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https://mathoverflow.net/questions/357427 | 3 | Let $X$ be a smooth projective connected surface of general type over $\mathbb{C}$ with $q(X) = 1$, where $q(X) = \mathrm{h}^1(X,\mathcal{O}\_X)$.
Let $E$ be the Albanese variety of $X$, and let $X\to E$ be the Albanese map (having fixed a point). Let $0$ be a closed point of $E$.
>
>
> >
> > Let $F$ be the schem... | https://mathoverflow.net/users/151501 | Surfaces of general type with $q=1$ | You can find plenty of examples with multiple Albanese fibres by considering surfaces *isogenous to a product*, namely of the form $S=(C \times F)/G$, where $G$ is a finite group acting faithfully on the smooth curves $C$, $F$ and whose diagonal action on the product is free.
For an explicit situation, you can look a... | 3 | https://mathoverflow.net/users/7460 | 357429 | 150,690 |
https://mathoverflow.net/questions/357232 | 2 | $\renewcommand{\J}{\mathrm{Jac}} \renewcommand{\F}{\mathbb{F}}$
I am reading [this paper](https://www.ams.org/journals/jams/1990-03-04/S0894-0347-1990-1071117-8/S0894-0347-1990-1071117-8.pdf) by B. Gross, and there is something I don't understand on p. 945. Here is the context: fix a prime $p \equiv 3 \pmod 4$, and def... | https://mathoverflow.net/users/84923 | Degree of morphisms and isogenies | It's easier if we forget about isogenies: $E\_1$ and $E\_2$ are isomorphic,
and $X\_1$ and $X\_2$ are isomorphic, so the cover $X\_2\to E\_2$
induces a cover $X\_1 \to E\_1$ of the same degree.
To make this more explicit:
let $d = (p+1)/4$,
and rewrite the curve equations in separate coordinate systems:
\begin{align... | 3 | https://mathoverflow.net/users/156215 | 357447 | 150,697 |
https://mathoverflow.net/questions/357445 | 4 | I'm interested in the dual question to:
[continuous images of open intervals](https://mathoverflow.net/questions/168084/continuous-images-of-open-intervals), about surjections onto open intervals.
Namely, if $X$ is a topological space, when can we guarantee that there exists a topological embedding of $X$ into some E... | https://mathoverflow.net/users/36886 | Which topological spaces admit embeddings into Euclidean spaces | There's an old theorem of Deák that gives an interesting characterization.
Given a topological space $X$, define a relation on subsets of $X$ as follows: we write $U \sqsubseteq V$ if and only if $\overline{U} \subseteq V$.
**Theorem** (Deák): A separable metrizable space is homeomorphic to a subset of $\mathbb R^n$... | 9 | https://mathoverflow.net/users/70618 | 357451 | 150,698 |
https://mathoverflow.net/questions/357455 | 4 | It is folklore that many number theoretic results on prime numbers have a simpler-to-prove finite field analog. For example, on the one hand, the proof of the Prime Number Theorem
$$\#\{\text{prime numbers}\leq x\} \sim \frac{x}{\log x}, \;\;\,\; x \to +\infty,$$
requires complex analysis and the study of the Riemann z... | https://mathoverflow.net/users/156235 | Reference / Survey for finite field analog number theory | One of the best references I am aware of, is the book by Rosen ‘Number theory in function fields’ <https://www.springer.com/gp/book/9780387953359>.
A good survey, which also concerns the important connection with random matrix theory, is the following paper by Rudnick, although you won’t find proofs there: <http://ww... | 7 | https://mathoverflow.net/users/152733 | 357457 | 150,699 |
https://mathoverflow.net/questions/357450 | 2 | An upper semi-continuous function $u : \Omega \to \mathbb{R}$, $\Omega \subseteq \mathbb{C}^n$ is said to be *subharmonic* if it satisfies the submean inequality $u(a) \leq \mu\_S(u;a,r)$, where $\mu\_S(u;a,r)$ is the mean value of $u$ on the sphere $S(a,r)$ with center $a$ and radius $r$ (for any $a$,$r$ such that the... | https://mathoverflow.net/users/49151 | Subharmonic in any holomorphic coordinates = Plurisubharmonic? | It is actually enough to assume that your function $u$ defined on a neighborhood of $0\in \mathbb C^n$ remains subharmonic after composing with any linear transformation.
Indeed, let $0\neq \lambda \in \mathbb C^n$ and set, for $\sigma \in \mathbb C^\*$,
$$A(\sigma) = \begin{pmatrix}
\lambda\_1 & \sigma a\_{12} & ... | 3 | https://mathoverflow.net/users/5659 | 357463 | 150,703 |
https://mathoverflow.net/questions/357465 | 1 | Let $R$ be a commutative (non-trivially) graded ring. By a non-archimedean valuation I mean a map $v: R \to \Gamma \cup {0}$ such that for all $x,y \in R$, we have $v(x+y) \leq \max\{v(x),v(y)\}$, $v(xy)=v(x)v(y)$ and $v(0)=0,v(1)=1$. Here $\Gamma$ is a totally ordered commputative group and $0 \leq g$ for all $g \in \... | https://mathoverflow.net/users/143607 | non-archimedean valuations on graded rings | Let $K$ be a field, and set $R=K[x]$ with the usual grading by degree. For each irreducible polynomial $p(x)\in R$, we get a valuation $v\_p$ on $R$ given by
$$
v\_p(f)=2^{-\operatorname{ord}\_p(f)},
$$
where
$\operatorname{ord}\_p(f)\in\mathbb{Z}\_{\geq 0}\cup\{\infty\}$ is the power to which $p(x)$ occurs in the prim... | 1 | https://mathoverflow.net/users/5263 | 357476 | 150,711 |
https://mathoverflow.net/questions/357475 | 1 | Given a metric space $(X,d),$ let us call a sequence $(x\_n)$ in $X$ to be **pseudo-Cauchy** if $\lim\limits\_{n\to\infty}d(x\_n,x\_{n+1})=0.$
For example, the sequence $\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)\_n$ is pseudo-Cauchy in $\mathbb R$ without being Cauchy.
