parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/234211 | 6 | In <http://arxiv.org/abs/math/0405089> Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was recently proved by Abouzaid and Smith in <http://arxiv.org/abs/1504.01230>. Their construction involves associating a Hamilto... | https://mathoverflow.net/users/89276 | Understanding "Decategorified" symplectic Khovanov homology | One thing you didn't mention is the manifold in which this calculation happens: the Slodowy slice to a nilpotent of type $(n,n)$. This manifold is the key to everything, since it is a geometric avatar of the invariants inside the representation $(\mathbb{C}^{2})^{\otimes 2n}$.
In what sense is this true? First, we ha... | 8 | https://mathoverflow.net/users/66 | 234214 | 108,601 |
https://mathoverflow.net/questions/234166 | 1 | I can't find any reference where the definition of Harris Ergodicity for Continuous time Markov processes is defined.
a) What would be exactly the definition?
b) What reference could be helpful?
EDIT: From what I've read From "Applied Probability and Queues(pg. 198-200)"(Asmussen) I understand that $(X(t))$ is ... | https://mathoverflow.net/users/86048 | Neat definition of Harris Ergodicity | A continuous time Markov process $X$ is Harris recurrent provided there exists a $\sigma$-finite measure $\mu$ on the state space $S$ of $X$ such that for all Borel sets $B\subset S$, if $\mu(B)>0$ then $\Bbb P^x(\int\_0^\infty 1\_B(X\_t)\,dt=\infty)=1$ for all $x\in S$. See [Mesure invariante sur les classes récurrent... | 0 | https://mathoverflow.net/users/42851 | 234216 | 108,603 |
https://mathoverflow.net/questions/233942 | 16 | The attaching map for the top cell of the torus $S^n \times S^n$ is a map
$$
[x,y]: S^{2n-1} \to S^n \vee S^n
$$
where the notation is such that
$x,y : S^n \to S^n \vee S^n$ are the two inclusions–––the map $[x,y]$ is the generalized Whitehead product of $x$ and $y$. (More generally, if $f: \Sigma X \to \Sigma Z$ and ... | https://mathoverflow.net/users/8032 | Whitehead products and Framed Manifolds | By Pontryagin's Theorem, you are asking for the preimages of any chosen points
in each wedge factor of $S^p \vee S^q \vee S^r$ for the iterated Whitehead product map $p : S^{p+q+r-2} \to S^p \vee S^q \vee S^r$. This can be obtained by chasing through the factorization of this map as
$$ S^{p+q+r-2} \to S^{p+q-1} \vee ... | 6 | https://mathoverflow.net/users/4991 | 234219 | 108,605 |
https://mathoverflow.net/questions/234141 | 3 | Let $f(x,y) = ax^2 + bxy + cy^2$ be an *indefinite* irreducible binary quadratic form with integer coefficients with non-zero discriminant. We can assume, without loss of generality, that $a \geq 1$. Can one categorize the solutions to the equation
$$\displaystyle f(x,y) = c, x,y \in \mathbb{Z} $$
as cleanly as in th... | https://mathoverflow.net/users/10898 | A class of quadratic equations | One could argue that this question is vague -- I mean, one can "categorise" the solutions as being the numbers that make the equation work, right? But let me give some sort of an idea as to what one would have to deal with in order to "categorise" the solutions to this sort of equation.
Let's start with $X^2-401Y^2$,... | 6 | https://mathoverflow.net/users/88614 | 234225 | 108,607 |
https://mathoverflow.net/questions/234234 | 3 | I'm looking for a reference for the following statement:
Let $P$ be a property of morphisms of schemes local on the target in the etale topology. Let $f : X\rightarrow Y$ be a morphism of schemes which is locally of finite presentation, and such that for all points $y\in Y$, the restriction of $f$ to the strict hense... | https://mathoverflow.net/users/88840 | etale localization reference request | That is not true without further hypotheses, but it is true with one additional hypothesis. First, here is a counterexample. Let $P$ be the property that the morphism $f$ is quasi-compact. This property is local for the étale topology, and even for the fpqc topology, cf. <http://stacks.math.columbia.edu/tag/02KQ> for i... | 4 | https://mathoverflow.net/users/13265 | 234255 | 108,614 |
https://mathoverflow.net/questions/234257 | 4 | Suppose a discrete group $\Gamma$ acts on a connected compact metrizable space $X$ by homeomorphisms. Denote such a topological dynamical system by $(X,\Gamma)$.
Question: is there any $(X,\Gamma)$ such that the set of ergodic $\Gamma$-invariant Borel probability measures on $X$ is infinite, countable and closed (equ... | https://mathoverflow.net/users/7360 | Examples of topological dynamical systems with countably infinitely many ergodic invariant measures | Let $T\colon [0,1] \to [0,1]$ be a homeomorphism such that $T(1/n)=1/n$ for all $n \geq 1$ and $T(x)<x$ for all other $x \in (0,1]$. If $\frac{1}{m+1}<x<\frac{1}{m}$ then $T^n(x)$ is monotone decreasing, hence convergent, and by continuity its limit must be fixed by $T$, so necessarily $\lim\_{n\to\infty} T^n(x)=1/(m+1... | 5 | https://mathoverflow.net/users/1840 | 234260 | 108,615 |
https://mathoverflow.net/questions/234221 | 3 | I am looking for a reference for the following folklore theorem, which is the Lorentzian analogue of the Bonnet fundamental theorem of surface in Euclidean space, hyperbolic space or $3$d sphere.
>
> If a Coddazzi tensor $b$ on a Riemannian surface $(S,g)$ of curvature $K$ satisfies the Gauss-equation $K=-det b+ k$... | https://mathoverflow.net/users/59238 | Reference for Bonnet Fundamental theorem of surfaces in Lorentzian spaces | See Section 7 (you have to scroll down a bit to get the corollary for the semi-Riemannian case) in
*Christian Bär, Paul Gauduchon, and Andrei Moroianu*, [**Generalized cylinders in semi-Riemannian and Spin geometry**](http://dx.doi.org/10.1007/s00209-004-0718-0), *Math. Z.* **249** (2005), no. 3, 545--580.
where t... | 4 | https://mathoverflow.net/users/3948 | 234268 | 108,616 |
https://mathoverflow.net/questions/234241 | 6 | Let $n$ be a positive integer and $S\_n$ be the symmetric group on $\{1,2,\ldots,n\}$.
For any $w\in S\_n$ and polynomial $f\in \mathbb{R}[x\_1,x\_2,\ldots,x\_n]$, denote $w(f)=f(x\_{w(1)},x\_{w(2)},\ldots,x\_{w(n)})$. Furthermore, define
$$\alpha(f)=\sum\limits\_{w\in S\_n}\epsilon(w)w(f),$$
where the sum is over all ... | https://mathoverflow.net/users/58096 | A question about homogenous polynomials of degree $\frac{n(n-1)}{2}$ | As for your polynomial $f=\prod\_{i=1}^5 (x\_i+x\_{i+1})^2$, you may do the following trick. At first, you replace your polynomial to $g=\prod\_{i=1}^5 (x\_i+x\_{i+1}-4)(x\_i+x\_{i+1}-3)$. Antisymmetrizations of $f$ and $g$ are the same, since their difference has degree less than 10. Next, we use points 0,1,2,3,4 as $... | 5 | https://mathoverflow.net/users/4312 | 234269 | 108,617 |
https://mathoverflow.net/questions/114177 | 4 | Let $A^\ast$ be an algebraic oriented cohomology theory (i.e. it is equipped with certain push-forwards for projective morphisms of smooth varieties over the base field $k$; see section 2 of <http://www.math.uiuc.edu/K-theory/0535/orient.pdf> for more detail); let $f:Y\to X$ be a finite morphism of smooth varieties who... | https://mathoverflow.net/users/2191 | For a finite flat (etale?) morphism $f:Y\to X$, is $f_*1_Y-\deg f . 1_X$ nilpotent in $A^0(X)$, where $A^*$ is the algebraic cobordism? | I think for such a statement to be true, you would need the oriented cohomology theory $A^\*$ to be generically constant in the sense of Levine-Morel and it should also satisfy the localization property. In this case, we can apply the Generalized degree formula of Levine-Morel (Theorem 4.4.7 in their book). It implies ... | 0 | https://mathoverflow.net/users/39193 | 234273 | 108,619 |
https://mathoverflow.net/questions/233879 | 2 | Consider a random variable $X\sim B(n,\frac 12)$. I'm trying to estimate the asymptotic behaviour of its central moments $E((X-\frac n2)^r)$, where $r$ is even and in the range $\Omega(1)\leq r\leq O(n)$, and $n$ goes to $\infty$.
I've looked at inequalities for central moments of sums of independent variables, but t... | https://mathoverflow.net/users/31092 | High order central moments of a symmetric binomial variable | Let $M:=(E(X-n/2)^r)^{1/r}$.
By Corollary 2 in **[[Latala](http://projecteuclid.org/euclid.aop/1024404522)]**,
$M\sim S$, where $A\sim B$ means that $\frac1C\,B\le A\le C B$ for some universal positive constant $C$ and
\begin{equation}
S:=r\sup\{t(2n/r)^t\colon1/r\le t\le t\_\*\},
\end{equation}
where $t\_\*:=\fra... | 1 | https://mathoverflow.net/users/36721 | 234276 | 108,621 |
https://mathoverflow.net/questions/234287 | 9 | Let $R$ be a smooth, integral, finite-type $\mathbb{Z}\_{(p)}$-algebra of relative dimension $n$ and $\overline{f} \colon R \to \mathbb{F}\_p$. Then Hensel's lemma tells us that this lifts to a map $R \to \mathbb{Z}\_p$. My understanding is that the space of lifts looks like an affine space, but I would like to underst... | https://mathoverflow.net/users/1355 | Tube of a mod p point on a smooth Z_(p)-scheme | Take $x\_1,\dots,x\_n$ in $R$ that lie in the kernel of $f$ and generate the tangent space a the point $f$. By Hensel's lemma, the map $(\frac{x\_1}{p},\dots,\frac{x\_n}{p})$ from the space of $\mathbb Z\_p$-points of $R$ to $\mathbb Z\_p^n$ is a bijection.
In fact the inverse function can be seen to be analytic. So ... | 7 | https://mathoverflow.net/users/18060 | 234289 | 108,626 |
https://mathoverflow.net/questions/234290 | 10 | R. Thompson introduced three groups $F\subset T\subset V$. The question concerning amenability of $F$ is still unanswered and has attracted much attention. I have read that Thompson group $V$ contains a copy of the free group $F\_2$ (with two generators), in particular it is not amenable. Does anyone know an explicit e... | https://mathoverflow.net/users/47294 | Explicit free subgroup in Thompson's group $V$ | Thompson's group $T$ of the circle itself contains a copy of $F\_2$: this is indeed easy to find a [ping-pong pair](https://en.wikipedia.org/wiki/Ping-pong_lemma) starting from 4 disjoint intervals. That $T$ is non-amenable is even easier, because it does not preserve any probability on Borel subsets on the circle (imm... | 11 | https://mathoverflow.net/users/14094 | 234293 | 108,628 |
https://mathoverflow.net/questions/234296 | 7 | Assume that $A$ is a $C^{\*}$ algebra with self adjoint elements $A\_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A\_{sa} \iff ba \in A\_{sa}$$
>
> Is $A$ necessarily a commutative algebra?
