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https://mathoverflow.net/questions/194635 | 2 | Let $P \subset \mathbf{R}^3$ be a convex polyhedron. Let $rP$ be the dilation of $P$ by the positive real number $r$.
The Dehn invariant of $P$ is an element of the weird real vector space $\mathbf{R} \otimes\_{\mathbf{Z}} \mathbf{R}/(\pi\mathbf{Z})$, given by the formula
$$
D(P) = \sum\_e (\text{length of }e) \otime... | https://mathoverflow.net/users/1048 | Is mean width a Dehn invariant? | Additivity in the sense of scissors congruence means that if $A$ and $B$ have disjoint interiors, then $$D(A \cup B) = D(A) + D(B).$$
Valuations such as the mean width satisfy $$W(A \cup B) = W(A) + W(B) - W(A \cap B).$$
That last term is not necessarily $0$ when $A$ and $B$ have disjoint interiors. For example, i... | 6 | https://mathoverflow.net/users/2954 | 194659 | 94,952 |
https://mathoverflow.net/questions/194656 | 2 | Let us consider the pseudo-euclidean space $\mathbb{R}^3\_1$; that is, the space $\mathbb{R}^3$ endowed with the metric
$$\langle x, y\rangle = x\_1y\_1+x\_2y\_2-x\_3y\_3.$$
I see in O'Neill's book that there does not exist any compact semi-riemannian hypersurface in this space. In fact, there does not exist any c... | https://mathoverflow.net/users/64649 | Semi-riemannian hypersurfaces | The first thing to understand is what kind of quadratic form $q$ is induced on a vector plane $V\subset \mathbb{R}^3$ by the Lorentz product $\langle\cdot,\cdot\rangle$: if $V$ is vertical enough, then $q$ has signature $(1,1)$; if $V$ is horizontal enough, then $q$ has signature $(2,0)$, i.e. is Riemannian; but if $V$... | 6 | https://mathoverflow.net/users/4961 | 194667 | 94,955 |
https://mathoverflow.net/questions/194671 | 0 | A $\sigma$-algebra $\mathcal F$ over $\Omega$ is generated by an countable partition if there exits a countable partition $\mathcal B = \{ B\_i \}$ of $\Omega$ such that $\mathcal F = \sigma(\mathcal B)$. Now let $\mathcal G$ be an arbitrary $\sigma$-algebra over $\Omega$. Is it possible to find $\sigma$-algebras $\mat... | https://mathoverflow.net/users/37580 | Approximating an arbitrary $\sigma$-algebra by simpler $\sigma$-algebras | It turns out that it is impossible to represent any $\sigma$-complete Boolean algebra as a countable union of Boolean subalgebras.
For example, in the paper, Boolean algebras as unions of chains of subalgebras by Sabine Koppelberg, the following result is proven.
>
> $\mathbf{Theorem}$ Suppose $B$ is a Boolean al... | 3 | https://mathoverflow.net/users/22277 | 194677 | 94,959 |
https://mathoverflow.net/questions/194676 | 2 | Assume $F, G : \mathbf C \to \mathbf D$ be functors. Denote by $\widehat{\mathbf C} = \mathrm{Fun}(\mathbf{C}^{\mathrm{op}}, \mathbf{Set})$ the category of presheaves of sets on $\mathbf C$. Then, $F$ and $G$ induce "restriction" functors, obtained by composition with $F$ and $G$:
\begin{align\*}
\mathrm{Res}\_F, \math... | https://mathoverflow.net/users/20883 | Given functors $F$ and $G$, does $\mathrm{Res}_F \cong \mathrm{Res}_G$ imply $F \cong G$? | You can recover $F$ from $\mathrm{Res}\_F$; this is an exercise in using the Yoneda lemma. Let $H\_A$ denote the presheaf represented by an object $A$ of $\mathbf C$ or $\mathbf D$. Then for $A\in \mathbf{C}$ and $B\in\mathbf{D}$ we have $$\operatorname{Hom}\_{\mathbf D}(FA,B)=H\_B(FA)=\mathrm{Res}\_FH\_B(A).$$
By Yone... | 11 | https://mathoverflow.net/users/75 | 194680 | 94,961 |
https://mathoverflow.net/questions/194647 | 4 | Given a poset $(P,\leq)$ the *interval topology* $\tau\_{\text{int}}(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.
Let $P, Q$ be posets and $e:P\to Q$ be order-preserving and... | https://mathoverflow.net/users/8628 | Image of poset with Hausdorff interval topology | The answer is no, not necessarily.
For a counterexample, let $Q$ be any atomless complete Boolean algebra, and let us view it via Stone's theorem as a field of sets, so that $Q$ is a subalgebra for some set $X$ of the power set algebra $P=P(X)$, which is atomic.
By [Proposition 5 in this paper of Northam](http://... | 7 | https://mathoverflow.net/users/1946 | 194685 | 94,962 |
https://mathoverflow.net/questions/194689 | 1 | In the context of KP, is the formula $\forall w(w\in x \leftrightarrow\forall y\exists z F(w,y,z))$ $\Pi\_3$ when $F(w,y,x)$ is $\Delta\_0$?
| https://mathoverflow.net/users/37385 | On whether a formula of KP is $\Pi_3$ | It seems so:
Unwinding the expression, we get $$G(x)\equiv\forall w [(w\in x\implies \forall y\exists z(F(w, y, z)))\wedge (\forall y\exists z(F(w, y, z))\implies w\in x)]$$ $$\iff \forall w[(\forall y\exists z(F(w, y, z))\vee w\not\in x)\wedge (\exists y\forall z(F(w, y, z))\vee w\in x)]$$ $$\iff [\forall w\forall y... | 3 | https://mathoverflow.net/users/8133 | 194692 | 94,965 |
https://mathoverflow.net/questions/194681 | 13 | Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard models offer "more" mathematical insight in some way about the standard model? Compare this with the use of the compactness t... | https://mathoverflow.net/users/43701 | nonstandard models and mathematical theorems | I view nonstandard models (particularly those produced by ultraproducts) as a convenient way to take limits of discrete structures to obtain continuous structures that capture all of the asymptotic first-order properties of the discrete structure. This of course can also be done by the compactness theorem, but ultrapro... | 17 | https://mathoverflow.net/users/766 | 194693 | 94,966 |
https://mathoverflow.net/questions/194683 | 3 | I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.
| https://mathoverflow.net/users/3124 | Are Carnot groups (as Carnot Caratheodory metric spaces) doubling? | Carnot groups as well as general (equiregular) SubRiemannian manifolds are typical examples of Ahlfors regular metric measure spaces, which are also geodesic. In particular, they are doubling as a metric space. You can find these facts in the lectures of Heinonen
Heinonen, Juha Calculus on Carnot groups. Fall School ... | 5 | https://mathoverflow.net/users/26608 | 194694 | 94,967 |
https://mathoverflow.net/questions/194690 | 3 | I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems fabulous, but it has way too much content on Measure Theory, which I do not know at all.
There is a wealth of profess... | https://mathoverflow.net/users/66278 | Book on Convergence Concepts in Probability without Measure Theory | Grimmett and Stirzaker's Probability and Random Processes fits the bill:
[http://www.amazon.com/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220](http://rads.stackoverflow.com/amzn/click/0198572220)
It covers everything up to stochastic calculus without measure theory.
| 3 | https://mathoverflow.net/users/934 | 194699 | 94,971 |
https://mathoverflow.net/questions/194523 | 26 | In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic.
What are some good examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?
Since limits and colimits can be characterized as Kan ex... | https://mathoverflow.net/users/56938 | Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry? | The following is really an adjunction between $2$-categories but I am going to ignore that subtlety. [This blog post](https://qchu.wordpress.com/2013/06/09/operations-and-lawvere-theories/) discusses everything in more detail and with a few more examples.
Consider on the one hand all concrete categories, by which I m... | 16 | https://mathoverflow.net/users/290 | 194708 | 94,977 |
https://mathoverflow.net/questions/194684 | 3 | Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function with $|f'(x)| \leq C$ for all $x$ and $f(0)=0$.
Suppose $u\_n \to u$ in $H^{\frac 12}(\partial\Omega)$, where $\Omega$ is a bounded domain of class $C^2$. Here, I use the norm
$$|u|\_{H^{\frac 12}(\partial \Omega)}^2 = |u|^2\_{L^2(\partial\Omega)} + \int\_{\parti... | https://mathoverflow.net/users/64264 | If $u_n \to u$ in $H^{\frac 12}(\partial\Omega)$, does $f(u_n) \to f(u)$ in $H^{\frac 12}(\partial\Omega)$ for $f$ Lipschitz? | **Yes.**
It might be more convenient to think of $H^{1/2}(\partial\Omega)$ as a quotient space:
$$
H^{1/2}(\partial\Omega)=H^1(\Omega)/H^1\_0(\Omega).
$$
Let $T:H^1(\Omega)\to H^{1/2}(\partial\Omega)$ be the corresponding quotient map (the trace map).
Since $u\_n\to u$ in $H^{1/2}(\partial\Omega)$, there is are functio... | 2 | https://mathoverflow.net/users/55893 | 194713 | 94,978 |
https://mathoverflow.net/questions/194709 | 1 | Let $(\Omega, \mathcal F, P)$ be a probability space, and let $\mathcal G \subseteq \mathcal F$ be a sub-$\sigma$-algebra of $\mathcal F$ and $X : \Omega \to \mathbb R$ a random variable. Then the conditional expection of $X$ conditioned on $\mathcal G$ is defined to be the a.e. unique random variable $Y$ such that
(... | https://mathoverflow.net/users/37580 | Conversion between condtional expection conditioned on $\sigma$-algebra and on r.v | If you want a *real-valued* random variables $Z$, then $\mathcal G$ must be countably generated. (At least up to null sets.)