*Now I would like to ask the f... | https://mathoverflow.net/users/115208 | For pseudo-Cauchy sequence, does there always exist a pseudo-Cauchy subsequence of $(x_n)$ having distinct terms? | The answer to the question is yes. I struggled for a while trying to find the right growth rate so that terms in sequences are sufficiently spread but also not too much, until I realized that the following basic trick is more suited to the problem:
The hypothesis of having no constant subsequence means that each valu... | 3 | https://mathoverflow.net/users/45005 | 357495 | 150,716 |
https://mathoverflow.net/questions/357434 | 2 | If $\ X\ $ is a set, we let $\ \binom X2\,=\, \big\{\{a,b\}: a \neq b \in X \big\}.\ $ Given a simple, undirected graph $\ G=(V,E),\ $ we let $\ \delta(G)\ $ be its minimum degree, and $\ \Delta(G)\ $ its maximum degree. We say that $\ G\ $ is *self-complementary* if $\ G \cong \bar{G}\ $ where $\ \bar{G} = \left(V, \b... | https://mathoverflow.net/users/8628 | Minimal and maximal degrees in self-complementary graph | Self-complementary graphs exist if $n\equiv0,1\pmod 4$.
Take a self-complementary graph $G$ with $n$ vertices. Append to it a path $u{-}v{-}w{-}x$ of 4 vertices and join $v$ and $w$ to all of $G$. Now you have a self-complementary graph with $n+4$ vertices, $\delta=1$ and $\Delta=n+2$. This gives all sizes from 4 on... | 7 | https://mathoverflow.net/users/9025 | 357499 | 150,717 |
https://mathoverflow.net/questions/357492 | 3 | [The incomplete elliptic integral of the second kind](http://functions.wolfram.com/EllipticIntegrals/EllipticE2/introductions/IncompleteEllipticIntegrals/ShowAll.html) $E(\varphi \, | \,k)$ is defined as follows:
$$E(\varphi \, | \,k) = \int\_0^\varphi \sqrt{1-k^2\sin^2\theta} \, \mathrm{d}\theta $$
Where $0<k^2 < ... | https://mathoverflow.net/users/128941 | Inverse of the incomplete elliptic integral of the second kind | For a numerical representation of the inverse in terms of the angle $\varphi$
where $E(\varphi \, | \,k) = \int\_0^\varphi \sqrt{1-k^2\sin^2\theta} \, \mathrm{d}\theta $ is the
elliptic integral for the second kind, one could expand $E(\varphi |k)$ in a power
series around $\varphi=0$,
$ E(k |\varphi) := \varphi
-1/... | 2 | https://mathoverflow.net/users/51189 | 357504 | 150,718 |
https://mathoverflow.net/questions/357444 | 4 | WillJagy answered a linear relation question on Pythagorean Triples in [Small linear relations between primitive Pythagorean triples $\mathsf I$](https://mathoverflow.net/questions/357219/small-linear-relations-between-primitive-pythagorean-triples).
Now let $a^2+b^2=c^2$ be a primitive Pythagorean triple and then co... | https://mathoverflow.net/users/136553 | Small linear relations between primitive Pythagorean triples $\mathsf{II}$ | 1. Yes, the minimal $\|(u,v,z)\|\_\infty$ is within a constant factor of
$\sqrt{|c|}$ (equivalently, of $\sqrt{\max(|a|,|b|)}$.
The orthogonal complement of $(a,b,c) = (m^2-n^2, 2mn, m^2+n^2)$
contains the independent integer vectors
$v\_1 := (n,-m,n)$ (which you found) and $v\_2 := (m,n,-m)$.
Their $\bf Z$-span is ... | 7 | https://mathoverflow.net/users/14830 | 357505 | 150,719 |
https://mathoverflow.net/questions/357509 | -2 | [Snevily's conjecture](http://www.openproblemgarden.org/op/snevilys_conjecture#comment) it is an open conjecture in Group theory for non cyclic Group and it were proved for abelian groups of prime order using a fairly standard application of the Alon-Tarsi polynomial technique .It states that:
**Snevily's conjecture*... | https://mathoverflow.net/users/51189 | What are the consequence of Snevily's conjecture to analytic number theory if really there is a connection between them? | Snevily's conjecture was [proved in 2009](https://link.springer.com/article/10.1007/s11856-011-0040-6) by Bodan Arsovski. He was a [high-school student](https://www.imo-official.org/participant_r.aspx?id=9164) at the time.
I don't know of any consequences in analytic number theory. On the other hand, there are severa... | 4 | https://mathoverflow.net/users/11919 | 357511 | 150,720 |
https://mathoverflow.net/questions/357500 | 2 | Are there explicit formulas for the Weyl symbol of $-f(x)D\_x^2 $ where $D\_x:=-i\partial\_x $ and $\partial\_x$ is the derivative and $f$ some sufficiently smooth function?
In the standard quantization the symbol would of course just be $w(x,\xi)=f(x) \xi^2$ but in Weyl quantization this seems to be no longer true.... | https://mathoverflow.net/users/150549 | Weyl symbol of product | By quantization, one is usually more trying to find the operator corresponding to a given symbol than the symbol corresponding to an operator. In your case, it seems the Weyl symbol you are talking about is the Wigner transform (the inverse of the Weyl trasnform). For an operator $A$ of kernel $a(x,y)$ it is defined by... | 2 | https://mathoverflow.net/users/153203 | 357518 | 150,724 |
https://mathoverflow.net/questions/357477 | 6 | It is well-known (due to [this](https://www.sciencedirect.com/science/article/pii/0040938366900024) work of Palais, I believe) that Banach manifolds are dominated by countable CW complexes. It then follows (due Whitehead, as indicated by Milnor in [this](https://www.ams.org/journals/tran/1959-090-02/S0002-9947-1959-010... | https://mathoverflow.net/users/153173 | CW structure on infinite-dimensional manifolds | Let $M$ be a Hilbert manifold. Then there exists a Riemannian metric $g$ on $M$ such that the induced metric $d$ is complete (See <https://arxiv.org/pdf/1610.01527.pdf> by Biliotti and Mercuri). Thus $M$ has the topology of a complete metric space. By the Baire category theorem $(M,d)$ is a Baire space: It cannot be wr... | 6 | https://mathoverflow.net/users/12156 | 357522 | 150,726 |
https://mathoverflow.net/questions/357259 | 6 | Let $G$ be a connected reductive algebraic group defined over an algebraically closed field and let $g\in G$ be semisimple. Write $C=\mathrm{C}\_G(g)$ and $C^\circ=\mathrm{C}\_G(g)^\circ$ for the centralizer of $g$ and for its identity component, respectively.
>
> **Question** Does there exist $h\in G$ such that $\... | https://mathoverflow.net/users/14443 | Is the connected centralizer of a semisimple element in a connected reductive group also a centralizer? | So after thinking about this for a few extra days, I think I found a counterexample. I'm recording it here, as community wiki, in case it would be of interest for anyone in the future.