>
>
>
This question is in line of [this post](https://math.stackexchange.com/questions/1687407/is-this-a-charact... | https://mathoverflow.net/users/36688 | Is this a characterization of commutative $C^{*}$ algebras? | Yes. I will show that any two positive elements of $A$ commute. Since every element is a linear combination of positive elements, this suffices.
Say $a$ and $b$ are positive. Then $a^{1/2}ba^{1/2} \in A\_{sa}$, so by hypothesis $ba^{1/2}a^{1/2} = ba \in A\_{sa}$. That is, $ba = (ba)^\* = a^\*b^\* = ab$. QED
| 17 | https://mathoverflow.net/users/23141 | 234300 | 108,633 |
https://mathoverflow.net/questions/233744 | 0 | I would like to ask if there is a good upper bound on the difference $$D\_2(n)=\sum\_{n^{1/3}<p,q\leq n^{1/2}} \left(\frac{n}{pq}-\left\lfloor \frac{n}{pq}\right\rfloor\right)\quad (1) $$where $p$ and $q$ range over primes in the given interval. I believe that the ratio $$R\_2(n)=\frac{\sum\_{n^{1/3}<p,q\leq n^{1/2}} \... | https://mathoverflow.net/users/17773 | An upper bound on $\sum_{n^{1/3}<p,q\leq n^{1/2}} \frac{n}{pq}-\lfloor \frac{n}{pq}\rfloor$ | Based on @GerhardPaseman and @alpoge's remarks I have written this answer, for $k=2.$ Consider
$$
D\_2(n)=\sum\_{n^{1/3}<p,q\leq n^{1/2}} \left(\frac{n}{pq}-\left\lfloor \frac{n}{pq}\right\rfloor\right)\quad (1)
$$
where $p,q$ are primes, and note that each term in (1) is positive, and in $[0,1),$ which gives
$$
D\_2(n... | 1 | https://mathoverflow.net/users/17773 | 234315 | 108,637 |
https://mathoverflow.net/questions/234249 | 4 | Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries.
Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) on Borel subsets of $B\_X$, the unit ball of $X$, that is invariant under this action? If not in general then under what... | https://mathoverflow.net/users/89292 | Invariant probability on a unit ball of a Banach space | The answer is: yes iff $X$ contains a finite dimensional invariant subspace of arbitrary large dimension (as you see, it has really nothing to do with the Banach structure of $X$ only with its quality as a $\Gamma$-representation).
Equivalently, the answer is: yes iff $X\_0$ (defined below) is of infinite dimension.
... | 3 | https://mathoverflow.net/users/89334 | 234321 | 108,639 |
https://mathoverflow.net/questions/234322 | 5 | Let's consider a separable Hilbert space $(\mathcal H, \langle\cdot, \cdot\rangle\_{\mathcal H})$ with Norm $||\cdot||\_{\mathcal H} := \langle\cdot, \cdot\rangle^{1/2}\_{\mathcal H},$ orthonomal basis $(e\_j)$ of $\mathcal H$ and let $s\colon \mathcal H \rightarrow \mathcal H$ be a Hilbert Schmidt operator, denoted by... | https://mathoverflow.net/users/66236 | Are Hilbert-Schmidt operators on separable Hilbert spaces "Hilbert Schmidt" on the space of Hilbert Schmidt Operators? | The answer is No, assuming of course that for you $s$ acts on $\mathcal{S}\_\mathcal{H}$ by left-multiplication. There is no need to assume that $\mathcal{S}\_\mathcal{H}$ is separable, since it actually is, with an explicit countable orthonormal basis given by the elementary operators $e\_{ij}$ that act as $e\_{ij}(e\... | 9 | https://mathoverflow.net/users/2622 | 234324 | 108,641 |
https://mathoverflow.net/questions/233546 | 13 | Let $\{v\_i\}\_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following inequality:
$$
\sum\_{i=1}^N \log \left(\frac{1}{N}\sum\_{j=1}^N \frac{v\_i^\top X^{1/2}v\_j v\_j^\top X^{1/2}v\_i}{v\_j... | https://mathoverflow.net/users/62673 | A log inequality for positive definite trace-one matrices | The proof of the general case, in a strong form suggested in the end of OP.
Denote $X^{1/4} v\_i=u\_i$, $X^{1/2}=S$, then we have ${\rm tr}\,S^2=1$ and need to prove that
$$
{\rm tr}\,S^2\geqslant
\sum\_i \frac{(Su\_i,u\_i) (Su\_j,u\_j)}{\sum\_j (u\_i,u\_j)^2},
$$
then the very original inequality follows by applyi... | 9 | https://mathoverflow.net/users/4312 | 234329 | 108,643 |
https://mathoverflow.net/questions/234270 | 6 | This question is about the state of current knowledge regarding Voevodsky's algebraic cobordism of a point $\mathrm{MGL}^{\*,\*}(\mathrm{Spec}\,k)$. That the geometric diagonal $\mathrm{MGL}^{2\*,\*}(\mathrm{Spec}\,k)$ is isomorphic to the Lazard ring $\mathbb{L}$ if $char(k)=0$ has been shown by Levine using the Hopki... | https://mathoverflow.net/users/39193 | Algebraic cobordism (of a point) outside the geometric diagonal | The answer to both question is yes, provided you invert $p$ in characteristic $p$ (though conjecturally this is not necessary). In fact, as far as I can see, all of Spitzweck's computations apply to fields as well. He only assumes mixed characteristic because some of the arguments are more difficult in that case (e.g. ... | 4 | https://mathoverflow.net/users/20233 | 234339 | 108,645 |
https://mathoverflow.net/questions/234245 | 14 | Let $X$ be an infinite set. Are there Hausdorff topologies $\tau\_1, \tau\_2$ on $X$ such that $\tau\_1\cap\tau\_2 = \{\emptyset\} \cup \{U\subseteq X: X\setminus U\text{ is finite}\}$?
(That is, the intersection is as small as it can get.)
And what about the special case $X=\mathbb{R}$ and $\tau\_1$ being the to... | https://mathoverflow.net/users/8628 | $T_2$ topologies that are "as disjoint as possible" | Let $\varepsilon$ denote the Euclidean topology on $\mathbb R$.
**Proposition:** *There is a 0-dimensional $T\_2$ topology $\tau$ on $\mathbb R$
such that $\tau$ and $\varepsilon$ intersect in only the co-finite sets.*
*Proof:*
We need the following lemma:
**Lemma:**
*Let $Y=(Y,\nu)$ be an infinite topological s... | 9 | https://mathoverflow.net/users/71011 | 234354 | 108,647 |
https://mathoverflow.net/questions/234337 | 6 | I have a strong suspicion that yes, but as I am not a specialist in o-minimal structures, I thought that I might have overlooked some corner case.
The precise statement is as follows: let $X \subset \mathbb{R}^n$ be a set definable in an o-minimal structure. Then there exists a point $P \in X$ and an open neighbourh... | https://mathoverflow.net/users/2234 | Are definable sets in an o-minimal expansion of the real field locally analytic? | No.
In their paper [Quasianalytic Denjoy-Carleman classes and o-minimality](http://www.ams.org/journals/jams/2003-16-04/S0894-0347-03-00427-2/home.html), *J. AMS*, vol. 16 (4), 2003, p. 751—777, Rolin, Speissegger and Wilkie show that there exists a function on $[-1,1]$ which belongs to an o-minimal structure but is ... | 7 | https://mathoverflow.net/users/10696 | 234357 | 108,648 |
https://mathoverflow.net/questions/234361 | 2 | Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form $\mathrm{Hom}(A,-)$ for some $A\in Ab$?
| https://mathoverflow.net/users/49822 | Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable? | The category of abelian groups is small-complete, well-powered, and has a cogenerator (e.g., $\mathbb{Q}/\mathbb{Z}$). It follows from the Special Adjoint Functor Theorem that any limit-preserving functor $G: Ab \to Ab$ has a left adjoint $F$. (A proof of the SAFT may be found on [this nLab page](https://ncatlab.org/nl... | 9 | https://mathoverflow.net/users/2926 | 234365 | 108,650 |
https://mathoverflow.net/questions/98868 | 7 | There are two generalizations of usual groups: groupoids, where the multiplication operation becomes "partial", and hypergroups, for which the result of multiplying two elements is a probability measure rather than a single element. I am interested in the structure combining both these features (it might have been call... | https://mathoverflow.net/users/8588 | Groupoids and hypergroups | Yes, the notion of hypergroupoids exits. See the following preprints:
<http://arxiv.org/abs/1403.3424>
<http://arxiv.org/abs/1402.0072>
Both the articles are published. The second article defines Haar systems for a locally compact hypergroupoids explains construction of $C^\*$-algebra of a hypergroupoid equipped ... | 1 | https://mathoverflow.net/users/89361 | 234366 | 108,651 |
https://mathoverflow.net/questions/234371 | 3 | Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that the restriction of $F$ to the special fiber is torsion free?
| https://mathoverflow.net/users/43198 | Torsion free sheaves in flat families | Not in general. For example, take $X=\mathbb{P}^1\_R$ and let $x\in X$ be a closed point on the special fiber. Then $I\_x$, the ideal sheaf of $x$ is torsion free (and thus flat over $R$), but the restriction to the special fiber is not torsion free.
| 5 | https://mathoverflow.net/users/9502 | 234374 | 108,653 |
https://mathoverflow.net/questions/234236 | 5 | Let $Q$ be a ADE type quiver and $s\_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$ is the preprojective algebra of $Q$.
In [Geiss, Leclerc, and Schröer's paper](https://projecteuclid.org/euclid.nmj... | https://mathoverflow.net/users/41979 | what is the injective hull of indecomposable module of preprojective algebra | The preprojective algebra of an ADE graph is non-symmetric Frobenius, so the injective hull of $s\_i$ is isomorphic to the indecomposable projective module $$\mathcal P e\_{\phi(i)},$$ where $\phi$ is the inverse of the Nakayama permutation.
For preprojective algebras of ADE graphs the Nakayama permutation is either ... | 3 | https://mathoverflow.net/users/18756 | 234379 | 108,654 |
https://mathoverflow.net/questions/234342 | 4 | Where can I find a list of all connected real Lie groups inside the 6-dimensional real Lie group $SL(2,C)$, up to conjugacy? How can one verify that a partial list is complete?
I found on wikipedia a [list of Lie subalgebras of so(1,3)](https://en.wikipedia.org/wiki/Lorentz_group#Subgroups_of_the_Lorentz_group), but ... | https://mathoverflow.net/users/56920 | Connected subgroups of $SL(2,C)$ | Let me describe the Lie subalgebras.
Up to conjugation by $\mathrm{PGL}\_2(\mathbf{C})$, the complex subalgebras are $\{0\}$, the diagonal subalgebra, the upper unipotent subalgebra, the upper triangular subalgebra, and the whole $\mathfrak{sl}\_2(\mathbf{C})$. Their real dimension are 0, 2, 2, 4, 6 respectively.