Suppose $\mathcal G$ is countably generated. We want to find a random variable $Z \colon \Omega \to \mathbb R$ such that $\sigma(Z) = \mathcal G$. Say $\mathcal G$ is generated by $A\_1, A\_2, ... | 2 | https://mathoverflow.net/users/454 | 194715 | 94,979 |
https://mathoverflow.net/questions/194714 | 14 | Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.
| https://mathoverflow.net/users/21956 | Elliptic curves and connected components | Yes. It is not hard to find an example: Take
$$E \colon y^2 = x^3 - 12 x - 1\,.$$
Then $E(\mathbb Q) \cong \mathbb Z$ and $P = (5, 8)$ is a generator (according
to Magma). Since $P$ is on the component of the identity, all rational points are on that component.
| 19 | https://mathoverflow.net/users/21146 | 194716 | 94,980 |
https://mathoverflow.net/questions/194710 | 3 | I am currently dealing with an unbounded operator
$T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim\_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow L^2(-2\pi,2\pi)$
$g \in C^{\infty}: g(-2\pi) = g(2\pi)=0$ and $g|\_{(-2\pi,2\pi)} >0.$
Then I want to show that $T(f)... | https://mathoverflow.net/users/66287 | adjoint of this closed (?) operator | Let's first of all recall that the minimal operator of differentiation
$$
D(S) = \{ f\in L^2\cap AC : f'\in L^2, f(\pm 2\pi)=0 \} , \quad\quad Sf = -if'
$$
is closed and has the maximal operator as its adjoint:
$$
D(S^\*) =\{ f\in L^2\cap AC: f'\in L^2\} , \quad\quad S^\*f=-if'
$$
Now since $g$ is smooth and $g>0$, you... | 2 | https://mathoverflow.net/users/48839 | 194722 | 94,983 |
https://mathoverflow.net/questions/194725 | 2 | Given $d\in \mathbb{N}$, let $X\_d:= \{(\ell\_1, \ldots , \ell\_d): 0 \le \ell\_1 \le \ldots \le \ell\_d \le d\}\subset \mathbb{Z}^d$, and endow $X\_d$ with the (usual) partial order, namely, $x\le y$ if and only if $x\_j \le y\_j$ for all $j=1,\ldots , d$. Note that if $Y\subset X\_d$ is linearly ordered then $\# Y \l... | https://mathoverflow.net/users/66292 | Counting linearly ordered subsets of maximal length in partially ordered $d$-tuples of nonnegative integers | Denote $x!!=1!2!\cdots x!$. The number is
$$N\_d=\frac{(d-1)!!(d^2)!}{(2d-1)!!}\prod\_{1\leqslant i<j\leqslant n}(j-i).$$
It can be interpreted as the number of lattice paths from $(0,\ldots,0)$ to $(d,\ldots,d)$ in $\mathbb{Z}^d$ such that all points on the path satisfy $x\_1\leqslant x\_2\leqslant\cdots\leqslant x\_d... | 1 | https://mathoverflow.net/users/12674 | 194736 | 94,989 |
https://mathoverflow.net/questions/194741 | 1 | The Newton's method that I know is defined as follows:
$$x\_{n+1}=x\_n-\frac{f(x\_n)}{f'(x\_n)}$$
However, I've recently encountered [a paper](http://www.emis.de/journals/HOA/JAM/Volume2012/294086.pdf) that talks about a ***one-parameter family of Newton's method*** (page 4, equation 2.8), defined as follows:
$$x... | https://mathoverflow.net/users/64121 | What is a one-parameter Newton's method? | "One-parameter family" simply means that one real parameter, $p$, appears in the definition. That method is simply a generalization of Newton's method exposed in the first equation; you can't derive it from the first.
The classical Newton's method has slower convergence when $f'(\alpha)=0$, $\alpha$ being the sought ... | 3 | https://mathoverflow.net/users/1898 | 194743 | 94,990 |
https://mathoverflow.net/questions/194497 | 15 | For simplicity, we restrict to constant coefficients. Let $A^{ij}\_{ab} \in \mathbb{R}$, $1 \le i, j \le n$ and $1 \le a, b\le m$, satisfy the Legendre-Hadamard condition:
$$
A^{ij}\_{ab}\xi\_i\xi\_jv^av^b \ge \lambda |\xi|^2|v|^2
$$
for some $\lambda > 0$ and any $\xi \in \mathbb{R}^n$ and $v \in \mathbb{R}^m$. Let $B... | https://mathoverflow.net/users/613 | Does the Legendre-Hadamard condition imply a generalized Gårding inequality? | The answer is **No**, and this is an interesting question.
As Terry Tao commented, the test case is when $B$ is replaced by a half-space $H$, so let us consider the latter case for a moment and fields $u$ with compact support in $\bar B$ (i.e. not vanishing at the boundary).
Because $H$ is dilation invariant, a Ga... | 14 | https://mathoverflow.net/users/8799 | 194745 | 94,991 |
https://mathoverflow.net/questions/194747 | 3 | I wonder whether hyperreal numbers isomorphic with formal Laurent series?
It seems that any hyperreal number can be represented in the form of Laurent series over $\omega$. For instance,
$e^{\omega}=\frac{\omega^0}{0!}+\frac {\omega^1}{1!}+\frac{\omega^2}{2!}+...+\frac{\omega^n}{n!}+...$
If so, it follows that an... | https://mathoverflow.net/users/10059 | Are hyperreal numbers isomorphic to formal power series? | One problem is that the set of formal Laurent series is not a [real closed field](http://en.wikipedia.org/wiki/Real_closed_field) (an ordered field where every positive element has a square root, and every polynomial of odd degree has a root). That particular problem would be fixed if one considered instead the real cl... | 14 | https://mathoverflow.net/users/2926 | 194752 | 94,992 |
https://mathoverflow.net/questions/194748 | 2 | For a given mean $\mu$, what is the entropy maximizing probability distribution on the nonnegative integers?
Different sources indicated either the geometric or the Poisson distribution for this. As I am new to the topic, it would be great if someone would give me a hint or point me to a source with a thorough expla... | https://mathoverflow.net/users/3816 | Discrete Maximum Entropy Distribution with given mean | maximizing $S=\sum\_{n=0}^\infty p\_n \log p\_n$ with the constraint $\sum\_n n p\_n=\mu$ and $\sum\_n p\_n =1$ gives $p\_n = a b^n$, with Lagrange multipliers $a=1/(\mu+1)$ and $b=\mu/(\mu+1)$ determined by the constraints, so this is indeed a geometric distribution.
what is the source you are referring to that says... | 6 | https://mathoverflow.net/users/11260 | 194753 | 94,993 |
https://mathoverflow.net/questions/194712 | 3 | Let $H = L^{2}(S^{1},\mathbb{C}^{n})$, $H\_{0}\subseteq H$ the subset of maps that extend holomorphically to the unit disc, and $H\_{m} = z^{m}H\_{0}$. Consider the affine Grassmannian for $GL\_{n}$ in the lattice model:
$\mathcal{Gr}\_{n} = \left\{ V\subseteq H:\, zV\subseteq V\right\}$
Let me describe two possib... | https://mathoverflow.net/users/26069 | Do the following two filtrations of the affine Grassmannian agree? | This statement isn't true; as you've currently defined everything, your $X\_i$ are infinite dimensional (in fact they are unions of connected components.) You want instead to take the $X\_i$ unions of $\lambda$ with $i\geq\lambda\_1$ and $\lambda\_n\geq -i$.
Let $B$ be the Borel for $GL\_n(\mathcal{K})$. Both $\mathc... | 4 | https://mathoverflow.net/users/51424 | 194757 | 94,994 |
https://mathoverflow.net/questions/194739 | 4 | I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture:
Conjecture. Suppose we have a finite group $G$ of order $n$, and $F\_p$ the field of size $p$, $p$ a prime. Let $s(k)$ be the number of non-i... | https://mathoverflow.net/users/8012 | the number of indecomposable modules of finite groups over finite fields of a fixed dimension | Are you sure you mean $p^k$, and not something like $p^{k^2}$?
If you look at indecomposable $d$-dimensional representations of the free algebra $\mathbb{F}\_p\langle x,y\rangle$, the number grows faster than $p^d$. For a lower bound consider those where $x$ acts as a fixed single Jordan block $X$ (just to ensure ind... | 8 | https://mathoverflow.net/users/22989 | 194773 | 95,001 |
https://mathoverflow.net/questions/194770 | 2 | As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as the properties thereof are. So, letting $n$ and $n'$ be two distinct positive integers, $\pi$ (respectively $\pi'$) an au... | https://mathoverflow.net/users/13625 | Rankin-Selberg convolution and product of degrees | I first disclaim being up-to-date on the precise issue in the question! Given that:
The only truly interesting example I know to have been definitely worked out, that exactly fits the question is Ramakrishnan's result from 2000 (Annals) which proves that the Rankin-Selberg convolutions for $GL\_2\times GL\_2$ (when t... | 17 | https://mathoverflow.net/users/15629 | 194778 | 95,002 |
https://mathoverflow.net/questions/194782 | 12 | Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum\_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that $$\sum\_{n=1}^{\infty} \frac{f(n)}{n} \neq 0?$$ I ask this question in attempt to understand how general of a statement ... | https://mathoverflow.net/users/50426 | Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$. | Not necessarily. The first counterexample might be
$q=14$ and $f(n)=1, -1, -1, -1, -1, 1, 0, -1, 1, 1, 1, 1, -1, 0$
for $n=1,2,3,\ldots,14$.
| 14 | https://mathoverflow.net/users/14830 | 194783 | 95,003 |
https://mathoverflow.net/questions/194588 | 6 | Let $K/k$ is a field extension and $G$ an affine group scheme over $K$. What are the Tannakian fundamental groups of these two $k$-tensor categories (with trivial fiber functors over $k$):
**1**. The category of pairs of finite vector spaces over $k$ with an isomorphism of their extensions to $K$,
**2**. The catego... | https://mathoverflow.net/users/51663 | Tannakian fundamental group of two explicit tensor categories | One possibly useful way of describing these groups is by their universal properties. The first group has the property that homomorphisms from it to any other pro-algebraic group $H$ are in bijection with $H(K) / H(k)$, and the second group has the property that homomomorphisms from it to $H$ are in bijection with homor... | 7 | https://mathoverflow.net/users/18060 | 194790 | 95,007 |
https://mathoverflow.net/questions/194776 | 0 | Before my question let me briefly describe a simplified version of the dynamical system I'm working with. Suppose that I have a density function $m(\boldsymbol{x},t)$, that describes the abundance of some quantity in a vector space $\Omega = [a,b]^{n}$. Now suppose that the density at each point in vector space changes... | https://mathoverflow.net/users/56169 | Help with notation for the state of a dynamical system defined by a PDE | For your first question, consider the following notation: At any given time, say $t$, the state of your system *is* the function $q\_t : \Omega \mapsto \mathbb{R}^{+}$, where $q\_t(x) \equiv m(x, t)$. How can we succinctly describe its time evolution? It is actually pretty simple:
$$ \frac{dq}{dt} = q \cdot (K q) $$
... | 1 | https://mathoverflow.net/users/24274 | 194792 | 95,008 |
https://mathoverflow.net/questions/194777 | 1 | Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise dense (and perhaps $S$-quasi-compact) open subscheme $U \subset A$ and an effective relative (to $S$) Cartier divisor $D \su... | https://mathoverflow.net/users/53197 | Schematic image of a relative Cartier divisor of a fiberwise dense open | Certainly it does not suffice to take the schematic closure if $S$ is nonreduced. For instance, let $S$ be $\text{Spec}\ k[x,y]/\langle x^2, xy \rangle$. Let $A$ be $E \times\_{\text{Spec} k} S$, where $E$ is an elliptic curve over $k$ with specified zero point $z$. Let $p \in S$ be the closed point with maximal ideal ... | 1 | https://mathoverflow.net/users/13265 | 194804 | 95,011 |
https://mathoverflow.net/questions/194796 | 7 | This question concerns a combinatorial identity obeyed by power series coefficients. Throughout we let $[x^{M}]\{\phi(x)\}$ denote the coefficient of $x^{M}$ in a power series $\phi(x)$.
