parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/328672 | 7 | What is the role played by BV functions in the study of (possibly nonlinear) wave equations?
I think that one would need assume small initial data in $L^1$ or $H^1$ to get a well-posedness result (is that correct?).
Has the case of initial data in BV been studied?
| https://mathoverflow.net/users/122620 | BV functions and wave equation | The answer to this question depends a lot on the space dimension $n$. It is true that if $n=1$, the Cauchy problem has been studied with data in either $L^\infty(R)$ or $BV(R)$. For superlinear wave equation, every $L^\infty$-data yields at least one bounded global-in-time "entropy" solution. This is done by Compensate... | 7 | https://mathoverflow.net/users/8799 | 328677 | 141,297 |
https://mathoverflow.net/questions/328676 | 3 | In section 3, pg 8 of [String theory on Elliptic curves](https://arxiv.org/pdf/1402.4885.pdf) they claim that for $X$ compact with $x\_0$ an involution fixed point, the map $i:x\_o\rightarrow X$ is equivariant and **equivariantly split**, which according to them implies
$KR^{j}(X) \approx KR^{j}(X - \{x\_0 \}) \oplus... | https://mathoverflow.net/users/73712 | Equivariantly split in an isomorphism of $KR$-groups | Equivariantly split just means [split](https://ncatlab.org/nlab/show/split+monomorphism) in the category of spaces with a $C\_2$-action, i.e. that there is an equivariant map $r:X→\{x\_0\}$ such that $r\circ i$ is equivalent to the identity. This map is usually called a *retraction* of $i$. Of course the splitting in t... | 3 | https://mathoverflow.net/users/43054 | 328678 | 141,298 |
https://mathoverflow.net/questions/328694 | 6 | For a positive definite diagonal matrix $A$, I want to prove that for any $x$:
$$\frac{x^T \sqrt{A} x}{\|\sqrt{A}x\|\_2} \geq \frac{x^T A x}{\|Ax\|\_2}$$
So far I cannot find any counterexamples, and it intuitively makes sense since the $\sqrt{\cdot}$ operator should bring the eigenvalues of $A$ closer to $1$, but ... | https://mathoverflow.net/users/88973 | Proving inequality for positive definite matrix | Your inequality says
$$\frac{\sum\sqrt{\lambda\_j}x\_j^2}{\left(\sum\lambda\_j x\_j^2\right)^{1/2}}\geq
\frac{\sum\lambda\_jx\_j^2}{\left(\sum\lambda\_j^2x\_j^2\right)^{1/2}},$$
or after a simple transformation
$$\sum\lambda\_j x\_j^2\leq\left(\sum\sqrt{\lambda\_j}x\_j^2\right)^{2/3}
\left(\sum\lambda\_j^2x\_j^2\rig... | 14 | https://mathoverflow.net/users/25510 | 328703 | 141,305 |
https://mathoverflow.net/questions/328686 | 2 | Let $X$ be a complex projective manifold and let $\phi \colon E \hookrightarrow F$ an injection of locally free sheaves. Then we have a sequence of coherent sheaves
$$
0 \to E \to F \to F/E \to 0
$$
that we can dualize:
$$
0 \to (F/E)^\vee \to F^\vee \to E^\vee \to \mathcal{Ext}^1\_{\mathcal{O}\_X}(F/E, \mathcal{O}\_X)... | https://mathoverflow.net/users/82672 | Ext sheaves as extension by zero of locally free sheaves | Assume $r(E) = e$, $r(F) = f$ (with $e \le f$). Then the natural scheme structure of $Z$ is given by the Fitting ideal, i.e., the image of the map
$$
\Lambda^e(E) \otimes \Lambda^e(F^\vee) \to \mathcal{O}\_X.
$$
Such $Z$ is usually called the degeneracy locus for the morphism of sheaves.
If $Z$ is defined like that t... | 6 | https://mathoverflow.net/users/4428 | 328705 | 141,306 |
https://mathoverflow.net/questions/328713 | 6 | Let $A$ be a finite dimensional central simple algebra over a field $F$ of characteristic $0$. So by Weddernburn's theorem, $A\cong M\_n(D)$ for some division algebra $D$ over $F$. Let $\dim\_F(D)=m^2$. Then $m$ is called the index of $A$.
Assume that $A$ is crossed product: there is a finite Galois extension $E$ of... | https://mathoverflow.net/users/6761 | Example of a central simple algebra | You can take: $F=\mathbb{C}(X\_1,Y\_1,\ldots,X\_n,Y\_n)$ and take the tensor product of quaternion algebras $$A=(X\_1,Y\_1)\_F\otimes\_F\cdots\otimes (X\_n,Y\_n)\_F.$$ Here $A$ contains a subfield $E$ isomorphic to $F(\sqrt{Y\_1},\ldots \sqrt{Y\_n})$ for example.
Now $A$ is a division algebra, so it has index $2^n$, ... | 7 | https://mathoverflow.net/users/36683 | 328714 | 141,308 |
https://mathoverflow.net/questions/328573 | 11 | Given a finite group $H$, define a *norm* on $H$ to be a function $f : H \rightarrow \mathbb{R}\_{\geq 0}$ satisfying:
* $f(x) = 0 \iff x = e$ is the identity;
* $\forall x \in H$, we have $f(x) = f(x^{-1})$;
* $\forall x, y \in H$, we have $f(xy) \leq f(x) + f(y)$.
This induces a metric $d : H \times H \rightarrow... | https://mathoverflow.net/users/39521 | Traveling Salesman Problem on finite group | Here is an observation which suggests that it might not be that easy:
There are a number of versions of the longstanding [Lovasz Conjecture](https://en.wikipedia.org/wiki/Lov%C3%A1sz_conjecture) among them
* Every finite connected vertex transitive graph contains a Hamilton path.
* Every finite connected vertex tra... | 3 | https://mathoverflow.net/users/8008 | 328717 | 141,309 |
https://mathoverflow.net/questions/328721 | 1 | We are in ZFC & CH. Given family $Y=\{y\_\alpha\}\_{\alpha<\omega\_1}$ of infinitesimal $\omega$-sequences (i.e. $\lim\_{n\to\infty}y\_{\alpha n}=0$) of rational numbers with the property: $\forall\alpha<\beta: \lim\_{n\to\infty}\frac{y\_{\beta n}}{y\_{\alpha n}}=0$. Can we prove that for any infinitesimal sequence $y$... | https://mathoverflow.net/users/118366 | Existence of $\alpha$-sequence of infinitesimal sequences with $\alpha>\omega_1$ | The answer is no. First, let be construct a family of $\omega\_1$ functions $f\_\alpha:\omega\to\omega$ with the following three properties: each $f\_\alpha$ tends to infinity, $f\_\alpha\ll f\_\beta$ whenever $\alpha<\beta$ (where $\ll$ is *eventual domination*: $f\_\alpha(n)<f\_\beta(n)$ for large enough $n$) and $f\... | 2 | https://mathoverflow.net/users/30186 | 328729 | 141,313 |
https://mathoverflow.net/questions/328728 | 1 | Let $x$ and $y$ be two correlated random variable (say, standard normal) with correlation coefficient $\rho>0$. Let $X= \{x\_1, x\_2, ..., x\_L\}$ and $Y= \{y\_1, y\_2, .. y\_L\}$ be samples of size $L$ from $x$ and $y$ respectively.
What is the probability that $\mbox{argmax}\ X = \mbox{argmax}\ Y$.
Alternativel... | https://mathoverflow.net/users/138745 | Probability that maximal elements has the same position in samples from correlated random variables | The question can apparently be clarified as follows:
>
> Let $(X\_1,Y\_1),\dots,(X\_n,Y\_n)$ be iid pairs of real-valued random variables with a common continuous joint cumulative distribution function (cdf) $F$, so that $F(x,y)=P(X\_i<x,Y\_i<y)$ for all $i$ and all real $x,y$. What is then the probability
> \be... | 1 | https://mathoverflow.net/users/36721 | 328736 | 141,314 |
https://mathoverflow.net/questions/328727 | 2 | **Question summary.** Does the Kolmogoroff condition $\sum\_{n=1}^\infty\frac{\mathbb V Y\_n}{n^2} < \infty$ hold for truncated random variables $Y\_n := X\_n \cdot 1\_{\{X\_n \le n\}}$ (see below for a more rigid definition)?
**General Definitions.** Let $(\Omega, \mathcal A, \mathbb P)$ be a probability measure spa... | https://mathoverflow.net/users/129831 | Kolmogoroff condition for truncated random variables | Welcome to MathOverflow! However, your conjecture is false. Indeed, let $P(X\_n=n)=1/n=1-P(X\_n=0)$. Then for all $n$ we have $EX\_n=1$, $Y\_n=X\_n$, $Var\, Y\_n=Var\,X\_n=n-1$. So, $\sum\_n Var\,Y\_n/n^2=\infty$.
---
Additional note: Your statement that $EM<\infty$ does not follow from the Beppo Levi theorem, a... | 5 | https://mathoverflow.net/users/36721 | 328738 | 141,315 |
https://mathoverflow.net/questions/328650 | 3 | Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric).
I am looking for solutions to $\Delta\_g f = 0$ where $f \to 1$ at infinity and $f=f\_0$ on $\partial M$ where $f\_0$ is some positive function on $∂M$.
1) What can... | https://mathoverflow.net/users/138705 | $\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on the boundary and at infinity | You will find a full treatement of this problem in: [The Mass of Asymptotically Flat Manifold](http://www.math.jhu.edu/~js/Math646/bartnik.mass.pdf), by Bartnik. Becarefull there is a small error, the decay rate of $f-1$ can't be better than $\frac{1}{r}$, that is to say the one of the Green function.
| 3 | https://mathoverflow.net/users/9253 | 328752 | 141,318 |
https://mathoverflow.net/questions/328568 | 1 | Let $x\_{n}$ be a sequence of operators in vN algebra $M$, $\Omega$ is a cyclic vector for $M$, if $x\_{n}\Omega$ converges in $\mathcal{H}$, can we say there exist a subsequence $\{y\_{n}\}$ of $\{x\_{n}\}$ are uniformly bounded in operator norm?
| https://mathoverflow.net/users/136400 | On boundedness of sequence of operators in vN algebra | An easy counterexample:
$M=M\_{2}(\mathbb C)$,
$H=\mathbb C^2$,
$\Omega=\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)$,
$x\_n=\left(\begin{smallmatrix}0 & 0\\0 & n\end{smallmatrix}\right)$.
| 2 | https://mathoverflow.net/users/5690 | 328756 | 141,320 |
https://mathoverflow.net/questions/328652 | 0 |
>
> Let $f:[0,1] \to \mathbb{R}, G = graph(f)$.
>
>
> If $\sum\_{j = 1}^m |f(x\_i)-f(x\_{i-1})|^s \le c$ for all partitions $0 = x\_0< \ldots < x\_m = 1 $ then $H^s(G) < \infty$
>
>
>
What technique can I use to prove this result?
Can it be reduced to the theorem stating that a rectifiable curve $\Gamma$ has... | https://mathoverflow.net/users/82839 | Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ | Take $s \ge 1$. We will use $(x+y)^s \le x^s+y^s$ for positive $x,y$.
Let $C=2^{1+s/2}(1+c)$. I claim: $\mathcal H^s(G) \le C$.
