipedia.org/wiki/Pointer_machine#Sch.C3.B6nhage.27s_Storage_Modification_Machine_.28SMM.29_model).
| 2 | https://mathoverflow.net/users/22795 | 129852 | 72,131 |
https://mathoverflow.net/questions/124604 | 3 | Let $N$ be the set of natural numbers, $S$ be the set of finite binary sequences, and
$Q = [N \rightarrow N] \times [N \rightarrow N],$
where $[N \rightarrow N]$ is the set of all computable functions on natural numbers.
Then, let us consider the family of *primitive blind automata*
$A\_p = (Q, \{p\}, Q, \delta... | https://mathoverflow.net/users/22795 | Turing-complete primitive blind automata | The construction is similar to that of [Schönhage's Storage Modification Machine (SMM) model](http://en.wikipedia.org/wiki/Pointer_machine#Sch.C3.B6nhage.27s_Storage_Modification_Machine_.28SMM.29_model).
| 0 | https://mathoverflow.net/users/22795 | 129853 | 72,132 |
https://mathoverflow.net/questions/129859 | 0 | Let $R$ be an associative and non-unital ring. (Suppose that $R$ is $s$-unital, i.e. for each $x\in R$ there is $u,v\in R$ such that $ux=xv=x$.)
It is not difficult to show that if $R$ is a simple ring, then $Z(R)=\{ 0 \}$. Thus, non-unital simple rings are in some sense "extremely" non-commutative.
Are there any (... | https://mathoverflow.net/users/11143 | Non-simple and non-unital rings with trivial centres | Take the semigroup ring $\mathbb{Z}S$ where $S=${ $a,b,c$} with multiplication $aS=bS=a, cS=c$. The elemwents $a$ and $c$ generate an ideal.
| 3 | https://mathoverflow.net/users/18814 | 129861 | 72,135 |
https://mathoverflow.net/questions/129858 | 0 | Hello
Call $Y^4$ a conifold which satisfies the following condition:
$\mathfrak{Y}(z):=\sum\_{\alpha=1}^{3}(z\_{\alpha})^{2}=0,$
where $z\_\alpha \in \mathbb{C}$. Now intersect $Y^4$ with $S^5$ to get a compact manifold called "base", $X^3$, which is $3$ dimensional just like $S^3$. We have a Hopf fibration rega... | https://mathoverflow.net/users/33483 | Homeomorphism between base of conifolds and spheres | Your space X is $\mathbf{RP}^3$. To see this, blow-up the origin. The proper transform of Y is then the total space of the bundle $\mathcal{O}(-2)\to\mathbf{CP}^1$. The unit circle bundle (your space X) is then $\mathbf{RP}^3$. One can see this by taking the fibrewise double cover to get the bundle $\mathcal{O}(-1)$. O... | 1 | https://mathoverflow.net/users/10839 | 129864 | 72,136 |
https://mathoverflow.net/questions/129866 | 2 | Assume we have a centrally symmetric convex set $K \subset \mathbb{R}^n$ such that Vol(K)=1. In addition, assume that for every direction $u$ we know that $Vol(K \Delta R\_u(K)) < \epsilon$, where $A \Delta B$ is the symmetric difference and $R\_u(K)$ denotes the reflection of $K$ with respect to $u^\perp$.
Does this i... | https://mathoverflow.net/users/33751 | Measuring the distance of a convex set from a ball (Nikodym distance) | Let $r$ be the radius of the maximal ball contained in $K$ and $R$ the radius of the minimal ball containing $K$. I claim that $\frac rR>1-2\epsilon^{1/n}$. It follows that the Hausdorff distance from $K$ to a ball is bounded by $C\_n\epsilon^{1/n}$. The argument does not use symmetry (although the constant can be impr... | 3 | https://mathoverflow.net/users/4354 | 129872 | 72,140 |
https://mathoverflow.net/questions/129720 | 5 | Let $\mathcal{P}\_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furthermore let $$ \mathcal{A} := \{ f \in X^{\mathcal{P}\_0(X)} : \ \forall A \in \mathcal{P}\_0(X): f(A) \in A\} $$ be the set of the functions mapping subsets of the ... | https://mathoverflow.net/users/31439 | Showing a filter with a certain property on the power set of $\mathbb{Z}$ is a one point filter | For this problem, we shall use $\mathbb{N}$ instead of $\mathbb{Z}$ since $\mathbb{N}$ is easier to work with in this case. We shall say that $A$ has property $P$ almost everywhere (or for almost all $A$) abbreviated a.e. if $\{A\in P\_{0}(X)|A\,\textrm{has property}\,P\}\in\varphi$.`
For $n>0$, let $f\_{n}\in\mathca... | 7 | https://mathoverflow.net/users/22277 | 129885 | 72,144 |
https://mathoverflow.net/questions/129850 | 6 | It is known that an intertwining (or even approximately intertwining) diagram implies the isomorphism of the limit algebras. Under what conditions the converse holds?
| https://mathoverflow.net/users/33724 | Inductive limit of C*-algebras | The converse holds if the inductive limits involve semiprojective building blocks. That is: suppose
$$\varinjlim (A\_i,\phi\_i^{i'}) \cong \varinjlim (B\_j,\psi\_j^{j'})$$
(here my notation is $\phi\_i^{i'}:A\_i \to A\_{i'}$, etc.), and each $A\_i$ and $B\_j$ are separable and semiprojective, then there exists subseque... | 7 | https://mathoverflow.net/users/22052 | 129887 | 72,145 |
https://mathoverflow.net/questions/129843 | 3 | Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true:
\begin{equation}
\frac{1}{b-a} \int\_a^b u(x) \ \rm{d} x = \frac{1}{u(a)-u(b)} \int\_{u(b)}^{u(a)} x \ \rm{d} x.
\end{equation}
I would like to prove that, if $f$ ... | https://mathoverflow.net/users/23017 | Integral inequality for convex function | The inequality is false in general. To see this, take $a=0,$ $b=1,$ $f(x)=x^2.$ Let $u(0)=0$
and $u(1)=1.$ Then our problem can be reformulated as follows: given that $\int\_{0}^1u(x)dx=\frac{1}{2}$ show that $\int\_{0}^1u^2(x)dx\ge\frac{1}{3}.$ Now take $u(x)=x+th(x)$ where $h(0)=h(1)=0$ and $\int\_{0}^1h(x)dx=0$ to e... | 3 | https://mathoverflow.net/users/17503 | 129892 | 72,146 |
https://mathoverflow.net/questions/129878 | 1 | Assume the $1 \times N$ vector
$\mathbf X = [X\_1, X\_2, \ldots , X\_N]$
contains i.i.d. normal samples such that $\mathbf X$ has a multivariate normal distribution. Now assume another random variable $Y$ also has a normal distribution, independent from the samples in $\mathbf X$, such that $Y$ has a mean and varia... | https://mathoverflow.net/users/23900 | Probability that one RV will exceed many others | Assume without loss of generality that each $X\_i$ is standard normal and that $Y$ is normal with mean $\mu$ and variance $\sigma^2$. By definition, for every $y$,
$$
P[y\gt\max(X\_1,\ldots,X\_N)]=\Phi(y)^N,
$$
hence, by the independence of $Y$ from $(X\_i)\_i$,
$$
P[Y\gt\max(X\_1,\ldots,X\_N)]=E[\Phi(Y)^N]=\int\_\math... | 4 | https://mathoverflow.net/users/4661 | 129894 | 72,148 |
https://mathoverflow.net/questions/129896 | 17 | Suppose
* $T$ is a [totally ordered set](http://en.wikipedia.org/wiki/Total_order) without a [maximal element](http://en.wikipedia.org/wiki/Maximal_element),
* $\tau$ is the [order type](http://en.wikipedia.org/wiki/Order_type) of $T$,
* $S$ is the set of order types of all proper [initial segments](http://en.wikiped... | https://mathoverflow.net/users/33754 | Is it possible to reconstruct an order type from its initial segments? | The answer is **No**.
Let $\eta$ be the order type of $\mathbb{Q}$, and $\omega\_1$ - the order type of the set of countable ordinals. The order types $\eta$ and $\eta \cdot \omega\_1$ are different (because they have different cardinality), but the set of order types of all proper initial segments of some instances ... | 15 | https://mathoverflow.net/users/9550 | 129898 | 72,151 |
https://mathoverflow.net/questions/129883 | 4 | Denote by $F\_n$ the free group of rank $n$. We say that an automorphism $\phi\in Aut(F\_n)$ is *geometric* if there exists a surface with boundary $M$ and a homeomorphism $h\colon M\to M$ such that $h$ induces $\phi$ on $\pi\_1$.