---
Let $k$ be an algebraically closed field of characteristic not $2$ and $G=\mathrm{PGSp}\_{2n}(k)=\mathrm{GSp}\_{2n}(k)/k^\tim... | 3 | https://mathoverflow.net/users/14443 | 357524 | 150,727 |
https://mathoverflow.net/questions/356369 | 3 | Suppose I have a graph $G$ (possibly with weights on edges), and I have a subset $S$ of $k$ vertices $s\_1, \dotsc, s\_k$. I want to solve the post office problem: that is, I want to partition the vertices of $G$ into subsets $D\_1, \dotsc, D\_k,$ so that $s\_i$ is the closest vertex of $S$ to every vertex in $D\_i.$ I... | https://mathoverflow.net/users/11142 | Voronoi diagram on (weighted) graphs | Once you decide that the elements of $S$ should look for the vertices in their part rather than the other way round the naive approach of exploring edges one at a time in increasing distance from $S$ is the best you can do (up to administrative overhead) because you might have to examine all of the edges anyway. This i... | 1 | https://mathoverflow.net/users/25485 | 357548 | 150,734 |
https://mathoverflow.net/questions/357333 | 4 | I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for.
For any $f \in L^2(\mathbb{R})$, let $\mathcal{Cf}:=\sup\_{N \in \mathbb{Z}} \left\vert P\_{-}(e^{iN\cdot}f)\right\vert$ be the maximal Carleson operator; wher... | https://mathoverflow.net/users/94414 | Idea behind Carleson's theorem modern proof "intitial reductions" | I have my own confusions here, but let me share my thoughts.
As you mention, there is a discretization here. If you want to decompose the operator $P\_-$, you use the standard decomposition $\sum\_k\hat{\varphi}\_k = 1\_{(-\infty,0]}$, where $\hat{\varphi}\_k(\xi) := \hat{\varphi}(\xi/2^k)$ is supported at frequencie... | 2 | https://mathoverflow.net/users/90189 | 357555 | 150,737 |
https://mathoverflow.net/questions/357452 | 2 | Suppose $B\_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F\_t\}\_t, \mathbb P\rangle$. Consider two SDEs below.
Suppose, $X\_0 = Y\_0 = 0$
\begin{align\*}
dX\_t =& sign(X\_t) dt + d B\_t\\
dY\_t =& \alpha\_t dt + dB\_t
\end{align\*}
where $\alpha\_t$ is som... | https://mathoverflow.net/users/78761 | Absolute value of a diffusion | Yes, square the processes then $$ dX\_t^2 = 2X\_t dX\_t + (dX\_t)^2 = 2|X\_t| dt + 1 + 2X\_t dB\_t$$ and similarly for the other. Then you see the the drift on this process must be bigger than that of the $Y\_t$ process at the same level. Then you can finish it off with a comparison theorem that says $X\_t^2$ will be s... | 3 | https://mathoverflow.net/users/143907 | 357556 | 150,738 |
https://mathoverflow.net/questions/357523 | 17 | It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see [here](https://mathoverflow.net/questions/324465/is-residual-finiteness-a-quasi-isometry-invariant-for-f-g-groups) for an example). Thus the following question makes sense:
**Question:**... | https://mathoverflow.net/users/147609 | Existence of a quasi-isometric residually finite group? | Take any finitely-presented group $G$ with undecidable word problem. Then $G$ is not quasi-isometric to any finitely generated group with decidable word problem, in particular, to any residually-finite group. (Note that finite presentability and decidability of the WP are quasi-isometry invariant. The latter is because... | 20 | https://mathoverflow.net/users/39654 | 357563 | 150,741 |
https://mathoverflow.net/questions/357359 | 10 | Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as
$$\text{co}(K) = \Big\{ \sum\_{i=1}^n a\_i x\_i : a\_i \geq 0,\, \sum\_{i=1}^n a\_i = 1,\, x\_i \in K \Big\},$$
define the $\sigma$-convex hull as
$$\sigma\text{-}\mathrm{co}(K) ... | https://mathoverflow.net/users/118647 | Equivalence of σ-convex hull and closed convex hull | Wlod AA gave a good counterexample for the case when $K$ is not required to be compact, here I give a counterexample $K$ compact, first in a locally convex space, and then for a(n infinite-dimensional) separable normed space, and (after an edit) for all infinite-dimensional Banach spaces.
There is a standard countere... | 5 | https://mathoverflow.net/users/61785 | 357567 | 150,742 |
https://mathoverflow.net/questions/357560 | 0 | An interesting combinatorial identity is the Vandermonde convolution identity:
$$ \sum\_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$
which can be proved by considering the coefficients in $(x+1)^{n+m} = (x+1)^n (x+1)^m$. I am interested in the behavior of the summands: which summands among them contribute the ... | https://mathoverflow.net/users/124549 | Local behavior of the Vandermonde convolution | $\newcommand\ep{\epsilon}$
Of course, there is no simple explicit expression for the smallest $\ep$. However, one can give upper bounds on or approximations of this $\ep$.
Indeed, $F$ is the probability mass function of a random variable, say $X$, with the [hypergeometric distribution](https://en.wikipedia.org/wiki/... | 2 | https://mathoverflow.net/users/36721 | 357571 | 150,743 |
https://mathoverflow.net/questions/357573 | 2 | Let $(X,w)$ be a compact kahler manifold, and $[\eta]$ be a class is on the boundary of the kahler cone. The claim is that one can find another class $[\beta]$ also on the boundary of the kahler cone such that $(1-t)[\beta] + t[\eta] > 0$ when $t \in (0, 1)$. How can one prove this?
Some thoughts:
according to [this ... | https://mathoverflow.net/users/156328 | Control the convex combination of two classes on the boundary of the kahler cone | Assume $[\eta]$ is on the boundary of the Kähler cone and that $[\eta] \neq 0.$
Define $k := -\sup\{t > 0 : [\omega]-t[\eta] \hskip4pt {\rm Kähler}\}$. We claim that $k > -\infty$.