Let $N \in \mathbb N$. Let $\eta > 0$ be so small that
$N2^s\eta^s \le c$.
For $j=1,2,\dots$ let
$$
M\_j = \sup\left\{f(x): \frac{j-1}{N} \le x \le \frac{j}{N}\right\},\qquad
m\_j = \in... | 1 | https://mathoverflow.net/users/454 | 328759 | 141,322 |
https://mathoverflow.net/questions/328753 | 10 | Consider the category of finitely presented commutative rings, and equip its opposite category with either the Étale or Zariski topology. These give rise to topoi $\operatorname{Sh}(\mathbf{Et})$ and $\operatorname{Sh}(\mathbf{Zar})$ by taking sheaves. By the descent theorem for schemes, the functors of points of schem... | https://mathoverflow.net/users/137348 | The 'gros' functor from schemes into (strictly) locally ringed topoi | So the final answer, is 'no', but there is still something interesting to say:
Given any topos $\mathcal{T}$, the construction you are describing produces an equivalence of category between $\mathcal{T}$ and the full subcategory of $Top\_{/\mathcal{T}}$ (where $Top$ is the 2-category of toposes) of toposes $\mathcal{... | 9 | https://mathoverflow.net/users/22131 | 328762 | 141,323 |
https://mathoverflow.net/questions/328761 | 0 | Let $(P, \le)$ be a [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) such that
$$
\forall a, b, c \in P: b \ge a \le c \implies
\exists d \in P: b \le d \ge c.
$$
I am looking for literature where such *confluent* partial orders are studied.
| https://mathoverflow.net/users/22795 | Confluent partial orders | In a context slightly more general than yours, this is called the [right Ore condition](https://ncatlab.org/nlab/show/Ore+condition). If you treat your poset as a category where there is a unique morphism from $p$ to $q$ if and only if $p \geq q$ and no other morphisms whatsoever, then your situation becomes a special ... | 1 | https://mathoverflow.net/users/18263 | 328765 | 141,324 |
https://mathoverflow.net/questions/328772 | 4 | Let $(E,d)$ be a Polish space equipped with the Borel $\sigma$-algebra $\mathcal{E}$. Let $\mathcal{P}(E)$ be the space of all probability measures on $(E,\mathcal{E})$. We eqiup this space with the topology of weak convergence.
Consider a transition kernel $P(x,\cdot)$, $x\in E$. By definition, for any fixed Borel s... | https://mathoverflow.net/users/7646 | Different type of measurability of transition kernel | Yes. The Borel $\sigma$-algebra of the weak topology on $\mathcal{P}(E)$ is generated by the maps $\mu \mapsto \mu(A)$ as $A$ ranges over the Borel sets of $E$. (See for instance Kechris, *Classical Descriptive Set Theory*, Theorem 17.24.) Since all the maps $x \mapsto P(x,A)$ are measurable by assumption, it follows t... | 3 | https://mathoverflow.net/users/4832 | 328785 | 141,329 |
https://mathoverflow.net/questions/328782 | 2 | It is obvious that for odd $n \in \Bbb N$, $a^n-b^n-(a-b)^n$ is divisible by $ab(a-b)$ (with $n=1$ being a special case in which $a^n-b^n-(a-b)^n$ is zero). This can be viewed as a fact about integers, but is more strongly a fact about these polynomials.
It is apparently also the case that for prime $p > 3$, it is a... | https://mathoverflow.net/users/82067 | Proving largest power of $(a^2-ab+b^2)$ that divides $a^p-b^p-(a-b)^p$ for odd prime $n$ | $\omega$ in what follows is the sixth root of unity.
As user44191 suggested, lets reduce this question to the 1-variable. For the first statement it is enough to prove that
$x^{6k+1} - 1 - (x-1)^{6k+1}$
has a root $\omega$ of order 2. It is clear that $\omega^6 = (\omega-1)^6 = 1$ (because $\omega-1$ is the third... | 5 | https://mathoverflow.net/users/33286 | 328793 | 141,331 |
https://mathoverflow.net/questions/328801 | 10 | *Added.* My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails even on the smaller interval $[1/3,1]$, where the function $\{1/t\}$ is piecewise glued by $1/t - 2$ (from $1/3$ to $1... | https://mathoverflow.net/users/26522 | Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$? | They are not. The function $g(t) = \begin{cases}\frac{-1}{(1-t)^2}& \frac{1}{3} < t < \frac{1}{2}\cr 1& \frac{1}{2} < t < 1\end{cases}$ is orthogonal to all of them. That is because $\{\frac{1}{t}\} = \frac{1}{t} - 2$ on the first interval and $\frac{1}{t} - 1$ on the second. So integrating $g$ against $\{\frac{1}{t}\}... | 11 | https://mathoverflow.net/users/23141 | 328804 | 141,336 |
https://mathoverflow.net/questions/328805 | 1 | I am looking the group of automorphisms $G$ of the curve defined over $\mathbb F\_3$ by (in projective coordinates) $Y^2Z=X(X-Z)(X-2Z)$.
Obviously, there are the automorphisms $X\mapsto X+\alpha Z$ (for $\alpha\in\mathbb F\_3$) and $Y\mapsto\pm Y$. But are they the only ones?
And a second question: What is the field $... | https://mathoverflow.net/users/33128 | Automorphisms of a curve | This is the elliptic curve $E:y^2=x^3-x$ over $\mathbb F\_3$ whose $j$-invariant is $0$. The automorphism groups of elliptic curves over finite fields are quite well known, so this question is probably better suited for MathStackExchange. Over $\overline{\mathbb F}\_3$ the automorphism group has order 12; it is the sem... | 6 | https://mathoverflow.net/users/11926 | 329804 | 141,337 |
https://mathoverflow.net/questions/329803 | 2 | The title is just about it. Assume we have a nontrivial knot $K$ in $S^3$ and the exterior of $K$, $E(K)$, is $S^3 \setminus N(K)$. Here $N(K)$ is a regular neighborhood.
1. Let $\tau$ be a properly embedded arc in $E(K)$ and let $M = E(K)\setminus N(\tau)$. Now, if we know that $\pi\_1(M) = \langle x,y\vert \rangle... | https://mathoverflow.net/users/36934 | Does a knot and a tunnel exterior having free fundamental group imply it's an unknotting tunnel? | The answer is "yes". This is because the manifold $M(K)$ is (in both cases) a handlebody of the correct genus. To see this, you will need to apply the disk theorem several times. The end of the proof requires Alexander's theorem: the three-sphere is irreducible.
| 3 | https://mathoverflow.net/users/1650 | 329815 | 141,340 |
https://mathoverflow.net/questions/329817 | 4 | Let $\Lambda$ be the set of all countable limit ordinals. Does there exist an injective function $f:\Lambda\to\omega\_1$ with the properties:
1. $\forall \lambda\in\Lambda:~f(\lambda)<\lambda$
2. $\forall\alpha<\omega\_1~~\exists\beta<\omega\_1~~\forall\lambda>\beta:~f(\lambda)>\alpha$ ?
| https://mathoverflow.net/users/118366 | Function on the set of limit countable ordinals | No. The first property is known as $f$ being *regressive*. [Fodor’s Lemma](https://en.m.wikipedia.org/wiki/Fodor%27s_lemma) says that any regressive function on a stationary set is constant on a stationary subset. In particular, because $\Lambda$ is club (and thus stationary), such $f$ cannot be injective.
| 10 | https://mathoverflow.net/users/11145 | 329819 | 141,341 |
https://mathoverflow.net/questions/329818 | 1 | Let $k$ be a field, $X$ be a connected smooth projective $k$-scheme. Let $f:X\rightarrow X$ be a finite $k$-morphism that is surjective on the underlying topological spaces. Suppose $f$ has degree $n$.
Is it possible that there does not exist a coherent locally free sheaf $F$ on $X$ such that $H^0(X, F)=n\,\mathrm{r... | https://mathoverflow.net/users/nan | Coherent locally free sheaves on projective varieties | You can always find such a sheaf.
Take a very ample line bundle $L$ on $X$, with $H^0(X, \, L)=a \geq n$, and set $$F:=L^{\oplus n-1} \oplus \mathcal{O}\_X^{\oplus a-n}.$$
Then $$H^0(X,\, F)=n(a-1)=n \; \mathrm{rank}(F).$$
Furthermore, the role of $f \colon X \to X$ is irrelevant here.
**Remark.** It seems to me... | 1 | https://mathoverflow.net/users/7460 | 329824 | 141,343 |
https://mathoverflow.net/questions/328723 | 5 | Perhaps an overly elementary question: let $\mathcal{E}$ be a topos and let $X, Y$ be non-isomorphic objects in $\mathcal{E}$. Is it always true that there exists a formula $\phi$ of $\mathcal{E}$'s Mitchell-Benabou language with one free variable which is true of one but not both of $X$ and $Y$ (on the Kripke-Joyal se... | https://mathoverflow.net/users/45570 | Discernible Objects in a Topos | Summary: The answer may depends on the choice of a precise interpretation of the question, which is too vague. But for all the interpretations I can think of the Kripke-Joyal semantics do not distinguishes between objects that are "locally isomorphic".
Conversely objects that satisfies the exact same formulas are loc... | 10 | https://mathoverflow.net/users/22131 | 329827 | 141,346 |
https://mathoverflow.net/questions/328800 | 2 | Let $C$ be a smooth projective curve of genus $g$, let $c$ be a point in $C$. Let $(n\_1,\dots, n\_{g-1})$ be a $(g-1)$ tuple of nonzero integers.
Consider the image $f\_{(n\_i)}\colon C^{g-1}\to \mathrm{Pic}^0(C)$ given by
$$(x\_i)\mapsto \mathcal{O}\_C\left(\sum n\_i(x\_i-c)\right),$$
the image $\mathrm{Im}(f\_{(... | https://mathoverflow.net/users/nan | Determining numerical class of divisors inside Jacobian | Let $C\_i$ be the image of $C$ by the multiplication map $JC \xrightarrow{\ \times n\_i\ }JC$. Your divisor is the image of the addition map $\mu : C\_1\times \ldots \times C\_{g-1}\rightarrow JC$. Its cohomology class is $(\deg \mu )^{-1}[C\_1]\*\ldots \* [C\_{g-1}]$, where "$\*$" is the Pontryagin product. Now the co... | 1 | https://mathoverflow.net/users/40297 | 329835 | 141,347 |
https://mathoverflow.net/questions/329814 | 2 | On $(\mathbb{R}, \tau)$ the euclidean space of real numbers, we define a new topology by letting $\tau^{\*}=\{X\subseteq \mathbb{R}: X=\emptyset \hspace{0.1cm}\mbox{or}\hspace{0.1cm}\mathbb{R}\setminus X\hspace{0.1cm}\mbox{is}\hspace{0.1cm} \mbox{compact} \hspace{0.1cm} \mbox{in}\hspace{0.1cm} (\mathbb{R}, \tau) \}$, i... | https://mathoverflow.net/users/138770 | Rothberger game and Meager in itself sets | The game is a win for Player I, not Player II. A winning strategy for Player I is to choose, in inning $n$, the open cover
$$\mathcal U\_n=\{(a,b)\cup(-\infty,-10)\cup(10,\infty):a,b\in\mathbb R,\ 0\lt b-a\lt2^{-n}\}.$$
If more details are wanted, see the answer I posted to the same question at math.SE:
<https://math... | 3 | https://mathoverflow.net/users/43266 | 329838 | 141,348 |
https://mathoverflow.net/questions/306199 | 9 | Given a (symmetric) convex body $K \subset \mathbb{R}^n$ (equivalently, given a norm on $\mathbb{R}^n$), there is a unique ellipsoid of maximal volume in $K$, called the *John ellipsoid*. The John ellipsoid can be described as a ``canonical ellipsoid'' associated to a convex body, and reading around there seem to be a ... | https://mathoverflow.net/users/126691 | Ellipsoid minimizing Banach-Mazur distance to convex body | 1. Yes, it has appeared in the works of several experts in Functional Analysis. In the book [Banach-Mazur distances and finite dimensional operator ideals](https://sites.ualberta.ca/~ntj/bm_book/index.html), by N. Tomczak-Jaegermann, it is proved that in finite dimensional Banach spaces that have enough symmetries, the... | 3 | https://mathoverflow.net/users/82382 | 329844 | 141,350 |
https://mathoverflow.net/questions/329837 | 2 | Robin showed that if $a\in(1/2, 1]$ is the supremum of the real parts of the zeros of the Riemann zeta function $\zeta(s)$, then $f(x)=\Omega\_{\pm} (x^{-b})$, where $b$ is some number on $(a-1/2, 1/2],$ $$f(x)=\log \Big(e^{\gamma}\log \theta(x)\prod\_{p\leq x} (1-p^{-1})\Big),$$ $\theta(x)=\sum\_{p\leq x} \log p$, the... | https://mathoverflow.net/users/480516 | Robin's inequality and the zeros of the Riemann zeta function | It is well-known that the Riemann hypothesis implies
$$
\theta(x)=x+O(\sqrt{x}\ln^2 x).