... | 12 | https://mathoverflow.net/users/14094 | 234388 | 108,657 |
https://mathoverflow.net/questions/234384 | 5 | in my research on dilations of contractions on Hilbert spaces and manifolds I have come across this nice publication concerning the classic Sz-Nagy theorem on the Arxiv by Levy and Shalit which states the classical Sz-Nagy theorem on the existence of unitary and isometric dilations for contractions and their minimality... | https://mathoverflow.net/users/69446 | Help in understanding result from publication on operator theory | There are probably many ways to see this, but here is a quick one. Suppose $T$ is a contractive matrix which dilates to a unitary matrix $U$, we wish to show that $T$ is already unitary. Now, to say that the matrix $U$ dilates $T$ means there is an isometric matrix $V$ so that
$$ T^n = V^\* U^n V $$
for all $n\geq 0$. ... | 8 | https://mathoverflow.net/users/13360 | 234391 | 108,658 |
https://mathoverflow.net/questions/53256 | 4 | I think that the definition of fractional Brownian Motion is widely [known](http://en.wikipedia.org/wiki/Fractional_Brownian_motion) (for example as a Gaussian Process with particular variance covariance stucture parametrized by the so-called Hurst index).
Heuristically, you can think of those processes as Gaussian ... | https://mathoverflow.net/users/2642 | What would be a fractional Poisson Process like | A standard Poisson process is a renewal process with exponential distributed waiting times. Fractional Poisson process (FPP) is also a renewal process with Mittag-Leffler waiting times. Note that Mittag-Leffler distribution is a heavy tailed generalization of exponential distribution. Further, let N(t) be a standard Po... | 4 | https://mathoverflow.net/users/82702 | 234402 | 108,662 |
https://mathoverflow.net/questions/234408 | 5 | Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ we set $N(v)=\{w\in V:\{v,w\}\in E\}$.
We say that $G$ is $k$-*common-neighbor-regular* if for all $v\neq w\in V$ we have $|N(v)\cap N(w)|=k$. (I haven't been able to find out whether there is a canonical term for this concept.)
Clearly the complete ... | https://mathoverflow.net/users/8628 | "Common-neighbor-regular" graphs | I claim that for $k>1$ any such graph is regular, in this case we get a well-known problem (subproblem of describing strongly regular graphs), which does not seem to be solved completely. Case $k=1$ is itself known, friendly vertex exists in this case by the friendship theorem of Paul Erdős, Alfréd Rényi, and Vera T. S... | 4 | https://mathoverflow.net/users/4312 | 234421 | 108,667 |
https://mathoverflow.net/questions/234180 | 7 | I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta\_\pi$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on $GL\_2(\mathbf{Q}\_p)$.
An idea, for instance appearing in Kin, Shin and Templier [1], is to use the Sally-Shalika ... | https://mathoverflow.net/users/43737 | Bounding $p$-adic characters and Jacquet-Langlands transfert | First some remarks on the Jacquet-Langlands correspondence. The image of the local Jacquet-Langlands transfer $\mathrm{JL}$ from $G$, the group of units in a non-split quaternion algebra, to $\mathrm{GL}\_2(\mathbb{Q}\_p)$ is the set of discrete series representations.
This set contains the supercuspidal representation... | 9 | https://mathoverflow.net/users/24773 | 234426 | 108,670 |
https://mathoverflow.net/questions/234411 | 3 | As the title says, let $S$ be the nonempty set of strongly regular graphs with given parameters. Must $S$ contain vertex transitive graph?
I suspect the most likely counterexample would be $|S|=1$.
| https://mathoverflow.net/users/12481 | Let $S$ be the nonempty set of strongly regular graphs with given parameters. Must $S$ contain vertex transitive graph? | There are exactly 10 strongly regular graphs with parameters (26,10,3,4), none of which are vertex-transitive. The graphs can be found on [Ted Spence's webpage](http://www.maths.gla.ac.uk/~es/srgraphs.php).
| 9 | https://mathoverflow.net/users/44433 | 234427 | 108,671 |
https://mathoverflow.net/questions/234348 | 6 | Cross-post: This very elementary question [was first posted](https://math.stackexchange.com/questions/1687641/) to Mathematics Stack Exchange but the response I got there (even after offering a bounty) was not useful.
For the purpose of this question, a *pseudoprime* is a composite number $n$ satisfying $2^{n-1} \equ... | https://mathoverflow.net/users/66308 | Can $b^4+1$ be a pseudoprime to base 2 (except for Fermat numbers)? | Carl Pomerance conjectured in
[On the Distribution of Pseudoprimes](http://www.ams.org/journals/mcom/1981-37-156/S0025-5718-1981-0628717-0/S0025-5718-1981-0628717-0.pdf), Math. Comput. 37, 587-593 (1981)
that for large $x$, the number of pseudoprimes $\leq x$ is
$$
\frac{x}{e^{(1+o(1))\log{x}\frac{\log{\log{\log{x... | 10 | https://mathoverflow.net/users/28104 | 234429 | 108,672 |
https://mathoverflow.net/questions/234434 | 1 | Given a value $p\in[0,1]$ (a probability of occurrence), I would like to bound the following expression:
$$ s\frac{1-(1-p)^{k+1}}{p(k+1)} + (1-s)\frac{1-(1-p)^{k}}{pk},\ \ \ \text{where $k=\lfloor 1/p \rfloor$ and $s=1/p-k$}.$$
I would like to prove that the latter is bigger than $1-e^{-1}$.
I already know that i... | https://mathoverflow.net/users/89337 | Averaged geometric series with floor function | For a given natural $k$, let $f(p)$ denote the expression in question. We need to show that $f(p)$ is increasing in $p$. Everywhere here $p\in[\frac1{k+1},\frac1k]$. Let
$$f\_1(p):= f'(p) k (1 + k) (1 - p) p^3,\quad f\_2(p):= f\_1'(p),\quad f\_3(p):= f\_2'(p),$$
$$f\_4(p):= \frac{f\_3'(p)}{k (1 + k) (1 - p)^{k-3}}
=1... | 1 | https://mathoverflow.net/users/36721 | 234452 | 108,674 |
https://mathoverflow.net/questions/234392 | 3 | Posted here too: <https://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition>
Let $q\_0, q\_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are $L^2[0,1]$, i.e. $q\_0, q\_1 \in L^2([0,1],\mathbb{R}^n)$. Denote by $||.||$ the Euclidean distance in $\mathbb{R}^n$... | https://mathoverflow.net/users/35936 | Two minimization problems using singular value decomposition | Problem **1** is essentially finite dimensional and quite geometric in nature.
For a real square matrix $M$ of order $n$ and $1\le p\le\infty$ denote $\big| M\big|\_p$ its [$p$-trace norm](https://en.wikipedia.org/wiki/Schatten_norm).
Consider the $n\times n$ matrix $Q$ with entries $Q\_{ij}:=(q\_0^i, q\_1^j)\_{ 2}... | 2 | https://mathoverflow.net/users/6101 | 234453 | 108,675 |
https://mathoverflow.net/questions/234406 | 3 | For sufficiently large $n$, it is known that most [tournaments](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) of size $n$ contains no nontrivial automorphism, though I forgot the reference.
For sufficiently large $n$, does there always exist a tournament of size $n$ such that
* any subtournament of size... | https://mathoverflow.net/users/89389 | Tournament whose large subtournaments contain no automorphism | In the edited version, how about just taking a transitive tournament and flipping the edge between the vertices of outdegree 0 and $(n-1)$? I think you can replace the $\sqrt{n}$ in the question by 4. The proof divides into cases based on whether the flipped edge is in the subgraph. If not, the subgraph is transitive. ... | 2 | https://mathoverflow.net/users/68848 | 234455 | 108,676 |
https://mathoverflow.net/questions/234333 | 4 | I know that the largest vector topology on countable dimensional vector space is sequential (i.e. every sequentially closed set is closed). Does it keep for the arbitrary vector space?
In countable dimensional case I can describe structure of largest vector topology (it coincides with the largest locally convex topol... | https://mathoverflow.net/users/89338 | Sequentiality of largest vector topology | For a linear space $X$ with a Hamel basis $H$ the largest vector topology seems to coincide with the topology of free linear topological space over the discrete space $H$. If this is true, then we can apply known results on the sequentiality of free linear topological spaces, see e.g. <http://arxiv.org/pdf/1602.04857>
... | 2 | https://mathoverflow.net/users/61536 | 234460 | 108,677 |
https://mathoverflow.net/questions/234471 | 10 | Illusie in "Grothendieck et la cohomologie étale" says Artin's Harvard notes on *Grothendieck Topologies* prove: The étale cohomology with coefficients in $Z/nZ$ of the variety $\mathrm{Spec}(k[X,Y])\backslash\langle0,0\rangle$ for any algebraically closed field $k$ agrees with the cohomology of the 3-sphere.
This s... | https://mathoverflow.net/users/38783 | Etale cohomology of $\mathrm{Spec}(k\{X,Y\})\backslash\langle0,0\rangle$ | I'm not sure what "easy" means in the context of etale cohomology but there is a way of passing from Artin's result to the stated one.
Let $j$ from $\mathbb A^2 \backslash \langle 0,0\rangle$ to to $\mathbb A^2$ be the open immersion. Then there is a Leray spectral sequence relating the etale cohomology of $\mathbb A... | 14 | https://mathoverflow.net/users/18060 | 234474 | 108,681 |
https://mathoverflow.net/questions/234476 | 2 | Is it possible to prove the **non-completeness** of the Borel-Lebesgue measure on $\mathbb{R}$ (restricted to the Borel $\sigma$-algebra) **without** the full axiom of choice, but still **with Countable Choice** ?
It seems to be the case when I read [Non-Borel sets without axiom of choice](https://mathoverflow.net/que... | https://mathoverflow.net/users/89425 | Non-completeness of the Borel-Lebesgue measure and countable choice | Countable choice is sufficient to prove that there is a non-Borel set, since under countable choice, the collection of sets of reals with a Borel code (which is a well-founded countable tree labeled with the instructions for building a Borel set, so that leaves are labeled with basic open sets and other nodes are label... | 5 | https://mathoverflow.net/users/1946 | 234479 | 108,684 |
https://mathoverflow.net/questions/234494 | 7 | I am trying to derive some basic relations for the height and width of the direct product and the coproduct of posets. I feel that these are very basic and should be written somewhere, however, I cannot find a reference.
Short question is: is there a short expression for the following quantities, representing height ... | https://mathoverflow.net/users/37129 | Is there a short expression for height and width of product and coproduct of posets? | Sticking first to finite sets, for the question of $h(P\times Q)$, one does in general have $h(P \times Q)=h(P)+h(Q)-1$. You've already proven the lower bound. For the upper bound, take a chain $(a\_1,b\_1),\ldots,(a\_n,b\_n)$ in $P\times Q$; let's assume this is written in increasing order. (Note this is strictly incr... | 6 | https://mathoverflow.net/users/5583 | 234496 | 108,689 |
https://mathoverflow.net/questions/234025 | 16 | I asked the question
[Why is multiplication on the space of smooth functions with compact support continuous?](https://math.stackexchange.com/questions/1680725/why-is-multiplication-on-the-space-of-smooth-functions-with-compact-support-cont) on M.SE
sometime ago but I didn't receive a satisfactory answer.