If $k = -\infty,$ then $\frac{1}{t}[\omega]-[\eta]$ is Kähler for all $t > 0,$ so that $-[\eta]$ is nef. The claim will be proved onc... | 2 | https://mathoverflow.net/users/5496 | 357581 | 150,745 |
https://mathoverflow.net/questions/357564 | 15 | Helmut Hasse has proved that for $s \in \mathbb{C}-\{1\}$ the Riemann zeta function can be written as:
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum\_{n=0}^\infty\frac{1}{2^{n+1}}\sum\_{k=0}^n(-1)^k\ {n \choose k}\ \frac{1}{(k+1)^{s}}$$
Let $K(a,b) = \frac{\gcd(a,b)}{a+b}$. Then we might define:
$$\hat{\zeta}(s,a)=\frac{1}{... | https://mathoverflow.net/users/nan | If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$? | This is just a simple calculation. Note that
$$
\sum\_{n=k}^{\infty} \frac{1}{2^{n+1}} \binom{n}{k} =1
$$
for any non-negative integer $k$, which immediately gives
$$
\frac{1}{1-2^{1-s}} \sum\_{n=0}^{\infty} \frac{1}{2^{n+1}} \sum\_{k=0}^{\infty} \frac{(-1)^k}{(k+1)^s} \binom{n}{k} = \frac{1}{1-2^{1-s}}\sum\_{k=0}... | 14 | https://mathoverflow.net/users/38624 | 357582 | 150,746 |
https://mathoverflow.net/questions/357446 | 7 | The kernel of a Hopf algebra map $\phi:H\_1 \to H\_2$ is in general not a Hopf
sub-algebra of $H\_1$. Is there some replacement or alteration of the notion
of a kernel in the Hopf algebra setting. Same question for cokernels. So can we construct an abelian category from the category of Hopf algebras (over a fixed fie... | https://mathoverflow.net/users/153228 | Abelian category from the category of Hopf algebras |
>
> $\DeclareMathOperator\Hker{Hker}\DeclareMathOperator\Hcoker{Hcoker}\DeclareMathOperator\Im{Im}\DeclareMathOperator\coIm{coIm}\DeclareMathOperator\Id{Id}$The category $\mathcal{H}$ of finite dimensional, commutative, cocommutative hopf algebras is an abelian category.
>
> The set $\mathcal{H}(F,G)$ of all hopf ... | 6 | https://mathoverflow.net/users/85967 | 357602 | 150,753 |
https://mathoverflow.net/questions/173066 | 10 | Is there an example of two complex projective complete intersections that are diffeomorphic but have different Hodge numbers?
Edit: as written by Daniel Loughran in the comments below, complete intersections with the same multidegree are diffeomorphic (apparently this result is attributed to R. Thom). And we know tha... | https://mathoverflow.net/users/27816 | Hodge numbers of diffeomorphic complete intersections | We got four pairs of diffeomorphic complete intersections but with Hodge numbers different. Please check the link:[here](https://arxiv.org/abs/2004.07142)
| 3 | https://mathoverflow.net/users/156352 | 357608 | 150,756 |
https://mathoverflow.net/questions/357609 | 1 | Is there example of a smooth, projective, complex $3$-fold $X$, having $b\_{2}(X)=2$ a Mori extremal contraction $\phi: X \rightarrow X'$ which contracts a smooth quadric surface $Q \subset X$?
It doesn't matter which of the two possible types it is, i.e. if $\phi(Q)$ is an ODP or $\{xy-z^2-t^3=0\} \subset \mathbb{C}... | https://mathoverflow.net/users/99732 | Mori extremal contraction with small Betti number | As abx mentioned, the simplest example is the blowup of a cubic 3-fold with an ODP. Alternatively, the same variety can be obtained as the blowup of $\mathbb{P}^3$ along a smooth complete intersection of a smooth quadric and a cubic. The strict transform of the quadric then can be contracted to an ODP.
| 2 | https://mathoverflow.net/users/4428 | 357611 | 150,757 |
https://mathoverflow.net/questions/201273 | 30 |
>
> Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces?
>
>
>
By a theorem of, independently, [Fontaine](http://www.ams.org/mathscinet-getitem?mr=1274493) and [Abrashkin](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1152&opti... | https://mathoverflow.net/users/18060 | Enriques surfaces over $\mathbb Z$ | A preprint by Stefan Schröer came out today with the answer to this question: [arXiv:2004.07025](https://arxiv.org/abs/2004.07025).
No such Enriques surface exists. In fact, there is no classical Enriques surface over $\mathbb F\_2$ with 25 $\mathbb F\_2$-points (and with the extension of $\mathbb Z^{10}$ by $\mathbb... | 14 | https://mathoverflow.net/users/43951 | 357618 | 150,759 |
https://mathoverflow.net/questions/357597 | 2 | Let $(M,g)$ be a compact connected smooth Riemannian manifold without boundary and let $\phi: M \rightarrow \mathbb{R}$ be a function on $M$. We say that $\phi$ is superdifferiantiable at $x$ with super-gradient $p$, if we have that
$$\phi(\exp\_{x}(v)) \leq \phi(x)+ g(p,v)\_{x}+o(|v|)\_{x}$$
for all small $v \in ... | https://mathoverflow.net/users/71233 | Superdifferentiable and subdifferentiable at $x$ implies differentiable at $x$ | Pick any arbitrary super- sub-gradients $p,q$, and an arbitrary direction $v\in T\_x \mathbb R^d\cong \mathbb R^d$.
By definition of sub- super-differentiability we get the double inequality
$$
\phi(x)+\langle q,v\rangle+o\_1(|v|) \leq \phi(x+v)\leq \phi(x)+\langle p,v\rangle+o\_2(|v|)
$$
for two negligible functions $... | 2 | https://mathoverflow.net/users/33741 | 357627 | 150,762 |
https://mathoverflow.net/questions/357557 | 4 | A family $\mathcal U$ of infinite subsets of $\omega$ is called an *ultrafamily* if for any sets $U,V\in\mathcal U$ one of the sets $U\setminus V$, $U\cap V$ or $V\setminus U$ is finite.
By the Kuratowski-Zorn Lemma, each ultrafamily $\mathcal U\subseteq [\omega]^\omega$ can be enlarged to a maximal ultrafamily.
Le... | https://mathoverflow.net/users/61536 | What is the smallest cardinality of a maximal ultrafamily of infinite subsets of $\omega$? | To my surprise, I found that this my ``new'' cardinal $\mathfrak{uf}$ is equal to $\mathfrak c$.