Let $k$ be a positive integer, and consider the function $F(k,x)$ defined as the following power series in $x$:
\begin{equation}
... | https://mathoverflow.net/users/5124 | Identity for Power Series and Binomial Coefficients | Seems that a general formula for the $x^M$ coefficient of $\exp NF(k,x)$ is
$$
\frac{N}{M} (k^2-k) \left( {(k^2-k) N - (k-1) M - 1 \atop M-1} \right),
$$
which agrees with your formula when $N=M+1$. This should follow from
an explicit formula for $dF(k,x)/dx$ as a degree-$k$ algebraic function of $x$
that's closely rel... | 10 | https://mathoverflow.net/users/14830 | 194808 | 95,012 |
https://mathoverflow.net/questions/14991 | 16 | Let $(M,g)$ be some smooth, Riemannian manifold. Let $d$ be the exterior derivative and $\delta$ the codifferential on forms. For a smooth vector field $X$, let $L\_X$ be the Lie derivative associated to $X$. We know from Cartan formula that $L\_X = d \iota\_X + \iota\_X d$ where $\iota\_X$ is the interior derivative a... | https://mathoverflow.net/users/3948 | Commutator of Lie derivative and codifferential? | Doing along what Deane and Jose suggest one can make the following observations. First is that the Hodge star operator can be expressed in terms of the inverse metric $g^{-1}$ and volume form $\epsilon$. That is to say, formally for a form $\omega$ we have
$$ \*\omega = \epsilon \cdot (g^{-1}) \cdot \omega $$
Take $L... | 7 | https://mathoverflow.net/users/3948 | 194809 | 95,013 |
https://mathoverflow.net/questions/194389 | 10 | in what follows all the rings are commutative, nontrivial, with unit.
Recall the following definitions:
1) $\pi\in A$ is **prime** if $(\pi)$ is a nonzero prime ideal
2) $\pi\in A$ is **irreducible** if $\pi$ is nonzero, non invertible, and for all $a,b\in A$, $\pi=ab$ implies that $a$ or $b$ is a unit.
3) $\pi\i... | https://mathoverflow.net/users/38862 | Irreducible/prime/indivisible elements | To answer Q2: Every noetherian (commutative) ring that is not a finite product of fields admits irreducible elements. Since any finite product of $\ge 2$ fields admits prime elements, this shows that the answer to Q2 is no.
In an arbitrary commutative ring $A$, define $x\in A$ to be quasi-irreducible if $Ax$ is maxim... | 8 | https://mathoverflow.net/users/14094 | 194824 | 95,019 |
https://mathoverflow.net/questions/194825 | 9 | Let $X,Y$ be Riemannian manifolds, and $f\colon X\to Y$ be a Riemannian submersion.
Let $\gamma$ be a geodesic on $X$ starting at a point $x\in X$ and which is orthogonal to the fiber $f^{-1}(f(x))$.
**Questions.** (1) Is it true that $\gamma$ is orthogonal to any fiber of $f$ which it intersects?
(2) If this is no... | https://mathoverflow.net/users/16183 | Geodesics and Riemannian submersions | Take horizontal lift $\gamma$ of the minimal geodesic $\bar\gamma$ in Y, connecting two points $p$ and $q$ close to each other. It is (minimal) geodesic, since any other curve connecting them is not shorter than its horizontal projection, which in turn not shorter than minimal $\bar\gamma$. Since in each direction we c... | 7 | https://mathoverflow.net/users/1988 | 194831 | 95,021 |
https://mathoverflow.net/questions/194823 | 7 | I asked this initially in [math.stackexchange](https://math.stackexchange.com/questions/1118632/trigonometric-polynomials-on-non-compact-and-non-abelian-groups), but it disappeared almost immediately, so I hope it will be proper to aks this here.
Hewitt and Ross define *trigonometric polynomial* on a locally compact ... | https://mathoverflow.net/users/18943 | Trigonometric polynomials on non-compact and non-abelian groups | It isn't an algebra: the tensor product of irreducibles does not always decompose as a finite sum of irreducibles, although in some of those cases it may decompose as a direct integral of irreducibles.
For instance, take $G=H\_3({\bf R})$, the group of upper-triangular matrices with real entries and with $1$ on the d... | 3 | https://mathoverflow.net/users/763 | 194837 | 95,024 |
https://mathoverflow.net/questions/194803 | 5 | Suppose that $\mu\_1$ and $\mu\_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma\_x$ denote the resulting distribution by translating the centre of $\gamma$ from $0$ to $x$, $d\_{TV}(\cdot,\cdot)$ denote the tota... | https://mathoverflow.net/users/48609 | 1-wasserstein distance v.s. total variation distance | Yes.
I presume that your "1-Wasserstein" distance is what is otherwise called the transportation metric.
Let $M$ be any measure with marginals $\mu\_1$ and $\mu\_2$, so that $\mu\_1=\int \delta\_x\,dM(x,y)$ and $\mu\_2=\int\delta\_y\,dM(x,y)$, whence $\mu\_1-\mu\_2=\int (\delta\_x-\delta\_y)\,dM(x,y)$, which, after... | 3 | https://mathoverflow.net/users/8588 | 194838 | 95,025 |
https://mathoverflow.net/questions/194852 | 9 | The following question appeared in my research:
Let $G\_1,G\_2,G\_3$ all be subgroups of $S\_n$, and consider the sum
$$
\sum\_{g\_i \in G\_i, g\_1g\_2g\_3 = id} \epsilon(g\_1)
$$
that is, we only consider triplets whose product is the identity permutation,
and we sum all the signs of $g\_1$.
The question is: *is... | https://mathoverflow.net/users/1056 | Certain signed sum over $S_n$ | There is an example in $S\_5$ where the sum is negative. Take $G\_1 = \langle (12345), (1325) \rangle$, $G\_2 = \langle (135) \rangle$, $G\_3 = \langle (35), (235) \rangle$. Then
$ \bigl\{(g\_2,g\_3) : g\_2 \in G\_2, g\_3 \in G\_3, g\_2g\_3 \in G\_1 \bigr\} = \bigl\{ (\mathrm{id}, \mathrm{id}), ((135), (25)), ((153),... | 16 | https://mathoverflow.net/users/7709 | 194854 | 95,028 |
https://mathoverflow.net/questions/194829 | 2 | I asked a similar question for the case of compact groups not long ago in [math.stackexchange](https://math.stackexchange.com/questions/1114047/is-a-matrix-element-of-a-norm-continuous-representation-always-a-trigonometric-p). Now I understand that the answer was "yes", and I want to modify that question. This is also ... | https://mathoverflow.net/users/18943 | Is a matrix element of a norm continuous representation always a trigonometric polynomial? | In general, the answer is "no".
Let $G=\Bbb Z$; then all trigonometric polynomials are linear combinations of $f\_\xi\colon t\mapsto\xi^t$, $\xi\in\Bbb C$, $|\xi|=1$. On the other hand, consider the space $L^2(\Bbb Z)$, i.e., the space of sequences $\{a\_i\,|\,i\in\Bbb Z\}$ such that $\sum\_i|a\_i|^2<\infty$. This sp... | 3 | https://mathoverflow.net/users/44953 | 194856 | 95,029 |
https://mathoverflow.net/questions/19195 | 12 | In the preprint "[Smooth toric DM stacks](http://arxiv.org/abs/0708.1254)", Fantechi, Mann and Nironi define the stacks of their title, and show that each of these can be obtained through the following sequence of steps:
1) start with a scheme (the coarse moduli scheme) with at worst finite quotient singularities, an... | https://mathoverflow.net/users/940 | Conditions for "bootstrapping" a smooth DM stack? | **Notation:** Let $x$ be a point of a smooth separated finite type DM stack $\mathcal X$ over a field. Suppose
• $G$ is the stabilizer of $x$,
• $V$ is the tangent space of $x$ (which comes equipped with an action of $G$),
• $G^\textrm{triv}\subseteq G$ is the subgroup which acts trivially on $V$,
• $H = G/... | 6 | https://mathoverflow.net/users/1 | 194861 | 95,031 |
https://mathoverflow.net/questions/194817 | 6 | The references listed at <http://en.wikipedia.org/wiki/Computable_analysis> have all been published 30-15 years ago. Are the approaches which these references expose still up-to-date and relevant to the current paths of research in computable analysis? (In case they are not, where can I find a good introduction to the ... | https://mathoverflow.net/users/nan | Recent trends in effective analysis | (At François's request, my comment in now an answer.)
Yes, it is still an active research area. It however is spread out throughout a number of camps (traditions): The Weihrauch camp, the reverse math camp, the computability theory camp, the randomness camp, the proof theory camp, and a few different constructive mat... | 6 | https://mathoverflow.net/users/12978 | 194879 | 95,039 |
https://mathoverflow.net/questions/194882 | 10 | Let $\mathcal{C}$ be a presentable category, and let $S$ be a set of objects such that $S$ generates $\mathcal{C}$ under colimits, i.e., such that the smallest cocomplete subcategory of $\mathcal{C}$ containing $S$ is all of $\mathcal{C}$. Under what conditions is it true that for every object $x \in \mathcal{C}$, ther... | https://mathoverflow.net/users/344 | Can any object in a presentable category be written as a colimit of generators? | [Here](http://home.sandiego.edu/~shulman/papers/generators.pdf) is a short note by Mike Shulman with some relevant material. He remarks that the arrow category $\mathbb{2}$ inside $Cat$ (which of course is presentable) is a colimit generator in your sense (see his definition 3.6), but it is not a colimit-dense generato... | 7 | https://mathoverflow.net/users/2926 | 194889 | 95,042 |
https://mathoverflow.net/questions/194840 | 6 | How can one compute each of the following matrices, explicitly:
$$\int\_{O(n)} e^{g}dg$$ or
$$\int\_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$
What is the explicite entries of the resulting matrices, for $n=2$?