I was re... | https://mathoverflow.net/users/43441 | Why is multiplication on the space of smooth functions with compact support continuous? | You can spare yourself the functional analytic abstract nonsense by using an explicit set of seminorms on $\mathcal{D}(\mathbb{R}^d)=C\_{c}^{\infty}(\mathbb{R}^d)$
which, unfortunately, are not well-known but can be found in the excellent book ["Topological Vector Spaces and Distributions"](https://books.google.com/boo... | 16 | https://mathoverflow.net/users/7410 | 234503 | 108,691 |
https://mathoverflow.net/questions/234501 | 0 | I see the remark that:
"Let $R$ be a Noetherian commutative ring, $M$ an $R$-module and $I$ an ideal of $R.$ Assume that $0 :\_M I$ is finitely generated. Then $0 :\_M I^n$ is finitely generated for all $i\ge 1.$" Could someone help me some ideas to prove this? Thank you very much.
| https://mathoverflow.net/users/111680 | $0 :_M I^n$ is finitely generated for all $i\ge 1$? | Let $I=(x\_1,..,x\_k)$. Then for $n \geq 1, (0 :\_M I^{n+1})/(0 :\_M I) \hookrightarrow \oplus x\_i(0 :\_M I^{n+1})\hookrightarrow \oplus(0 :\_M I^{n})$, so by induction on $n, (0 :\_M I^{n+1})/(0 :\_M I)$ and thus also $(0 :\_M I^{n+1})$ is finitely generated.
| 0 | https://mathoverflow.net/users/59248 | 234509 | 108,694 |
https://mathoverflow.net/questions/234518 | 2 | Any help with this problem would be appreciated. Thanks
Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does there exist a **global** smooth function $f:M \to \mathbb{R}$ such that $df(X)|\_{\gamma} \neq 0$ along $\gamma$ and $|df... | https://mathoverflow.net/users/50438 | Finding a specific Global Smooth Function | **Edit.** Surely $f$ has at least two critical points if $M$ is closed, so we cannot have $|df|\_g\ne 0$ everywhere in that case. This arguments fails if we assume that each component of $M$ has a nonempty boundary. To the contrary, now one can assume that $f$ is generic (after a $C^1$-small perturbation, which would n... | 2 | https://mathoverflow.net/users/70808 | 234521 | 108,698 |
https://mathoverflow.net/questions/234351 | 3 | We have a continuous and differentiable function $f(\cdot)$ that maps from $R^n$ to $R^n$. We are trying to solve a nonlinear least square problem:
Minimize $J(x)=\Vert f(x)-z\Vert^2$
subject to box constraints: $l\_i \leq x\_i \leq u\_i$.
This function $f(\cdot)$ has a beautiful property that
$\forall x\_1, ... | https://mathoverflow.net/users/89358 | Global minimum of nonlinear least square | Any function $f$ that satisfies $(x-y)^T(f(x)-f(y)) \ge 0$ is a *monotone* function, and has the interpretation of a subgradient for a certain convex function. If $f$ is continuous and monotone, then the solution set $S=\{x:f(x)=0\}$ is convex.
The global solution of monotone equations is a well-studied problem that... | 1 | https://mathoverflow.net/users/60984 | 234522 | 108,699 |
https://mathoverflow.net/questions/234489 | 18 | This is a question on nomenclature of $K$-theory in the topological category.
The $K$-theory of a compact space $X$ is defined as the Grothendieck group of the vectorbundles on $X$. The Atiyah-Jänich Theorem states that this is the same thing as the homotopy classes of maps $X\rightarrow \Phi(\mathbb{H})$, where $\Ph... | https://mathoverflow.net/users/12156 | K-theory of non-compact spaces | 2 and 3 are not equivalent, because of a phenomenon known as "phantom maps".You can have a map of a CW complex $X$ to $Y$ which is non-trivial in homotopy, but homotopy trivial when restricted to every finite subcomplex of $X$.
That this actually occurs for K-cohomology is shown in an old paper of Anderson and Hodgki... | 8 | https://mathoverflow.net/users/4648 | 234531 | 108,701 |
https://mathoverflow.net/questions/234435 | 1 | This is probably easy, but I can't think of an answer. Assume $X$ is a Banach space and $A$ is a (not assumed closed) subspace of $X$. Let $T:X \to X$ be a bounded linear operator, which is also injective. If $T(A)$ is dense in $X$, does it follow that $A$ is dense in $X$?
| https://mathoverflow.net/users/20484 | Density of sets whose image is dense | Here's an explicit example for what Bill's answer proposes.
Let $T\_1$ be the operator which maps $e\_n$ to $4^{-n} e\_{n-1}$ for $n \ge 2$ and maps $e\_1$ to $0$. That is, $T\_1 \left(\sum\_{n \ge 1} a\_n e\_n\right) = \sum\_{n \ge 1} a\_{n+1} 4^{-(n+1)} e\_n$. Note $T\_1$ is injective on $A$, and $T\_1 A$ is dense ... | 2 | https://mathoverflow.net/users/4832 | 234532 | 108,702 |
https://mathoverflow.net/questions/232137 | 5 | I'm interested in the "size" of the roots of a sequence of Taylor Polynomials of an entire function.
**For example**, consider $\mathrm f(z) = \mathrm e^z$. The Taylor Polynomials, or $k$-jets, are
$$\mathrm P\_k(z) = 1 + z + \frac{1}{2!}z^2 + \cdots + \frac{1}{k!}z^k$$
I want to find the best possible lower boun... | https://mathoverflow.net/users/44642 | Lower Bounds for the Roots of Polynomials | Your question is related to the Szego curve, which has been much studied (just google it). A lower bound for the roots is given by $W(1/e)n=0.278..n$ where $W$ is the Lambert function.
Theorem 4 of <http://www.math.kent.edu/~varga/pub/paper_184.pdf> should give you an expansion of a lower bound, up to order $O(1/n)$... | 2 | https://mathoverflow.net/users/89429 | 234540 | 108,705 |
https://mathoverflow.net/questions/233608 | 9 | Given $m\geq 1$, let $I=(a\_1,\ldots,a\_{3m})$ be a sequence such that $I$ contains exactly $m$ zeros, $m$ ones, and $m$ twos.
Given $i=1,2$ and $j\leq 3m,k\leq m$ we can define $$U\_{i,j}(k)=\text{number of $i$'s before finding $k$ zeros, starting from position $j$}.$$
(moving to the right, in a cyclic way)
For ... | https://mathoverflow.net/users/57519 | combinatorics on cyclic sequences |
>
> ***Answer:*** *No this is not true.*
>
>
>
For $m=5$, a counter example is: $(0, 2, 0, 2, 0, 2, 1, 2, 1, 1, 1, 0, 1, 2, 0)$. In this case we have:
$$\begin{align}
U(1) &= \left(\begin{array}{rrrrrrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 4 & 4 & 3 & 3 & 2 & 1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 & 2 & 1 & 1 & 0 & 0 & ... | 6 | https://mathoverflow.net/users/39495 | 234547 | 108,707 |
https://mathoverflow.net/questions/234523 | 16 | I had asked this [question](https://math.stackexchange.com/questions/1712414/classification-of-o2-bundles-in-terms-of-characteristic-classes) in stackexchange but there seems to be no consensus in the answer
It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by their f... | https://mathoverflow.net/users/66688 | Classification of $O(2)$-bundles in terms of characteristic classes | The $O(2)$ bundles $\xi$ over a manifold $M$ are classified by their first Stiefel-Whitney class $w\_1(\xi)\in H^1(M;\mathbb{Z}/2)$ and their twisted Euler class $e(\xi)\in H^2(M;\mathbb{Z}\_{w\_1(\xi)})$.
This is because the space $BO(2)$ is a generalized Eilenberg--Mac Lane space $L\_{w\_1}(\mathbb{Z},2)$ in the se... | 16 | https://mathoverflow.net/users/8103 | 234548 | 108,708 |
https://mathoverflow.net/questions/233452 | 4 | Consider large tridiagonal matrix (where $a$ and $b$ are real numbers):
$$M =
\begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
0 & b & (a+2)^2 & b & \cdots \\
\vdots & \vdots & \vdots & \vdots
\end{pmatrix}$$
What can be said about eigenvalues? Are analytic expressions known?
O... | https://mathoverflow.net/users/78657 | Eigenvalues of large tridiagonal matrix | Since $M\_n(a,b)$ and $M\_n(a,-b)$ have same real spectrum, we may assume that $b\geq 0$. Let $\lambda\_n$ be the smallest eigenvalue of $M\_n$. Since there exist hidden othgonal polynomials, the real sequence $(\lambda\_n)\_n$ is non-increasing.
Assume that $a\geq 0$. Note that $e\_1^TM\_ne\_1=a^2$; then $\lambda\_... | 5 | https://mathoverflow.net/users/9091 | 234549 | 108,709 |
https://mathoverflow.net/questions/234552 | 2 | Let $R$ be a discrete valuation ring with residue field $k$, an algebraically closed field of characteristic zero and $\pi:X\to \mbox{spec}(R)$ a smooth, projective family of surfaces. Denote by $X\_0$ the special fiber in the family. Suppose there exists a curve $C$ in $X\_0$ which does not lift as a subcurve of $X$, ... | https://mathoverflow.net/users/43198 | Families of smooth projective varieties over dvr | That can happen. Here is one source of examples. Consider $\mathbb{P}^3\_k = \text{Proj}\ k[s,t,u,v]$. Let $d\geq 4$ be an integer, and consider hypersurfaces $X\subset \mathbb{P}^3$ of degree $d$, i.e., $X$ is the zero locus of a degree $d$ homogeneous polynomial, $$F(s,t,u,v) = \sum\_{\alpha+\beta+\gamma+\delta=d} c\... | 6 | https://mathoverflow.net/users/13265 | 234554 | 108,710 |
https://mathoverflow.net/questions/234451 | 3 | Suppose $x\_1\ge x\_2\ge \cdots \ge x\_n\ge 0$ and $y\_1\ge y\_2\ge\cdots\ge y\_n\ge0$ be reals such that for any positive **integer** $p$,
$$
\sum\_{i=1}^n x\_i^p \geq \sum\_{i=1}^n y\_i^p.
$$
>
> **Question:** Is there a constant $c\ge 1$ such that for all $x,y$ satisfying the above,
> we have $cx\succ\_w y$.
> ... | https://mathoverflow.net/users/58456 | Sufficient conditions for weak majorization | I think there is no such $c$. In fact, for $k$ and $\epsilon \gt 0$ given, there are $n \gg k$ and $x\_1\ge x\_2\ge \cdots \ge x\_n\ge 0$ and $y\_1\ge y\_2\ge\cdots\ge y\_n\ge0$ with
$$\sum\_{i=1}^n x\_i^p \geq \sum\_{i=1}^n y\_i^p.$$ for all $p\ge 1$ but
$$k\cdot \sum\_{i=1}^k (x\_i - \epsilon) \le\sum\_{i=1}^k y\_i$$... | 5 | https://mathoverflow.net/users/68822 | 234560 | 108,712 |
https://mathoverflow.net/questions/234558 | 8 | This is a refinement of a [question asked on MSE](https://math.stackexchange.com/questions/1703749/can-the-quotient-by-a-nonabelian-group-yield-an-abelian-singularity).