**Theorem.** $\mathfrak{uf}=\mathfrak{c}$.
*Proof.* Fix any maximal ultrafamily $\mathcal U\subseteq[\omega]^{\omega}$. For two sets $A,B$ we write $A\subset^\* B$ if $A\setminus B$ is finite but $B\setminus A$ is infi... | 1 | https://mathoverflow.net/users/61536 | 357634 | 150,765 |
https://mathoverflow.net/questions/357626 | 1 | Let there be two random variables and with a certain joint copula. Is it always true that there is another random variable independent from such as the vectors $(X,Y)$ and $(X,Z)$ have the same law?
| https://mathoverflow.net/users/3898 | The existence of a copy of a random variable with conditional expectation constraint | No, suppose $X=Y$ a.s. and that they are non-degenerate. If we want $(X, Z)$ to have the same joint distribution as $(X, Y)$, we must also have $X=Z$ a.s. and hence $Y=Z$ a.s. Then $Y$ and $Z$ can obviously not be independent.
| 3 | https://mathoverflow.net/users/154137 | 357636 | 150,766 |
https://mathoverflow.net/questions/357622 | 4 | **Mertens' third theorem** states that:
$$\prod\_{\substack{
p \leq x \\
\text{p prime}
}} \left( 1 - \dfrac{1}{p} \right) \sim \dfrac{e^{-\gamma}}{\log(x)}$$
**Question:** what is the best functions (unconditionally and conditionally) satisfying:
$$\prod\_{\substack{
p \leq x \\
\text{p prime}
}} \left( 1 - \dfrac{1}{... | https://mathoverflow.net/users/164630 | Error term in Mertens' third theorem | There's been a lot of work on unconditional results of this sort.
Rosser and Schoenfeld showed in a 1962 paper that one can take
$$\dfrac{e^{-\gamma}}{\log x} \left(1- \frac{1}{2\log^2 x} \right) < \prod\_{\substack{
p \leq x \\
\text{p prime}
}} \left( 1 - \dfrac{1}{p} \right) < \dfrac{e^{-\gamma}}{\log x} \left(... | 9 | https://mathoverflow.net/users/127690 | 357656 | 150,772 |
https://mathoverflow.net/questions/355891 | 10 | I recently started reading up on simple homotopy theory. Here is a question I stumbled upon.
In his 1938 Paper [Simplicial Spaces, Nuclei and m-Groups](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s2-45.1.243) Whitehead introduced the notion of elementary expansions and elementary collapses of sim... | https://mathoverflow.net/users/103344 | Simplicial simple homotopy vs. cellular simple homotopy | Turns out I should have read the original material [Simplicial Spaces, Nuclei and m-groups](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s2-45.1.243) a little more carefully. It was in there all along. I'm still supprised nobody ever explicitly stated this though. But I guess it was such common know... | 4 | https://mathoverflow.net/users/103344 | 357676 | 150,779 |
https://mathoverflow.net/questions/357588 | 4 | I know hardly anything about quantum field theory (QFT) but I'm giving a try to understand some ideas of it. As far as I understand, in QFT one is interested in studying measures such as:
\begin{eqnarray}
d\mu(\varphi) \propto e^{-S(\varphi)}d\varphi \tag{1}\label{1}
\end{eqnarray}
where $S$ is a given action and $d\va... | https://mathoverflow.net/users/150264 | QFT and its notations | The quick answer is that (3) written by physicists is not to be taken too seriously by mathematicians. However, it is a statement of a goal or research problem which is to find a rigorous definition/construction of what (3) is trying to say.
One has an *injective* continuous linear map $\iota:\mathscr{S}(\mathbb{R}^d... | 5 | https://mathoverflow.net/users/7410 | 357681 | 150,782 |
https://mathoverflow.net/questions/351268 | 12 | Call a group presentation $\langle X \,\|\,R \rangle$ *minimal* if no relator from $R$ is a consequence of the remaining relators, i.e., no $r \in R$ belongs to the normal closure of $R\setminus \{r\}$ in the free group $F(X)$.
**Question:** Does every finitely generated group have a minimal presentation (with $X$ fi... | https://mathoverflow.net/users/7644 | Does every f.g. group have a minimal presentation? | The answer is **no**.
In order to see this, you may combine Theorem 3.9 and Remark 5.3 of [1]. A counter-example is given by the nilpotent-by-Abelian group $B$ of Equation (3.2). Further examples are provided by Remark 5.15.
---
[1] R. Bieri, Y. de Cornulier, L. Guyot and R. Strebel, ["Infinite presentability ... | 9 | https://mathoverflow.net/users/84349 | 357693 | 150,785 |
https://mathoverflow.net/questions/357630 | 7 | Let $\mathcal{E}'(\mathbb{R})$ be algebra of all compactly supported distributions on $\mathbb{R}$, equipped with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and with the usual operations of addition and convolution.
Is the set ${{\textrm{GL}}}\_1(\mathcal{E}'(\mathbb{R}))$ of invertible elements open... | https://mathoverflow.net/users/156379 | $GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$? | Perhaps Jochen Wengenroth's comments already give the answer, but here's a direct argument.
By [Paley-Wiener](https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem) for distributions, the Fourier transform $\tilde{\Delta}(k)$ of a distribution of compact support $\Delta(x) \in \mathcal{E}'(\mathbb{R})$ is entire... | 3 | https://mathoverflow.net/users/2622 | 357700 | 150,786 |
https://mathoverflow.net/questions/357707 | 2 | Define $F(x) = \sum\_{n\geq 1} f\_{n}x^n$ and $G(x) = \sum\_{n\geq 1} g\_{n}x^n$. Then the Hadamard product of $F$ and $G$ is
$$H(x):=(F\*G)(x) = \sum\_{n\geq 1} f\_{n}g\_{n}x^n.$$
The author of [Riesz equivalent of Riemann hypothesis and Hadamard product](https://mathoverflow.net/q/351443) claims that
$$H(x) = ... | https://mathoverflow.net/users/156413 | Reference request for the integral representation of the Hadamard product of two infinite series | E.C. Titchmarsh, The theory of functions, Oxford University Press
Section 4.6 Hadamard multiplication theorem, p.158
| 2 | https://mathoverflow.net/users/89429 | 357710 | 150,789 |
https://mathoverflow.net/questions/357709 | 1 | I came across an odd circumstance where it appears as though the poisson summation formula fails to yield a correct answer (involving Hermite Functions), and I don't quite understand why this happens.