Every automorphism of $F\_2$ is geometric, but in higher rank geometric automorphisms a... | https://mathoverflow.net/users/12996 | Periodic automorphisms of free groups and surface homeomorphisms | Assuming that you mean outer automorphisms of $F\_n$ (as per my comment), the answer is that if $n \ge 3$ then there is a nongeometric finite order element of $Out(F\_n)$, for example the outer automorphism class of $a\_1 \mapsto a\_1^{-1}$ and $a\_i \mapsto a\_i$ for $2 \le i \le n$.
For the proof, given $h : M \to... | 7 | https://mathoverflow.net/users/20787 | 129899 | 72,152 |
https://mathoverflow.net/questions/129890 | 3 | Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that $\rho((hI-S)^{-1}T) < 1$.
A hint is $hI-S$ are invertible, and $hI-(S+T)=(hI-S)(I-(hI-S)^{-1}T)$. Since $hI-(S+T)$ is invertible, $I-(hI-S)^... | https://mathoverflow.net/users/32660 | A spectral radius inequality | Since $(hI-S)^{-1} = \frac{1}{h} \sum\_{k=0}^{\infty}\left(\frac{S}{h}\right)^k$, $(hI-S)^{-1}T$ is non-negative. From the Perron-Frobenius theorem spectral radius is equal to the greatest (positive) eigenvalue. It is then enough to prove that for $\lambda \geq 1$ the matrix $\lambda I - (hI-S)^{-1}T$ is invertible. Bu... | 2 | https://mathoverflow.net/users/24953 | 129900 | 72,153 |
https://mathoverflow.net/questions/129886 | 8 | All,
I'm wondering if anyone can point me to a reference on how to address the following problem.
In my thesis work on lattice QCD many years ago I had to enumerate all possible paths of a given length on a
2D lattice, up to the symmetries of reflection and rotation (plus some other internal symmetries we won't
con... | https://mathoverflow.net/users/8955 | Enumerating/counting paths of a given length on a 2D lattice | I am trying to make sense of your problem. From what I can gather you are talking about walks with steps of size $1$ in each of the cardinal directions East, West, North, South, (E,W,N,S). To use your notation $E=x$, $W=X$, $N=y$, $S=Y$. You are not allowed to immediately backtrack, i.e., if say you take an $E$-step, t... | 5 | https://mathoverflow.net/users/20302 | 129903 | 72,155 |
https://mathoverflow.net/questions/129854 | 4 | In connection with string theory I encountered the following problem:
Given the set M\_N of all semi-standard Young tableaux
of size N (i.e. all fillings of Ferrers diagrams with natural
numbers with weakly increasing rows and strictly increasing columns,
the content of which sum to N).
The conjecture, which I cann... | https://mathoverflow.net/users/33746 | Semi-Standard Young Diagrams and Families | I use standard facts about symmetric functions that can be found, e.g,
in Chapter 7 of *Enumerative Combinatorics*, vol. 2. Let $s\_\lambda$
denote a Schur function and $p\_1=s\_1=x\_1+x\_2+\cdots$. Then
$\frac{\partial s\_\lambda}{\partial p\_1}=\sum\_\mu s\_\mu$,
where the $\mu$'s are obtained by removing a single bo... | 10 | https://mathoverflow.net/users/2807 | 129913 | 72,161 |
https://mathoverflow.net/questions/129914 | 13 | Define harmonic numbers for a complex argument $z$ as $H\_z=\frac{\Gamma'(z+1)}{\Gamma(z+1)}-\Gamma'(1)$.
For $n\in\mathbb{N}$, $H\_n$ are usual harmonic numbers $\sum^n\_{k=1} k^{-1}$ . They are obviously rational and are known (Taeisinger 1915) to be non-integers for $n>1$.
*Question:* Is there a non-integer rati... | https://mathoverflow.net/users/33664 | Can a harmonic number be a rational number for non-integer rational argument? | The answer is "no". Your function $H\_z$ which is the same as $\psi(z+1)+\gamma$, where $\psi$ is the [digamma function](http://en.wikipedia.org/wiki/Digamma_function), and $\gamma$ is the Euler-Mascheroni constant, takes transcendental values at non-integer rationals. This is a theorem of M. Ram Murty and N. Saradha, ... | 11 | https://mathoverflow.net/users/2384 | 129916 | 72,162 |
https://mathoverflow.net/questions/129912 | 4 | Does
$$\frac{1}{N^2}\sum \_{d=1}^N \log d \sum \_{n=1}^{N/d} \frac{\phi(n)}{\log (dn)},$$
converges or not when $N$ goes to infinity?
| https://mathoverflow.net/users/33766 | Average involving the Euler phi function | The expression converges to $0$, even when $\phi(n)$ is replaced by the larger $n$. The contribution from $n\le\sqrt N/d$ can be given by ignoring the logarithm in the denominator:
\begin{align\*}
\frac1{N^2} \sum\_{d=1}^N \log d \sum\_{n=1}^{\sqrt N/d} \frac{\phi(n)}{\log dn} &\le \frac1{N^2} \sum\_{d=1}^N \log d \sum... | 9 | https://mathoverflow.net/users/5091 | 129921 | 72,164 |
https://mathoverflow.net/questions/69467 | 22 | There are probably dozens of [ways of defining "ultrafilter"](http://golem.ph.utexas.edu/category/2011/07/definitions_of_ultrafilter.html). The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter.
But there's another, shorter way to state the definit... | https://mathoverflow.net/users/586 | An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request | You can find that characterization (even with n = 3), as well as the generalization to $\kappa$-complete ultrafilters, in: Fred Galvin and Alfred Horn, Operations preserving all equivalence relations, Proc. Amer. Math. Soc. 24 (1970), 521-523.
| 12 | https://mathoverflow.net/users/nan | 129930 | 72,167 |
https://mathoverflow.net/questions/129915 | 3 | In usual formulation of conjugate gradient algorithm initial search direction is taken to be the residual (so residual and search direction spans Krylov subspace). However, in cases where inexact Krylov is used, search direction is not anymore in span of Krylov subspace. I am in process of analyzing convergence of one ... | https://mathoverflow.net/users/22429 | Conjugate gradient algorithm where first search direction is not equal to residual | I recommend Simoncini and Szyld, Theory of inexact Krylov subspace methods and applications to scientific computing, SIAM J. Sci. Comput., 25 (2003) pp. 454-477. It looks like it discusses exactly what you want.
| 1 | https://mathoverflow.net/users/17113 | 129937 | 72,170 |
https://mathoverflow.net/questions/129938 | 3 | Probably this is well know and elementary and will delete it, but couldn't find it on the web.
Got a sketch of proof and numerical evidence that
$\zeta(2k+1)$ is a rational multiple of $\pi^{2k} \zeta'(-2k)$
An identity from [Derivatives of the Hurwitz Zeta Function for Rational Arguments](http://citeseerx.ist.psu... | https://mathoverflow.net/users/12481 | zeta(2k+1) is a rational multiple of pi^{2k} zeta'(-2 k) ? | I am also sure that this is all known, but here is a quick proof that
>
> $$ \zeta(2k+1)=\frac{(-1)^k2^{2k+1}}{(2k)!}\pi^{2k}\zeta'(-2k). $$
>
>
>
Following Carl Dettmann's suggestion, let us start from the functional equation
$$ \pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\pi^{-\frac{1-s}{2}}\Gam... | 12 | https://mathoverflow.net/users/11919 | 129950 | 72,173 |
https://mathoverflow.net/questions/129939 | 1 | Let $E$ be a Banach space, $X\_1, X\_2, \ldots $ be a numerable collection of
finite dimensional subspaces $X\_1\subset X\_2$ with dimension
tending to infinity, denote by $S^n$ the unit sphere in $X\_n$ .
Denote by $X$ the union of the $S^n$ and let Y be the
completion of the union $\cup X\_n$ in the ambient norm i... | https://mathoverflow.net/users/21985 | Homotopy of Unitary sphere in a Banach space and finite dimensional spheres | Well, it is well-known that both of $X$ and $Y$ are contractible (hence they are homotopy equivalent, this should answer your question). Notice that $Y$ is exactly the set of unit vectors of $E$.
To see that $X$ is contractible consider the map $H : X \times [0, 1] \to X$ defined by $$H((x\_1, \dots, x\_n, 0,\dots), ... | 1 | https://mathoverflow.net/users/23193 | 129951 | 72,174 |
https://mathoverflow.net/questions/129942 | 3 | Hi,
Let $S = R[T\_1,\dots,T\_n]/(f\_1,\dots,f\_r)$ where $\det(\partial f\_i/\partial T\_j)\_{i,j=1,\dots,r}\in S^\times$. Then $S$ is flat over $R$. How to prove it?