$$
Therefore, under the Riemann hypothesis we have
$$
\ln\theta(x)=\ln x+O\left(\frac{\ln^2 x}{\sqrt{x}}\right).
$$
Also, from the partial summation we get
$$
\sum\_{p\leq x}\frac{1}{p}=\int\_{1.5}^x \frac{d\theta(t)}{t\ln t}=\ln\ln... | 4 | https://mathoverflow.net/users/101078 | 329846 | 141,351 |
https://mathoverflow.net/questions/329830 | 2 | If one considers how the particle's energy grows in the (idealized) cyclotron, one gets the following sequence of numbers
$$E\_1=1+2V, \;\;E\_n=E\_{n-1}+2V\cos{[2\pi(E\_1+\ldots E\_{n-1})]},\;n\ge 2. \tag{1}$$
For $V\ll 1$, one can find an approximate expression
$$E\_n\approx 1+\sqrt{\frac{V}{\pi}}\,\frac{\mathrm{sn}(2... | https://mathoverflow.net/users/32389 | Cyclotron sequences | This would appear to be identical to <https://en.m.wikipedia.org/wiki/Standard_map>, with the identifications: $x\_n = 2 \pi \Sigma E\_n + \pi$, $p\_n = 2\pi E\_n$, $ K = V/\pi$.
Physically, your idealized cyclotron may be same as the kicked rotator, so generalizations you are interested in might already be explored ... | 2 | https://mathoverflow.net/users/139791 | 329852 | 141,354 |
https://mathoverflow.net/questions/329841 | 6 | It is claimed [here](https://mathoverflow.net/a/148650/138661) that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O\_X$-module of rank 1 is trivial.
Do there exist proper schemes over a field such that every locally free $O... | https://mathoverflow.net/users/nan | Proper scheme such that every vector bundle is trivial | According to [this paper](http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/resolution.pdf) it is not known if every proper algebraic scheme admits nontrivial vector bundles. Partial results can be found [here](https://www.ams.org/journals/tran/2017-369-07/S0002-9947-2016-06813-0/S0002-9947-2016-06813-0.pdf... | 5 | https://mathoverflow.net/users/104669 | 329859 | 141,357 |
https://mathoverflow.net/questions/329858 | 3 | Let $X$ be an integral Noetherian scheme that has no automorphisms other than the identity. Is it possible to characterize (in a not completely tautological way) $X$ such that
* there is no finite morphism $X\rightarrow X$ surjective on the underlying topological spaces (other than the identity)?
* there is no finite... | https://mathoverflow.net/users/nan | Schemes with no finite morphisms onto themselves | The question as stated seems to me too broad. Nevertheless, in some cases it is actually possible to provide a complete characterization. For instance, ruled surfaces admitting non-trivial, surjective endomorphisms are classified in
N. Nakayama: [*Ruled surfaces with non-trivial surjective endomorphisms*](http://dx.d... | 2 | https://mathoverflow.net/users/7460 | 329863 | 141,359 |
https://mathoverflow.net/questions/329845 | 4 | Let $C$ is a perimeter of a convex hull (plane geometry) and $d\_{max}$ is the largest distance of two arbitrary points in the convex hull. I am looking for a proof that:
$$\frac{C}{d\_{max}} \le \pi $$
What is a generalization of the inequality for higher dimension?
| https://mathoverflow.net/users/122662 | Inequality $\frac{C}{d_{max}} \le \pi $ relating perimeter and diameter of planar convex body | Bonnesen and Fenchel’s *Theorie der konvexen Körper* ([1934](//zbmath.org/?q=an:60.0673.01), [(6) p. 77](//books.google.com/books?id=cvGoBgAAQBAJ&pg=PA77)) ([translation](//zbmath.org/?q=an:0628.52001)) gives the generalization to a convex body $K\subset\mathbf R^n$ as
$$
\mathrm{vol}(\partial K)\leqslant \omega\_n\lef... | 5 | https://mathoverflow.net/users/19276 | 329865 | 141,360 |
https://mathoverflow.net/questions/328692 | 2 | Let $U\subset \mathbb R^n$ be an open.
Let $f: S^1 \times U \to \mathbb C$ be a smooth map (i.e. $C^\infty$) s.t. for every $x\in U$ the map $f\_x:=f|\_{S^1\times \{x\}}:S^1\to \mathbb C$ is a smooth embedding (i.e. with everywhere non-zero derivative) that goes around the origin with winding number $+1$.
For every... | https://mathoverflow.net/users/5690 | families of Riemann mappings | The conformal map $\phi$ between the unit disk $D$ and the simply connected domain $\Omega$ with a smooth boundary, subject to normalization $\phi(0)=a$ and $\phi'(0)>0$, depends smoothly on the domain $\Omega$. It's proved, for example, in Bell's book "The Cauchy Transform, Potential Theory and Conformal Mapping", see... | 4 | https://mathoverflow.net/users/1811 | 329869 | 141,361 |
https://mathoverflow.net/questions/329868 | 1 | Let be $X$ and $Y$ two random variables which are respectively continuous and binary. Assume that we have a sample $(X\_i, Y\_i)\_{1\leq i\leq n}$. We define the point-biserial correlation coefficient as below :
$$ r\_{X,Y} = \frac{m\_1 - m\_0}{\hat{s}\_X} \sqrt{\frac{n\_0n\_1}{n(n-1)}} $$
$m\_k$ is the mean of the $n\... | https://mathoverflow.net/users/139796 | Hypothesis test and point-biserial correlation coefficient | First here, when the null hypothesis is that the population correlation is zero, the choice of the sign before $m\_1-m\_0$ does not really matter.
Next, as pointed out in [this Wikipedia article](https://en.wikipedia.org/wiki/Point-biserial_correlation_coefficient) (where the point-biserial correlation coefficient i... | 0 | https://mathoverflow.net/users/36721 | 329876 | 141,365 |
https://mathoverflow.net/questions/329877 | 7 | Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?
| https://mathoverflow.net/users/85926 | Additive group of local rings | Any nonzero finite abelian $ p $ - group is the additive group of a commutative local ring.
Proof : Let $ G $ be such a group, by the structure theorem you can write it as $ \mathbb{Z}/p^k \mathbb{Z} \times M $ where $ p^k = \exp (G) $. Then $ M $ naturally has the structure of a $ \mathbb{Z}/p^k \mathbb{Z} $ - modu... | 9 | https://mathoverflow.net/users/102343 | 329882 | 141,367 |
https://mathoverflow.net/questions/329881 | 1 | Let $a,b \in (0, 1)$ and $N \ge 1$, and consider the incomplete gamma function $x \mapsto \Gamma(1-a,x)$.
Question
========
Is there a simple bound (involving 'simple function's) for the expression $\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))$ ?
Motivation
==========
Ultimately, I'm interesting in bounding the sum... | https://mathoverflow.net/users/78539 | Good upper bound for $\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))$, where $a,b \in (0, 1)$ and $N \ge 1$ | Let us go for your ultimate goal and provide a tight upper bound on
\begin{equation}
s:=\sum\_{n=1}^N a^{-n} n^{-b}=\sum\_{n=1}^N c^n n^{-b},
\end{equation}
where $c:=1/a>1$ and $b>0$. We assume that $N\to\infty$. Take any natural $M$ such that $1<M<N$ and write
\begin{equation}
s=s\_1+s\_2,
\end{equation}
where
\... | 2 | https://mathoverflow.net/users/36721 | 329893 | 141,370 |
https://mathoverflow.net/questions/329885 | 4 | Let $X$ be a Noetherian integral affine scheme. Let $U\subset X$ be an open subscheme whose complement has irreducible components of codimension $1$. Is $U$ affine?
Some remarks:
* By EGA 4, Cor. 21.12.7, the complement of a codimension 2 closed subset is not affine.
* if $X$ is the spectrum of a UFD, then I belie... | https://mathoverflow.net/users/nan | Complement of codimension 1 subset of affine scheme not affine | Quoting from [a paper](https://arxiv.org/abs/math/0406384) by Roth and Vakil ($\mathrm{cd}$ stands for cohomological dimension):
>
> Let $S$ be $\mathbb{P}^2$ blown up at a point, and let $X$ be the
> affine cone over some projective embedding of $S$. Let $Z\subset X$ be
> the affine cone over the exceptional div... | 1 | https://mathoverflow.net/users/nan | 329894 | 141,371 |
https://mathoverflow.net/questions/329896 | 2 | In the classical divisor problems, for $k\geq 2$, $\alpha\_k$ usually denotes the infimum of real numbers $\sigma<1$ such that
$$\Delta\_k(x)=\sum\_{n\leq x}d\_k(n)-\textrm{Res}\left(\frac{\zeta^k(z)x^z}{z},z=1\right)=O(x^{\sigma})$$
as $x\rightarrow\infty$.
As often happens in analytic number theory, one might exp... | https://mathoverflow.net/users/10980 | A simple question about the classical divisor problems | It is known (see Titchmarsh Chapter 12) that if you define $\gamma\_k$ the lower bound of $\sigma > 0, \int\_{-\infty}^{\infty}\frac{|\zeta(\sigma+it)|^{2k}}{|\sigma+it|^2}dt < \infty$, then $\frac{k-1}{2k}\le \gamma\_k=\beta\_k \le \alpha\_k$, where $\beta\_k$ is the usual mean bound of $\Delta\_k$ ( lowest bound of o... | 3 | https://mathoverflow.net/users/133811 | 329906 | 141,375 |
https://mathoverflow.net/questions/329811 | 9 | It may be better to move this to a separate question.