Define the family of Hermite Functions by:
$$
\psi\_n(x) = \frac{(-1)^n}{\sqrt{2^n n! \sqrt{\pi}}}e^{\frac{x^2}{2}}\... | https://mathoverflow.net/users/156408 | Poisson Summation Formula appears to fail when applied to Hermite Functions (why?) | Your mistake is that you are not using the correct form of the Poisson summation formula based on your choice of how to define the Fourier transform. There are multiple conventions on how to define the Fourier transform in terms of where you stick factors of $2\pi$ (in the exponent or as a coefficient) and this leads t... | 12 | https://mathoverflow.net/users/3272 | 357719 | 150,792 |
https://mathoverflow.net/questions/357614 | 2 | Let $(A\_\alpha)\_{\alpha\in\mathfrak c}$ be an almost disjoint family of infinite subsets of $\omega$. The almost disjointness of the family means that $A\_\alpha\cap A\_\beta$ is finite for any ordinals $\alpha,\beta\in\mathfrak c$. This almost disjoint family generates a filter $$\mathcal F=\{F\subseteq \omega:|\{\a... | https://mathoverflow.net/users/61536 | What is the smallest cardinality of a base of an ultrafilter on $\omega$ related to an almost disjoint family of cardinality $\mathfrak c$? | Lyubomyr Zdomskyy informed me that the answer to my question is affirmative and follows from a recent (still unpublished) result of [Osvaldo Guzman and Damjan Kalajdzievski](https://lc2019.cz/static/public/slides/Friday/room-0/guzman/Logic-Colloquim.pdf?b8f311b8201af564aceb) who proved the consistency of $\mathfrak u<\... | 1 | https://mathoverflow.net/users/61536 | 357724 | 150,794 |
https://mathoverflow.net/questions/357711 | 13 | For $K$ a field, is it known what the abelianization of $GL\_2(K[X])$ is?
| https://mathoverflow.net/users/91826 | Abelianization of general linear group of a polynomial ring | There is a discussion on whether $K$ has two elements or is larger, which strongly affects the conclusion.
One has the determinant map $\mathrm{GL}\_2(K[X])\to K^\*$. To show that it's the abelianization, it's enough to show that the kernel $\mathrm{SL}\_2(K[X])$ is contained in the derived subgroup of $\mathrm{GL}\_... | 21 | https://mathoverflow.net/users/14094 | 357732 | 150,797 |
https://mathoverflow.net/questions/231598 | 3 | From pari's implementation of [Coppersmith method](https://en.wikipedia.org/wiki/Coppersmith_method)
>
> zncoppersmith(P, N, X, {B=N}): finds all integers $x$ with $|x| \le X$ such that
> $\gcd(N, P(x)) \ge B$. $X$ should be smaller than
>
>
>
$$\exp((\log B)^2 / (\deg(P) \log N)) \qquad (1) $$
Observe th... | https://mathoverflow.net/users/12481 | When is Coppersmith method polynomial? (Factorization related) | We asked on the pari-dev mailing list and the developers
replied that the documentation was incorrect if the leading
coefficient is not coprime to $N$.
This is fixed in pari-master.
Discussion:
<http://pari.math.u-bordeaux1.fr/archives/pari-dev-1912/msg00002.html>
| 3 | https://mathoverflow.net/users/12481 | 357734 | 150,798 |
https://mathoverflow.net/questions/357685 | 5 | Suppose $\pi:C'\to C$ is a branched cover of compact Riemann surfaces such that the associated extension of function fields is Galois with group $G$ -- so that $\pi$ presents $C$ as the quotient $C'$ by the action of $G = \text{Aut}(C'/C)$. Now, let $\rho:G\to GL(W)$ be a finite dimensional complex representation of $G... | https://mathoverflow.net/users/110236 | Descent of vector bundle along branched cover of curve | This computation is related to the well-known semiorthogonal decomposition of the $G$-equivariant derived category of $C'$, or equivalently, of the quotient stack $C'/G$. The latter can be thought of as the curve $C$ with a root stack of order $n\_p$ structure at each of the branch point $p \in C$.
The semiorthogonal... | 7 | https://mathoverflow.net/users/4428 | 357743 | 150,799 |
https://mathoverflow.net/questions/357730 | 11 | I came across this integral involving the derivative $f'(x)$ of the Fermi function $f(x)=(1+e^x)^{-1}$:
$$I(\phi)=-\int\_{-\infty}^\infty \log\left[1+2f'(x)(1-\cos\phi)\right]\,dx.$$
I'm pretty certain that $I(\phi)=\phi^2$ for $|\phi|<\pi$, periodically repeated, as in the plot. It means that only the ${\cal O}(\phi^2... | https://mathoverflow.net/users/11260 | Why is $-\int_{-\infty}^\infty \log\left[1+2f'(x)(1-\cos\phi)\right]\,dx$ equal to $\phi^2$? | It is clear that $I(0)=0$, hence by [Leibniz's integration rule](https://en.wikipedia.org/wiki/Leibniz_integral_rule), it suffices to show that
$$\int\_{-\infty}^\infty\frac{-2f'(x)\sin\phi}{1+2f'(x)(1-\cos\phi)}\,dx=2\phi,\qquad |\phi|<\pi.$$
We can assume, without loss of generality, that $0<\phi<\pi$. By a bit of al... | 9 | https://mathoverflow.net/users/11919 | 357747 | 150,800 |
https://mathoverflow.net/questions/357544 | 3 | Let $F$ be a non-archimedean local field, $G = \operatorname{GL}\_n\left(F\right)$, $B$ the standard Borel subgroup, $K = \operatorname{GL}\_n\left( \mathcal{O}\_F \right)$. We have the Iwasawa decomposition $G = BK$. Let $\pi$ be an unramified principal series $\left( \pi, I\left({\underline{s}}\right) \right) = \oper... | https://mathoverflow.net/users/103908 | Are polynomial sections for unramified principal series generated by the spherical vector? | This can't be true, because for some choices of $\underline{s}$, the spherical vector does not generate $I\left( \underline{s} \right)$ (see [Paul Garrett - Representations with Iwahori-fixed vectors](http://www-users.math.umn.edu/~garrett/m/v/iwahori_fixed.pdf), page 10). However, we can write $f^{\circ}\_{\underline{... | 1 | https://mathoverflow.net/users/103908 | 357750 | 150,803 |
https://mathoverflow.net/questions/357539 | 3 | Let $A$ be a finite, nonempty subset of an abelian group, and let $2A:=\{a+b\colon a,b\in A\}$ and $A-A:=\{a-b\colon a,b\in A\}$ denote the sumset and the difference set of $A$, respectively.
If every non-zero element of $A-A$ has a unique representation as $a-b$ with $a,b\in A$, then all sums $a+b$ are pairwise dist... | https://mathoverflow.net/users/9924 | Unique representation and sumsets | One can have $|2A|$ as small as $2|A|$. Take $A = H \cup \{g\}$ where $H$ is a subgroup, $g \notin H$ and$g \neq -g$. Then $|A+A| = 2|A| + O(1)$ while
$g+H$ and $H - g$ all have a unique representative in $A-A$.