Moreover, for $n,m\in \mathbb{Z}$ define the linear operator $T\_{n,m}$ on $M\_{n}(\mathbb{R})$ as ... | https://mathoverflow.net/users/36688 | Some calculus in the orthogonal group $O(n)$ | let me work out the comments a bit further, starting from the identity (equation 8.2 from [Diaconis and Evans](http://statweb.stanford.edu/~cgates/PERSI/papers/functionals.pdf), correcting an earlier [paper](http://statweb.stanford.edu/~cgates/PERSI/papers/random_matrices.pdf) by Diaconis and Shahshahani)
$$\int\_{{\... | 13 | https://mathoverflow.net/users/11260 | 194900 | 95,048 |
https://mathoverflow.net/questions/194907 | 7 | Maybe this doubt is silly, but I do not understand the final step of the proof of Lemma 5.2 in Hamkins' paper *Fragile measurability,* Journal of Symbolic Logic 59 (1994) 262-282.
There, $\mathbb P\_\lambda$ is the $\lambda$-th step of an iterated forcing construction which is the inverse limit of the previous steps,... | https://mathoverflow.net/users/41274 | Generic filters of inverse limits | We can justify this with a general fact about forcing.
**Fact:** Let $G$ be $\mathbb P$-generic, where $\mathbb P$ is separative. If $X$ is a subset of $\mathbb P$ in the ground model, $X \subseteq G$, and $m = \inf X$, then $m \in G$.
**Proof:** Consider the set $\{ p \in \mathbb P : p \leq m$ or $(\exists x \in X... | 7 | https://mathoverflow.net/users/11145 | 194908 | 95,050 |
https://mathoverflow.net/questions/194911 | 13 | Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion.
Does $\Bbb P$ collapse the continuum?
| https://mathoverflow.net/users/7206 | Adding a real with infinite conditions | This is the forcing to add a Prikry-Silver real, discussed on [page 17 of Jech's book Multiple Forcing](https://books.google.com/books?id=qYqEYg6TMScC&pg=PA17&lpg=PA17&dq=prikry+silver+forcing&source=bl&ots=vts83a-YWY&sig=q4b6vmf78pwOTfN3HWGc2CJwapQ&hl=en&sa=X&ei=qD3GVI_iJuXIsAT2-ILwCw&ved=0CDgQ6AEwAw#v=onepage&q=prikr... | 11 | https://mathoverflow.net/users/1946 | 194913 | 95,052 |
https://mathoverflow.net/questions/114654 | 8 | By Hopkins Theorem it is well-known that every right (resp. left) artinian unitary ring is right (left) noetherian. Suppose that a noncommutative unitary ring R satisfies the descending chain condition on its two-sided ideals. Does R satisfy the ascending chain condition on two-sided ideals?
| https://mathoverflow.net/users/17582 | Descending chain condition in noncommutative rings | I'm somewhat surprised this question hasn't been answered previously. It turns out DCC on two-sided ideals does *not* imply ACC on two-sided ideals.
Let $V$ be a $k$-vector space with a basis of size $\aleph\_{\omega}$, and put $R={\rm End}\_k(V)$. Then, by Exercise 3.16 in Lam's "A First Course in Noncommutative Rin... | 6 | https://mathoverflow.net/users/3199 | 194928 | 95,057 |
https://mathoverflow.net/questions/194881 | 18 | A graph $G$ is described as a *unit-distance graph* if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can necessarily find an embedding into the algebraic numbers $\bar{\mathbb{Q}}$. In other words, if $G$ is a unit-distance g... | https://mathoverflow.net/users/39521 | Can all unit-distance graphs have their vertices at algebraic integers? | $\let\eps\varepsilon$No. I will present a graph whose realization necessarily contains a pair of vertices at distance $1/2$. THis cannot happen if the vertices are algebraic integers.
Firstly, we note that we may force a graph to contain a given piece of triangular lattice. It will be clear after we understand how to... | 13 | https://mathoverflow.net/users/17581 | 194930 | 95,058 |
https://mathoverflow.net/questions/194918 | 7 | For $n\ge 1$, let $g(x\_1,x\_2,\ldots,x\_n)$ be an irreducible *homogeneous* polynomial in $n$ variables over a field $k$ and $f(x)$ an irreducible polynomial of $k[x]$. Is $f(g(x\_1,x\_2,\ldots,x\_n))$ necessarily irreducible?
For instance this holds when $n=1$ (since then $g(x\_1)=\lambda x\_1$), or when $f$ has de... | https://mathoverflow.net/users/66365 | Irreducibility of a polynomial | I believe that the answer is yes. Put $c:=g(x\_1,\ldots, x\_n)$, which is irreducible in the UFD $R:=k[x\_1,\ldots, x\_n]$. Assume $f$ is irreducible, but also assume by way of contradiction that $f(c)=\alpha\beta$ in $R$, with $\alpha,\beta\notin k$. Write $\alpha=\sum \alpha\_i$ and $\beta=\sum \beta\_j$, where the $... | 5 | https://mathoverflow.net/users/3199 | 194937 | 95,060 |
https://mathoverflow.net/questions/194935 | 0 | consider the function given by $f(t):=\sum\limits\_{n=0}^{\infty}e^{-\left(n+\frac{1}{2}\right)^2t}$ for $t\in (0,\infty)$.
This function can be continued holomorphically for all complex numbers with positive real part $\Re(z)>0$ by the same formula.
My Question is: How can I prove that there must be an holomorphi... | https://mathoverflow.net/users/21870 | holomorphic continuation | On the contrary: the Jacobi theta function
$$\theta\_2(0,q) = 2 q^{1/4}\sum\_{n=0}^\infty q^{n(n+1)}$$
has a natural boundary at $|q|=1$.
Your function is $f(t) = (1/2) \theta\_2(0,\exp(-t))$, so you can't continue into the left half-plane.
| 8 | https://mathoverflow.net/users/13650 | 194939 | 95,062 |
https://mathoverflow.net/questions/194926 | 4 | Some authors use the term "continuous piecewise-linear" where other authors use the shorter term "piecewise-linear" (with continuity tacit).
I'd be interested in people's thoughts about this nomenclatural issue. I'm sympathetic to the use of the term "cpl" in work of Kirillov and Berenstein (see e.g. <http://math.uor... | https://mathoverflow.net/users/3621 | Continuous-piecewise-linear versus piecewise-linear | If "piecewise linear " means, linear (affine) on each member of some finite cover by *closed* intervals, then continuity is already there.
| 3 | https://mathoverflow.net/users/6101 | 194944 | 95,065 |
https://mathoverflow.net/questions/194952 | 2 | Is there a cardinal $\kappa \neq \emptyset$ and a connected poset $P$ of cardinality $\leq \kappa$ such that there is no surjective order-preserving map from $(\mathcal{P}(\kappa),\subseteq)$ onto $P$?
(We say that a poset is connected if it is connected as a directed graph.)
| https://mathoverflow.net/users/8628 | Order-preserving images of $(\mathcal{P}(\kappa),\subseteq)$ | A trivial necessary condition for such a surjection to exist is that $P$ is bounded, and this is sufficient. For any set $X$, let $F(X)$ be the free bounded poset on $X$ (i.e., $F(X)=X\sqcup \{0,1\}$ with any two elements of $X$ incomparable). For any bounded poset $P$ with underlying set $X$, there is a canonical orde... | 6 | https://mathoverflow.net/users/75 | 194954 | 95,069 |
https://mathoverflow.net/questions/194946 | 1 | I've come across this convex optimization problem in my research where I need to project a matrix $X\_0$ onto a non-negative affine space constraint and box constraints. Concretely,
$X \in \mathbb{R}^{m \times n}, ~ A \in \mathbb{R}^{q \times m}, C > 0, ~ m > q >> n > 1$.
$\displaystyle \min\_X || X - X\_0 ||\_F$, ... | https://mathoverflow.net/users/66375 | What's the most efficient way to solve this euclidean projection on non-negative affine space constraint? | By simple inspection, this is a simple quadratic programming problem which is convex and can be solved via interior point methods. CVX should work.
The KKT conditions can provide some insights about the analytical solution but I do not see and easy closed-form solution.
| 0 | https://mathoverflow.net/users/11825 | 194956 | 95,070 |
https://mathoverflow.net/questions/194955 | 5 | Let the permutation group $S\_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S\_4$. It is a projective variety by a theorem of Mumford. Does the line bundle $\mathcal O(1)^{\otimes 12}$ on $\mathbb P(\mathbb C^4)$ descend to the quotient $\mathbb P(\mathb... | https://mathoverflow.net/users/64495 | line bundle descends? | Yes, and the good news are that there isn't anything to compute. By a result of Kempf a line bundle $L$ on $\mathbb{P}^3$ descends to the quotient if and only if the stabilizer $S\_x$ of each point $x\in \mathbb{P}^3$ acts trivially on the fiber $L\_x$. Now $S\_4$ acts on $\mathcal{O}(1)$, and $S\_x$ act on $\mathcal{O... | 11 | https://mathoverflow.net/users/40297 | 194960 | 95,073 |
https://mathoverflow.net/questions/194910 | 29 | Given a positive integer $n$ which is not a perfect square, it is well-known that
[Pell's equation](http://en.wikipedia.org/wiki/Pell%27s_equation) $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$.
>
> **Question:** Let $n$ be a positive integer which is not a perfect square.
> Is there always ... | https://mathoverflow.net/users/28104 | Parametric solutions of Pell's equation | Let $n$ be a positive integer which is not a square and consider a fundamental solution $(a,b)$ of Pell's equation
$$a^2-nb^2=1.$$
Setting
$$\begin{cases}
D=(a+1)^2b^2X^2+2(a+1)^2X+n,\\
P=b^4(a+1)X^2+2b^2(a+1)X+a,\\
Q=b^3X+b,
\end{cases}$$
we have the identity
$$P^2-DQ^2=1,$$
with $D(0)=n,P(0)=a$ and $Q(0)=b$. This exp... | 19 | https://mathoverflow.net/users/14809 | 194966 | 95,074 |
https://mathoverflow.net/questions/194968 | 1 | If $L$ is a complete lattice and $P$ is a poset and $f: L\to P$ is an order preserving surjective map, does this imply that $P$ is a (complete) lattice?
| https://mathoverflow.net/users/nan | Order-preserving image of a complete lattice | Not necessarily - there is even a finite counterexample.
Let $X = \{1,2,3\}$ and set $L := \mathcal{P}(X)$.