Let $G$ be a finite group and let $V$ be a finite-dimensional faithful complex representation of $G$. Consider $V$ as an affine complex variety. In g... | https://mathoverflow.net/users/12419 | How fine an invariant of a representation is its quotient singularity? | To cherry-pick the answer from Jason Starr's comment above: the condition you want is that $G$ acts on $V$ without any element giving a pseudo-reflection. Chevalley-Shephard-Todd tells you that you must lose all information about pseudo-reflections, but you don't lose anything else. When there are no pseudo-reflections... | 6 | https://mathoverflow.net/users/66 | 234561 | 108,713 |
https://mathoverflow.net/questions/234559 | 3 | A subvariety $V$ of an abelian variety $X$ is *geometrically nondegenerate* if it meets any subvariety of $X$ of dimension bigger than or equal $codim(V)$.
My question is about the Prym varieties as subvarieties of the Jacobians of curves. Are they geometrically nondegenerate?
| https://mathoverflow.net/users/66528 | Are the Prym varieties geometrcally nondegenerate subvarieties of the Jacobians? | Pryms are geometrically degenerate. Let $f:Y\to X$ be an étale double cover of a genus $g>1$ curve $X$. Up to taking isogenous quotients, which will not affect "geometric degeneracy", the Prym of $f$ is the kernel of $\text{Alb}\_f : \text{Alb}\_Y\to \text{Alb}\_X$, where the domain is an Abelian variety of dimension $... | 4 | https://mathoverflow.net/users/13265 | 234563 | 108,714 |
https://mathoverflow.net/questions/234569 | 4 | I am presently doing research concerned with operator algebras and operator theory and I thought to write here in the hopes of seeking expert advice on an idea I had here. The classic Sz.-Nagy dilation theorem says
>
> Given a contraction T on a Hilbert space, we are guaranteed the existence of a lager Hilbert spac... | https://mathoverflow.net/users/89375 | On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces | This is a much-studied problem. If you do not require the $V$'s to commute, then a dilation (even a unitary dilation) always exists, this is a theorem of Bozejko. For commuting operators $T$ (and seeking a commuting dilation) the problem is more subtle. A theorem of Ando says that two commuting contractions $T\_1, T\_2... | 6 | https://mathoverflow.net/users/13360 | 234571 | 108,717 |
https://mathoverflow.net/questions/233615 | 1 | Let $\langle\mathbf{R}^n,\mathscr{O}\rangle$ be the $n$-dimensional Euclidean space. Define $\mathbf{Q}\subseteq\mathcal{P}(\mathscr{O})$ to consist of all sets $\mathsf{Q}$ which simultanously satisfy the following four conditions:
1. $\mathsf{Q}\neq\emptyset$ and $\mathsf{Q}\neq\{\emptyset\}$
2. $\forall\_{U\in\mat... | https://mathoverflow.net/users/22019 | Intersections of families of open sets ordered by well-inside relation in Euclidean space | Yes. (Assuming $\varnothing$ isn't allowed in your $Q$'s, as Joel pointed out.)
First note that for such $Q$, $\bigcap Q$ cannot contain two points. Indeed, if $a,b \in \bigcap Q$ are distinct, we can pick open neghborhoods $a \in A$, $b \in B$ such that $\operatorname{Cl} A \cap \operatorname{Cl} B = \varnothing$ an... | 2 | https://mathoverflow.net/users/2000 | 234572 | 108,718 |
https://mathoverflow.net/questions/234473 | 10 | **Background:** Let $M$ be an $n\times n$ matrix with nonnegative entries. It is immediate that for any integer $k$, $M^k$ has nonnegative entries.
Suppose now that, on top of having nonnegative entries, $M$ is a positive semi-definite matrix (i.e., it is symmetric and has nonnegative eigenvalues). Now one may ask w... | https://mathoverflow.net/users/58456 | Positivity of power of positive PSD matrices | There is an $f$ such that $f\ge 0$, $Hf\ge 0$ however $Hf^{p}$ contains negative values for some $p\ge 1$. Take
$$
f = \begin{bmatrix} 727 & 200 & 163 & 234 & 429 & 448 & 437 & 6 \end{bmatrix}^T
$$
The vector $Hf$ is positive, however $(Hf^{8/7})\_{100}\approx -3.35$. Here the subscript $100\in \mathbb F\_2^3$ is the 5... | 3 | https://mathoverflow.net/users/58456 | 234583 | 108,721 |
https://mathoverflow.net/questions/233745 | 1 | We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation:
$z\_m = \sum\_{i,j :\, (i \cdot j) = m} x\_i \cdot y\_j$.
That is, we are given two finite input vectors $x$ and $y$ and are trying to calculate
the full vector $z$, where each element $z\_m$ in $z$ consists of the sum of t... | https://mathoverflow.net/users/74852 | Subquadratic multiplication of probability mass functions (with log-convolution?) | There is little room for improvement over brute force because the average number of collisions is a very slowly growing function.
The [Erdős multiplication table problem](http://oeis.org/A027424) asks how many numbers are products $ab$ with $1\le a,b \le n$. It is known that the count is $$\Theta\left(\frac{n^2}{(\l... | 2 | https://mathoverflow.net/users/2954 | 234592 | 108,725 |
https://mathoverflow.net/questions/234564 | 7 | Let the real polynomial
$$f\_{a,b,c}(x\_1,x\_2,x\_3)=(x\_1-x\_2)^{2a+1}(x\_2-x\_3)^{2b+1}(x\_3-x\_1)^{2c+1},$$
where $a,b,c$ are nonnegative integers.
Let $m\_{a,b,c}$ be the coefficient of the monomial $x\_1^{a+c+1}x\_2^{a+b+1}x\_3^{b+c+1}$ in the expansion of $f\_{a,b,c}(x\_1,x\_2,x\_3)$.
It is easy to see $m\_{... | https://mathoverflow.net/users/58096 | The coefficient of a specific monomial of the following polynomial | This coefficient $L$ is a constant term of the Laurent polynomial $g(x\_1,x\_2,x\_3)=f(x\_1,x\_2,x\_3)/x\_1^{a+c+1}x\_2^{a+b+1}x\_3^{b+c+1}$, this guy $g$ satisfies $g(x\_1,x\_2,x\_3)=-g(1/x\_1,1/x\_2,1/x\_3)$, thus $L=-L$.
| 17 | https://mathoverflow.net/users/4312 | 234602 | 108,726 |
https://mathoverflow.net/questions/234588 | 0 | I have a question about the effect of applying a linear transformation $M$ in $\mathbb{R}^{n \times n}$ to a vector $v \in \mathbb{R^n}$.
I know that if $M$ has p-norm $\|M\|\_p = \lambda$, then by definition I can guarantee that for every vector $v \in \mathbb{R^n}$
$$\|Mv\|\_p \leq \lambda \|v\|\_p.$$
Is there a... | https://mathoverflow.net/users/51134 | The effect of linear transformation on generic vectors | Note that your constraint (2) is not relevant, because an inequality
$\|Mv\|\_p\ge C\|v\|\_p$ is preserved by scalar multiplication. So we
might as well assume $\|v\|\_p=1$. This is a compact set, so any
continuous real-valued function on it will have a maximum value. If
$M$ is nonsingular, then $\frac{\|v\|\_p}{\|Mv\|... | 1 | https://mathoverflow.net/users/89506 | 234610 | 108,729 |
https://mathoverflow.net/questions/234538 | 6 | Let $A$ and $B$ be arbitrary nonempty subsets of a group $G$. Then
the product $AB$ is called direct, and we denot it by $A \cdot B$,
if the representation of every its element by $x=ab$ with $a\in A$, $b\in B$ is unique.
It is obvious that if $AB=A \cdot B$ then $|A B|=|A||B|$, and the converse is true if both are f... | https://mathoverflow.net/users/40520 | Cardinality of factors of infinite non-abelian groups | Let $G=C\_2\ast C\_3$ be the free product of cyclic groups $C\_2=\langle a\rangle$ and $C\_3=\langle b\rangle$.
Let $A$ be the subset of $G$ consisting of
$$b,bab,babab,\dots$$
together with all reduced words ending with $a$ except for
$$ba,baba,bababa,\dots.$$
Then $A$ is a set of left coset representatives for bo... | 6 | https://mathoverflow.net/users/22989 | 234614 | 108,732 |
https://mathoverflow.net/questions/234570 | 7 | For a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[\sqrt{-6}]$ which does not divide $2$, does $\mathfrak{p}$ decompose completely in the extension $\mathbb{Q}(\zeta\_{24})/\mathbb{Q}(\sqrt{-6})$ of degree $3$ if and only if $\mathfrak{p} = (\alpha)$ for some $\alpha \in \mathbb{Z}[\sqrt{-6}]$ such that $a \equiv ... | https://mathoverflow.net/users/nan | A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ iff it is generated by $\alpha\in1+2\Bbb{Z}[\sqrt{-6}]$ | OK, so here are the details. The decomposition law in cyclotomic extensions tells you that a prime $p$ coprime to $m$ splits completely in ${\mathbb Q}(\zeta\_{m})$ if and only if $p \equiv 1 \bmod m$. Since ${\mathbb Q}(\zeta\_{24}) = {\mathbb Q}(\sqrt{-1},\sqrt{2},\sqrt{-3})$ this is equivalent to the condition $(-1/... | 3 | https://mathoverflow.net/users/3503 | 234622 | 108,736 |
https://mathoverflow.net/questions/234621 | 5 | Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $TM \otimes\mathbb{C}$ as what there exists and how can we do the differentiation from the sections of this bundle by usi... | https://mathoverflow.net/users/86401 | On the complexification of a Riemannian manifold | Each section of $TM \otimes \mathbb{C}$ has a unique decomposition $Z=X+iY$ as a sum with $X$ and $Y$ sections of $TM$. Define your metric using this, for example as $\left<Z\_1,Z\_2\right>=\left<X\_1,X\_2\right>+\left<Y\_1,Y\_2\right>$. Use an affine connection as $\nabla\_{X+iY} U+iV=\nabla\_X U - \nabla\_Y V + i \le... | 7 | https://mathoverflow.net/users/13268 | 234625 | 108,738 |
https://mathoverflow.net/questions/234623 | 2 | Is there a set of triple-primes satisfying the following equation?
$p\_1p\_2+p\_2p\_3+p\_3p\_1+p\_1+p\_2+p\_3=2^β,p\_1p\_2p\_3=2^α-1,α>β.$
I have checked the first 11 numbers that no one satisfy the above condition. It's easy to check that when all the prime numbers $p\_i$ are in the form of $4k+3,$ then we have $p\_... | https://mathoverflow.net/users/89519 | Set of triple-primes satisfying a certain equation | Yes, there is precisely one such triple of distinct primes
with $\alpha \leq 500$, namely $(3, 11, 31)$.