On the other hand, I can show if the number of uniquely representable elements of $A-A$ is at least $|A|$... | 1 | https://mathoverflow.net/users/50426 | 357770 | 150,809 |
https://mathoverflow.net/questions/357769 | 10 | I am aware that the Spelling Theorem of B. B. Newman implies that one-relator groups with torsion are hyperbolic, and thus have a solvable conjugacy problem. My understanding is that for one-relator groups without torsion, the conjugacy problem is still open, though the most recent reference I have for this is over 20 ... | https://mathoverflow.net/users/135406 | Reference request: Recent progress on the conjugacy problem for torsion-free one-relator groups? | As mentioned in the comments, this is still considered an open problem. I thought I'd flesh out a few aspects. A solution was claimed in 1992 by [Juhasz](https://link.springer.com/chapter/10.1007%2F978-1-4613-9730-4_3), but it seems to have failed to convince experts. The small cancellation theory involved in the proof... | 11 | https://mathoverflow.net/users/120914 | 357773 | 150,810 |
https://mathoverflow.net/questions/357757 | 2 | Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that
$$ \int\_\Omega \left(f(x\_1,x\_2)- \frac{m}{(n+1)}g(x\_1,x\_2)\right) x\_1^n \,x\_2^m \,dx\_1\,dx\_2 = 0 \quad \text{for all $n,m=0,1,\ldots$}.$$
Does it follow that $f \equiv g \equiv 0$ on $\Omega$?
| https://mathoverflow.net/users/50438 | A density question | Certainly not. Assume $\partial h/\partial x\_1=g/x\_1$, and assume that $h$ and $g$ have compact support. Then integrate by parts to obtain
$$\int\int (f-x\_2{\partial h\over\partial x\_2})x\_1^nx\_2^m=0.$$
This implies $f=x\_2(\partial h/\partial x\_2)$ and nothing more.
| 3 | https://mathoverflow.net/users/12120 | 357774 | 150,811 |
https://mathoverflow.net/questions/357703 | 6 | Let $A$ be a non-empty set of regular cardinals such that $\vert A\vert <\text{min}\ A$, and $\{\nu\_i\mid i<i\_0\}\subseteq \text{pcf}\ A$ be a strict increasing sequence having limit length $i\_0$.Then $\mathcal{J}\_{<\nu\_{i\_0}:=\text{sup}\{\nu\_i\mid i<i\_0\}}A=\bigcup\{\mathcal{J}\_{<\nu\_i}A\mid i<i\_0\}$ hence ... | https://mathoverflow.net/users/156285 | Regular limit points of possible cofinalities | This is open. In fact, the situation you describe is related to one of the most important open problems in pcf theory:
Can there can be a progressive set of regular cardinals $A$ (that is, $|A|<\min(A)$) with the property that $pcf(A)$ contains a regular limit point?
This is essentially equivalent to the question... | 5 | https://mathoverflow.net/users/18128 | 357779 | 150,813 |
https://mathoverflow.net/questions/357780 | 4 | Sorry if this is trivial, but I could not find any reference.
Let $k,a,b$ be integers. The space of modular forms of integer weight $M\_k(\text{SL}\_2(\mathbb{Z}))$ admits a basis of the form $\{ E\_4^aE\_6^b : 4a+6b = k \}$ where $E\_4$ and $E\_6$ are normalized Eisenstein series of weight 4 and 6 respectively. Sim... | https://mathoverflow.net/users/127239 | (Explicit) Basis for Kohnen's plus-space of modular forms of half integral weight | There does exist an explicit basis when $k$ is EVEN: denote by
$E\_{k,4}$ the Eisenstein series $E\_k(4\tau)$. Then the Rankin-Cohen
brackets $[\theta,E\_{k-2j,4}]\_j$ for $0\le j\le\lfloor k/6\rfloor$ (with a small modification for $k=2$) form a basis of $M^+\_{k+1/2}(4)$, where of course $\theta$
is the usual generat... | 6 | https://mathoverflow.net/users/81776 | 357781 | 150,814 |
https://mathoverflow.net/questions/357755 | 2 | I would like to lift an arbitrary one-parameter subgroup $e^{t K}$ with $K\in\mathfrak{sp}(2N,\mathbb{R})$ to the universal cover $\widetilde{\mathrm{Sp}}(2N,\mathbb{R})$ (or at least its two-fold cover, i.e., the metaplectic group).
I follow the paper of John Rawnsley [On the universal covering group of the real sym... | https://mathoverflow.net/users/80903 | Lifting one parameter subgroup $e^{t K}$ to the universal cover of $\mathrm{Sp}(2N,\mathbb{R})$ | I made some progress in the sense that I believe that I could reduce it to a more standard problem: Morally speaking, I have
\begin{align}
c\_K(t)=\mathrm{Im}\log\det\left(\frac{e^{tK}-Je^{tK}J}{2}\right)=\mathrm{Im}\mathrm{Tr}\log\left(\frac{e^{tK}-Je^{tK}J}{2}\right)\,.
\end{align}
The tricky thing is that $\log{(e^x... | 1 | https://mathoverflow.net/users/80903 | 357787 | 150,817 |
https://mathoverflow.net/questions/357772 | 1 | Let $\mathcal{B}(x,r)$ the ball of center $x \in \mathbb{R}^n$ and radius $r>0$ (so $\mathcal{B}(x,r) = \{y \in \mathbb{R}^n : \|y-x\| \leq r\}$, where all norms are $\ell^2$-norms).
I would like to express the following integral analytically:
\begin{equation}
\int\_{\mathcal{B}(x,r)} \exp\left(\frac{-\|y-\mu\|^2}{... | https://mathoverflow.net/users/156454 | How to compute integral of a gaussian over a noncentered ball? | Without a real loss of generality, I can assume that $x=0$, $2r^2=1/π$, and I check
$$
J(x)=\int\_{\vert y\vert\le 1} e^{-π\vert x-y\vert^2} dy=(G\ast\mathbf 1\_{\mathbb B^n})(x),\qquad G(x)=e^{-π\vert x\vert^2}.
$$
The function $J$ is radial (if $A\in O(n)$, just calculate $J(Ax)$ by the change of variables $y=Az$) so... | 2 | https://mathoverflow.net/users/21907 | 357796 | 150,823 |
https://mathoverflow.net/questions/357663 | 4 | Let $\mathfrak{g}$ be a simple Lie algebra with Cartan subalgebra $\mathfrak{h}$, $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak g$, $Z(\mathfrak g)$ the center of $U(\mathfrak{g})$, $\lambda$ a complex weight and $\chi$ the central character of the Verma module with highest weight $\lambda$. Let $A\_... | https://mathoverflow.net/users/155568 | Maximal quotients of the enveloping algebra of a simple Lie algebra | This is proved in Chapter 8 of [J. Dixmier, Enveloping algebras](http://www.ams.org/books/gsm/011) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=1393197)).