Let me call a linear subspace $V \subset L^2(0,1)$ to be *tame* if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or else there exists a $c > 0$ such that already the restriction $W|\_{[c,1)}$ is not dense in $L^2(c,1)$. (We think of... | https://mathoverflow.net/users/26522 | Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$ | *It is not a priori clear to me whether there is any tame and dense subspace of $L^2(0,1)$.*
Indeed, no such space exists. To see it, choose any sequence of functions $f\_k\in L^2([0,1])$ such that $f\_k|\_{[c,1]}$ are dense in $L^2([c,1])$ for all $c>0$ but $\int\_0^1 f\_k=0$ for all $k$. Now if $V$ is dense in $L^2... | 8 | https://mathoverflow.net/users/1131 | 329919 | 141,376 |
https://mathoverflow.net/questions/328749 | 1 | Suppose $M$ is $II\_{1}$ factor but need not be in standard form. Under what condition (on $M$ or Hilbert space) is the commutant $M'$ of $M$ again $II\_{1}$ factor on the Hilbert space acted by $M$?
| https://mathoverflow.net/users/136400 | On commutant of $II_{1}$ factors | Look at V. Jones 2015 von Neumann notes
<https://math.vanderbilt.edu/jonesvf/VONNEUMANNALGEBRAS2015/VonNeumann2015.pdf>
Theorem 10.2.1(1).
You shall learn the coupling constant first.
| 1 | https://mathoverflow.net/users/121051 | 329924 | 141,378 |
https://mathoverflow.net/questions/329931 | 7 | Is there an embedding (i.e. injective continuous map)
$$\phi:\Bbb R \Bbb P^2\hookrightarrow S^4\subseteq\Bbb R^5$$
of the projective plane $\Bbb R\Bbb P^2$ into the $4$-sphere, that is *transitive*, i.e. for any two $x,y\in \Bbb R\Bbb P^2$ there is an orthogonal transformation $T\in\mathrm{O}(\Bbb R^5)$ that fixe... | https://mathoverflow.net/users/108884 | Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere | Yes. You take each vector $v \in \mathbb{R}^3$ to the vector $v \cdot v \in \operatorname{Sym}^2\mathbb{R}^3=\mathbb{R}^6$. This takes each unit vector $v$ to the same place as $-v$. So it descends to $S^2/\pm 1=\mathbb{RP}^2$.
If we identify each element of $\operatorname{Sym}^2\mathbb{R}^3$ with a symmetric matrix b... | 18 | https://mathoverflow.net/users/13268 | 329933 | 141,380 |
https://mathoverflow.net/questions/329950 | 5 | In [this paper](https://arxiv.org/abs/1811.03338) by Nondas Kechagias, the Steenrod algebra and the Dyer–Lashof algebra are compared. The rough difference ist:
1. The Steenrod algebra arises by dividing out the “cohomological” Ádem relations and $Q^0=1$.
2. The Dyer–Lashof algebra arises by dividing out the “homologi... | https://mathoverflow.net/users/124042 | Ádem relations for the Steenrod and the Dyer–Lashof algebra | See Peter May's "A General Algebraic approach to Steenrod Operations". Both arise from considering the inclusion of a Sylow $p$-subgroup into $\Sigma\_{p^2}$. There is a universal $\mathbb{Z}$-indexed set of operations with the Steenrod algebra and Dyer-Lashof algebra arising as the quotients acting on the homology of ... | 12 | https://mathoverflow.net/users/6872 | 329952 | 141,383 |
https://mathoverflow.net/questions/329842 | 3 | I am trying to understand the proof of Lemma 1 in [this paper (Section 9.2)](https://arxiv.org/pdf/1809.03113.pdf).
The proof shows that given a discrete probability distribution $P=(p\_1,p\_2,...,p\_k)$ where $p\_1 \geq p\_2 \geq ... \geq p\_k$, and a discrete probability distribution $Q=(q\_0,q\_0,q\_3,q\_4...,q\_k... | https://mathoverflow.net/users/139786 | Lower bound Renyi divergence between two discrete probability distributions | The author is here.
We did make a mistake in the formula. $q^\*=\left(\frac{p\_1^{1-\alpha}+p\_2^{1-\alpha}}{2}\right)^{\frac{1}{1-\alpha}}$ is not the minimizer for $q\_0$. It should be $\frac{q^\*}{1-p\_1-p\_2+2q^\*}$. The corresponding minimizer for $q\_i$ should be $\frac{p\_i}{1-p\_1-p\_2+2q^\*}$.
I was misled b... | 3 | https://mathoverflow.net/users/139830 | 329953 | 141,384 |
https://mathoverflow.net/questions/329856 | 2 | I am wondering whether there are estimates for the Euclidean length of vertical hyperbolic geodesics for annuli with good geometry.
More precisely:
Take an annulus $A$, whose outer boundary $\gamma\_{ext} $ is the circle $|z|=1+\epsilon$ and the inner boundary $\gamma\_{int}$ is an analytic curve of length at most $... | https://mathoverflow.net/users/44316 | Euclidean length of hyperbolic geodesics for annuli with bounded geometry | Yes, you can say things like this.
The easiest way to get these kind of results is probably the Gehring-Hayman theorem. It states that, for two points $z$ and $w$ in a simply-connected domain $D$ or its closure, if there is a curve in $D$ of Euclidean length at most $\ell$ connecting $z$ and $w$, then the hyperbolic... | 2 | https://mathoverflow.net/users/3651 | 329956 | 141,385 |
https://mathoverflow.net/questions/329942 | 6 | Let's fix a field $k$. First, consider $Aff\_k$ to be the category of affine finite type $k$ schemes. On this category, one can define the etale topology and thus consider the site $Aff\_k^{et}$, then define the category of sheaves:
$Sh(Aff^{et}\_k)$
Similarly, it seems to me that one could also take the category $... | https://mathoverflow.net/users/38075 | Etale sheaves on algebraic spaces vs. Etale sheaves on affines | Yes, the two $\infty$-topoi are equivalent. Let $u: \mathrm{Aff} \to \mathrm{AlgSp}$ be the inclusion. Then $u$ preserves étale covering families and pullbacks, hence commutes with the formation of the Čech nerve of such a covering. This implies that $u^\*$ preserves sheaves. The functor $u$ is also cocontinuous: if $X... | 3 | https://mathoverflow.net/users/20233 | 329966 | 141,388 |
https://mathoverflow.net/questions/329963 | 2 | Let $F\colon\mathbb R^3\to\mathbb R$ be a compactly supported smooth function. I want to compute
$$ \int\_{\mathbb R^n}\int\_{\mathbb R^n} F(\lVert x\rVert^2,\lVert y\rVert^2,\langle x,y\rangle)~\mathrm{d}x~\mathrm{d}y,$$
where $\mathrm{d}x$ is the Lebesgue measure in $\mathbb R^n$, as a three dimensional integral. Do... | https://mathoverflow.net/users/89934 | Integral substitution involving the length and angle of two vectors | I'm not sure about a clever way, but the straightforward Calculus 3 type computation is not hard at all. Making the integration in $x$ outer and switching to polar coordinates in $x$, then switching in the inner integral with respect to $y$ to the cylindrical coordinates with the axis of the cylinder parallel to $x$, w... | 2 | https://mathoverflow.net/users/1131 | 329978 | 141,390 |
https://mathoverflow.net/questions/329980 | 7 | [Cauchy's integral formula](https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula) is a powerful method to extract the $n$'th power series coefficient of an analytic function by evaluating a single complex integral. Is there any such analytic method to extract (ordinary) Dirichlet series coefficients? That is, assu... | https://mathoverflow.net/users/7581 | Extracting Dirichlet series coefficients | Even for more general Dirichlet series
$$f(z)=\sum\_{0}^\infty a\_n e^{-\lambda\_nz}$$
there is the formula
$$a\_ne^{-\lambda\_n\sigma}=\lim\_{T\to\infty}\frac{1}{T}\int\_{t\_0}^Tf(\sigma+it)e^{\lambda\_n it}dt,$$
where $t\_0$ is arbitrary (real) and $\sigma>\sigma\_u$, the abscissa of uniform convergence.
This formu... | 7 | https://mathoverflow.net/users/25510 | 329988 | 141,394 |
https://mathoverflow.net/questions/329987 | 14 | In $\sf ZF$, we have that the axiom of choice is equivalent to:
>
> For all sets $X$, and for all proper classes $Y$, $X$ inject into $Y$
>
>
>
and
>
> For all sets $X$, and for all proper classes $Y$, $Y$ surject onto $X$
>
>
>
To see that those are indeed equivalent to choice we have for one directi... | https://mathoverflow.net/users/113405 | Injection into a proper class and choice without regularity | The results appear in Jech's "The Axiom of Choice" in the problem section of Chapter 9 (Problems 2,3, and 4).
Indeed, it is easy to see that the injections into classes imply the surjections from classes which imply choice. Exactly by means of the class of ordinals. So the point is to separate the others.
And if we... | 8 | https://mathoverflow.net/users/7206 | 329994 | 141,398 |
https://mathoverflow.net/questions/330004 | 5 | Let $F:[0,1]\to\mathbb{R}$ be a measurable function such that
$$
\mu\_n(F)=\int\_0^1F(t)t^ndt\sim\frac1{(\ln n)^n}\quad\mbox{as}\quad n\to\infty.
$$
More precisely,
$$
0<c<|\mu\_n(F)|(\ln n)^n<C<\infty,\quad\forall n\in\mathbb{N}.
$$
Note that in this case the series
$$
\sum\_{n=0}^\infty\mu\_n(F)z^n
$$
represents an ... | https://mathoverflow.net/users/89313 | Find a function $F$ on $[0,1]$ with moments decaying as $(\ln n)^{-n}$ | $\newcommand{\R}{\mathbb{R}}
\newcommand{\si}{\sigma}
\newcommand{\supp}{\operatorname{\mathrm supp}}
\newcommand{\cch}{\operatorname{\mathrm cch}}
$
If $F\in L^2$, then the condition
\begin{equation\*}
|\mu\_n(F)|(\ln n)^n<C<\infty\quad\forall n\in\mathbb{N} \tag{1}
\end{equation\*}
implies that $F=0$ almost everywh... | 5 | https://mathoverflow.net/users/36721 | 330006 | 141,402 |
https://mathoverflow.net/questions/330007 | 7 | In 1960, R. Hermann showed the following:
**Theorem** Let $M$ be a manifold with a foliation $F$ and a bundle-like metric, if all leaves are compact and the holonomy group of each leaf is trivial, then $M/F$ is a smooth manifold.
(It is the partially result of the main theorem on *Hermann, R.*, [**On the differentia... | https://mathoverflow.net/users/95296 | Foliation with trivial leaf holonomy | This follows from Theorem 2 in Thurston's 1974 paper "A generalization of the Reeb stability theorem", at least if $H^1(L,R)=0$.
<https://core.ac.uk/download/pdf/82172971.pdf>
| 4 | https://mathoverflow.net/users/39082 | 330012 | 141,403 |
https://mathoverflow.net/questions/330010 | 5 | I am currently trying to learn Patterson-Sullivan theory, but I am getting stuck on basic questions about ergodic theory. Here is one of them, given as an exercise in one of the texts I am trying to read. If you need wider context, the text (in French) is [here](http://www.numdam.org/issue/MSMF_2003_2_95__1_0.pdf), wit... | https://mathoverflow.net/users/39348 | Ergodic without atoms implies completely conservative? | The desired statement is entirely measure-theoretic, so it's not really necessary to think about the topology of $X$. By standard descriptive set theory, there exists a Borel linear ordering $\preceq$ on $X$. (This means that the graph of $\preceq$ is a Borel subset of $X \times X$.)