It would be a surprising coincidence if there are further
such triples for larger $\alpha$.
| 1 | https://mathoverflow.net/users/28104 | 234636 | 108,741 |
https://mathoverflow.net/questions/234627 | 5 | Consider an algebraically closed field $k$, a finite field extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, and the integral closure $A'$ of $k[T^{-1}]$ in $K$. Does it follow that $A \cap A' = k$?
| https://mathoverflow.net/users/83496 | Intersection between integral closures is algebraically closed field | Yes. Each element of $K$ satisfies a unique irreducible monic polynomial over $k(T)$. It is integral over $k[T]$ if and only if the coefficients lie in $k[T]$ and integral over $k[T^{-1}]$ if and only if the coefficients lie in $k[T^{-1}]$. If it is integral over both, the coefficients lie in $k[T] \cap k[T^{-1}]=k$. B... | 9 | https://mathoverflow.net/users/18060 | 234638 | 108,742 |
https://mathoverflow.net/questions/234648 | 3 | I asked this at [math.stackexchange](https://math.stackexchange.com/questions/1715947/a-linear-functional-on-ck-continuous-on-each-l-1-mu?noredirect=1#comment3502382_1715947), but nobody answered.
Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K... | https://mathoverflow.net/users/18943 | A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$ | Yes.
Suppose $p$ were not continuous. Then we could find a sequence of signed Radon measures $\mu\_n$ with norms $\|\mu\_n\| \le 2^{-n}$ but $|p(\mu\_n)| \ge 1$. Let $|\mu\_n|$ denote the total variation measure of $\mu\_n$, which is still Radon and has the same norm as $\mu\_n$. Set $\mu = \sum\_n |\mu\_n|$; this s... | 8 | https://mathoverflow.net/users/4832 | 234652 | 108,746 |
https://mathoverflow.net/questions/234618 | 10 | Assume that $M$ is a manifold.
Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x\_{1},x\_{2},\ldots x\_{i},\ldots,x\_{n}) \mapsto (x\_{1},x\_{2},\ldots , -x\_{i},\ldots,x\_{n})$, for all $i\in \{1,2,\ldots ,n\}$?
| https://mathoverflow.net/users/36688 | A symmetric embedding of manifolds | No. There exist closed manifolds which do not admit any compact group actions, in particular, no $Z\_2$ actions. For example, Shultz showed in ["Group actions on hypertoral manifolds. II."](https://eudml.org/doc/152358) that in dimensions $\ge 4$ every oriented cobordism class contains such a manifold.
| 20 | https://mathoverflow.net/users/18050 | 234660 | 108,749 |
https://mathoverflow.net/questions/234242 | 3 | In [this question](https://mathoverflow.net/questions/39386/number-of-unique-sortings-of-subset-sums) the number of unique sortings has been discussed.
As a follow-up, I would like to know, whether the problem of sorting the sequence of subset sums has ever been studied.
There should different algorithms for the ... | https://mathoverflow.net/users/31310 | Algorithms for Sorting Subset Sums | One standard situation where sorted subset sums comes up is in the [subset sum problem](https://en.wikipedia.org/wiki/Subset_sum_problem). (I.e. is zero among the subset sums of a given set?) By splitting the input into two equinumerous subsets, sorting the subset sums of each subset, and comparing the two lists of sor... | 1 | https://mathoverflow.net/users/440 | 234668 | 108,751 |
https://mathoverflow.net/questions/234647 | 8 | Let $(A\_{n},\*)$ denote the $n$-th classical Laver table. Let
$X\_{n}$ be the set of all finite sequences of elements from $A\_{n}$.
Define a function $E\_{n}:X\_{n}\rightarrow X\_{n}$ by letting
1. $E\_{n}((x))=(x)$.
2. $E\_{n}((2^{n},x\_{1},...,x\_{k}))=(x\_{1},...,x\_{k})$
3. $E\_{n}((x,1,x\_{1},...,x\_{k}))=(x... | https://mathoverflow.net/users/22277 | How long does the slow inefficient algorithm for computing the product in classical Laver tables take? | Let $m=2^n$ for simplicity. Then the worst input pair seems to be $(1,m-1)$, in which case the number of steps is roughly on the order of $m!/2^m$. Meaning, although the algorithm is quite inefficient, still the number of steps is boringly small compared to non-primitive recursive functions.
Analysis:
If we start w... | 3 | https://mathoverflow.net/users/27742 | 234676 | 108,752 |
https://mathoverflow.net/questions/232184 | 12 | Let $G$ be a finitely generated amenable group.
Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability?
Being more formal, note that $G^n$ is amenable, so it carries a translation invariant finitely additive probability measure $\mu$. I am asking whether there is ... | https://mathoverflow.net/users/38889 | Are finitely generated amenable groups positively finitely generated? | The answer is already no for $\mathbb{Z}$, assuming the question is whether this holds for every meaure. Let $n\in\mathbb{N}$, and let
$$
S \,=\, \{(a\_1,\ldots,a\_n)\in\mathbb{Z}^n \mid \gcd(a\_1,\ldots,a\_n)=1\}.
$$
I will demonstrate a Følner sequence in $\mathbb{Z}^n$ that does not intersect $S$, which leads to a t... | 10 | https://mathoverflow.net/users/6514 | 234683 | 108,755 |
https://mathoverflow.net/questions/234686 | 0 | Suppose that $L/k$ is a Galois extension of number fields and that $G$
is the corresponding Galois group. Further, for $\frak p$ a prime ideal
of $\cal O$$\_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$] = $p$ is prime. Suppose $m$ is the highest integer such that $\mu\_{p^m}\subseteq L$, and $\mu\_{p^n}\subseteq K$ wi... | https://mathoverflow.net/users/57771 | How to calculate $N_{L/k}$(roots of unity)? | The answer should be: if $n = m$, then the image is the roots of unity of order $p^{n-1}$. If $n = m - 1$ then the norm map should be surjective on the roots of unity.
In the first case, since all roots of unity $\zeta$ are already contained in $K$ and $[L:K] = p$, then $N\_{L/K}(\zeta) = \zeta^p$, and $\zeta \mapsto... | 3 | https://mathoverflow.net/users/30726 | 234689 | 108,756 |
https://mathoverflow.net/questions/234688 | 9 | Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U\_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special maximal subgroups of such a group are.
More specifically, let $\mathbf{K}$ be the subgroup consisting of unitary matri... | https://mathoverflow.net/users/30726 | What are the special parahoric subgroups in unitary groups? | There's an explicit description of maximal compact subgroups of all unitary groups over local fields (not necessarily quasi-split) in section 3 of this paper:
Gan, Hanke, and Yu, ["On an exact mass formula of Shimura"](http://www.math.nus.edu.sg/~matgwt/Mass.pdf) , Duke Math Journal 107(1) (2001).
This doesn't quit... | 8 | https://mathoverflow.net/users/2481 | 234692 | 108,758 |
https://mathoverflow.net/questions/233595 | 2 | I'm trying to prove global wellposedness for the Klein-Gordon-Equation with **radial** initial data. I'm therefore searching for/trying to prove strichartz estimates of the form: $$ ||e^{it\langle D\rangle}u||\_{L^q\_{t}L^r\_{x}}\lesssim||\langle D\rangle^lu||\_{L^{2}}$$ or $$||P\_{L}((e^{it\langle D\rangle}u\_{M})(e^{... | https://mathoverflow.net/users/88808 | Strichartz Estimates for radial Klein-Gordon equation | You can find a complete set of Strichartz estimates for the nonradial KG equation in Lemma 3 in [this paper](https://projecteuclid.org/euclid.rmi/1049123084) by Machihara, Nakanishi and Ozawa. Let me add that, usually, in proofs of local or global well posedness for the wave or KG equation, the non-endpoint Strichartz ... | 3 | https://mathoverflow.net/users/7294 | 234695 | 108,759 |
https://mathoverflow.net/questions/234649 | 0 | Suppose $\Omega\_1, \Omega\_2 \subset R^2$ are bounded open regions with $\Omega\_1 \Subset \Omega\_2$. Let $f\_1\in C(\partial \Omega\_1)$ and $f\_2\in C(\partial \Omega\_2)$. Is there a function $h\in H^1(\Omega\_2)$ such that
(a) $h=f\_1$ on $\partial \Omega\_1$ and $h=f\_2$ on $\partial \Omega\_2$,
(b) $-\Delt... | https://mathoverflow.net/users/42326 | Harmonic/Subharmonic lifting of functions on an annulus | By the maximum principle for subharmonic function, a necessary condition is that the max of $f\_1$ on $\partial\Omega\_1$ is less than the max of $f\_2$ on $\partial\Omega\_2$. But even in this case, there is no reason, in general, that the function $h$ exists. Indeed, by (c), it should be equal on the annulus $\Omega\... | 0 | https://mathoverflow.net/users/89429 | 234696 | 108,760 |
https://mathoverflow.net/questions/234666 | 6 | For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $u$. Moreover, by the Gauss-Bonnet theorem the sign of the curvature of $\overline{g}$ is uniquely determined by the top... | https://mathoverflow.net/users/70615 | Criterion for deciding the conformal class of a metric on a complete surface | This is a classical problem which is called the Type Problem of a simply connected Riemann surface: If you have a metric on an open simply connected surface, to determine whether it is conformally equivalent to the plane or to the disk.
The general situation is the following: there are necessary and sufficient condit... | 4 | https://mathoverflow.net/users/25510 | 234697 | 108,761 |
https://mathoverflow.net/questions/234579 | 4 | Hardy-Littlewood-Sobolev inequality states that if $1<p<q<\infty$, $1/r=1-1/p+1/q$, then we have
$$\left\|\frac{1}{|x|^{n/r}}\ast f\right\|\_{L^q(\mathbb R^n)}\le\|f\|\_{L^p(\mathbb R^n).}$$
Note that here $q=\infty$ is not allowed. My question is, is it possible to get some bounds for $q=\infty$, if we weaken the RH... | https://mathoverflow.net/users/85168 | Hardy-Littlewood-Sobolev inequality in Lorentz spaces | The Young inequality in Lorentz spaces covers these cases: if $p\_1,p\_2,p\in]1,\infty[$, $q\_1,q\_2,q\in[1,\infty]$,
\begin{equation}
\|f\ast g\|\_{L^{p,q}}\leq C\|f\|\_{L^{p\_1,q\_1}}
\|g\|\_{L^{p\_2,q\_2}},\qquad p\_1^{-1}+p\_2^{-1}
=1+p^{-1},\ q\_1^{-1}+q\_2^{-1}\geq q^{-1};
\end{equation}
and if $p\_{1},p\_2\... | 3 | https://mathoverflow.net/users/7294 | 234702 | 108,764 |
https://mathoverflow.net/questions/234678 | 7 | I'm reading the article
*M. Hochster*, [**Prime ideal structure in commutative rings**](http://www.ams.org/mathscinet-getitem?mr=251026), *Trans. Amer. Math. Soc.* **142** (1969), 43--60. Freely available [here on the journal's website](http://www.ams.org/journals/tran/1969-142-00/S0002-9947-1969-0251026-X/).