I am not looking for an answer like: "$Spec(S)$ is smooth over $Spec(R)$, hence flat.".
Thanks!
| https://mathoverflow.net/users/36285 | How to prove this algebra is flat? | The algebra $S$ is etale, i.e., smooth of relative dimension $0$. A proof, that $S$ is
flat over $R$ can be found in
www.math.purdue.edu/~dvb/preprints/etale.pdf,
see Proposition $1.3.5$ on page $17$. The proof uses induction, and the
following lemma: Let $S$ be an $R$-algebra. Suppose that $M$ is an $S$-module,
$... | 5 | https://mathoverflow.net/users/32332 | 129952 | 72,175 |
https://mathoverflow.net/questions/129949 | 2 | Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $Ann\_R(Ann\_R(r))=Rr$ or $\sqrt{Ann\_R(Ann\_R(r))}=\sqrt{Rr}$ hold?
I already know that it holds for Artinian Gorenstein rings (due to an exercise in Bruns-Herzog) and it seems to be true for $R=\Bbb {Z}/n\Bbb {Z}$.
The question is more interesting... | https://mathoverflow.net/users/21992 | When does "second annihilator" of a (principal) ideal equal the ideal itself , ie $Ann_R(Ann_R(r))=Rr$? | For what it's worth, it suffices for $(r)$ to be an interesection of minimal primes; in fact, more generally if $J$ is any intersection of minimal primes then $J=Annih(Annih(J))$. (This requires only that $R$ be commutative and noetherian; you don't need local.)
This is Lemma 2.17 in an old paper of mine called "Patc... | 2 | https://mathoverflow.net/users/10503 | 129956 | 72,176 |
https://mathoverflow.net/questions/129927 | 6 | Let $G = SL\_2, \mathfrak{g} = \mathfrak{sl}\_2$, $B$ the Borel subgroup, and $\mathfrak{u}$ the unipotent radical; so that $G/B = \mathbb{P}^1$; how does $\widetilde{\mathfrak{g}}$ decompose as a vector bundle over $\mathbb{P}^1$? Recall the definition:
$\widetilde{\mathfrak{g}} = $ {$(X, gB) \in \mathfrak{g}^\* \ti... | https://mathoverflow.net/users/2623 | Computing the Grothendieck-Springer resolution for $G = SL_2$ | Let $\frak{b}\subseteq\frak{g}$ denote the Lie algebra of your Borel $B$. There is a natural $G$-equivariant isomorphism $\tilde{\frak{g}}\cong G\times\_B\frak{b}$ of vector bundles over $G/B$, where $G\times\_B\frak{b}$ is the associated bundle arising from the adjoint representation of $B$ on $\frak{b}$. Let us take ... | 4 | https://mathoverflow.net/users/25358 | 129958 | 72,177 |
https://mathoverflow.net/questions/129857 | 15 | Let $\mathfrak g$ be a finite-dimensional simple Lie algebra over $\mathbb C$.
Theorem 1. The highest root is perpendicular to all but one simple root, except in the case ${\mathfrak g}={\mathfrak sl}\_{n > 2}$, in which case it is perpendicular to all but two (the first and last).
Theorem 2. The space $Hom({\mathf... | https://mathoverflow.net/users/391 | Relating two characterizations of ${\mathfrak sl}_{n > 2}$ among simple Lie algebras | As Evan points out, "modern" technology (including Littelmann paths and canonical bases) provides an improved way to think about tensor product decompositions for simple Lie algebras. But your Theorems 1 and 2 could also be understood in purely classical terms, though I'm not sure how far anyone looked at these. The as... | 5 | https://mathoverflow.net/users/4231 | 129960 | 72,178 |
https://mathoverflow.net/questions/129963 | 0 | Suppose $R$ is a regular local ring and $I$ is a non-zero ideal such that $I$ is a radical ideal and $I$ is height unmixed. Suppose $J$ is any radical ideal contained in $I$ and with the same height as $I$. Can we always find a prime $P\in \operatorname{Ass}(R/J)$ such that $P$ is not minimal.
| https://mathoverflow.net/users/33782 | Height unmixed ideal | $I=(x) \cap (y)$, $J = (x) \cap (y) \cap (z)$.
| 0 | https://mathoverflow.net/users/460 | 129965 | 72,180 |
https://mathoverflow.net/questions/129970 | 1 | Consider three 2-torus ($S^1\*S^1$) living in four space. Can I have links of these objects, which is generalization of links of circles in 3D? If so, how can I judge whether three 2-torus are linked or not?
| https://mathoverflow.net/users/33455 | Questions about knot (link) of surface in four dimension | Yes.
There are a lot of ways to approach this.
Following Roseman, you can take a generic projection of the torus into R^3, with overcrossing and undercrossing data. It will
have double points, triple points and branched points. There is a presentation of
the fundamental group of the complement of the torus using m... | 4 | https://mathoverflow.net/users/4304 | 129973 | 72,182 |
https://mathoverflow.net/questions/119235 | 0 | This is a similar question from the book "Valued Fields by Antonio J. Engler and Alexander Prestel, Springer, 2005 " page 82, Exercise 3.5.4.(b).
Let $(K\_{1}, V\_{1})\subseteq (K\_{2}, V\_{2})$ be finite extension of valued fields. Assume that $[K\_{2} : K\_{1}] = n$. Let $G\_{1}$ and $G\_{2}$ be value groups of $V\... | https://mathoverflow.net/users/30267 | Finite extension of valuation | As remarked before, such a statement will not hold in general. One has the usual formula $n=\sum\_{w|v\_1\ on\ K\_2} d(w/v\_1) e(w/v\_1) f(w/v)$ where $n$ is the degree, $d(w/v\_1)$ is the defect of $w/v\_i$, $e(w\_i/v)$ is the ramification degree of $w/v\_1$ and $f(w/v\_1)$ is the residue field extension degree of $w/... | 3 | https://mathoverflow.net/users/33792 | 129989 | 72,188 |
https://mathoverflow.net/questions/129988 | 3 | Let $G$ be a connected, simply-connected, complex semisimple Lie group with Lie algebra $\frak{g}$. Let $\mu:T^\*\mathcal{B}\rightarrow\mathcal{N}$ be the Springer resolution of $\mathcal{N}$. If $G=\operatorname{SL}\_n(\mathbb{C})$, then the Springer fibers $\mu^{-1}(e)$, $e\in\mathcal{N}$, are known to be connected p... | https://mathoverflow.net/users/25358 | Connectedness of Springer Fibers | The answer is yes. I'm not sure what sources you are working with, but much of this theory originates in the work of Spaltenstein (Lecture Notes in Math. 946, Springer, 1982). While the fibers are connected, they are not irreducible as varieties but the irreducible components are shown to be of equal dimension, etc. Fo... | 3 | https://mathoverflow.net/users/4231 | 129997 | 72,190 |
https://mathoverflow.net/questions/129926 | 6 | Given a Fano threefold $X$, its index $ind(X)$ is the largest integer $r$ such that there exists a divisor $H$ such that $rH \cong -K\_X$. Let $\mathcal{L}$ be the associated (ample) line bundle and define the twisted forms $\Omega^q(k) = \Omega^q \otimes \mathcal{L}^k$.
When do the twisted cohomology groups $H^p(X, \O... | https://mathoverflow.net/users/874 | Cohomology of twisted holomorphic forms on Fano threefolds | The Hodge groups $H^1(X,\Omega^2\_X)$ and $H^2(X,\Omega^1\_X)$ are usually *not* zero. These are the groups used to construct the Griffiths intermediate Jacobian of $X$. For many types of Fano threefolds, e.g., smooth cubic hypersurfaces in $\mathbb{C}P^4$, the Torelli theorem holds -- one can uniquely reconstruct $X$ ... | 5 | https://mathoverflow.net/users/13265 | 130001 | 72,193 |
https://mathoverflow.net/questions/129078 | 2 | Hello members, let's consider the following equation
$$X=F\_{1}XF\_{1}^{T}+...+F\_{p}XF\_{p}^{T}+C$$
where $p$ is an positive integer and $C$ is a known positive semidefinite matrix. If we augment $F=[F\_{1}...F\_{p}]$ and $Y=diag (X...X)$, then the equation becomes
$$FYF^{T}−[I...0]Y[I...0]^{T}+C=0$$
seems like a gene... | https://mathoverflow.net/users/25957 | On solution of a class of discrete-time Lyapunov equation | A trivial way (not necessarily the one you'd use, unless the matrices were very large) to solve this problem is to rewrite it using Kronecker product notation. We use the observation
\begin{equation\*}
\text{vec}(AXB) = (B^T \otimes A)\text{vec}(X),
\end{equation\*}
where the $\text{vec}(\cdot)$ stacks columns of $X... | 2 | https://mathoverflow.net/users/8430 | 130002 | 72,194 |
https://mathoverflow.net/questions/129992 | 6 | I am wondering whether $\ell^\infty(\mathbb N)$ has the Radon-Nikodým property. Of course $\ell^1(\mathbb N)$ does, but I was unable to find out whether (e.g.) duals of spaces with the R-N property have the R-N property themselves.