Let $P := \{b, t\} \cup \{(i,j): i,j \in \{0,1\}\}$, where $b$ will be the botton (least) element, $t$ is the top (greatest) element, and $\{(i,j): i,j \in \{0,1\}\}$ is ordered by $$(i, j) < (k, l) \text{ ... | 0 | https://mathoverflow.net/users/8628 | 194970 | 95,077 |
https://mathoverflow.net/questions/194971 | 2 | Here is an extract of the doctoral thesis of C. Lewis under the supervision of D. Joyce (<https://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf>, 1998):
---
---
**2.6 Spin Bundles and the Dirac Operator**
To consider spin bundles over a $Spin(7)$ manifold $M$, it is usually best to first consider Cliffo... | https://mathoverflow.net/users/62367 | Isomorphisms of Positive and Negative Spinor Bundles | You're really asking an algebra question about how the various representations of $\mathrm{Spin}(8)$ interact. There are lots of places where you can read about this, but [here](http://www.math.duke.edu/~bryant/Spinors.pdf "here") is a set of notes that I wrote, *Remarks on Spinors in Low Dimensions*, that explains thi... | 3 | https://mathoverflow.net/users/13972 | 194973 | 95,078 |
https://mathoverflow.net/questions/194979 | 20 | Let $Con(\mathtt{ZFC}, n)$ denote the statement "$\mathtt{ZFC}$ cannot prove the contradiction within $n$ steps (or better within $n$ symbols) within a given proof system (say a natural deduction to avoid trivialities)". Suppose that ($\mathtt{ZFC}$ is consistent and) $\mathtt{ZFC} \models Con(\mathtt{ZFC}, n)$, then s... | https://mathoverflow.net/users/21152 | Can ZFC prove it cannot derive an inconsistency in $n$ steps? | It’s not very clear to me what you mean by “steps”. One might interpret it as the number of lines in a Hilbert/natural deduction proof, but then there are infinitely many proofs with a fixed number of steps, so this is inconsistent with the argument outlined in the question.
So, let me use a measure that has the prop... | 44 | https://mathoverflow.net/users/12705 | 194984 | 95,082 |
https://mathoverflow.net/questions/194962 | 2 | I've read somewhere that the cut rule in sequent calculus
$$\frac{A \vdash \mathbf{C}, B \qquad A',\mathbf{C} \vdash B'}{A,A' \vdash B,B'} (\text{cut})$$
states that the $\mathbf{C}$ on the right is stronger than $\mathbf{C}$ on the left.
I would like to know what is this notion of strongness and how is $\mathbf{C}$ ... | https://mathoverflow.net/users/66386 | Notion of strongness in cut rule | First, by "stronger than", Girard means "at least as strong as". So the identity rule can be read as saying "If you have a C on the left side, you can have C on the right side as well (because you can use the identity rule to derive them)", and in that sense "a C on the left is as good as having a C on the right".
An... | 4 | https://mathoverflow.net/users/8991 | 194988 | 95,083 |
https://mathoverflow.net/questions/191708 | 5 | I want to find the solution with most zero-components for the following problem:
$Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional constraints.
It's a known fact that it's NP-Complete. So I want to find
1) a heuristic for it, or
2) another NP-Complete Pr... | https://mathoverflow.net/users/64413 | Finding sparsest solution of a linear system | I think you'll find attempts at solving this problem mostly in dictionary learning papers.
The most popular approach today is solving the double optimization problem of learning a dictionary and providing a "good" sparse representation by performing multiple iterations that make use of a sparse coding algorithm in th... | 2 | https://mathoverflow.net/users/58424 | 194995 | 95,085 |
https://mathoverflow.net/questions/194994 | 2 | It is well-known that the symmetric group S4 has two Schur covering groups, S4-tilde and S4-hat. There are explicit presentations for both groups, and we know that S4-hat is isomorphic to GL(2,3). Question: Is there an alternative, explicit description/realization of S4-tilde?
The lowest degree d for which S4-tilde c... | https://mathoverflow.net/users/66397 | Schur covering group for S4 | What you call $\tilde{S}\_{4}$ is also known as the binary octahedral group. Probably the easiest way to construct it is to take the natural matrix representation of $G = {\rm GL}(2,3)$ and leave the elements of ${\rm SL}(2,3)$ as they are, but replace the extra generator $\left(\begin{array}{clcr} 1 & 0\\0& -1 \end{ar... | 6 | https://mathoverflow.net/users/14450 | 194998 | 95,087 |
https://mathoverflow.net/questions/195011 | 14 | I'm studiyng [Higman's Embedding Theorem](http://en.wikipedia.org/wiki/Higman's_embedding_theorem), and a fundamental part of the proof is the following lemma:
*If R is a benign normal subgroup of finitely generated group F, then F/R can be embedded in a finitely presented group.*
A proof of the lemma in context ... | https://mathoverflow.net/users/66413 | Why is "The Higman Rope Trick" thus named? | [Too long for a comment] No mathematical content to offer, but the name is almost certainly a reference to the Indian Rope Trick (see <http://en.wikipedia.org/wiki/Indian_rope_trick>), which is a famous magic trick in which a boy climbs up a rope and disappears. Perhaps the lemma has something to do with "climbing up" ... | 3 | https://mathoverflow.net/users/24993 | 195015 | 95,095 |
https://mathoverflow.net/questions/195018 | 20 |
>
> This is a crossport of [this question](https://math.stackexchange.com/questions/1099950/categorical-proof-that-subgroups-of-free-groups-are-free) from MSE.
>
>
>
Is there a categorical proof that subgroups of free groups are free?
How about the result that subgroups of free *abelian* groups are free abelian?... | https://mathoverflow.net/users/53127 | Categorical proof subgroups of free groups are free? | "Subobjects of free algebras are free" is satisfied comparatively rarely for algebraic theories. I'm going to start a list and people should feel free to add. I'm making it CW.
Before starting, let me say that IMHO a more interesting general question to consider is: when are retracts of free objects free? That can b... | 12 | https://mathoverflow.net/users/2926 | 195023 | 95,099 |
https://mathoverflow.net/questions/195020 | 0 | Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that
$$K\_X = f^{\*}K\_S+\sum\_ia\_iE\_i$$
with $a\_i>0$. By Grauert-Mumford theorem the intersection matrix of the $E\_i$'s is negative definite. I found the following claim: there exists an $E\_j$ such that
$$E\... | https://mathoverflow.net/users/nan | Intersection Matrix of a resolution | You can argue also in the following way (let us do the case of two componets $E\_1$, $E\_2$ for simplicity of notation. The general case will be clear):
the intersection matrix
$$
I = \left(\begin{array}{cc}
E\_1^2 & E\_1E\_2 \\
E\_1E\_2 & E\_2^2
\end{array}\right)
$$
is negative definite. In particular if you take ... | 1 | https://mathoverflow.net/users/14514 | 195024 | 95,100 |
https://mathoverflow.net/questions/194965 | 8 | Given a regular set $E\subset \mathbb R^2$ define
$$
R(E) = \sup\{r\colon \exists x,\ B(x,r)\subseteq E\}
$$
to be the radius of the largest circle contained in $E$ and let $|\partial E|$ be the length of the perimeter of $E$.
Among all planar sets $E$ with fixed area, what is the one which minimizes the product $|\... | https://mathoverflow.net/users/36826 | Set with small internal radius, small perimeter and prescribed area | For the simply connected version of your question: yes, such an estimate exists. Namely: take a radius $R=R(E)$ (closed) "coloring" disk, and let us move its center along the boundary $\partial E$. Note that while the center of the disk makes a path of length $l=|\partial E|$, the disk "colors" the area that is at most... | 4 | https://mathoverflow.net/users/31371 | 195028 | 95,101 |
https://mathoverflow.net/questions/195030 | 3 | An orientable manifold can have torsion in its integer homology. But I believe by Poincare duality the manifold must be at least 4-dimensional -- isn't that right? Anyway are there simple examples of such torsion?
| https://mathoverflow.net/users/38783 | Example of torsion in orientable manifolds? | Consider $PSU(2)$, the three-dimensional projective special unitary group (or just $\mathbb{R}\mathbb{P}^3$). It is a Lie group, and therefore orientable. Yet, it is the quotient of the simply-connected group $SU(2)$ by its center $\mathbb{Z}\_2$, so it has 2-torsion in homology.
| 3 | https://mathoverflow.net/users/25358 | 195033 | 95,102 |
https://mathoverflow.net/questions/184347 | 12 | Let $K$ be a quadratic field. Let $f\in\mathbb{Z}\_{\geq 1}$. Let $\mathcal{O}\_f=\mathbf{Z}+f\mathcal{O}\_K$ be the unique order of $K$ of index $f$ in $\mathcal{O}\_K$. Let $H\_f^{ring}$ denote the ring class field in the narrow sense
(so we allow ramification at infinite real places of $K$, if such places exist)
ass... | https://mathoverflow.net/users/11765 | Intersection of a ring class field of a quadratic field K with the cyclotomic extension of K | The answer is no. I'll only focus on imaginary quadratic fields since they are easier to deal with than real ones. Great reference for all this theory is Cox's book *Primes of the form $x^2+ny^2$*.
So let $K$ be imagimary, thus $d<0$, and let $D=f^2d$. Since $L=\bigcup\_{n\in\mathbb{N}}\mathbb{Q}(\zeta\_n)$ is the ma... | 6 | https://mathoverflow.net/users/32216 | 195041 | 95,108 |
https://mathoverflow.net/questions/195031 | 15 | Given a matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$,
its transpose, obviously, is $A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$.
But is there a conventional way of notating the matrix
$\begin{pmatrix}d&b\\c&a\end{pmatrix}$
in terms of $A$?
This generalizes in an obvious way to larger matrices.
Also, do you know of so... | https://mathoverflow.net/users/14835 | Is there a standard notation for off-diagonal transpose? | In <http://arxiv.org/abs/math/0701936> (Fuchsian equations of type DN, by Vasily Golyshev and Jan Stienstra) the transpose of the matrix $A$ with respect to the anti-diagonal is denoted by $A^\tau$. It relates to the ordinary transpose $A^T$
(or $A^t$ as used in the paper), as follows:
$$A^\tau=JA^TJ$$
where $J=(J\_{ij... | 19 | https://mathoverflow.net/users/32389 | 195044 | 95,110 |
https://mathoverflow.net/questions/188840 | 8 | Let $d \geq 2$ be an integer, and let $\mathcal{F}\_d$ be the family of finite groups such that $G \in \mathcal{F}\_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements.
**Is there a finite nontrivial word $w = w(x\_1, \dots, x\_n)$ which is trivial on $\mathcal{F}\_d$?**
That is, can ... | https://mathoverflow.net/users/38889 | Is the free abstract group residually of rank d > 2? | No, there is not such a word, by the following two facts.
1. The two elements
$$A=\left(\begin{array}{cc}1&2\\0&1\end{array}\right),\quad B=\left(\begin{array}{cc}1&0\\2&1\end{array}\right)$$ of $\text{SL}\_2(\mathbf{F}\_p)$ satisfy no nontrivial relation of length less than $c\log p$.