But,... | https://mathoverflow.net/users/nan | What does the notation $[b_1,b_2]$ in M. Hochster's "Prime Ideal Structure in Commutative Rings" mean? | My best guess for what $[b\_1,b\_2]$ is supposed to mean is the subring of $A$ generated by $b\_1$ and $b\_2$. However, even with this interpretation, there are some statements that aren't quite right (though as I recall, this particular paper has a lot of minor errors of this sort, so this shouldn't be too surprising)... | 4 | https://mathoverflow.net/users/75 | 234715 | 108,768 |
https://mathoverflow.net/questions/234720 | 1 | I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using,
H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term $\alpha$ is defined as, $0<\alpha < \frac{2}{trace(A^{T} A)}$.
Can I claim that each one of therm in my matrix H will b... | https://mathoverflow.net/users/89565 | Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1? | Yes. Your matrix $H$ is symmetric, so the spectral radius is the same as the maximal eigenvalue is the same as the maximal singular value. From the singular value description, for any unit vectors $\vec{u}$ and $\vec{v}$, we deduce that $|\vec{u}^T H \vec{v}| < 1$. Taking $\vec{u}$ and $\vec{v}$ to be the $i$-th and $j... | 3 | https://mathoverflow.net/users/297 | 234721 | 108,770 |
https://mathoverflow.net/questions/234731 | -6 | Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v\in V$ we have $$\text{deg}\_G(v) = \deg\_H(\varphi(v)).$$
Does this imply that $\chi(G) = \chi(H)$?
| https://mathoverflow.net/users/8628 | Do degrees determine the chromatic number? | Let $G$ be a 6-cycle and let $H$ be two 3-cycles.
Or, if you want them connected, let $G$ be a 6-cycle with an extra edge between vertices 1 and 3, and $H$ a 6-cycle with an extra edge between vertices 1 and 4.
| 5 | https://mathoverflow.net/users/4832 | 234732 | 108,772 |
https://mathoverflow.net/questions/234726 | 2 | I am interested in intransitive irreducible linear subgroups $G\subseteq\mathrm{GL}\_n(\mathbb{F}\_p)$ acting on $V-\{0\}=\mathbb{F}\_p^n-\{0\}$ in the natural way, such that all of the orbits are very large, with constant density in $V$.
Actually I am wondering if there exists an infinite family of groups satisfyin... | https://mathoverflow.net/users/4162 | Intransitive finite irreducible linear groups whose orbits are all large | Take an odd prime $p$, let $q=p^n$ and consider $V=\mathbb{F}\_q$ as a vector space over $\mathbb{F}\_p$. For $G\_n=(\mathbb{F}\_q^\*)^2$ you get $c=1/2$.
| 2 | https://mathoverflow.net/users/89334 | 234737 | 108,773 |
https://mathoverflow.net/questions/234404 | 6 | Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach spaces", a function $f:U\to \mathbb C ^M $ is called $holomorphic$ if for every $p \in U$ there is a bounded linear map ... | https://mathoverflow.net/users/83937 | Holomorphy of a function with values in a Hilbert space | No. There are non-holomorphic functions $f:\mathbb D\to \ell^2$ such that all components $f\_n=\pi\_n\circ f$ are holomorphic. This follows from a general result of Arendt and Nikolski [Vector-valued holomorphic functions revisited. Math. Z. 234 (2000), no. 4, 777–805]:
Theorem 1.5 Let $X$ be a Banach space and $W$ ... | 7 | https://mathoverflow.net/users/21051 | 234740 | 108,774 |
https://mathoverflow.net/questions/234735 | -2 | For any simple, finite, undirected graph $G=(V,E)$ and $v\in V$ let $N(v) = \{w\in V:\{v,w\}\in E\}$.
Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v,w\in V$ (not necessarily distinct) we have
$$|N(v)\cap N(w)| = |N... | https://mathoverflow.net/users/8628 | Degrees and common neighbors | **No**. And they can in fact be arbitrarily far apart. To see this let $G$ be two disjoint copies of $K\_n$, and let $H$ be $K\_{n,n}$ minus the edges of a perfect matching. Let $\varphi$ be a map that sends the two copies of $K\_n$ in $G$ to the two sets in the bipartition of $H$.
Let $v$ and $w$ be vertices of $G$... | 4 | https://mathoverflow.net/users/2233 | 234745 | 108,776 |
https://mathoverflow.net/questions/234741 | 2 | Given a set $X\neq \emptyset$ it is well-known that the collection $\text{Top}(X)$ of all topologies on $X$ is a (complete) lattice with respect to $\subseteq$.
Let $0$ denote the smallest element of the lattice - in our case it is the indiscrete topology $\{\emptyset, X\}$. Given a lattice $L$ with a bottom element ... | https://mathoverflow.net/users/8628 | Pseudocomplements in the lattice of topologies | Yes, and in fact, most familiar topologies do not have a pseudo-complement.
To see this, notice that that it often happens with a topology
$\tau$ on a set $X$ that there are non-open sets $A$ and $B$ with
$A\cap B$ open but not empty or $X$. For example, in the usual
topology on the reals, construct $A$ and $B$ by ad... | 9 | https://mathoverflow.net/users/1946 | 234752 | 108,777 |
https://mathoverflow.net/questions/234750 | 7 | Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an element of the unit ball of $C\_0(\mathbb{R})^\*$. By Banach-Alaoglu, the unit ball is weak-$^\*$ compact and since $C\_0(... | https://mathoverflow.net/users/50426 | Tightness and Functional Analysis | The functional analysis setting for this was established by R.C. Buck in the $1950$'s. He introduced a natural complete locally convex topology on $C^b(S)$, the space of bounded, continuous functionss on a locally compact space S$, the so- called strict topology with the properties
1) the dual space is the space of t... | 5 | https://mathoverflow.net/users/88761 | 234758 | 108,779 |
https://mathoverflow.net/questions/234718 | 1 | Let $k$ be a field of characteristic $ \neq 2$, and let $f \in k[T]$ be a polynomial of degree $\ge 1$ which is square free. Let $K$ be the quadratic extension $k(T)(\sqrt{f})$ of $k(T)$. I know that the integral closure of $k[T]$ in $K$ is $k[T, \sqrt{f}]$. My question is as follows.
>
> Does a prime ideal $(g)$ o... | https://mathoverflow.net/users/83593 | Prime ideal ramified in extension if and only if certain polynomial divides another one? | Set $A=k[T,\sqrt f]$. In fact, $A=k[T,U]/(U^2-f)$.
Then $A/gA\simeq k[T,U]/(g,U^2-f)$. If we set $L=k[T]/(g)$, then $A/gA\simeq L[U]/(U^2-\bar f)$.
If $g$ ramifies in $A$ then there is $h\in k[T]$ such that $U^2-\bar f=(U-\bar h)^2$, so $g\mid h$ and $g\mid f-h^2$ hence $g\mid f$.
| 2 | https://mathoverflow.net/users/23950 | 234776 | 108,784 |
https://mathoverflow.net/questions/234751 | 3 | Say $\mathscr{A}$ is a reflective subcategory of $\mathscr{B}$, meaning the inclusion functor $i: \mathscr{A} \to \mathscr{B}$ is fully faithful and admits a left adjoint, and $\mathscr{B}$ is enriched over itself, where for sake of simplicity we assume the tensor bifunctor to be the product.
Moreover, assume $\maths... | https://mathoverflow.net/users/57280 | Self-enrichment of reflective subcategories of self-enriched categories | The most natural question to ask is whether $\mathcal{A}$ is closed under the internal-hom of $\mathcal{B}$, i.e. that $[A,B]\in \mathcal{A}$ whenever $A,B\in\mathcal{A}$. This is an extra assumption, but it's sometimes easier to verify by reformulating it in terms of the tensor product; the resulting condition is that... | 5 | https://mathoverflow.net/users/49 | 234782 | 108,786 |
https://mathoverflow.net/questions/234770 | 2 | Let $S = \oplus\_{d \geq 0} S\_d$ be a graded (Noetherian) ring, let $I \subset S$ be a homogeneous ideal, and let $f \in S$ be a homogeneous element. Denote by $S\_{(f)}$ the subring of degree-$0$ elements of the localization $S\_f$.
>
> Do we always have $(H^i\_{IS\_f}(S\_f))\_0 \simeq H^i\_{IS\_f \cap S\_{(f)}}(... | https://mathoverflow.net/users/31771 | Does local cohomology commute with taking the degree-zero component? | The answer to your more general question is indeed positive. This (and its generalisation to $\mathbb{Z}^n$-graduations and to arbitrary degrees) is proven in Lemma 14.1.12 in the second edition of the book by Brodmann and Sharp (*Local cohomology (second edition),* Cambridge University Press, 2013) mentioned in the qu... | 2 | https://mathoverflow.net/users/11025 | 234786 | 108,789 |
https://mathoverflow.net/questions/234755 | 4 | I have asked this question in [math.stackexchange](https://math.stackexchange.com/questions/1711517/a-particular-diophantine-approximation-of-pi-2) without any answer, so I have decided to post it here too.
Recently I was playing around with the sequence $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$
After some computati... | https://mathoverflow.net/users/53175 | A particular Diophantine approximation of $\pi/2$ | The highlighted question is not quite what I expected. If there are infinitely many convergents to $\pi/2$ with odd denominators, then you could make the sequence consist of just these, and the answer is "no," you can't necessarily pass to an infinite subsequence with all even denominators. Every irrational number has ... | 4 | https://mathoverflow.net/users/2954 | 234789 | 108,790 |
https://mathoverflow.net/questions/234777 | 18 | My question concerns the argument given by Gauss in his "geometric proof" of the fundamental theorem of Algebra. At one point he says (I am reformulating) :
*A branch (a component) of any algebraic curve either comes back on itself (I suppose that means : it is a closed curve) or it goes to infinity on both sides.*
... | https://mathoverflow.net/users/89592 | Gauss proof of fundamental theorem of algebra | Gauss actually appends a footnote to this statement "if a branch of an algebraic curve enters a limited space, it necessarily has to leave it again" (Latin original follows below), in which he argues that:
>
> It seems to be well demonstrated that an algebraic curve neither ends
> abruptly (as it happens in the tra... | 30 | https://mathoverflow.net/users/11260 | 234794 | 108,792 |
https://mathoverflow.net/questions/234795 | 5 | I could find many resources on generating random unitary matrices, usually citing [F. Mezzadri, *Notices of the AMS* 54 (2007), 592-604](http://arxiv.org/abs/math-ph/0609050) for a method which generates unitaries random with respect to the Haar measure, but none of them mentioned the special unitary subgroup. In parti... | https://mathoverflow.net/users/82441 | Generating a random special unitary matrix | yes, this normalization will produce a uniform distribution in ${\rm SU}(N)$, however, it might be more efficient to generate directly random matrices with unit determinant (you'll need one fewer parameter and no need to calculate the determinant), as explained in [Composite parameterization and Haar measure for all un... | 5 | https://mathoverflow.net/users/11260 | 234797 | 108,794 |
https://mathoverflow.net/questions/234492 | 100 | In 1991, Kapranov and Voevodsky [published a proof](https://eudml.org/doc/91469) of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity categories that are weak infinity groupoid.