Suppose toward a contradiction t... | 4 | https://mathoverflow.net/users/30721 | 330015 | 141,404 |
https://mathoverflow.net/questions/329927 | 4 | Let $G$ and $H$ be subgroups of the symmetric groups $\mathfrak S\_m$ and $\mathfrak S\_n$. Assume that $n>1$ and that $H$ is a 'connected' permutation group, that is, there is no non-trivial $H$-stable subset $X\subset [n]$ (with $[n]=\{1,\dots,n\}$) such that $H$ is isomorphic to the direct product of the image of it... | https://mathoverflow.net/users/3032 | Connected permutation groups and wreath product | I am afraid that I would have no idea where to look for a reference for this statement, but here is a very rough sketch proof. I can fill in details if necessary.
Let $A\_1,\ldots,A\_s$ be the orbits of $G$ on $[m]$ and $B\_1,\ldots,B\_t$ the orbits of $H$ on $[n]$. We can assume that each $|B\_j| > 1$, since otherwi... | 3 | https://mathoverflow.net/users/35840 | 330017 | 141,405 |
https://mathoverflow.net/questions/330024 | 8 | In regular ZF, AC, WO, and Zorn's Lemma are equivalent, but every proof I know (of the implication AC -> WO and AC -> Zorn) uses the axiom of choice on the powerset of X (where X is the Set which is to be well-ordered).
My question is, whether or not there is a proof of this equivalence that doesn't use the axiom of ... | https://mathoverflow.net/users/138274 | Relationship between AC, WO, and Zorn's lemma in ZF-Powerset | This is a classic theorem of Zarach, that it is consistent that ${\sf ZF}^-$ holds with the Axiom of Choice, but not every set can be well-ordered.
>
> *Zarach, Andrzej*, **Unions of ${\sf ZF}^-$models which are themselves ${\sf ZF}^-$ models**, Logic colloquium ’80, Eur. Summer Meet., Prague 1980, Stud. Logic Foun... | 13 | https://mathoverflow.net/users/7206 | 330027 | 141,408 |
https://mathoverflow.net/questions/330019 | 12 | I just want to know what properties of valuations extend to $\mathbb R$...
Denote an extension of the 2-adic [valuation](https://en.wikipedia.org/wiki/P-adic_order) from $\mathbb Q$ to $\mathbb R$ by $\nu$.
Suppose $\nu(x)=\nu(y)=0$.
Is it true that $\nu(x+y)\ne 0$?
What about $\nu(x^2+y^2)\le 1$?
I'm intereste... | https://mathoverflow.net/users/955 | Extension of 2-adic valuation to the real numbers | No. The important thing to know is that, if $K \subseteq L$ is a field extension and $v: K \to \mathbb{R}$ is a valuation, then $v$ can be extended to $L$. So I can answer all of your questions by working in some easy to handle subfield of $\mathbb{R}$. I'll work in $K = \mathbb{Q}(\sqrt{5})$ for the first question and... | 16 | https://mathoverflow.net/users/297 | 330028 | 141,409 |
https://mathoverflow.net/questions/329991 | 5 | I call etale a finite-type flat $R$-algebra $A$ such that $\Omega\_A =0$ (I hope this is the standard definition).
In the case where $R=k$ is a field, any such algebra $A$ decomposes as a finite product of finite separable extensions of $k$.
What about algebras over more general rings? Do we get a decomposition of $A... | https://mathoverflow.net/users/139854 | Structure theorem for etale algebras over a more general ring than a field | I would recommend you to look at EGA IV$\_4$ (18.4.5) and (18.4.6). In brief: Let $A \to B$ be a locally finite presentation algebra with $A$ local with maximal $\mathfrak{m}$. Then $B$ is étale at a point corresponding to a maximal $\mathfrak{n}$ If and only if there is a polynomial $F \in A[T]$ such that $B = A[T]/\l... | 2 | https://mathoverflow.net/users/6348 | 330035 | 141,412 |
https://mathoverflow.net/questions/329890 | 1 | **Setup:**
I have a model of a biological process described by two ODEs as follows:
$$\dot{X\_1} = (\beta\_1-d-1)X\_1 + 2X\_1^2 - X\_1^3 + dX\_2$$
$$\dot{X\_2} = (\beta\_2-d-1)X\_2 + 2X\_2^2 - X\_2^3 + dX\_1$$
I want to analyze the stochastic version of this system using an appropriate underlying mechanistic proces... | https://mathoverflow.net/users/nan | Obtaining generator matrix and first-passage time distribution for CTMC? | The infinitesimal generator $\mathscr{A}$ corresponding to the OP's chemical reaction network can defined by its action on a function $f: \mathbb{Z}^2\_{\ge 0} \to \mathbb{R}$ as follows
$$ \mathscr{A}f(x) = \sum\_{\ell} a\_{\ell}(x) ( f(x+\nu\_{\ell}) - f(x) )
$$ where we introduced the propensity functions and re... | 1 | https://mathoverflow.net/users/64449 | 330042 | 141,413 |
https://mathoverflow.net/questions/330038 | 1 | [This post](https://mathoverflow.net/questions/194683/are-carnot-groups-as-carnot-caratheodory-metric-spaces-doubling) shows that every Carnot group is a [doubling metric space](https://en.wikipedia.org/wiki/Doubling_space). However, what is its doubling constant?
| https://mathoverflow.net/users/36886 | Doubling constant of Carnot group | If the homogeneous dimension of the Carnot group is $s$, then the $s$-dimensional Hausdorff measure satisfies $\mathcal{H}^s(B(x,r))=Cr^s$ with a fixed constant $C$ independent of the center and the radius of the ball. Therefore
$\mathcal{H}^s(B(x,2r))=C(2r)^s=2^sCr^s=2^s\mathcal{H}^s(B(x,r))$ and the doubling constant... | 5 | https://mathoverflow.net/users/121665 | 330043 | 141,414 |
https://mathoverflow.net/questions/328625 | 5 | Let $X$ and $Y$ be two simplicial groups such that there exists $f:X\rightarrow Y$ and $g:Y\rightarrow X $ two injective morphisms of simplicial groups. Suppose that $X$ is contractible, does it follow that $Y$ is contractible ?
| https://mathoverflow.net/users/136128 | pair of injective morphisms of simplicial groups | Pick pointed topological spaces $A$ and $B$ which admit pointed injective continuous maps $A \to B$ and $B \to A$ for which $A$ is contractible but $B$ has nonvanishing reduced homology. For example, we could take $A = [0,1]$ and $B = [0,1] \cup \{2\}$ pointed at $0$.
Now apply the functor $\mathbb{Z}[\mathrm{Sing}(-... | 2 | https://mathoverflow.net/users/126667 | 330047 | 141,415 |
https://mathoverflow.net/questions/330057 | 4 | Suppose we have a Banach space $X$ and have chosen a set $\Sigma$ consisting of some sequences whose members are in $X$. We can then say that $(x\_n)\_{n=1}^\infty\in X^\mathbb{N}$ is $\Sigma$-*convergent* to $x\in X$ if $(x\_n-x)\_{n=1}^\infty\in \Sigma$. We can say $C\subset X$ is $\Sigma$-*closed* if whenever $(x\_n... | https://mathoverflow.net/users/nan | Defining a topology by sequences | What you are describing is very close to what is called a convergence space. Strictly speaking, these are quite a lot more general than a topology, but with some additional conditions on the convergent filters, they form a topology and vice-versa. Several good references are "Foundations of Topology", by Gerhard Preuss... | 2 | https://mathoverflow.net/users/137147 | 330060 | 141,419 |
https://mathoverflow.net/questions/330054 | 4 | How do I show the following bounds on the mills ratio :
$\frac{1}{x}- \frac{1}{x^3} < \frac{1-\Phi(x)}{\phi(x)} < \frac{1}{x}- \frac{1}{x^3} +\frac{3}{x^5} \ \ \ \ \ \ \ $ for $ \ \ \ x>0$ where $\Phi()$ is the CDF of the Normal distribution , and $\phi()$ is the density function of the Normal distribution ?
Also ,... | https://mathoverflow.net/users/137618 | Bounds on the mills ratio | Here's a sketch and a link for how I prove it. Let
$$ f(x) = - \left( \frac{1}{x} - \frac{1}{x^3} + \frac{3}{x^5}\right) \phi(x) .$$
Now show (lemma): $\frac{df}{dx} = \left(1 + \frac{15}{x^6}\right)\phi(x)$.
(To prove this, use that $\frac{d\phi}{dx} = - x \phi(x)$, the product rule, and some cancellation.)
Now... | 3 | https://mathoverflow.net/users/29697 | 330069 | 141,421 |
https://mathoverflow.net/questions/330068 | 3 | Suppose $\mu\_n\implies\mu$, i.e. $\mu\_n$ converges weakly to $\mu$ where $\mu\_n$, $\mu$ are probability measures on some metric space $(X,d)$. Given a Borel set $B$, define $\mu^B$ to be the conditional probability given $B$, i.e. $\mu^B(A)=\mu(A\cap B)/\mu(B)$, and similarly for $\mu\_n^B$.
Under what conditions ... | https://mathoverflow.net/users/99132 | Weak convergence of conditional probabilities | For $\mu\_n^B\Longrightarrow\mu^B$, it is enough that $\mu(\partial B)=0$ (and $\mu(B)>0$), where $\partial B$ denotes the boundary of $B$.
Indeed, then for any Borel set $A$ such that $\mu(\partial A)=0$ we have $\mu(\partial (A\cap B))=0$, because
$\partial(A\cap B)\subseteq(\partial A)\cup(\partial B)$. So, by t... | 3 | https://mathoverflow.net/users/36721 | 330073 | 141,423 |
https://mathoverflow.net/questions/330058 | 5 | Let us have two symplectic manifolds $(M, \,\omega)$ and $(N, \,\omega')$ and morphism between them:
$$ \varphi \ :\ M \to N.$$
Then we geometrically quantize these systems:
1. we add a prequantum line bundle, $ L\_M \to M$ and $L\_N \to N$ correspondingly;
2. choose some real polarizations $P\_M$ and $P\_N$ such tha... | https://mathoverflow.net/users/125724 | Does symplectic morphism after geometric quantization induce Hilbert spaces morphism? | The slogan to keep in mind is that "quantization is not a functor". For details on this slogan, see the (decade old!) Mathoverflow question [What does “quantization is not a functor” really mean?](https://mathoverflow.net/questions/8606/what-does-quantization-is-not-a-functor-really-mean). Basically, "is not a functor"... | 2 | https://mathoverflow.net/users/78 | 330079 | 141,425 |
https://mathoverflow.net/questions/329939 | 5 | Let $(G\rightrightarrows M)$ be a Lie groupoid (i.e. a groupoid with source map $s$ and target map $t$ such that $G,M$ are smooth manifolds and the structural maps are all smooth (and $s$,$t$ are submersions).