2. Every subgroup of $\text{SL}... | 4 | https://mathoverflow.net/users/20598 | 195062 | 95,115 |
https://mathoverflow.net/questions/195027 | 4 | Suppose I am given a morphism $f:BG\to BGL\_1(R)$ for $R$ some at least $E\_1$-ring spectrum and $G$ a loop space. Then This corresponds, I believe, to an action of $G$ on $R$, coming from a morphism $G\to GL\_1(R)$. The first part of my question is: does this imply a map of spectra $R[G]\wedge R\to R$ or something lik... | https://mathoverflow.net/users/11546 | Bar Construction Model of Ring Spectrum Quotient | For your first question, the answer is yes. A map $BG\to BGL\_1(R)$ gives you a map $G\to GL\_1(R)$. Since $GL\_1(R)$ is a set of component of $\Omega^{\infty}(R)$, this gives you a map $G\to \Omega^{\infty}(R)$ or equivalently a map $\mathbb{S}[G]=\Sigma\_+^\infty G\to R$. The multiplication map $R\wedge R\to R$ gives... | 5 | https://mathoverflow.net/users/10707 | 195068 | 95,116 |
https://mathoverflow.net/questions/195078 | 9 | Is it consistent that there is some partial order $\mathbb P$ and some inaccessible cardinal $\kappa$, which is the least inaccessible, such that $\mathbb P$ forces $\kappa$ to be singular while preserving all cardinals?
| https://mathoverflow.net/users/11145 | singularize the least inaccessible? | In the paper "On Lowenheim-Skolem-Tarski numbers for extension of first order logic", by Magidor and Vaananen, in Theorem 21 they state that it is consistent, relative to the existence of a supercompact cardinal, that the Lowenheim-Skolem-Tarski number of $L(I)$ is the first inaccessible cardinal, were $L(I)$ is the ex... | 10 | https://mathoverflow.net/users/41953 | 195083 | 95,119 |
https://mathoverflow.net/questions/195067 | 6 | There are several invariants whose "natural" domain is a category of **disoriented** tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the orientation is reversed.
*For example*:
* Khovanov homology [HERE](http://arxiv.org/pdf/math/0701339v2.pdf) and [... | https://mathoverflow.net/users/2051 | What is the original reference for disorientations on tangle diagrams? | The *term* "disoriented tangle" first appeared in arxiv:0701339v2 (your first link). That is also the earliest reference I know for bordisms of disoriented tangles.
But the *idea* of disoriented tangles is much older. It emerges naturally when one considers string diagrams for $Rep(U\_q sl\_2)$. See, for example, Fig... | 3 | https://mathoverflow.net/users/284 | 195084 | 95,120 |
https://mathoverflow.net/questions/194983 | 3 | I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the integrand. Is it possible to use Monte Carlo technique to do this integration?
To make the problem more specific, I write it in... | https://mathoverflow.net/users/66393 | Monte Carlo integration of Gaussian integrals | If you are computing an integral, and it has a single sharp peak, then the simplest solution is to use the [Laplace approximation](http://en.wikipedia.org/wiki/Laplace%27s_method), which provides a good approximation for this exact type of problem. Monte Carlo methods in general have trouble with sharply-peaked integra... | 1 | https://mathoverflow.net/users/3711 | 195098 | 95,123 |
https://mathoverflow.net/questions/195100 | 0 | Let $M\_g$ and $M\_h$ be closed orientable 3-manifolds of genus $g$ and $h$ respectively and suppose that $M\_g$ is an $n$-sheeted cover of $M\_h$. Is there a formula that would allow us to compute $g$ if we knew the values of $h$ and $n$?
I know there is a formula for closed orientable surfaces and I was wondering i... | https://mathoverflow.net/users/42049 | Genus of Covering Space of 3-Manifold | Heegaard genus is not very constructive and is very difficult to control. Obviously, there cannot be a formula in terms of just $n$ and $h$. It suffices to consider the double coverings $S^3\to\Bbb R\mathrm{p}^3$ and $S^1\times S^2\to S^1\times S^2$.
| 6 | https://mathoverflow.net/users/44953 | 195101 | 95,124 |
https://mathoverflow.net/questions/195105 | 4 | In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated,
For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction map on the N-torsion, $$E[N] \rightarrow \tilde{E}[N]$$ is surjective for all N.
The authors then state that this is be... | https://mathoverflow.net/users/66443 | Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves | Let $p$ be the characteristic of $\mathfrak{p}$.
If $p\nmid N$, then the map is an isomorphism, since both groups have order $N^2$ and the kernel is in the formal group, which has no prime-to-$p$ torsion. [1, Prop VII.3.1] So by the Chinese remainder theorem, we're reduced to the case that $N=p^k$. If $\tilde E$ is sup... | 8 | https://mathoverflow.net/users/11926 | 195111 | 95,127 |
https://mathoverflow.net/questions/195110 | 5 | I'm reading Alexeev's "Complete moduli in the presence of semiabelian group action" and I just started to study toric geometry.
In chapter 2 in order to obtain an affine toric variety he takes $P:=Spec(k[\omega\cap X])$ where $\omega$ a cone in $X\otimes \mathbb{R}$ and $X$ is a lattice. Then the torus $T=Hom(X,k^\*)... | https://mathoverflow.net/users/60675 | Alexeev's projective torus embeddings | There are two ways to understand it.
**(1) A projective variety is a quotient of affine one.**
Let $V$ be denote the projective toric variety defined by polytope $\delta$ and $n=\mathrm{dim}(T)=\mathrm{rk}(X)$ be its dimension.
While we have $\mathrm{cone}(\delta)$ in lattice $X\oplus\mathbb Z$ we have an affine to... | 1 | https://mathoverflow.net/users/19436 | 195113 | 95,128 |
https://mathoverflow.net/questions/25178 | 14 | In studying presentations of pro-$p$-groups via generators and relations, one is led (via the so-called Magnus embedding) to questions involving power series in non-commuting variables. Results from local algebraic geometry occasionally shed some insight on how to make progress, but more often that not, I find myself l... | https://mathoverflow.net/users/35575 | A Non-Commutative Nullstellensatz | Let $F$ be a field, and let $f\_1,f\_2,\ldots, f\_k\in R:=F\langle\langle x,y\rangle\rangle$ with $k\in \mathbb{N}$. Order monomials in $R$ by degree, and then lexicographically. Since the question concerns computability, assume that there is an algorithm which spits out the coefficients of the monomials in $f\_i$ (in ... | 4 | https://mathoverflow.net/users/3199 | 195116 | 95,129 |
https://mathoverflow.net/questions/195092 | 2 | I'm trying to find the limit of the ratio of two functions
$ \lim\_{t \rightarrow \infty} \frac{f(t)}{g(t)} $ but only have the initial conditions and the differential equations they solve, but the equations don't have easy closed form solutions:
$f'(t) = -\beta f(t)^2 + (\beta - \gamma) f(t) + \gamma e^{-\delta t} \... | https://mathoverflow.net/users/66436 | Limiting Ratio of Solutions to Ordinary Differential Equations | Your differential equations do have closed-form solutions. $f(t)$ is a rather complicated expression involving Bessel functions, while
$$ g \left( t \right) ={\frac {{{\rm e}^{- \left( \gamma-\beta \right) t
}} \left( \gamma-\beta \right) }{{\gamma-{\rm e}^{- \left( \gamma-\beta
\right) t}}\beta}}
$$
Since you want ... | 3 | https://mathoverflow.net/users/13650 | 195124 | 95,132 |
https://mathoverflow.net/questions/194284 | 4 | Let $k$ be a field with char $k \neq 2$. For $a,b \in k^{\times}$, let $(a,b)$ denote the quaternion algebra with $i^2=a$ and $j^{2}=b$, and let $C(a,b)$ denote the projective plane conic given by $ax^2+by^2=cz^2$. A theorem of Witt (Theorem 1.4.2 in [1]) says that $(a\_{1},b\_{1})$ is $k$-isomorphic to $(a\_{2},b\_{2}... | https://mathoverflow.net/users/66133 | Are quaternion algebras from Witt's theorem endomorphism rings of vector bundles? | Let $S$ denote an indecomposable bundle over $C(a,b)$ of rank two and degree two. By the proof of Proposition 3.5 [2], $S \otimes \Omega^{1}$ corresponds to a nonsplit extension of $\mathcal{O}\_{C(a,b)}$ by $\Omega^{1}$. The answer now follows from Bhargav's answer to
[Explicit Bijection between Central Simple Algeb... | 1 | https://mathoverflow.net/users/66133 | 195127 | 95,134 |
https://mathoverflow.net/questions/195103 | 9 | It is known that orders in quaternion algebras (over a number field) are used for constructing geometric objects like hyperbolic orbifolds and Shimura curves. Moreover, if one knows embedding properties (selectivity) of the orders one can construct isospectral but non isometric manifolds. I wonder what happen in higher... | https://mathoverflow.net/users/66441 | Orders in Central Simple Algebras. Applications | I believe that the answer to your question about whether or not connections are known in higher dimensions is yes. Before I get into the details however, it might be useful to recall what happens for quaternion algebras.
Let $k$ be a totally real field and $B$ be a quaternion division algebra over $k$ which is split ... | 6 | https://mathoverflow.net/users/nan | 195133 | 95,139 |
https://mathoverflow.net/questions/195081 | 3 | Let $P$ be a normal, $\mathbb{Q}$-Gorestein variety with terminal singularities. Let $X \subseteq P$ be a normal, irreducible Weil divisor such that $X \sim\_{\mathbb{Q}} - K\_P$, that is $\mathbb{Q}$-linearly equivalent to the anticanonical divisor of $P$ (which is $\mathbb{Q}$-Cartier) .
My question is, does $X$ ha... | https://mathoverflow.net/users/29730 | Is an anticanonical Weil divisor in $\mathbb{Q}$-Gorenstein variety Calabi-Yau? | How about this:
As you assumed, $P$ has terminal singularities, which means that the singular locus of $P$ has codimension at least 3 (see Corollary 5.18 in Kollár-Mori's book). Hence, the singular locus has codimension at most 2 in $X$. We can consider everything outside the singular locus because we work on the level... | 5 | https://mathoverflow.net/users/42636 | 195142 | 95,144 |
https://mathoverflow.net/questions/194675 | 4 | Suppose that $\Phi\_1$ and $\Phi\_2$ represents two independent Poisson point processes respectively with intensity $\lambda\_1$ and $\lambda\_2$ (therefore). We know very well different operations on Poisson processes preserving the Poisson law like superposition, dilation and thinning.
I would like to know if there... | https://mathoverflow.net/users/39212 | On Minkowski sum of two independent Poisson point processes | Indeed the process is not locally finite if your processes are homogeneous on $R^d$ (that can be deduced from the computations below).