UPDATE: A Banach space is Asplund if and only if its dual has the Radon-Nikodým proper... | https://mathoverflow.net/users/26039 | Radon-Nikodým property of $\ell^\infty$ | Since a closed subspace of a space with RNP clearly also has RNP, in order to get the requested elementary example, it suffices to display a measure on the unit interval with values in $c\_0$ which is absolutely continuous with respect to Lebesgue measure but whose
derivative does not take its values there. This can b... | 10 | https://mathoverflow.net/users/26013 | 130003 | 72,195 |
https://mathoverflow.net/questions/128269 | 7 | Let me recall that any small category $\mathbb{A}$ enriched in a complete and cocomplete symmetric monoidal closed category $\mathbb{V}$ admits embedding (the Yoneda embedding):
$$y\_\mathbb{A} \colon \mathbb{A} \rightarrow \mathbb{V}^{\mathbb{A}^{op}}$$
into a complete and cocomplete $\mathbb{V}$-enriched category. An... | https://mathoverflow.net/users/13480 | Internal Day convolution | After some research, I think it has not been observed until now. However, all of the bricks needed to make the argument are almost ready.
In paper "Monoidal bicategories and Hopf algebroids" Brain Day and Ross Street defined a notion of convolution in the context of Gray monoids. For a reason that shall become clear ... | 5 | https://mathoverflow.net/users/13480 | 130014 | 72,197 |
https://mathoverflow.net/questions/130010 | 10 | Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as columns.
A moduli problem for elliptic curves is then just a contravariant functor $\textbf{Ell}\rightarrow\textbf{Sets}$.... | https://mathoverflow.net/users/15242 | examples of "exotic" moduli problems for elliptic curves? | Sure -- try the set of homomorphisms from $\pi\_1^{\mathrm{et}}(E - O)$ to a fixed finite group G. This is a "non-abelian level structure" of the sort considered by de Jong and Pikaart
<http://arxiv.org/abs/alg-geom/9501003>
Rachel Davis, a 2013 Wisconsin Ph.D. working with Nigel Boston, wrote her thesis about this... | 11 | https://mathoverflow.net/users/431 | 130020 | 72,198 |
https://mathoverflow.net/questions/130015 | 8 | Suppose $X$ is a "good enough" Hausdorff topological space; we assume that $X$ is *not* compact. Now, for a natural number $k$ and an abelian group $G$, consider the group $\varinjlim\_{\substack{U\subseteq X\\ \text{$X\setminus U$ is compact}}} H^k(U,G)$.
Could you, please, give me a reference to a text where this ... | https://mathoverflow.net/users/29992 | "Cohomology at the infinity": what does one call it | This is the *cohomology of $X$ at $\infty$*. You'll find a discussion of (singular) homology and cohomology at $\infty$ in Hughes & Ranicki, [*Ends of Complexes*](http://www.maths.ed.ac.uk/~aar/books/ends.pdf) (CUP, 1996).
| 11 | https://mathoverflow.net/users/2622 | 130025 | 72,202 |
https://mathoverflow.net/questions/129999 | 3 | I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, as I've really struggled with this.
Suppose we have a system with two states, say $+$ and $-$. The waiting time distrib... | https://mathoverflow.net/users/33794 | Probability distribution for two-state system that depends on residence time | first simple case: $p\_+=p\_-\equiv p\_0$ and $\kappa\_+=\kappa\_-\equiv \kappa$; then all you need to know is the time $\delta t$ since the last switching event, which has an exponential distribution, hence:
$$p(x,t)=(1-e^{-\kappa t})^{-1}\int\_0^t d\delta t\; \kappa e^{-\kappa \delta t}p\_0(x,\delta t)$$
now the ... | 0 | https://mathoverflow.net/users/11260 | 130026 | 72,203 |
https://mathoverflow.net/questions/130019 | 9 | A cardinal $\theta$ is worldly if $V\_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$, but is there a less catastrophic way to do so? I'd like to know about a notion of forcing which makes $V\_{\theta}$ no longer a model of Z... | https://mathoverflow.net/users/nan | Forcing mildly over a worldly cardinal. | I've got it! We can kill the worldliness of a singular worldly
cardinals as softly as we like.
**Theorem.** If $\theta$ is any singular worldly cardinal, then
for any natural number $n$ there is a forcing extension $V[G]$ in
which $\theta$ remains $\Sigma\_n$ worldly, but not worldly,
meaning that $V\_\theta^{V[G]}$ ... | 10 | https://mathoverflow.net/users/1946 | 130028 | 72,204 |
https://mathoverflow.net/questions/130022 | 5 | Given a (splitting) $p$-modular system $(K, \mathcal{O}, k)$ for a finite group $G$, any given simple character $\chi$ is afforded by some $KG$ module $V\_\chi$, and there in general many non-isomorphic $\mathcal{O}G$ modules $B$ (free as an $\mathcal{O}$ module) such that $B\otimes\_{\mathcal{O}} K\cong V\_\chi$ (call... | https://mathoverflow.net/users/26223 | Modular reductions of simple characters | The answer to your basic question is certainly no, though it would take me some time to provide convincing examples. Decades ago I raised a similar question with experts like Walter Feit and Jon Alperin, who assured me that almost anything is possible when choosing the form $B$: the modular reduction might be indecompo... | 4 | https://mathoverflow.net/users/4231 | 130032 | 72,205 |
https://mathoverflow.net/questions/129812 | 5 | It is stated throughout the computational complexity literature that the Dominating Set problem is NP-hard to approximate within a factor of $\Omega(\log n)$. To my knowledge, the first and only proof available ([Lund and Yannakakis, 1994](ftp://ftp-sop.inria.fr/coati/personnel/Stephane.Perennes/LuYa94.pdf)), relies on... | https://mathoverflow.net/users/33736 | Hardness of approximation of Dominating Set | I had a private conversation with Dana Moshkovitz (whom I thank), who confirmed that, in [Alon, Moshkovitz, and Safra (2006)](http://www.tau.ac.il/~nogaa/PDFS/GPGames.pdf), the hard instances of Set Cover resulting from a rather involved gap-preserving reduction are all such that $m\leqslant {\rm poly}(n)$. Hence, afte... | 5 | https://mathoverflow.net/users/33736 | 130042 | 72,208 |
https://mathoverflow.net/questions/128816 | 13 | In a [blog post](http://ldtopology.wordpress.com/2013/04/23/when-are-two-hyperbolic-3-manifolds-homeomorphic/#more-3385) on ldtopology, a recent arxiv posting of [Lins-Lins](http://arxiv.org/pdf/1304.5964v2.pdf) is discussed. The main argument of that paper is difficult to algorithmically distinguish two 3-manifolds an... | https://mathoverflow.net/users/27453 | Distinguishing 3-manifolds by homologies of covers | To atone for my comment above, I'm going to write out the Sol
example. A Sol lattice takes the form $\mathbb Z^2\rtimes \mathbb Z$,
where the action is hyperbolic. The rank of the abelianization is
always 1, though the amount of torsion varies. The profinite
completion is $\hat{\mathbb Z}^2\rtimes \hat{\mathbb Z}$.
We ... | 3 | https://mathoverflow.net/users/4639 | 130048 | 72,211 |
https://mathoverflow.net/questions/130050 | 3 | We say that a group $G$ is union of $k$ proper subgroups $H\_1,H\_2,\cdots, H\_k$, if $\cup\_{i=1}^k H\_i=G$, and union of any $k-1$ subgroups among $H\_i$'s is proper subset of $G$.
It is a theorem of M. J. Tomkinson which says, "*there is no group which is union of seven proper subgroups*". The question, with which... | https://mathoverflow.net/users/6761 | Covering of a group by seven proper subgroups: Counterexample | The only theorem of Tomkinson I could find about this was in
>
> Tomkinson, M.J.. "Groups as the union of proper subgroups." Mathematica Scandinavica 81.2 (1998): 191-198.