Defining
$$IG := \{g \in G \mid s(g) = t(g) \}$$
we obtain the so called isotropy subgroupoid $IG \rightrigh... | https://mathoverflow.net/users/46510 | Isotropy subgroupoid of a regular Lie groupoid | This is not exactly an answer to your question, but I hope it helps.
If it would be fine to first pass to the connected components of the identity of each isotropy group, I believe you find the proof of the analogous statement (that you then obtain an embedded subgroupoid) in Proposition 2.5 of
I. Moerdijk, [On t... | 4 | https://mathoverflow.net/users/104042 | 330091 | 141,427 |
https://mathoverflow.net/questions/330089 | 4 | Let $(X,\|\cdot\|)$ be a Banach space and $B\subset X$ be a bounded subset. Now define a function $f\_B:X\to [0,\infty)$ by $f\_B(x)=\sup\_{b\in B}\|b-x\|$.
My question is in which kind of Banach space, $f\_B$ can always reach its minimum at a unique point of $X$ for any given bounded subset $B$.
I find that the n... | https://mathoverflow.net/users/38714 | Condition for Banach space that the distance function can always reach its minimum at a unique point | I think the condition you need is the uniformly convexity:
For any fixed $M>0$, $\forall \epsilon>0$, $\exists \delta>0$ such that
$$\|x\|\le M,\|y\|\le M,\|x-y\|\ge \epsilon\Rightarrow\left\|\frac{x+y}{2}\right\|\le M-\delta.$$
Since $B$ is bounded, suppose $B\subset B\_M=\{x\in X:\|x\|\le M\}$ for some $M>0$. It i... | 4 | https://mathoverflow.net/users/121051 | 330094 | 141,429 |
https://mathoverflow.net/questions/330077 | 11 | Let $G$ be a finite group, $S \subset G$ a generating set. Set $\sigma(G):=\sum\_{U \subset G} |U| $, where the sum runs over all subgroups $U$ of $G$. Set $H\_G := \sum\_{g \in G} \frac{1}{|g|+1}$, where $|g|:= $ word length (with respect to $S$).
For $G:=\mathbb{Z}/(n)$ we get $\sigma(G) = \sigma(n)=$ sum of divisor... | https://mathoverflow.net/users/nan | A group theoretic interpretation of Lagarias inequality | The general inequality is interesting, but the special case you want is easy to prove -- at least if $n$ is large, but the proof can easily be quantified.
Note that every finite group $G$ of size $n$ can be generated by at most $\lfloor \log n/\log 2 \rfloor$ elements. You can see this greedily. Suppose a set $S$ ge... | 10 | https://mathoverflow.net/users/38624 | 330112 | 141,432 |
https://mathoverflow.net/questions/330127 | 1 | Before asking my question I like to admit that i am not an expert in the foundation of mathematics and I am interested in this issue more from a philosophical perspective. So my question may be competently naive. If we assume $ZFC$ and the negation of the continuum hypothesis, what is know about the set $$\{|A|~|~A\sub... | https://mathoverflow.net/users/23542 | Cardinals in $ZFC+\neg CH$ | By a theorem of Solovay, $|\mathbb R|$ can consistently be $\aleph\_\alpha$ for any ordinal number $\alpha>0$ that does not have countable cofinality. Then the set $\{|A|:A\subseteq\mathbb R, |\mathbb N|<|A|<|\mathbb R|$ in your question would have cardinality $\alpha-1$ if $\alpha$ is finite, and it would have cardina... | 10 | https://mathoverflow.net/users/6794 | 330135 | 141,443 |
https://mathoverflow.net/questions/330138 | 11 | Let $X,Y$ be objects of some category $\mathcal C$, and suppose I want to study homotopy classes of maps from $X$ to $Y$ (almost everything one does in algebraic topology can be viewed this way). It seems as though there are two classes of methods available to me: I can approach this via some kind of *obstruction theor... | https://mathoverflow.net/users/2362 | What is the relationship between spectral sequences and obstruction theory? | This is a partial answer, but every obstruction theory (in some precise sense) provides you with a spectral sequence (in fact several). Let me clarify what do I mean with obstruction theory. All this material is a modern reinterpretation of Bousfield's amazing paper [*Homotopy Spectral Sequence and Obstructions*](https... | 15 | https://mathoverflow.net/users/43054 | 330142 | 141,446 |
https://mathoverflow.net/questions/330064 | 1 | Suppose we have a probability distribution $\pi : X \rightarrow [0,1]$ where $X$ is finite and let $Q : X \times X \rightarrow [0,1]$ be a Markov kernel that is reversible with respect to $\pi$. That is,
$$\pi(x) Q(x,y) = \pi(y) Q (y,x) $$
for all $x,y \in X$. Suppose we know the mixing time $t\_x(Q, \varepsilon)$ ... | https://mathoverflow.net/users/139897 | Comparing mixing time of lazy and non-lazy Markov chains | **Edit [following the comment by R W]**: As *R W* pointed out, the stationary distribution may change if you remove the laziness as you suggested. Still your motivation makes sense: can making the chain less lazy reduce the mixing significantly faster? What I wrote below answers that.
---
The "non-lazy version" o... | 1 | https://mathoverflow.net/users/23297 | 330147 | 141,448 |
https://mathoverflow.net/questions/329977 | 2 | Let $X$ be a connected scheme, smooth and proper over $\mathbb{C}$. Let $F$ be a locally free $\mathcal{O}\_X$-module of finite rank $r>1$. Suppose on a non-empty affine open $U\subset X$ whose complement is irreducible we have an isomorphism of $\mathcal{O}\_X$-modules $f:\mathcal{O}^{\oplus r}\_X|\_{U}\rightarrow F|\... | https://mathoverflow.net/users/nan | Map to a given vector bundle from a split vector bundle | Yes. Slightly more precisely, we have the following result.
>
> **Proposition:** Let $X$ be a normal, integral, finite-type scheme over a field.
> Let $F$ be a locally free $\mathcal O\_X$-module of finite rank.
> Let $U\subseteq X$ be a nonempty open subset.
> Let $r$ be a nonnegative integer.
> Let $\phi : \m... | 2 | https://mathoverflow.net/users/20562 | 330163 | 141,453 |
https://mathoverflow.net/questions/330169 | 2 | Suppose *G* is a graph on *n* vertices. Then the ***closure*** of *G*, written [*G*], is constructed by adding edges that connect pairs of non-adjacent vertices *u* and *v* for which
$\deg(u) + \deg(v) \geq n.$
One continues recursively, adding new edges until all non-adjacent pairs *u*, *v* satisfy
$\deg(u) + \... | https://mathoverflow.net/users/138301 | The uniqueness of graph closure | Many years ago Staszek Radziszowski and I published a very elementary lemma that covers this case and many similar cases.
Let $(X,\le)$ be a partially ordered set and let $\varPhi$ be a family of functions from $X$ to $X$. Suppose that, for $x,x'\in X$ and $\phi\in\varPhi$ we have $\phi(x)\le x$ and $x\le x'\implies ... | 4 | https://mathoverflow.net/users/9025 | 330177 | 141,459 |
https://mathoverflow.net/questions/330162 | 1 | Suppose I have $X,Y$ bivariate normal with correlation coefficient $\rho \in (0,1)$ . Then , what is the correlation between $X^2 $ and $Y^2$ ?
I am aware of the fact that the square of the normal follows a chi square distribution . So , I can find out $Var(X^2)$ and $Var(Y^2)$ . However , I am unable to calculate th... | https://mathoverflow.net/users/137618 | Correlation between square of normal random variables | You can do what Nate Eldredge suggested. Otherwise, you can use the moment generating function $M$ of the bivariate normal distribution $N(\mu\_1,\mu\_2,\sigma\_1^2,\sigma\_2^2,\rho)$ of $(X,Y)$ given by
$$M(t\_1,t\_2)=\exp(\boldsymbol\mu'\boldsymbol t+\tfrac12\,\boldsymbol t'\Sigma\boldsymbol t)
$$
for $\boldsymbol t... | 5 | https://mathoverflow.net/users/36721 | 330191 | 141,462 |
https://mathoverflow.net/questions/329828 | 5 | Let $f:X\rightarrow Y$ be a finite morphism between Noetherian integral schemes that is surjective on the underlying topological spaces. Does there exist an integer $n>0$ and a coherent $O\_X$-module $F$ such that there is a morphism of $O\_Y$-modules $O\_Y^n\rightarrow f\_\*F$ inducing an isomorphism on the stalks at ... | https://mathoverflow.net/users/nan | Map of coherent sheaves inducing isomorphism on the stalks at the generic point | The following is basically a copy of dhy's comments with links (any mistakes are mine).
First, assume you know that for a finite surjective morphism $f:X\rightarrow Y$ between Noetherian integral affine schemes, there exists a positive integer $n$ and a morphism of $\mathcal{O}\_Y$-modules $\mathcal{O}\_Y^{\oplus n}\... | 0 | https://mathoverflow.net/users/nan | 330194 | 141,464 |
https://mathoverflow.net/questions/330036 | 6 | A bornologigal topological vector space is such that any bounded linear function on it is continuous. It is a standard result [Jarchow, Locally convex spaces, 1981] that if the dual $E'$ of a Mackey space $E$ is complete for the topology $\mathcal{T}\_{\mathcal{B}\_0}$ of uniform convergence on bipolars of null sequenc... | https://mathoverflow.net/users/109133 | Complete dual of bornological space | If I understand properly, I doubt very much that this is true.
I the article *On different types of non-distinguished Frechet spaes* (Note di Mat. 10 (1990), 149-165), Bonet, Dierolf, and Fernandez write that for a Frechet space $X$ with dual $(X',\beta(X',X))$ the Mackey topology $\mu(X',X'')$ is bornological if and... | 3 | https://mathoverflow.net/users/21051 | 330200 | 141,466 |
https://mathoverflow.net/questions/330196 | 2 | When I am studying some paper dealing with dispersive PDE(e.g. Wave, Schrödinger and Klein-Gordon equations), the potential $\frac {1}{|x|^2}$ which is called critical decay (or inverse square potential) is lots of handled.
They also consider Morrey-Campanato or Fefferman-Phong class to deal with that potential since... | https://mathoverflow.net/users/137669 | Why is critical decay important? | "Critical" in the context of the Schrödinger equation (which seems to be what the OP is after) refers to the number of bound states confined by the potential. The Schrödinger equation with potential $-C|x|^{-2+\epsilon}$ and $C>0$ has a discrete spectrum for $E<0$, with finitely many eigenvalues
if $\epsilon < 0$ and i... | 0 | https://mathoverflow.net/users/11260 | 330203 | 141,468 |
https://mathoverflow.net/questions/330174 | 3 | This is a conceptually easier version of a box packing problem I stated [earlier](https://mathoverflow.net/questions/330023/box-stacking-problem).
Let $n$ be a positive integer and let $r\_1, \ldots, r\_n$ be positive integers. We take $r\_i$ to be the radius of a sphere in $\mathbb{R}^3$ for each $i\in \{1,\ldots, n... | https://mathoverflow.net/users/8628 | Computabillity of packing of spheres with different radii | Yes, it is computable. Use the decidability of the theory of the real numbers $(\mathbb{R}, 0, 1, \times, +, <)$. With a very little standard work, you can define $\mathbb{R}^3$, vector addition, and the norm function $|x|$. Then using the decidability of real-closed fields search for the minimum $R \in \mathbb{N}$ suc... | 12 | https://mathoverflow.net/users/12978 | 330211 | 141,471 |
https://mathoverflow.net/questions/330100 | 6 | This question asks for the necessity of a noetherian hypothesis in a certain relation between properties of rings concerning chains of prime ideals. We use the following definitions.