If on the other hand your intensity measures $\lambda\_1$ and $\lambda\_2$ are finite, then you have a point process that is not Poisson.
Let $K$ be some compact set. Then $$\Phi\_3... | 1 | https://mathoverflow.net/users/16934 | 195147 | 95,146 |
https://mathoverflow.net/questions/195155 | 5 | I have the following question:
The usual construction of the [Antoine's Necklace](http://en.wikipedia.org/wiki/Antoine%27s_necklace) produces a Cantor of $1$-dimensional Hausdorff measure in $\mathbb R^3$.
I would like to know whether one could adapt the construction to produce a larger Cantor set, namely a antoin... | https://mathoverflow.net/users/26608 | Antoine's Necklace and positive Hausdorff/Lebesgue measure | It can be done for the very same reason you mention: given a torus, one can find inside it four linked tori of arbitrarily large relative measure. So one can proceed as in the standard construction of a Cantor set of positive Lebesgue measure.
More explicitly, fix a sequence $r\_n\in (0,1)$ such that $\prod\_{n=1}^\i... | 2 | https://mathoverflow.net/users/11009 | 195158 | 95,150 |
https://mathoverflow.net/questions/102331 | 10 | It is known that for any graph H and all $k∈N$, there exists a graph $G$ such that any $k$-coloring of the edges of $G$ yields a monochromatic copy of H and ω(G)=ω(H) (the two graphs have the same clique numbers).
My question is: Given any graph $H$ with finite girth, is there a $G$ with the same girth as $H$ such t... | https://mathoverflow.net/users/14875 | Sparse ramsey theory | [Here](http://jdc41.user.srcf.net/useful/sparse-ramsey.pdf) there is a set of lecture notes from a course given by Imre Leader in 2003. On page 18 of these notes this is given as an open problem.
| 8 | https://mathoverflow.net/users/1098 | 195159 | 95,151 |
https://mathoverflow.net/questions/195114 | 3 | Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions:
1) is $X(\mathbb{Q})$ an empty set ?
2) is $X(\mathbb{Q})$ a finite (non empty) set ?
3) is $X(\mathbb{Q})$ an infinite set ?
thanks.
| https://mathoverflow.net/users/66195 | cohomological obstructions and rational points | Let me just give some pointers to further literature.
In the curve case, there is a cohomological obstruction to a curve over $\mathbb{Q}$ having infinitely many points, it is given by Faltings' solution of the Mordell conjecture (I know that this is almost certainly a deliberate misinterpretation of the question). ... | 4 | https://mathoverflow.net/users/50846 | 195164 | 95,152 |
https://mathoverflow.net/questions/195076 | 3 | A *linear hypergraph* is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties:
1. for $e\in L$ we have $|e|\geq 2$;
2. if $e\_1\neq e\_2 \in L$ then $|e\_1\cap e\_2|\leq 1$.
We set $X(\pi)=X$ and $L(\pi)=L$. The *graph* $G\_\pi$ *associated to* a linear hype... | https://mathoverflow.net/users/8628 | Linear intersection number and vertex covering number | Let $H$ be a graph on $n$ vertices, thought of as the hypergraph in the question. Then a vertex cover in $G\_H$ is precisely a subgraph of $H$ that contains all but at most one edge at each vertex of $H$. Such a subgraph is obtained from $H$ by deleting a subgraph of maximum degree at most $1$, which contains at most $... | 2 | https://mathoverflow.net/users/25485 | 195172 | 95,155 |
https://mathoverflow.net/questions/195152 | 3 | $\require{AMScd}$ I am interested in detecting using a lifting property when a map $f:X\to Y$ in $sSet$ (with the standard Kan model structure) is a weak equivalence. In the paper *Weak Equivalences and Simplicial Presheaves* (<http://www.math.uiuc.edu/K-theory/0564/wesp.pdf>) Daniel Dugger and Daniel Isaksen give the ... | https://mathoverflow.net/users/56531 | recognising weak equivalences of simplicial sets | $\newcommand{\Ex}{\mathrm{Ex}}\newcommand{\Sd}{\mathrm{Sd}}$Consider the square
$$\begin{CD}
X @>{\sim}>> \Ex^\infty X \\
@V{f}VV @VV{\Ex^\infty f}V \\
Y @>>{\sim}> \Ex^\infty Y \textrm{.} \\
\end{CD}$$
The map $f$ is a weak equivalence if and only if $\Ex^\infty f$ is if and only if there is a relative lift in eve... | 6 | https://mathoverflow.net/users/12547 | 195176 | 95,156 |
https://mathoverflow.net/questions/195180 | 3 | Let $M$ ba a compact Kaehler Fano manifold. Under which conditions $M$ admit a smooth anti-canonical divisor $D$
| https://mathoverflow.net/users/nan | Fano manifold admit an smooth anti-canonical divisor? | If $M$ is *smooth* and $\dim M =3$, then the answer is *always*. This is a theorem due to Shokurov, see
V.V. Shokurov: [*Smoothness of a general anticanonical divisor on a Fano variety*](http://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCMQFjAA&url=http%3A%2F%2Fwww.maths.ed.ac.uk%2Fch... | 8 | https://mathoverflow.net/users/7460 | 195187 | 95,159 |
https://mathoverflow.net/questions/195175 | 2 | For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a [vertex cover](http://en.wikipedia.org/wiki/Vertex_cover) of $G$.
As noted [here](https://math.stackexchange.com/questions/1123167/chromatic-number-and-vertex-covering-number), we have $\tau(G) \geq \chi(G) - 1$ for all finite graphs $G$. I'm inte... | https://mathoverflow.net/users/8628 | When the vertex covering number is smaller than the chromatic number | Expanding a bit the construction given by Leen Droogendijk on MSE, let $C$ be a vertex cover of minimal size. Then we can colour $G$ by colouring $C$ (considered as an induced subgraph), and giving all vertices of $V-C$ an extra colour. This shows
$$\chi(G)\le1+\chi(C),$$
hence $\tau(G)=\chi(G)-1$ can only happen if $\... | 4 | https://mathoverflow.net/users/12705 | 195200 | 95,167 |
https://mathoverflow.net/questions/195186 | 8 | I am trying to find the answer of
$$\int dU \ |Tr(U^m)|^2$$
where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the answer seems to be $1$.
I don't know where to start. Does anyone have an idea? Is there a clean answer for $m>1$ ?
| https://mathoverflow.net/users/66471 | Expectation of trace of nth power of unitary matrices | $$\int\_{{\rm U}(n)} dU\,|{\rm Tr}\,(U^m)|^2={\rm min}\,(n,m).$$
see Theorem 2.1.b of [Diaconis and Evans](http://statweb.stanford.edu/~cgates/PERSI/papers/functionals.pdf) (2001). [\*]
[\*] This 2001 reference corrects an earlier paper by [Diaconis and Shahshahani](http://statweb.stanford.edu/~cgates/PERSI/papers/... | 10 | https://mathoverflow.net/users/11260 | 195201 | 95,168 |
https://mathoverflow.net/questions/195203 | 3 | Let $f(t)\in\mathbb{C}[t]$, and let $I\_f$ be the ideal in $\mathbb{C}[t]$ generated by $f(t)$.
The ideal $I\_f$ has a natural $\mathbb{C}$-algebra structure. My question is the following:
>
> For which polynomials $f\in \mathbb{C}[t]$ is the group of ($\mathbb{C}$-algebra) automorphisms of $I\_f$ trivial?
>
>
... | https://mathoverflow.net/users/20105 | Automorphisms of ideals of $\mathbb{C}[t]$ | We may assume that $f$ is not a unit. Let $A\_f=I\_f+\mathbb{C}\cdot 1$, the free unital algebra on $I\_f$; this embeds naturally in $\mathbb{C}[t]$. Furthermore, $A\_f$ and $\mathbb{C}[t]$ have the same fraction field, and $\mathbb{C}[t]$ is the integral closure of $A\_f$ in its fraction field. Thus every automorphism... | 11 | https://mathoverflow.net/users/75 | 195205 | 95,171 |
https://mathoverflow.net/questions/194899 | 1 | Given $A\in\{0,1\}^{n\times n}$ with $\operatorname{rank}(A)=r=2^{O((\log\_2n)^{\frac{1}{c+1}})}$.
Denote $\mathsf{1\_n}\in\{0,1\}^n$ as vector with $1$s.
Does
$$\lim\_{n\rightarrow\infty}\mathsf{P\_{A\in\{0,1\}^{n\times n}}}\Bigg(\mathsf{1\_n}'A\mathsf{1\_n}>\frac{(r-1)r^{(\log\_2r)^c}}{2\log\_2r}\Bigg)=1\quad?$$
... | https://mathoverflow.net/users/10035 | Probabilistic statement on matrix ranks | Consider the set of $n\times n$-matrices with entries in $\{0,1\}$ which have at most $r$ distinct rows. The number of such matrices is $2^{rn}r^n$. As long as $n$ and $n-r$ tend to infinity, we have that such a matrix almost surely has rank $r$, and contains close to $n^2/2$ entries equal to 1.
Next the number of $... | 1 | https://mathoverflow.net/users/37555 | 195206 | 95,172 |
https://mathoverflow.net/questions/195099 | 6 | Let $K\subset \mathbb{R}^N$ be a compact set. We say
$K$ is "good" if the following property holds:
Given a set of open neighborhoods $\{x\in U\_x\subset \mathbb{R}^N\}\_{x\in K}$ there exists a finite set $S\subset K$ and relatively compact open subsets $x\in \tilde{U}\_x\subset U\_x$ such that
1. $\{\tilde{U}\_x\... | https://mathoverflow.net/users/5259 | A question on compact sets | This is an attempt to formalize Christian Remling´s idea.
Cover $K$ by finitely many balls $\{B(x,r\_x): x \in S \}$ such that $\overline{B(x,2r\_x)} \subseteq U\_x$. Let $\delta$ be the minimum of the $r\_x$'s, and for each pair $x,y \in S$, find a positive real $\lambda\_{x,y}<\delta$ such that: $$\overline{B(x,r\_... | 0 | https://mathoverflow.net/users/17836 | 195208 | 95,173 |
https://mathoverflow.net/questions/195153 | 1 | Let $Q^{-1}$ be the inverse function of a standard normal CDF. For $0 < \epsilon < p,p' < 1 - \epsilon$, how much does the function $Q^{-1}$ change as a function of $|p - p'|$? Any useful upper bounds would be helpful.
| https://mathoverflow.net/users/37537 | Sensitivity of inverse normal cdf | If $F$ is the standard normal CDF,
$$(Q^{-1})'(p) = \dfrac{1}{F'(t)} = \sqrt{2\pi} \exp(t^2/2)$$
where $p = F(t)$. The maximum for $\epsilon \le p \le 1-\epsilon$ is at the endpoints. So
$$|Q^{-1}(p) - Q^{-1}(p')| \le \sqrt{2\pi} \exp(t^2/2) |p - p'|$$
where $t = Q^{-1}(1-\epsilon)$.