In 1998 Carlos Simpson [showed](http://arxiv.org/ab... | https://mathoverflow.net/users/22131 | What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky? | Here is my guess. To compare spaces with their notion of strict $\infty$-groupoids (in which everything is strict except inverses) Kapranov and Voevodsky use an intermediate category of Kan diagrammatic sets, which they show to be equivalent to both spaces and strict $\infty$-groupoids (after inverting a suitable colle... | 51 | https://mathoverflow.net/users/51164 | 234798 | 108,795 |
https://mathoverflow.net/questions/234806 | 7 | I am trying to find a problem which appeared years ago in the American Mathematical Monthly. It went something like this: There was a Putnam Competition question which asked to show that there is a number $x$ with the property that $[x^n]$ has the same parity as $n$ for all positive integers $n$. The square brackets in... | https://mathoverflow.net/users/40145 | On the parity of $[x^n]$ | Google turns up [this "Mock Putnam Exam" from the U[niversity] of I[llinois]](http://www.math.illinois.edu/~hildebr/putnam/problems/mock11.pdf), whose unattributed second problem asks to show that $[(\sqrt2+1)^n]$ has parity *opposite* to that of $n$ for each $n=1,2,3,\ldots\,$. Perhaps that's what you remember? (It sh... | 9 | https://mathoverflow.net/users/14830 | 234809 | 108,800 |
https://mathoverflow.net/questions/234807 | 3 | According to Springer's Encyclopedia of Math entry on [differential inequalities](https://www.encyclopediaofmath.org/index.php/Differential_inequality), T. Wazewski proved in 1950 the following theorem:
>
> Consider the system of differential inequalities given by
> $$ \dot{x}\_i (t) > f\_i(t,x\_1, \ldots, x\_n), ... | https://mathoverflow.net/users/22389 | On Wazewski's theorem on system of differential inequalities | In dimension 1 it is true without the ordered condition, assuming that $x$ and $z$ are suitably differentiable.
*Proof*:
At $t = t\_0$, we have $x(t\_0) = z(t\_0)$, and $x'(t\_0) > z'(t\_0)$ by assumption. This implies that $x > z$ on some interval $(t\_0, t\_m)$ for $t\_m$ sufficiently close to $t\_0$. Let $T$ b... | 6 | https://mathoverflow.net/users/3948 | 234813 | 108,801 |
https://mathoverflow.net/questions/234815 | 1 | In algebraic number theory we come across following formula:
$n= e\_1f\_1+\cdots+e\_rf\_r$
where all $e\_i$ and $f\_i$ are positive integers. I am sure writing a positive integer n as above must be studied.
For example :
1=1.1
2=1.1+1.1 = 1.2 = 2.1
3=1.1+1.1+1.1 = 1.1+1.2 = 1.1 +2.1 = 1.3 = 3.1
I am lookin... | https://mathoverflow.net/users/69977 | Reference : Partition of integer | 1) If we do not care on the order of summands (it looks to be the case).
For each ordered pair $(e,f)$ take a sum $1+x^{ef}+x^{2ef}+\dots$, generating function is a product of these guys over all $(e,f)$. This may be further rewritten in different ways. At first, we may fix $e$, then we have $\prod\_f (1-x^{ef})^{-1... | 4 | https://mathoverflow.net/users/4312 | 234821 | 108,804 |
https://mathoverflow.net/questions/233407 | 4 |
>
> *The question turns out quite long and maybe a bit vague, I apologize in advance for that.*
>
>
>
I am currently trying to understand Cheeger and Colding proof of the almost splitting theorem. Currently I am having trouble understanding how they use the so-called segment inequality. Recall that the segment i... | https://mathoverflow.net/users/8887 | Applying Cheeger and Colding segment inequality | I think that the point is that you need to use the Bishop-Gromov inequality (assuming $\mathrm{ricci}\_g\ge -(n-1)\kappa^2g$ :
$$\frac{\mathrm{vol}\_g B(x,R)}{\mathrm{vol}\_g B(x,\epsilon)}\le \frac{v\_n(\kappa, R)}{v\_n(\kappa,\epsilon)}$$
Hence you will get some points $x^\*\in B(x\_0,\epsilon), y^\*\in B(y\_0,\epsil... | 4 | https://mathoverflow.net/users/48525 | 234827 | 108,807 |
https://mathoverflow.net/questions/234816 | 2 | Consider arbitrary unit vectors $w,x,y,z \in \mathbb{C}^d$. Is there an explicit formula for what this average is?
$$
\int \mathrm{Tr}( \psi \psi^\* \, \, w x^\* \,\, \psi \psi^\* \,\, y z^\*) d\psi
$$
where the average is over a Haar-random unit vector $\psi \in \mathbb{C}^d$. Here, $\psi \psi^\*$ is the rank-one mat... | https://mathoverflow.net/users/5534 | Averages of vector inner products over the Haar measure | $\psi\_\alpha$, $\alpha=1,2,\ldots d$, is a column vector of a $d\times d$ unitary matrix $U$; averaging over the Haar measure gives
$$\int d\psi\, \psi\_{\alpha} \psi^\ast\_\beta \psi\_{\alpha'}\psi^\ast\_{\beta'}=\frac{1}{d+d^2}\left(\delta\_{\alpha\beta}\delta\_{\alpha'\beta'}+\delta\_{\alpha\beta'}\delta\_{\alpha'\... | 5 | https://mathoverflow.net/users/11260 | 234831 | 108,808 |
https://mathoverflow.net/questions/234818 | 4 | Let $X$ be a smooth proper curve over a field $k$, with function field $K$. Let $L$ be a finite separable tamely ramified extension of $K$, and let $Y$ be the normalization of $X$ in $L$. Is $Y\rightarrow \text{Spec }k$ smooth?
This seems to be true if $k$ is perfect, though I'd appreciate if someone could explain wh... | https://mathoverflow.net/users/88840 | is the normalization of a smooth curve in a tamely ramified finite separable extension of the function field also smooth? | No, but we can *exactly characterize* all counterexamples: $Y$ is non-smooth over $k$ at precisely those non-etale points of $Y \rightarrow X$ at which the residue field is not separable over $k$ (or equivalently the image point in $X$ has residue field not separable over $k$, since $f$ is separable between all residue... | 7 | https://mathoverflow.net/users/81332 | 234834 | 108,810 |
https://mathoverflow.net/questions/234801 | 1 | As we know the number of different substrings has the upper bound $O(n^{2})$.
Consider the strings on $\{0,1\}$ alphabet. Can I build a string with $\Omega(n^{2})$ different substrings?
Actually I was thinking about the Thue-Morse sequence. We could use the fact that there are no two $s\_1,s\_2\in$ the Thue-Morse s... | https://mathoverflow.net/users/71060 | Building the string on $\{0,1\}$ alphabet with $\Omega(n^{2})$ different substrings | Consider the $2n$-bit string $0^n1^n$. There is an injective map $\{1,\dotsc,n\}^2 \ni (i,j) \mapsto 0^i1^j \in \text{Sub}(0^n1^n)$, which shows that there are at least $n^2$ different substrings.
| 1 | https://mathoverflow.net/users/62519 | 234839 | 108,814 |
https://mathoverflow.net/questions/234793 | 4 | For which values of $n$ does the following inequality hold for?
>
> $$2(n-2) < Ap\_n\prod\_{i=3}^n \left(\frac{p\_i-1}{p\_i}\right)$$
>
>
>
$p$ are prime numbers and the notation $p\_i$ indicates the $i$-th prime number as per convention.
$A$ is defined by the following relationship $p\_n = 6A + r$ and $0\... | https://mathoverflow.net/users/41928 | The values of $n$ which satisfy an inequality about prime numbers | Here is a proof that the inequality holds for $n\geq 9$. I let you verify the remaining cases $n\leq 8$.
Let $x:=p\_n$ so that $x\geq 23$ by $n\geq 9$. Then the inequality can be rewritten as
$$ 2\bigl(\pi(x)-2\bigr)<\left\lfloor\frac{x}{6}\right\rfloor x \prod\_{3\leq p\leq x}\frac{p-1}{p}.$$
By increasing the LHS a... | 8 | https://mathoverflow.net/users/11919 | 234840 | 108,815 |
https://mathoverflow.net/questions/234835 | 10 | Let $C$ be a smooth curve of genus $g$, and let us consider its $d$-th symmetric product $\textrm{Sym}^d(C)$ and its Jacobian $J(C)$. Fixing a point $p\_0 \in C,$ there are two maps $$u\_d\colon C\_d \to J(C), \quad u\_d(D):=\mathcal{O}\_C(D-dp\_0),$$ $$i\_{d-1} \colon \textrm{Sym}^{d-1}(C) \to \textrm{Sym}^d(C), \quad... | https://mathoverflow.net/users/7460 | The class of the diagonal in the symmetric product of a smooth curve | This identity might be explicitly given in one of the following articles of Arthur Mattuck (Mattuck does prove many identities), but I could not find it.
MR0142553 (26 #122)
Mattuck, Arthur
Symmetric products and Jacobians.
Amer. J. Math. 83 1961 189–206.
14.20 (14.51)
MR0136608 (25 #76)
Mattuck... | 8 | https://mathoverflow.net/users/13265 | 234849 | 108,818 |
https://mathoverflow.net/questions/234829 | 5 | Fix a field $\Bbbk$. Let $Q$ be a Dynkin quiver and let $\Pi(Q)$ be its preprojective algebra. It is well-known that in this case $\Pi(Q)$ is finite-dimensional, but I've been unable to find a reference for what the dimension actually is for each type. I'm also interested in knowing the dimension of $e\_i \Pi(Q)$, wher... | https://mathoverflow.net/users/78095 | Dimension of preprojective algebra of Dynkin type | It is something well-known but I do not know where it is written. q-deform the Cartan matrix into
$$C(q)=I+qB+q^2I.$$
$B$ has zeroes on the main diagonal; substituting $q=1$ gives you Cartan. Invert the thing:
$$C(q)^{-1}=I+qA\_1+\ldots q^nA\_n+\ldots.$$
Let $k$ be the smallest integer such that $A\_k$ has negative coe... | 3 | https://mathoverflow.net/users/5301 | 234851 | 108,819 |
https://mathoverflow.net/questions/234759 | 7 | For every natural number $c \geq 2$, let $f(c)$ denote the least natural number $f$ with the following property : every union-closed family of sets with at least $f$ members has $c$ members whose intersection is nonempty. If I'm not wrong, it is easy to prove that $f(c)$ always exists. Frankl's conjecture amounts to sa... | https://mathoverflow.net/users/82840 | Frankl's conjecture and Oeis sequence A188163 | Let me elaborate a bit on what I [said](https://gowers.wordpress.com/2016/02/22/func4-further-variants/#comment-155662) over at Polymath11.
There is a nice [survey](http://arxiv.org/abs/1309.3297) on Frankl's conjecture, where the following similar question is discussed in Section 8. Let $\phi:\mathbb{N}\to\mathbb{N}... | 7 | https://mathoverflow.net/users/27013 | 234863 | 108,823 |
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