1. A ring $R$ is called **universally catenary** if every $R$-algebra of finite type is catenary. (Note that $R$ need not be noetherian... | https://mathoverflow.net/users/11025 | Is every universally catenary ring a going-between ring? | OK, let $R \subset S$ be an integral ring extension with $R$ universally catenary. Let $\mathfrak q \subset \mathfrak q'$ be primes in $S$ such that there is no prime strictly in between them. We have to show that the same is true for the corresponding primes $\mathfrak p \subset \mathfrak p'$ of $R$.
Reduction to th... | 4 | https://mathoverflow.net/users/139974 | 330214 | 141,473 |
https://mathoverflow.net/questions/329945 | 9 | For which fields $k$ do there exist two finite-dimensional $k$-algebras of different dimension which are isomorphic as commutative unital rings? Some thoughts:
* for a finite field this can not happen, because the cardinality of the algebra determines its dimension.
* consider any field $K$ and the function field $k=... | https://mathoverflow.net/users/nan | Finite-dimensional algebras isomorphic as commutative unital rings | To find examples of such fields, it is sufficient to look for fields $k$ having isomorphic finite extensions of differing degrees, just like the example given by schematic\_boi.
>
> **Claim:** For a field $k$, the following are equivalent:
>
>
> 1. Finite dimensional commutative $k$-algebras that are isomorphic a... | 3 | https://mathoverflow.net/users/778 | 330223 | 141,476 |
https://mathoverflow.net/questions/330126 | 3 | I am looking at Usha Bhosle's paper on [Moduli of Parabolic $G$-bundles](https://projecteuclid.org/download/pdf_1/euclid.bams/1183554904). She calculated the [dimension of the moduli space](https://i.stack.imgur.com/04eer.png).
Now I want to know the the dimension of the moduli of parabolic $G$-Higgs bundles over a ... | https://mathoverflow.net/users/139928 | Dimension of Moduli space of Parabolic $G$- Higgs bundles | Let $G$ be a connected reductive complex affine algebraic group. Let $\mathcal{N}$ be the moduli space of holomophic $G$-bundles over a Riemann surface of genus $g\geq 2$ with $n$ punctures (and fixed generic boundary data). Let $\mathcal{M}$ be the moduli space of $G$-Higgs bundles over a Riemann surface of genus $g\g... | 4 | https://mathoverflow.net/users/12218 | 330231 | 141,479 |
https://mathoverflow.net/questions/328730 | 0 | When is it possible to properly color the vertices of a given regular graph uniformly? By uniformly, it is implied that between any two color classes, there exist at least one adjacency(edge) from a vertex in one color class to some vertex in the other color class.
I think this is possible in most Cayley grpahs, and... | https://mathoverflow.net/users/100231 | Uniformity in colorings of a graph | (Not a full answer, but too long for a comment)
**Fact:** there are regular graphs where every equitable coloring has color classes with no edges between them. (As mentioned in comments, if we remove the restriction that the coloring be equitable, the question is no longer interesting.)
Moreover, there are regular,... | 2 | https://mathoverflow.net/users/22512 | 330239 | 141,484 |
https://mathoverflow.net/questions/330202 | 3 | In this paper [SOME EXTREMAL QUESTIONS FOR SIMPLICIAL COMPLEXES](https://link.springer.com/content/pdf/10.1007/BF02851262.pdf), the author discussed about minimal area of bounded by multiples of a curve. Say we have a (well-behaved) curve $\Gamma$, the solution to the minimal area problem is $S$. Now e consider $2\Gamm... | https://mathoverflow.net/users/90295 | Least area bounded by multiple of curves | As @alesia points out $2\Gamma$ means to take the curve ``with multiplicity two" (this is usually understood to be in the context of currents or flat chains -- for an oriented curve you can think about it as tracing out the curve twice). The reason this can give a lower area for the minimizer is that the space of compe... | 4 | https://mathoverflow.net/users/127803 | 330241 | 141,485 |
https://mathoverflow.net/questions/330209 | 5 | There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable?
That is, for each small abelian category $A$ does there exist a small stable $\infty$-category $\underline A$ and a homological functor $\underline A \to ... | https://mathoverflow.net/users/2362 | Are there universal homological functors? | Here is half an answer (i.e. an answer to the last question). Given a small stable $\infty$-category $C$ and a small abelian category $A$, any functor $C\to A$ factors through the homotopy category $ho(C)$ of $C$. Hence the category of homological functors from $C$ to $A$ is equivalent to the category of homological fu... | 8 | https://mathoverflow.net/users/1017 | 330243 | 141,486 |
https://mathoverflow.net/questions/330248 | 2 | Let $X$ be a (finite if you want) set and form the simplicial set $F^{\bullet}(X)$ with
$$
F^{n}(X) = \mathrm{Hom}\_{\mathrm{set}} ([n], X)
$$
where the right hand side denotes arbitrary maps of sets (of course
it wouldn't make sense to say order preserving as $X$ doesn't come
with an order).
I'm wondering about a d... | https://mathoverflow.net/users/113296 | Simplicial set represented by an (unordered) set | You are overlooking some nondegenerate simplices. For example, when $X={0,1}$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^\bullet(X)$ is infinite dimensional if $X$ has more than one element.
It is contractible whenever $X$ is non-empty; this can be seen by identifying it with the... | 8 | https://mathoverflow.net/users/6666 | 330250 | 141,489 |
https://mathoverflow.net/questions/330179 | 3 | It is known that a graph is 2-vertex-connected iff it has an [(open) ear decomposition](https://en.wikipedia.org/wiki/Ear_decomposition), and there is a linear-time algorithm for finding an ear decomposition.
I think it is also true that if a graph is 2-vertex-connected, we can find an ear decomposition starting wit... | https://mathoverflow.net/users/139952 | Ear decomposition with initial cycle | The [paper of Schmidt](https://arxiv.org/ftp/arxiv/papers/1209/1209.0700.pdf) that you found answers the question in the positive, both existence and algorithm. Start the DFS on $C$ and preferentially follow edges of $C$ until the final edge, which becomes a back-edge to the root. Then in the second stage you can choos... | 1 | https://mathoverflow.net/users/9025 | 330251 | 141,490 |
https://mathoverflow.net/questions/330272 | 8 | Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}(3)$ over a number field $F$. I am wondering how general are Gelbart-Jacquet lifts of automorphic representations of $\mathrm{GL}(2)$ among these.
I have a very limited understanding of Gelbart-Jacquet lifts: they are the cuspidal automorphic represent... | https://mathoverflow.net/users/43737 | How strong is the requirement of being a Gelbart-Jacquet lift? | The [Gelbart-Jacquet lift](https://doi.org/10.24033/asens.1355) of a $\mathrm{GL}\_2$ cuspidal automorphic representation $\pi$ is an automorphic representation $\Pi$ of $\mathrm{GL}\_3$. More generally, the Gelbart-Jacquet lift $\Pi$ is a $\mathrm{GL}\_3$ automorphic representation associated to a $\mathrm{GL}\_2$ aut... | 12 | https://mathoverflow.net/users/3803 | 330278 | 141,497 |
https://mathoverflow.net/questions/330259 | -1 | Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous spectrum) other than unitary $VTV^{\*}$ also have continuous spectrum?
| https://mathoverflow.net/users/136400 | Invariance of spectrum under conjugation | In any infinite-dimensional Hilbert space, the only such operators $V$ that work for all $T$ are scalar multiples of the identity.
Suppose $V$ is not a scalar multiple of a unitary. Then there are linearly independent vectors $v$, $w$ such that $V^\* V v = w$.
Let $T$ be a self-adjoint invertible operator such that $... | 5 | https://mathoverflow.net/users/13650 | 330280 | 141,498 |
https://mathoverflow.net/questions/265217 | 5 | In Kriegl/Michor's "Convenient Setting for Global Analysis", they put on the set $L(E, F)$ of bounded linear operators between locally convex spaces $E$, $F$ the subspace topology induced by the inclusion $L(E, F) \subset C^\infty(E, F)$, where smooth maps are those that map smooth curves in $E$ to smooth curves in $F$... | https://mathoverflow.net/users/16702 | $c^\infty$ topology on $L(E, F)$ | Since this was a bit long for a comment, I am posting it here as an answer (though it unfortunately does not answer your question per se).
It is not clear to me in general how the topologies are related to each other. In some case one can say something though:
Lemma 5.3. in the book states that a subset is bounded ... | 2 | https://mathoverflow.net/users/46510 | 330284 | 141,500 |
https://mathoverflow.net/questions/330244 | 4 | At first, I figured that an automorphism of a scheme $X$ would be a homeomorphism $f:|X| \to |X|$ of topological and an isomorphism of sheaves $f^{\#}: \mathcal{O}\_X \to f\_\*(\mathcal{O}\_X)$.
However, if $X= \mathbb{P}\_k^n$, then the automorphisms of $X$ are actually defined as follows:
Let $\textbf{Aut}(\mat... | https://mathoverflow.net/users/100155 | Automorphisms of Schemes and their $A$-points | I just organized my comments into an answer/review (though the existing answers are already perfect).
**Schemes over $k$.**
The category $Sch\_S$ of schemes over a given scheme $S$ is by definition the comma category over $S$, i.e. objects are morphisms of schemes $\pi:X\to S$ and arrows are morphisms of schemes ma... | 3 | https://mathoverflow.net/users/4721 | 330290 | 141,503 |
https://mathoverflow.net/questions/330292 | 0 | Let $(X,d)$ be a metric space. Suppose that $\{A^n\}\_{n \in \mathbb{N}}$ is a sequence of closed, non-empty subsets of $X$.
Is there a Hausdorff topology on the space of closed subsets of $X$, guaranteeing that if $A^n$ converges in this space to a $A\subseteq X$, then for any continuous function $f:X \rightarrow \... | https://mathoverflow.net/users/36886 | Topology on closed subsets characterized by sup on continuous functions? | The answer to the original version of the problem, with the opposite inequality, is clearly "no": if $X$ contains more than one point then there is no such topology. Let $A^1 = \{x\}$ and $A^n = \{y\}$ for $n \geq 2$, where $x,y \in X$ are distinct. Define $f(z) = d(y,z)$. Since the sequence $(A^n)$ is eventually const... | 1 | https://mathoverflow.net/users/23141 | 330295 | 141,504 |
https://mathoverflow.net/questions/225738 | 20 | I'm familiar with some examples of pairs of derived equivalent varieties, for example an abelian variety and its dual, a K3 surface and certain moduli schemes on it, or the [Pfaffian-Grassmannian derived equivalence](http://arxiv.org/pdf/math/0608404v3.pdf).
However, when I looked for other known examples, I could on... | https://mathoverflow.net/users/37059 | List of known Fourier Mukai partners? | There are actually several known Fourier Mukai partners.