>
>
>
In section 3, he proves that any group which is the union of seven proper subgroups can be given as the union of fewer than seven pro... | 10 | https://mathoverflow.net/users/121 | 130051 | 72,212 |
https://mathoverflow.net/questions/129931 | 5 | I would like to approximate the following when $n \gg k$.
$\sum\_{y = k + 1}^n \frac{\sum\_{m = 0}^{k - 1} {y - 2 \choose m} (y - 1)}{\sum\_{m = 0}^k {y - 1 \choose m}}.$
The formula can be re-written as
$\sum\_{y = k + 1}^n \frac{(y - 1) + \sum\_{m = 1}^{k - 1} {y - 1 \choose m + 1} (m + 1)}{\sum\_{m = 0}^k {y -... | https://mathoverflow.net/users/33767 | Summing ratio of ratio of partial sums of binomial coefficients | $$ \sum\_{m=0}^k \binom{z+1}{m} = \sum\_{m=0}^k [ \binom{z}{m-1} + \binom{z}{m} ]
= \binom{z}{k} + 2 \sum\_{m=0}^{k-1}\binom{z}{m},$$ when $z \geq k$. So the sum
can be rewritten as
$$ \sum\_{z=k}^{n-1} \frac{z}{2 + \binom{z-1}{k}/\sum\_{m=0}^{k-1}\binom{z-1}{m}}. $$ Let's
call this summand $a\_z$ . Note that $a\_k =... | 3 | https://mathoverflow.net/users/3528 | 130058 | 72,214 |
https://mathoverflow.net/questions/130062 | 11 | A subgroup of $SL\_2(\mathbb{R})$ is called **arithmetic** if it is commensurable with $SL\_2(\mathbb{Z})$.
An arithmetic subgroup is called **congruence** if it contains a subgroup of type $\Gamma(N)$ for some $N\in \mathbb{N}$.
>
> Question: What are concrete examples of subgroups of $SL\_2(\mathbb{R})$, which ... | https://mathoverflow.net/users/10400 | Concrete examples of noncongruence, arithmetic subgroups of SL(2,R) | I would quibble with your definition of "arithmetic". There are discrete subgroups of $SL(2, \mathbf{R})$ coming from quaternion algebras, which are not commensurable with $SL(2, \mathbf{Z})$, but are commensurable with $G(\mathbf{Z})$ for some twisted form $G$ of $SL(2)$ which becomes isomorphic to the usual form over... | 17 | https://mathoverflow.net/users/2481 | 130063 | 72,215 |
https://mathoverflow.net/questions/130068 | 3 | Let $D$ and $S$ be two regular schemes and let $D$ be a divisor of $S$. Let $C \to S$ be a finite flat morphism, branched along $D$. Is $C$ regular as well?
| https://mathoverflow.net/users/4096 | regularity of finite flat branched covers | The answer is **yes** when $\dim D=2$ and the variety upstairs (i.e. $C$ in your notation) is *normal*:
see
Bas Edixhoven, Robin de Jong, Jan Schepers, [*Covers of surfaces with fixed branch locus*](http://arxiv.org/pdf/0807.0184v2.pdf), Lemma 2.1.
It seems plausible that this proof can be extended in any dimensio... | 4 | https://mathoverflow.net/users/7460 | 130077 | 72,222 |
https://mathoverflow.net/questions/129879 | 5 | Quick version: given natural $n$ and a row of $n$ integers such that the sum of the squares is another square, call it $m^2.$ For $n=5,6,7$ is it always possible to fill in the rest of an $n$ by $n$ matrix of integers, call it $M,$ so that $M M^T = m^2 I? $ If so, $M/m$ is rational orthogonal.
Notes: this is true for... | https://mathoverflow.net/users/3324 | Filling in a rational orthogonal matrix given one row | Yes, it is possible to fill in. Your problem is a particular case of a completion problem and is treated in the following paper:
Hsia, J.S.
Two theorems on integral matrices. Linear Multilinear Algebra 5, 257-264 (1978).
| 1 | https://mathoverflow.net/users/30062 | 130082 | 72,223 |
https://mathoverflow.net/questions/129805 | 14 | Are there known (preferably ``concrete'') examples of a ring $R$ (commutative, with 1) such that:
$\bullet$ the first order theory of $R$ is undecidable, but
$\bullet$ the positive existential (= Diophantine) theory of $R$ is decidable?
The Diophantine theory consists of formulas of the form $\exists x S(x)$ wh... | https://mathoverflow.net/users/7666 | First order decidability of rings vs Diophantine decidability | Let $F=\mathbb{R}(t)$ be the field of rational functions in the variable $t$ with real coefficients. We regard $F$ as a structure of type $(+ \times -\,\, 0\,\, 1)$. Then
1. The (positive) existential theory of $F$ is effectively computable (e.c.)
2. The full first-order theory of $F$ is not e.c.
Proof of 1: Suppos... | 11 | https://mathoverflow.net/users/5229 | 130087 | 72,227 |
https://mathoverflow.net/questions/127589 | 6 | For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define
$$
U\_t(\mathfrak{g}):=\text{T}(\mathfrak{g})/(X\otimes Y-Y\otimes X-t[X,Y]).
$$
Then $S(\mathfrak{g})=U\_0(\mathfrak{g... | https://mathoverflow.net/users/24965 | Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the disover of Duflo's isomorphism theorem? |
>
> My question is: Does the vanishing of
> the Poisson bracket plays an important
> role in finding and proving Duflo's
> isomorphism theorem? Or it is just an
> literally first step?
>
>
>
Let $A\_0$ be a Poisson algebra and $A$ a deformation quantization of $A\_0$ (assume we are in a context when it exist... | 8 | https://mathoverflow.net/users/7031 | 130091 | 72,229 |
https://mathoverflow.net/questions/130081 | 8 | Disclaimer : I suspect the question I am about to ask is really hard, but I just want to know the status of such questions.
Thanks to Milnor, we know that the $\pi\_1$ any compact manifold with nonnegative Ricci curvature has polynomial growth.
I want to know if anything is known about the opposite direction : whi... | https://mathoverflow.net/users/8887 | Converse to Milnor's theorem on manifolds with nonnegative Ricci curvature | For a compact counterexample, take any nilmanifold $N/H$ modulo the action of a freely acting cocompact lattice $\Lambda$, assuming $N/H$ is not just Euclidean space and $\Lambda$ is not just virtually abelian. For instance, $N=N/H$ is the $3 \times 3$ real Heisenberg group and $\Lambda$ is the $3 \times 3$ integer Hei... | 6 | https://mathoverflow.net/users/20787 | 130094 | 72,231 |
https://mathoverflow.net/questions/129983 | 10 | Is there a covering of $\mathbb{Z}^n$ by disjoint translates of the ~~basis-and-origin~~ minimal integer $n$-simplex? By haphazard I have such coverings for $\mathbb{Z}$, $\mathbb{Z}^2$ and $\mathbb{Z}^3$, where the wanted translations are lattices spanned by $\{2\}$, $\{(2,-1),(-1,2)\}$, and $\{(1,1,-1),(1,-1,1),(-1,1... | https://mathoverflow.net/users/1631 | Discrete disjoint covering of integer lattices | Let $S$ be the set of integer points $(x\_1,x\_2,\dots,x\_n)$ satisfying
$$x\_1+2x\_2+3x\_3+\dots+nx\_n \equiv 0 \mod n+1,$$
and $T$ be the basis-and-origin simplex as described in Ben's comment.
Then translates of $T$ by $S$ disjointly cover $\mathbb{Z}^n$ (since decreasing the $x\_i$ coordinate by $1$ changes... | 6 | https://mathoverflow.net/users/405 | 130109 | 72,235 |
https://mathoverflow.net/questions/128725 | 0 | ***Intuitive description***: In the 2D plane, there are $m$ bars connected by $n$ joints. The length of each bar is fixed. These joints and bars can be viewed as a graph (see the figures below). Denote $s\_i$ as the static stress of bar $i$.
1. For some graphs (see figure 1), it is clear all $s\_i$ must be zero. Oth... | https://mathoverflow.net/users/12734 | When does the rigidity matrix of a graph have full row rank? | The term used by Connelly and Whiteley is self-stressed, and they discuss this notion extensively in their paper "[Second-Order Rigidity and Prestress Stability for Tensegrity Frameworks](http://dx.doi.org/10.1137/S0895480192229236)."
| 3 | https://mathoverflow.net/users/20186 | 130118 | 72,239 |
https://mathoverflow.net/questions/130110 | 7 | I want to know if there is a $\sigma$-algebra such that for every countable ordinal $\alpha$ the $\sigma$-algebra can be generated in more than $\alpha$ steps but less than $\omega\_{1}$ steps.