Asymptotically as $t \to +\infty$,... | 2 | https://mathoverflow.net/users/13650 | 195211 | 95,175 |
https://mathoverflow.net/questions/195179 | 5 | After pondering this MO question > [Location of maximum of Brownian motion with rough drift](https://mathoverflow.net/questions/193848) <, I wonder whether a Brownian motion can be fast (i.e. beats the law of the iterated logarithm) at its extrema? Is it necessarily fast?
| https://mathoverflow.net/users/57941 | Can a Brownian motion be fast at its extrema? | Heuristically, a point $(t,B\_t)$ being a extremum is antithetical to fast oscillation, since there is no oscillation on one side of (above/below) $B\_t$.
However,
1. This is only a heuristic, and
2. One may wonder whether there are so many fast times and so many maxima as to force an overlap.
At least we can ru... | 4 | https://mathoverflow.net/users/4600 | 195228 | 95,181 |
https://mathoverflow.net/questions/185527 | 42 | I would love to understand the famous formula $g\_{ij}(x) = \delta\_{ij} + \frac{1}{3}R\_{kijl}x^kx^l +O(\|x\|^3)$, which is valid in Riemannian normal coordinates and possibly more general situations.
I'm aware of 2 proofs: One using Jacobi fields [cf. e.g. S.Sternberg's "Curvature in Mathematics and Physics" from w... | https://mathoverflow.net/users/9161 | Riemann's formula for the metric in a normal neighborhood | My question has been answered in comments by Liviu Nicolaescu:
The (almost) ultimate proof (for my taste) is via A. Gray's formula(e) for (symmetric) higher covariant derivative(s) of normal coordinate vector fields. **Any exposition of normal coordinates lacking this formula is severely lacking.** (I'd prefer symmet... | 4 | https://mathoverflow.net/users/9161 | 195238 | 95,183 |
https://mathoverflow.net/questions/195239 | 1 | Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the same true in general? Something like this seems to be used without explanation on top of p. 263 of "Neron models," and I... | https://mathoverflow.net/users/53197 | Does a line bundle on a normal Noetherian algebraic space come from a Weil divisor? | Same proof as the scheme case works: Choose a meromorphic section and take the associated Weil divisor. This makes sense because algebraic spaces are schemes in codimension one, see [Lemma Tag 0ADD](http://stacks.math.columbia.edu/tag/0ADD).
| 3 | https://mathoverflow.net/users/nan | 195240 | 95,184 |
https://mathoverflow.net/questions/195232 | 15 | I thought they were the same, just different names. Let me make question more precise:
Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}\_p$, is $G$ a p-adic Lie group w.r.t. the analytic topology from $\mathbb{Q}\_p$ in the sense of Peter Schneider? If this is the case, Does the Lie algebra from ... | https://mathoverflow.net/users/9401 | What is the difference between p-adic Lie groups and linear algebraic groups over p-adic fields? | Consider the map $x\mapsto (x,e^x)$ from $p^2{\mathbb Z}\_p$ into ${\mathbb Z}\_p\times {\mathbb Z}\_p^\*$, the latter being the ${\mathbb Z}\_p$ rational points of the algebraic group ${\mathbb G}\_a\times {\mathbb G}\_m$. The image of this map is Zariski dense and hence $p^2{\mathbb Z}\_p$ is not an algebraic subgrou... | 9 | https://mathoverflow.net/users/23291 | 195242 | 95,185 |
https://mathoverflow.net/questions/195215 | 3 | I want to find the answer of
$$\int dU \ U^m X \ U^{\dagger m}$$
Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint matrix.
I used Schur lemma and found that the answer is of the form
*(Edit: the correct form is )*
$$pX+(1-p)tr(X)\f... | https://mathoverflow.net/users/66471 | An expectation of the product of random unitaries | Let me try to work this out, along the lines of a [similar calculation](http://mathoverflow.net/questions/194840/some-calculus-in-the-orthogonal-group-on/194900#194900) in the orthogonal (rather than unitary) group.
We need the fourth-order tensor
$$\int\_{{\rm U}(n)}(U^m)\_{ij}(\bar{U}^m)\_{kl}\,dU=a\_{m}(n)\delta\_... | 6 | https://mathoverflow.net/users/11260 | 195247 | 95,188 |
https://mathoverflow.net/questions/195254 | 2 | Let $\alpha$ be a non vanishing one form on a manifold which which defines a codimension one foliation. With this $\alpha$ we define the following complex:
$$\phi:\Omega^{i}(M)\to \Omega^{i+2}(M)\;\;\;\phi(\beta)=d(\alpha\wedge \beta)$$ Obviously $\phi$ satisfies $\phi \circ \phi=0$, so we have cohomologies associate... | https://mathoverflow.net/users/36688 | A cohomology associated with a codimension one foliation | First, it's not finite dimensional, even in the case of a torus. Just let $x,y$ be the $2\pi$-periodic functions on the torus and take $\alpha = \mathrm{d} x$, and you'll see that $H^0$ is all the functions of the form $f(x)$. On the other hand, if you let $\alpha = \mathrm{d} x + \sqrt{2}\,\mathrm{d} y$, then $H^0$ ju... | 2 | https://mathoverflow.net/users/13972 | 195256 | 95,192 |
https://mathoverflow.net/questions/195249 | 5 | This question has something to do with [that one](https://mathoverflow.net/q/194169).
Let $n\ge1$ and $d\ge1$ be two given integers. Consider the polynomial vector fields $v=(v\_1,\ldots,v\_n)$ whose components $v\_j$ are homogeneous of degree $d$ in the indeterminates $X\_1,\ldots,X\_n$.
>
> What is the dimensio... | https://mathoverflow.net/users/8799 | Homogeneous polynomial vector fields tangent to the unit sphere | Isn't the answer just
$$
D(n;d) = n{{n+d-1}\choose{d}}- {{n+d}\choose{d+1}}= d{{n+d-1}\choose{d+1}}\quad ?
$$
| 4 | https://mathoverflow.net/users/13972 | 195257 | 95,193 |
https://mathoverflow.net/questions/195248 | 8 | Given a topological space $(X,\tau)$ we can define the "$T\_2$-ification" of $X$ by setting $T\_2(X,\tau) = X/\simeq$ where $x\simeq y$ in $X$ if and only if for every open neighborhood of $x$ has nonempty intersection with every open neighborhood of $y$. Then $T\_2(X,\tau)$ has the following universal property:
For ... | https://mathoverflow.net/users/8628 | Co-Hausdorffification | The procedure you describe does not produce a $T\_2$-space in general. You can read [here](https://mathoverflow.net/questions/11191/nonhausdorff-dimension) and [here](https://mathoverflow.net/questions/78175/largest-hausdorff-quotient) on MO, and also [this nice recent bachelor thesis by Bart van Munster](http://www.ma... | 17 | https://mathoverflow.net/users/36502 | 195259 | 95,194 |
https://mathoverflow.net/questions/119501 | 9 | Suppose that I want to force to add a "single" new subset of $\omega$ and not much else. For example, consider the Cohen forcing consisting of finite partial functions from $\omega$ to 2. The forcing I am interested in is different (in fact not CCC, but Proper), but still the generic set codes a subset of $\omega$.
... | https://mathoverflow.net/users/4241 | Forcing to "minimally" add new reals. | There is no forcing that satisfies $(\*)$. Let $r$ be a new real, $r \in M[G]\cap \mathcal{P}(\omega) \setminus M$. Take $t = \{n < \omega \mid n + 1\in r\}$, and let $a, b\in M$ such that $t = (a\cap r)\cup (b\setminus r)$.
I will show that it is possible to reconstruct $r$ from $a, b$ and the bit $0\in r$. This im... | 6 | https://mathoverflow.net/users/41953 | 195266 | 95,197 |
https://mathoverflow.net/questions/195207 | 17 | Let $S$ be the set of polynomials defined as follows: $0$ is in $S$, and if $p$ is in $S$, then $p + 1$ is in $S$ and $x \cdot p$ is in $S$, so that $S$ "grows" in generations: $g(0)=\{0\}$, $g(1)=\{1\}$, $g(2)=\{2,x\}$, $g(3)=\{3,2x,x+1,x^2\}$, and so on, with $|g(n)|=2^{n-1}$. Let $S^\*$ be the set obtained from $S$ ... | https://mathoverflow.net/users/61426 | A possibly surprising appearance of Lucas numbers | For a number of the form $a + b\sqrt{2}$ with nonnegative integers $a$ and $b$, define its *length* to be the minimal number of steps needed to obtain it from zero, where the allowed steps are $x \mapsto x+1$ and $x \mapsto \sqrt{2}x$.
Then the problem asks to count the numbers of given length.
Now it should be easy ... | 15 | https://mathoverflow.net/users/21146 | 195272 | 95,198 |
https://mathoverflow.net/questions/195270 | 3 | I'm not sure if this question is suited for MO, but it does seem quite challenging to me, and is required for a research problem in chemistry I'm working on. I did try getting help from elsewhere ([Cross-post on MSE](https://math.stackexchange.com/questions/1126285/how-to-count-the-number-of-substrings-in-this-combinat... | https://mathoverflow.net/users/23202 | Combinatorics problem involving counting the number of certain substrings | The number of sequences with ABA or BAB at one specific place is
$$\binom{a+b-3}{a-2}+\binom{a+b-3}{a-1}=\binom{a+b-2}{a-1}.$$
Hence the total number of ABA and BAB is
$$(a+b-2)\binom{a+b-2}{a-1}.$$
| 6 | https://mathoverflow.net/users/35593 | 195279 | 95,201 |
https://mathoverflow.net/questions/195234 | 2 | Let $G$ be a Lie group (not necessarily connected) and let $H$ be a closed subgroup of $G$. I am after an algebraic (group theoretic) characterization of when the homogeneous space $G/H$ is connected.
I found the following necessary condition in Onishchik/Vinberg, Lie groups and algebraic groups: Let $G\_0$ denote th... | https://mathoverflow.net/users/24389 | When is a homogeneous space connected? | As Ryan says, the idea is in the first sentence of the fourth paragraph, so it's just a matter of formalizing it into a proof.
There is a natural isomorphism $\pi\_0(G) \cong G/G\_0$ for any Lie group $G$. The inclusion $i: H \to G$ induces by functoriality a function $\pi\_0(i): \pi\_0(H) \to \pi\_0(G)$; by the nat... | 5 | https://mathoverflow.net/users/2926 | 195285 | 95,206 |
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