1. Standard flop/Atiyah flop. See Chapter 11 of Fourier-Mukai Transforms in Algebraic Geometry by Huybrechts.
2. Mukai flops (Chapter 11 of Fourier-Mukai Transforms in Algebraic Geometry by Huybrechts), stratified Mukai flops <https://arxiv.org/abs/1111.0688> ... | 8 | https://mathoverflow.net/users/58609 | 330327 | 141,512 |
https://mathoverflow.net/questions/327529 | 1 | I need to simulate a process of the form
$$X\_t=\int\_0^t f(s,t)\mathop{dW\_s}$$
where $f$ is deterministic and the integral is an Itô integral. I know I can simply take finite Itô sums of discrete increments of the Brownian motion driver, but I am wondering if there are more sophisticated approaches. Common method... | https://mathoverflow.net/users/106451 | Simulation of Itô integral processes where integrand depends on terminal (Volterra process) | Here is an approximation scheme that uses a chain of independent Brownian bridges. For $t>0$ fixed, consider the following partition of the time interval $[0,t]$ $$
t\_0 = 0 < t\_1 < t\_2 < \dots < t\_{n} = t \;.
$$ At these discrete values, compute a discretized Brownian motion $W\_i = W(t\_i)$ in the standard way $$
... | 1 | https://mathoverflow.net/users/64449 | 330328 | 141,513 |
https://mathoverflow.net/questions/330225 | 2 | Let $G(V, E)$ be a simple graph with $|V|=n$, and let $h$ be an integer in $[n]$.
We repeat $h$-many times the following operation in a sequential fashion, where the graph may change at each round. We denote by $G\_t(V\_t, E\_t)$ the graph obtained just after round $t\in [h]$ (we have thereofore $G(V,E)=G\_0(V\_0,E\... | https://mathoverflow.net/users/115803 | Combinatorial optimization for a sequential random process on graphs | The conjectured upper bound is true and rather simple. Note that if we have a vertex $v$ of degree $d(v)=d>\frac nh$ in $G(t)$, then if we choose it first, the conditional probability that we remove it and its neighbors is $1-(1-\frac dn)^{h-1}\ge 1-(1-\frac 1h)^{h-1}\ge \frac 12$ (assuming $h\ge 2$; otherwise there is... | 1 | https://mathoverflow.net/users/1131 | 330338 | 141,515 |
https://mathoverflow.net/questions/330137 | 0 | I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x\_i$ of a finite domain $X$.
For example, the j-th such preimage list on an $n$ element domain would be the list of j-th inverse sets
$P\_j=\left[f^{-j}(x\_1), f^{-j}(x\_2), f^{-... | https://mathoverflow.net/users/76735 | Primage structures: induced domain partitioning by itterated inverse (reference request) | This is a slightly different perspective on transformation semigroups.
You have reviewed a lot of the literature. I only recall dimly some related work by Dietmar Schweigert who, in preparation for work on certain kinds of hyperidentities, presented some results (from Frobenius perhaps?) on periods of selfmaps of a f... | 0 | https://mathoverflow.net/users/3402 | 330340 | 141,516 |
https://mathoverflow.net/questions/330347 | 2 | Let $M$ be an infinite dimensional non-type $I$ factor, given $\xi$ in $\mathcal{H}$, does there exist a not identify operator $x$ in $M$ such that $x\xi=\xi$, I have tried with taking projection $P\_{\xi}:\mathcal{H}\rightarrow [M'\xi]$, this works unless $P\_{\xi}\neq I$, but how to tackle the case when $P\_{\xi}=I$.... | https://mathoverflow.net/users/136400 | On existence of fixed point operator | No, there does not necessarily exist such an $x$. For example, if $M$ is a $II\_1$ factor with trace $\tau$, $\mathcal{H} = L^2(M,\tau)$ and $\xi = 1$ (the identity of $M$, seen in $L^2(M,\tau)$), then $x\xi=\xi$ if and only if $x=1$.
| 1 | https://mathoverflow.net/users/10265 | 330349 | 141,520 |
https://mathoverflow.net/questions/330348 | 1 | Given I have a random variable $a$ that can be realised in the domain $D$ and has a finite variance $\sigma^2$. Furthermore I have a function $f$ which is differentiable(hence continuous) with an absolute derivative in the domain $D$ always less than $K$.
Is there then a way to prove that the variance of $f(a)$ is bo... | https://mathoverflow.net/users/129067 | Proof of variance bounds for transformed random variables | Assume wlog that $E[a]=0$. Then
$$\big|f(a)-f(0)\big|\le \left|\int\_0^a|f'(x)|dx\right|
\le \left|\int\_0^a Kdx\right|
\le K|a|\ \ \ \ \ \ \ \ \ \ $$
So $$E[(f(a)-f(0))^2]<K^2 E[a^2]$$
On the left, we can subtract $(E[f(a)]-f(0))^2$, which is non-negative; on the right, we can subtract $K^2E[a]^2$, which is zero. This... | 2 | https://mathoverflow.net/users/nan | 330357 | 141,522 |
https://mathoverflow.net/questions/330302 | 11 | I am interested in (as explicit as possible) descriptions of non-Gorenstein integral projective curves. Most of the literature on singular curves appears to be focused around the Gorenstein case, with a notable exception being the paper of Kleiman, *The Canonical Model of a Singular Curve*, and a handful of others. Any... | https://mathoverflow.net/users/65875 | Non-Gorenstein Curves | I don't think there is a reasonably short explicit description of non-Gorenstein curves. They could be described as lacking some of those conditions that define Gorenstein curves.
In any case, here is a simple example: Let $C$ be the union of three coordinate axis in $\mathbb A^3$. Then $C$ is not Gorenstein. It is p... | 7 | https://mathoverflow.net/users/10076 | 330358 | 141,523 |
https://mathoverflow.net/questions/330344 | 6 | The [classifying topos for local rings](https://ncatlab.org/nlab/show/classifying+topos#for_local_rings) is the big Zariski topos of $\text{Spec}(\mathbb{Z})$. Call this topos $T$. Geometric maps of topoi from a topos $T'$ to $T$ are in correspondence with sheaves of rings on $T'$ which are locally local rings. This is... | https://mathoverflow.net/users/30211 | Classifying Space of "Valuation Ringed Spaces over a Topos" | Since the axioms describing what a valuation ring can be put as what's called *geometric sequents* [\*], by the fundamental theorem on classifying toposes, there is a topos $T\_{val}$ with precisely the universal property you're asking for. (See for instance Section 2.1.2 and more specifically Theorem 2.1.8 in Olivia C... | 7 | https://mathoverflow.net/users/31233 | 330363 | 141,525 |
https://mathoverflow.net/questions/330351 | 4 | In my research I came across the following problem. Let $A$ be a symmetric and $\Gamma$ be a diagonal $n\times n$ matrices. The eigenvalues of $A$ are known $\lambda\_1,\ldots\lambda\_n$. The traces $\mathrm{Tr}(A^k\,\Gamma)=t\_k$, $k\in\mathbb{N}$ are also known, $\Gamma$ is given. Can $A$ be found based on this infor... | https://mathoverflow.net/users/41145 | Reconstruct a matrix from its traces | Unfortunately even the generic situation is bad. Since we know the eigenvalues, we should search for the orthonormal system of eigenvectors $v\_i$ of $A$. We have ($e\_i$ is the standard basis)
$$
Tr(A^k\Gamma)=\sum\_i\left[\sum\_j\gamma\_j \langle v\_i,e\_j\rangle^2\right]\lambda\_i^k
$$
so, in effect, you have the kn... | 12 | https://mathoverflow.net/users/1131 | 330366 | 141,526 |
https://mathoverflow.net/questions/330367 | 6 | I have frequently come across the statement "Any two triangulations of a compact n-manifold are related by bistellar moves" attributed to Pachner via Lickorish's paper 'Simplicial moves on complexes and manifolds'. The theorem this refer to is the following: Closed combinatorial n-manifolds are PL homeomorphic if and o... | https://mathoverflow.net/users/136604 | Are triangulations of compact manifolds PL homeomorphic? | Pachner‘s Theorem is about PL-homeomorphic manifolds, while the Hauptvermutung is asking for a PL-homeomorphism between manifolds which a priori are only homeomorphic.
| 11 | https://mathoverflow.net/users/39082 | 330369 | 141,528 |
https://mathoverflow.net/questions/324996 | 4 | Call a function $f: [0, \infty) \to \mathbb R$ **nearly eventually almost periodic** with period $p > 0$ if for a.e. $x \in [0, p)$, the sequence ${f(x + np)}\_{n \in \mathbb N}$ converges.
Suppose $f: [0, \infty) \to \mathbb R$ is continuous and nearly eventually almost periodic of periods $1$ and $a$, where $a$ is ... | https://mathoverflow.net/users/132446 | Nearly eventually almost periodic functions | I tried to prove a positive answer by copying and modifying a bit [Dap](https://mathoverflow.net/users/112284/dap)’s answer to your very similar [question](https://mathoverflow.net/q/324513), so a main contribution to this answer belongs to @Dap.
Let $$Z\_a=\{x\in[0,a)| \mbox{ the sequence }\{f(x + np)\}\_{n \in \mat... | 3 | https://mathoverflow.net/users/43954 | 330370 | 141,529 |
https://mathoverflow.net/questions/189222 | 21 | Is there a "Cauchy-Schwarz proof" of the following inequality?
**Theorem.** Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int\_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int\_{[0,1]^2} f(x,y) \, dxdy\right)^3.
$$
*Background.*
This inequality is due to [Blakley and Roy (1965)](http://www.ams.org/mathsci... | https://mathoverflow.net/users/8297 | Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality) | Before giving a very short Cauchy-Schwarz inequality proof for the 3-edge path (can be done in a similar fashion for any tree), let me comment on the authorship of the inequality in question.
In 1959, Mulholland and Smith <https://doi.org/10.2307/2309342> proved that for any symmetric non-negative matrix $A$ and any ... | 15 | https://mathoverflow.net/users/140072 | 330384 | 141,534 |
https://mathoverflow.net/questions/330289 | 9 | Let $F: C \to D$ be an infinity functor. Is it true that the homotopy fiber at $y$ can be described as $C \times\_D D^{\simeq}\_{/y}$? If not, is there a simple formula resembling this one?
Beside the infinity structure, its points are pairs $(x \in C, s: F(x) \to y \text{ equivalence})$.
Models I am using are the ... | https://mathoverflow.net/users/140013 | Homotopy fibers of infinity functors | In general the homotopy pullback of the diagram given by $i:\{y\} \to \mathcal{D}$ and $f:\mathcal{C} \to \mathcal{D}$ is given by first replacing $i$ and $f$ by fibrations between fibrant objects (so that the diagram formed by $f$ and $i$ is fibrant in the injective model structure), and then taking the actual pullbac... | 10 | https://mathoverflow.net/users/51164 | 330394 | 141,536 |
https://mathoverflow.net/questions/330399 | 2 | If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, with derivative given Lebesgue almost everywhere by the function $$z\ni\mathbb R\mapsto\begin{cases}g'(z)e^{g(z)}&\text... | https://mathoverflow.net/users/91890 | If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set | Yes, it is true. Since $2\min(a,b)=a+b-|b-a|$ for positive numbers $a,b$, and $F(z)=e^{g(z)}-1$ is differentiable, it suffices to prove that $|F|$ is differentiable except at a countable set of points. Namely, this exceptional set is a set of isolated zeroes of $F$ (or, better to say, a subset of the set of isolated ze... | 7 | https://mathoverflow.net/users/4312 | 330400 | 141,539 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.