Given an algebra of sets $(X,\mathcal{A})$, let $\mathcal{A}\_{0}=\mathcal{A}$, and for all ordinals $0<\alpha\leq\omega\_{1... | https://mathoverflow.net/users/22277 | How long can it take to generate a $\sigma$-algebra? | The answer to the first question is in the positive; thanks to a corollary drawn by Ken Kunen from a key theorem (about Boolean algebras) of Arnie Miller. More specifically, as shown in Theorem 9.2 of [Arnie Miller's beautiful monograph](http://projecteuclid.org/euclid.lnl/1235423343), we have the following:
>
> **... | 9 | https://mathoverflow.net/users/9269 | 130122 | 72,240 |
https://mathoverflow.net/questions/53596 | 21 | It's "well known" that, for any weight $k$ and level $N$, the space $S\_k(\Gamma\_1(N))$ of cusp forms of that weight and level has a basis in which all the Hecke operators act by matrices with entries in $\mathbb{Z}$; consequently all the Hecke eigenvalues are algebraic numbers (indeed algebraic integers).
I was ref... | https://mathoverflow.net/users/2481 | Elementary proof of algebraicity of Hecke eigenvalues in weight 1 | Let $S = S\_{\mathbf{Q}} = M\_{13}(\Gamma\_1(N),\mathbf{Q})$, and $S\_{\mathbf{C}} = S \otimes \mathbf{C}$ denote the corresponding space of modular forms over $\mathbf{C}$.
Let $V \subset S \times S$ be the subspace cut out by pairs of forms $(A,B)$ satisfying the following equation:
$$A \cdot E\_{12} = B \cdot \D... | 14 | https://mathoverflow.net/users/33127 | 130124 | 72,241 |
https://mathoverflow.net/questions/127889 | 77 | "No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was one of the panelists of a discussion on "proof" this afternoon, alongside two of my mathematician colleagues, and in front of about 100 peopl... | https://mathoverflow.net/users/29316 | Is rigour just a ritual that most mathematicians wish to get rid of if they could? | I was not going to write anything, as I am a latecomer to this masterful troll question and not many are likely going to scroll all the way down, but Paul Taylor's call for Proof mining and Realizability (or Realisability as the Queen would write it) was irresistible.
Nobody asks whether numbers are just a ritual, or... | 87 | https://mathoverflow.net/users/1176 | 130125 | 72,242 |
https://mathoverflow.net/questions/130107 | 15 | From memories of a quantum mechanics class and [Wikipedia](http://en.wikipedia.org/wiki/Uncertainty_principle):
>
> In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a par... | https://mathoverflow.net/users/33640 | Is there an equivalent of Heisenberg's uncertainty principle in the decision sciences ? | Let me answer in terms of operator theory. The uncertainty principle can be interpreted as some particular inequality (as you say) such as
$$
\frac{\hbar}{2}\Vert{u}\Vert^2\le \Vert{D\_xu}\Vert\Vert{xu}\Vert,
$$
inequality due to the identity
$
2\Re\langle\hbar\frac{1}{i}\partial\_x u, ixu\rangle=\hbar\Vert{u}\Vert^2,
... | 7 | https://mathoverflow.net/users/21907 | 130126 | 72,243 |
https://mathoverflow.net/questions/130127 | 8 | Let $R=\mathbb{C}[[x,y]]$ be a power series ring in two variables (or maybe more generally a strictly Henselian local ring) with maximal ideal $\mathfrak{m}$.
>
> What is $H^\*\_{et}(\operatorname{Spec}(R)\setminus\{\mathfrak{m}\}, \mathbb{G}\_m)$?
>
>
>
My motivation is this: I'm trying to understand the ext... | https://mathoverflow.net/users/6950 | Etale Cohomology of Punctured Spectra of Local Rings | With your particular choice of $R$, the $H^2$ is $0$. More generally, if $R$ is a strictly Henselian *regular* local ring of dimension $2$, then by the purity for the Brauer group (in this particular case it is known and due to, I believe, Grothendieck; for a proof see Grothendieck "Le groupe de Brauer II", Prop. 2.3) ... | 13 | https://mathoverflow.net/users/5498 | 130130 | 72,244 |
https://mathoverflow.net/questions/130078 | 13 | Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by
$$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = g^{-1}f(p)g\ (\forall g \in G, p \in P) \rbrace.$$
I want to understand the following statement which I found in Atiyah-Bott ([... | https://mathoverflow.net/users/5206 | The classifying space of a gauge group | **Proof of (1):**
(a). Suppose $X$ and $Y$ are $G$-spaces, the action of $G$ on $X$ is free, and $X\to X/G$ is
a principal bundle, then the space of $G$-equivariant maps
$$
F(X,Y)^G
$$
is the same thing as the space of sections of the fibration $X\times\_G Y \to X/G$.
(b). If $E\to B$ is a Hurewicz fibration, wit... | 17 | https://mathoverflow.net/users/8032 | 130135 | 72,246 |
https://mathoverflow.net/questions/107073 | 5 | Has anyone translated Erich Kähler's "[Über eine bemerkenswerte Hermitesche Metrik](https://doi.org/10.1007/BF02940642)" into English or French? (Preferably, but I'll take anything.)
| https://mathoverflow.net/users/4054 | Translation of Kähler's "Über eine bemerkenswerte Hermitesche Metrik" | If anyone is still interested:
<http://www.woflmao.net/publications/Kaehler1932.pdf> ([Wayback Machine](https://web.archive.org/web/20160603045141/http://www.woflmao.net/publications/Kaehler1932.pdf))
| 7 | https://mathoverflow.net/users/33837 | 130136 | 72,247 |
https://mathoverflow.net/questions/130112 | 3 | Is there somewhere a database of incidence matrices of generalized quadrangles that one can download?
| https://mathoverflow.net/users/22051 | Incidence matrices of generalized quadrangles | <http://www.uwyo.edu/moorhouse/pub/genpoly/>
| 5 | https://mathoverflow.net/users/nan | 130137 | 72,248 |
https://mathoverflow.net/questions/130142 | 1 | I encountered the following situation:
Let C be an abelian category and X, Y be objects of C verifying
$
Hom\_C(X, Z)\cong Hom\_C(Y, Z)
$
For any object Z of C.
Does it follow that $ X \cong Y$?
My belief is that in general this not true. The real question is what additional hypothesis have to be satisfied by C i... | https://mathoverflow.net/users/33326 | A question in category theory | Let $C$ be a category, and $X$ an object in $C$. Then $h\_{X} = \textrm{Hom}(X, \\_)$ is a functor from $C$ to $\textrm{Set}$.
If for two objects $X,Y$ in $C$ the functors $h\_{X}$ and $h\_{Y}$ are naturally isomorphic, then so are $X$ and $Y$. This is called the Yoneda lemma.
In fancy categorical terms it says tha... | 4 | https://mathoverflow.net/users/21815 | 130143 | 72,250 |
https://mathoverflow.net/questions/130071 | 3 | I hope this question is not too simple, but I would like to know the asymptotic behaviour of the following function $f: \mathbb{N}^{+} \rightarrow \mathbb{Q}$ where
$$
f(n) = \sum\_{i=1}^{n} \frac{i^n}{n^{4i}}
$$
Any references, pointers, or answers would be most appreciated.
| https://mathoverflow.net/users/15056 | Asymptotics of a function | From the comments so far (including mine above) it follows that
$$ f(n) = \sum\_{i=1}^{n} \frac{i^n}{n^{4i}}=\frac{n!}{(4\ln n)^{n+1}}\left(1+O(n^{-1/2}\ln n)\right). $$
| 6 | https://mathoverflow.net/users/11919 | 130145 | 72,251 |
https://mathoverflow.net/questions/130039 | 6 | I was recently reading a paper titled "Model Categories of Diagram Spectra" and it was mentioned in the paper that the contents of the paper were also useful in algebraic geometry. I'm really interested in this subject and I'm vaguely aware that techniques from stable homotopy are finding ways into algebraic geometry. ... | https://mathoverflow.net/users/17121 | Diagram spectra and Algebraic Geometry | A nice paper to understand the connection between Spectra and Motivic Homotopy Theory is Mark Hovey's [Spectra and symmetric spectra in general model categories](http://www.math.uiuc.edu/K-theory/0402/). To form the classical category of spectra you start with Topological Spaces and an endofunctor $\Sigma$ which you wi... | 4 | https://mathoverflow.net/users/11540 | 130166 | 72,257 |
https://mathoverflow.net/questions/130171 | 9 | This should really be well-known, but I was not able to find a definite answer to this question:
*Is the Fourier transform of a bounded function always a borel measure (i.e. an order 0 distribution)?*
In some sense, the distributional order corresponds to the order of a bounding polynomial and a bounded function ca... | https://mathoverflow.net/users/16702 | Fourier transform of a bounded function | The answer is no : Fourier transform of the signum function is a constant times the distribution $\lim\_{\epsilon\to 0} 1\_{|x|>\epsilon}/x$, of order $1$, also known as the [principal value](http://en.wikipedia.org/wiki/Cauchy_principal_value) of $1/x$. Another keyword is [Hilbert transform](http://en.wikipedia.org/wi... | 15 | https://mathoverflow.net/users/6451 | 130174 | 72,261 |
https://mathoverflow.net/questions/130157 | 2 | Given a free group $F$ on $d$ generators and a normal subgroup $H$ of $F$ whose index is finite of prime power order, is there a systematic way to find the numbers of generators of $H/[H,F]$ and of $H/[H,F]H^p$?
| https://mathoverflow.net/users/31883 | Generators of sections of free groups | This is really a cohomological question and has a simple cohomological answer. Recall that if a group $G$ acts on an abelian group $M$, then $M\_G$ denotes the *coinvariants* of the action, that is, the quotient of $M$ by the subgroup generated by $\{\text{$m-g(m)$ $|$ $m \in $M, $g \in G$}\}$. The group $F$ acts on $H... | 5 | https://mathoverflow.net/users/317 | 130188 | 72,268 |
https://mathoverflow.net/questions/20683 | 27 | The most elementary construction I know of quantum groups associated to a finite dimensional simple Hopf algebra is to construct an algebra with generators $E\_i$ and $F\_i$ corresponding to the simple positive roots, and invertible $K\_j$'s generating a copy of the weight lattice. Then one has a flurry of relations be... | https://mathoverflow.net/users/1040 | Quantum group as (relative) Drinfeld double? | There is a detailed exposition of this in Majid's paper *Double-bosonization of braided groups and the construction of* $U\_q(\mathfrak{g})$, Math Proc Cambridge Phil Soc 125(1). Especially appendix B where quantum group is obtained by a version of Tannaka-Krein duality for braided monoidal categories applied to the ca... | 8 | https://mathoverflow.net/users/33854 | 130190 | 72,270 |
https://mathoverflow.net/questions/130180 | 6 |
>
> Suppose $S \subset \omega\_{1}$ is stationary co-stationary. Then there is a forcing notion $P\_{S}$ which shoots a closed unbounded $C \subset S$ without collapsing cardinals (or
> changing cofinalities).
>
>
>
> >
> > Let $P$ = $Levy$($\aleph\_{0}$， $< \aleph\_{1}$). So P is essentially adding $\aleph\_{1}... | https://mathoverflow.net/users/29231 | Question about Shelah's version of "Shooting a club" found in PIF | Sorry this is sketchy, but I hope it helps!
A Cohen condition $p$ of length $n$ is going to determine $\langle C\_m:m<n\rangle$. We know there's going to be an $i<\omega$ such that $q\_i$ extends both $C\_{n-1}$ and a member of $N[G\_P]\cap A$. Extend $p$ to a Cohen condition $p'$ with $p'(n)=i$. If $r^\*$ is any Coh... | 3 | https://mathoverflow.net/users/18128 | 130202 | 72,274 |
https://mathoverflow.net/questions/130191 | 3 | Noether's second variational theorem gives a correspondence between symmetries of a Lagrangian and Noether identities, which are relations among the Euler–Lagrange equations.
How about relations among relations among the Euler–Lagrange equations, cf. syzygies?
| https://mathoverflow.net/users/36067 | higher order Noether identities | Identities among Noether identities (lets call them *higher stage* Noether identities, for lack of a better standard name) are mapped to gauge symmetries of gauge symmetries (lets call them *higher stage* gauge symmetries, also sometimes known as *reducibility identities*). As far as I know, references to the notions o... | 2 | https://mathoverflow.net/users/2622 | 130203 | 72,275 |
https://mathoverflow.net/questions/130178 | 2 | A well-known feature used in PDE's is the regularization by convolution with a mollifier sequence $\rho\_n$, i.e. $\rho\_n(x) := n^d \rho(nx)$ with $x \in \mathbb R^d$, $\rho \in C^\infty\_c(\mathbb R^d)$ and $\int \rho = 1$. If $f\in L^p(\mathbb R^d)$ for $p\in[1,\infty)$, then $f^n := f \* \rho\_n \in C^\infty(\mathb... | https://mathoverflow.net/users/33851 | Regularization by mollifier sequence | If you assume $f \in L^\infty(0,T; L^1(\mathbb R^d))$ then you get convergence
$$
f\_n\to f \text{ in } L^r(0,T;L^1(\mathbb{R}^d))
$$
for all $r$.
This follows by Lebesgue dominated convergence: for a.e $t\in(0,T)$ you have pointwise convergence $||f\_n(t,.)-f(t,.)||\_{L^1}\to 0$ when $n\to\infty$, and the $L^1-L^1... | 1 | https://mathoverflow.net/users/33741 | 130205 | 72,277 |
https://mathoverflow.net/questions/130170 | 1 | Let $X$ be a smooth, projective complex algebraic variety. Let $D$ be a simple normal crossings divisor on $X$, with irreducible components $D\_i$, for $i \in I$. For each non-empty subset $J \subset I$, put
$$
D\_J=\bigcap\_{i \in J} D\_i
$$ Define also $D(0)=X$ and, for $m \geq 1$,
$$
D(m)=\bigsqcup\_{|J|=m} D\_J ... | https://mathoverflow.net/users/33847 | a question on Euler characteristic of normal crossing divisors | Your formula is a little bit wrong. You want $D(0)= X$, not $D$, and your summation should start at $m=0$.
This shouldn't work. Let $X$ be $\mathbb P^3$ with $2$ lines on a quadric surface blown up, becoming copies of $\mathbb P^1 \times \mathbb P^1$. Let $D$ be the union of the quadric surface and the exceptional di... | 1 | https://mathoverflow.net/users/18060 | 130206 | 72,278 |
https://mathoverflow.net/questions/130159 | 4 | Hi,
consider a simple situation in quantum mechanics: Your Hilbert space is $\mathcal{H}=L^2(\mathbb{R}^3)$ and you use the obvious unitary representation $\pi\colon G=O(3)\times\mathbb{R}^3\to U(\mathcal{H})$ given by acting on the underlying space. From the group representation you get a representation of the Lie a... | https://mathoverflow.net/users/33842 | Quantized conserved quantities appearing from the Lie-algebra | Suppose we have a $G$-group action on a configuration manifold $Q$ (in your case $G=O(3)\ltimes\mathbb{R}^3$ and $Q=\mathbb{R}^3$). Then for each element $\xi\in\mathfrak{g}$ in the Lie algebra of $G$, there is a corresponding vector field $\xi\_Q$ on $Q$. Following approach 2: the conserved quantity (i.e. momentum map... | 4 | https://mathoverflow.net/users/17945 | 130215 | 72,282 |
https://mathoverflow.net/questions/130221 | 3 | An elementary question of algebraic geometry which was posted on math.stackexchange but received no answer there.
Let $f: X \rightarrow Y$ be a morphism of schemes.
Assume that
(i) $f: X \rightarrow Y$ is locally finite, in this sense: $Y$ can be covered by affine open
sets $U\_i=spec\ A\_i$, and $f^{-1}(U\_i)$ ca... | https://mathoverflow.net/users/9317 | locally finite + quasi compact vs finite | $\newcommand\Spec{\mathrm{Spec}}$
$ $
Since finiteness is local in the base, we may
assume that $Y = \Spec(A)$ is affine.
The assumption then becomes that $f: X \rightarrow \Spec(A)$ is
quasi-compact, and that $X$ may be covered by finitely many
(by quasi-compactness)
open affines $\Spec(B)$ which are finite over... | 4 | https://mathoverflow.net/users/33865 | 130225 | 72,286 |
https://mathoverflow.net/questions/129462 | 9 | I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two matrices and I would like to show the following result:
>
> Let $M \in \mathbb R^{n \times n}$ be a *M-matrix* and $S \in \mathbb R^{n \ti... | https://mathoverflow.net/users/10203 | M-matrix plus S-matrix is P-matrix? | This is not true in general.
$\left(\begin{array}{ccc} 4 & 0 & -16 \\ 2 & 4 & 0 \\ 0 & 2 & 4 \end{array}\right)= \left(\begin{array}{ccc} 1 & -2 & -16 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right) +\left(\begin{array}{ccc} 3 & 2 & 0 \\ 2 & 3 & 2 \\ 0 & 2 & 3 \end{array}\right)$
A matrix with determinant $0$ is $M+S... | 7 | https://mathoverflow.net/users/18060 | 130229 | 72,288 |
https://mathoverflow.net/questions/130232 | 0 | It's a fact proven by Pendleton, Gilmer, and Ohm (as an obvious corollary of their work, anyways) that PIDs are QR-domains, meaning every overring (ring between the domain and the quotient field) is a ring of quotients. I'm trying to find a counterexample for something else not being a QR-domain, and so understanding h... | https://mathoverflow.net/users/27445 | Explicit representation of $R[\frac{x}{y}]$ where $x, y\in R$ for non-Euclidean PIDs $R$? | By PID, we have $(x,y)=(z)$, so $z=ax+by$ and $x=cz$, $y=dz$ for some $a,b,c,d$, then
$$\frac{1}{d} = \frac{z}{y} = a \frac{x}{y} + b$$
$$ \frac{x}{y} =c \frac{1}{d}$$
so $R[\frac{x}{y}]=R[\frac{1}{d}]$. It's not $R[\frac{1}{y}]$ in general because some prime ideal could divide both $x$ and $y$ in equal quantitie... | 3 | https://mathoverflow.net/users/18060 | 130233 | 72,289 |
https://mathoverflow.net/questions/130241 | 1 | By using the Iwasawa decomposition, one obtains a (bi-invariant) Haar measure on $G:=\mathrm{SL}(2,\mathbb{R})$ which can be symbolically written as $\mathrm{d}x=\mathrm{d}a\,\mathrm{d}n\,\mathrm{d}k$, the measures appearing on the right-hand side being the usual ones. This actually means $$\int\_G f(x)\,\mathrm{d}x=\i... | https://mathoverflow.net/users/19142 | How do these two Haar measures on SL(2,R) compare? | Yes, since you get one from the other by applying the inversion and there is a general rule for the inversion that says $\int\_Gf(x^{-1})dx=\int\_Gf(x)dx\Delta(x)$, where $\Delta(x)$ is the modular function.
| 4 | https://mathoverflow.net/users/nan | 130246 | 72,298 |
https://mathoverflow.net/questions/130228 | 1 | Is there any test to tell me whether a straight line in a 3D euclidean space passes through a bounded closed convex region? To focus on a more specialised version of the problem, you can assume that the convex region is described by intersection of closed half spaces? In the end, it will be great if you can generalize ... | https://mathoverflow.net/users/38193 | Straight Line Passing Through a Convex Region | It so happens that Wikipedia contains an article entitled, "[Intersection of a polyhedron with a line](http://en.wikipedia.org/wiki/Intersection_of_a_polyhedron_with_a_line)," but I doubt that answers your question.
A better answer is provided by another MO question, "[Intersection points of straight line segment wit... | 3 | https://mathoverflow.net/users/6094 | 130269 | 72,305 |
https://mathoverflow.net/questions/130256 | 19 | In 1965 Jean Giraud published two Comptes Rendus notes titled "Cohomologie non abélienne", and in 1971 he published a book with the same title.
In 1966 Tonny A. Springer's paper "Nonabelian $H^2$ in Galois cohomology" appeared, where he, in particular,
constructs nonabelian $H^2$ of a group in terms of group extensions... | https://mathoverflow.net/users/4149 | Second nonabelian group cohomology: cocycles vs. gerbes | The cocycle data which you review together is a map of [2-groupoids](http://ncatlab.org/nlab/show/2-groupoid) $B \Gamma \to B Aut(B G)$ to the [delooping](http://ncatlab.org/nlab/show/delooping) of the [automorphism 2-group](http://ncatlab.org/nlab/show/automorphism%202-group) "of $G$" (really: of $BG$). As for any coc... | 14 | https://mathoverflow.net/users/381 | 130272 | 72,307 |
https://mathoverflow.net/questions/130287 | 25 | What is the intuition behind using compact open topology for eg. in the case of Pontryagin dual ?
| https://mathoverflow.net/users/30999 | Compact open topology | Given two spaces $X$ and $Y$, how to define the mapping space betweeen them, i.e. what topology should we put on the set of maps between them?
If $X$ is compact and $Y$ a metric space, this is quite easy as one can put a metric on $Map(X,Y)$: For $f,g\in Map(X,Y)$ define their distance just to be the maximum of the ... | 49 | https://mathoverflow.net/users/2039 | 130300 | 72,319 |
https://mathoverflow.net/questions/130295 | 7 | **Theorem**: Let be $f$ a homeomorphism of a compact metric space $X$, then
$$
h\_{top}(f)=\sup\_{\mu\in \mathcal{M}\_{f}}~ h \_\mu (f)
$$
**Question:** The above theorem is the famous variational principle for compact spaces, I'm looking for an example to see that the hypothesis $ f $ be a homeomorphism is really n... | https://mathoverflow.net/users/nan | Variational Principle for the Entropy | The theorem is true not only for homeomorphisms but also for continuous maps that are not necessarily invertible.
For discontinuous maps $f$, I'm not sure if there's any problem beyond the fact that the definition of topological entropy is generally made under the assumption that $f$ is continuous, and so one needs t... | 4 | https://mathoverflow.net/users/5701 | 130303 | 72,322 |
https://mathoverflow.net/questions/130194 | 16 | My question concerns a technical step in the proof of Linnik's theorem on the least prime in an arithmetic progression, as presented in Chapter 18 of Iwaniec-Kowalski: Analytic number theory.
The proof uses certain weights $\theta\_b$ coming from the theory of the Selberg sieve. The sequence is supported on square-f... | https://mathoverflow.net/users/11919 | Least prime in an arithmetic progression and the Selberg sieve | The proof given in Iwaniec-Kowalski is, as it stands, wrong. It can be easily fixed, as I explain below.
In general, one can think of $\nu(n)^2$ as the characteristic function of integers with $P^-(n):=\min(p|n)\ge y$. So
$$
\sum\_{n\le x} \frac{\nu(n)^2 f(n)}{n} \approx \sum\_{ n\le x,\ P^-(n)\ge y } \frac{f(n)}{n}... | 12 | https://mathoverflow.net/users/4003 | 130308 | 72,326 |
https://mathoverflow.net/questions/130319 | 38 | show the formula always gives an integer
$$\frac{(2m)!(2n)!}{m!n!(m+n)!}$$
I don't remember where I read this problem, but it said this can be proved using a simple counting argument (like observing that $\frac{(3m)!}{m!m!m!}$ is just the number of ways of permuting m identical things of type 1, m of type-2 and m o... | https://mathoverflow.net/users/33902 | Show that this ratio of factorials is always an integer | I found this paper
[I. M. Gessel, G. Xin, A Combinatorial Interpretation of the
Numbers $6(2n!)/n!(n+2)!$, *Journal of Integer Sequences* **8** (2005) Article 05.2.3](https://cs.uwaterloo.ca/journals/JIS/VOL8/Gessel/xin.pdf)
whose abstract says:
>
> It is well known that the numbers $\frac{(2m)!(2n)!}{m!n!(m+n)... | 45 | https://mathoverflow.net/users/27829 | 130320 | 72,331 |
https://mathoverflow.net/questions/130322 | 1 | I know that for general linear group $\mathrm{GL}(n,p^r)$, one Sylow $p$-subgroup is the set of all unitriangular matrices. I need a reference for this theorem.
Thank you.
| https://mathoverflow.net/users/33209 | Sylow $p$-subgroup of GL | Steinberg's "Lecture Notes on Chevalley Groups" the Corollary of Lemma 54 on page 132. There is possibly a more modern reference.
EDIT: Sorry the reference to Steinberg is not sufficient as he does not treat arbitrary finite reductive groups. However all that is needed is to obtain a slightly more general formula for... | 5 | https://mathoverflow.net/users/22846 | 130328 | 72,335 |
https://mathoverflow.net/questions/130326 | 3 | Suppose $D$ is a non-empty set and $\{ R\_i : i \in \mathbb{N} \}$ is a family of binary relations on sequences over $D$ so that $R\_i \subseteq D^i \times D^i$. Let $R\_\omega \subseteq D^\omega \times D^\omega$ be a relation that holds between two denumerable sequences $x$ and $y$ over $D$ iff for all $n\in \mathbb{N... | https://mathoverflow.net/users/33903 | Validity in Kripke frames whose points are finite or infinite sequences | It is an attractive idea, but unfortunately, it seems not to be true.
The reason is that we can have that every $R\_n$ is nontrivial, in the sense that the relation sometimes holds between different two different sequences, but there is no path through these relations so that $R^\omega$ never holds between two differ... | 3 | https://mathoverflow.net/users/1946 | 130330 | 72,337 |
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