See also 3.1. in [J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren](https://link.springer.com/book/10.1007%2F978-3-642-68955-0) ([MSN](https... | 5 | https://mathoverflow.net/users/15292 | 357798 | 150,824 |
https://mathoverflow.net/questions/357713 | 0 | $r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$.
1. Is it true that there are constants $$\alpha,\beta,\gamma,\delta>0$$ such that
$$|u|,|v|<\alpha r^2\implies|z|>\beta r^6$$
$$|z|<\gamma r^6\implie... | https://mathoverflow.net/users/136553 | Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples | Consider the Linear Diophantine Equation $$a^{2t}u+b^{2t}v+c^2z=0$$ where $t\geq2$.
There is always a $(u,v,z)\neq(0,0,0)$ solution with $\|(u,v,z)\|\_\infty=O(r^{4(t-1)})$ since $c^2|(a^{2t}-b^{2t})$ and we can take $(u,v,z)=(-1,1,\frac{a^{2t}-b^{2t}}{c^2})=(-1,1,\frac{a^{2t}-b^{2t}}{a^2+b^2})$.
Thus for $a^{4}u+b... | 0 | https://mathoverflow.net/users/136553 | 357812 | 150,828 |
https://mathoverflow.net/questions/357392 | 4 | Let $K$ be a finite $d$-dimensional simplicial complex embedded in $\mathbb{R}^{d+1}$. The setting of this question is simplicial homology with coefficients over $\mathbb{Z}\_2$. By Alexander duality $K$ partitions $\mathbb{R}^{d+1}$ into $\beta\_d + 1$ connected components, where $\beta\_d$ is the $d$th Betti number o... | https://mathoverflow.net/users/156106 | Planar duality generalized to embedded simplicial complexes | Like you guessed, the dual of a cycle in $G$ is a set of $d$-simplices in $K$ whose removal increases the rank of the $(d-1)$-th homology group.
In the following I use that $H\_i(X;\mathbb{Z}\_2) \cong H^i(X;\mathbb{Z}\_2)$. It might have been better to keep track of the distinction...
Consider the closures of the ... | 1 | https://mathoverflow.net/users/75344 | 357833 | 150,835 |
https://mathoverflow.net/questions/337158 | 5 | I want to solve a linear program but with a subset of the variables taken from a unit sphere.
That is, given fixed $\textbf{c} \in \mathbb{R}^{n}$, $\textbf{A} \in \mathbb{R}^{m \times (n+k)}$,
I want to find variables $\left[ \begin{array}{c} \textbf{x} \\ \textbf{y} \end{array} \right]$ with $\textbf{x} \in \mathbb{R... | https://mathoverflow.net/users/143718 | Solving a linear program, but over the unit sphere | Going by the first comment, the optimal solution to the convex problem (= replaced by $\leq$) must give a solution on the unit sphere. Firstly, Since $\{0,0\}$ is a feasible point, the optimal value cannot be positive. It can either be $0$ or negative. If its zero, an optimal solution can be scaled to lie on the unit s... | 1 | https://mathoverflow.net/users/155380 | 357837 | 150,837 |
https://mathoverflow.net/questions/357789 | 4 | The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.
Let $G$ be a locally compact Abelian group and let $bG$ denote its Bohr compactification (the Pontryagin dual of $\widehat{G}$ with the discrete topology). Denote by $\mathfrak{... | https://mathoverflow.net/users/15129 | Measure algebra on the Bohr compactification vs the bidual algebras | I'm going to say no. The "canonical" pairing of $M(bG)$ with $L^\infty(G)$ is to integrate a function in $L^\infty(G)$ against the restriction to $G$ of a measure in $M(bG)$. But this is not faithful: any measure supported on $bG\setminus G$ would go to zero in $L^\infty(G)^\*$. To pick up mass on this corona we would ... | 3 | https://mathoverflow.net/users/23141 | 357844 | 150,839 |
https://mathoverflow.net/questions/357741 | 3 | Suppose that $X$ is a complex, rational, $\mathbb{Q}$-Gorenstein $3$-fold with at most terminal singularities, and $H\_{2}(X,\mathbb{R})=\mathbb{R}$. Is $X$ Fano?
By Kleiman ampleness criterion this is equivalent to $(-K\_{X})^3>0$, but the standard argument from the smooth case of using Kodaira dimension doesn't qui... | https://mathoverflow.net/users/99732 | Singular $3$-fold with $b_{2}=1$ | I guess this just follows from exercise 4.1.9 here <https://www.dpmms.cam.ac.uk/~cb496/birgeom.pdf>.
| 0 | https://mathoverflow.net/users/99732 | 357846 | 150,840 |
https://mathoverflow.net/questions/357751 | 6 | I have a question about the combinatorial Laplacian $\Delta$ which is defined by
$$\Delta(u,v)=c(u)1\_{u=v}-c(u,v)$$
where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a conductance function defined on the edge $uv$ (i.e. weighted functions).
>
> If I define a function $F: V\to \mathbb{R}$, we... | https://mathoverflow.net/users/168083 | How to understand the combinatorial Laplacian $\Delta$ which is defined on the graph? | Just to add an (in my opinion important) piece of information. Say $F$ is a function on the vertices of a graph, so $F:V \to \mathbb{R}$. Then $\nabla F$ is a function from the edges to $\mathbb{R}$ (here I see an edge as a pair of vertices $(x,y)$, so edges are oriented):
$$\nabla F (x,y) := F(y) - F(x)$$
Now this def... | 10 | https://mathoverflow.net/users/18974 | 357850 | 150,841 |
https://mathoverflow.net/questions/357862 | -3 |
>
> Question - I am thinking to present one or two papers on Sieve Theory in my masters thesis. I will also present 3 other papers on Riemann Zeta Function which I have studied earlier . But I have no previous knowledge of Sieve Theory. Although I know about some books in sieve theory like Montgomery ( Topics in mult... | https://mathoverflow.net/users/151209 | References of research papers which lead to starting of Sieve Theory | Sieve theory as such is generally considered to have started with Brun's 1915 and 1919 papers. The titles are "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare" and ""La série $1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+\cdots$, où les dénominateurs sont nombres premiers jumeaux est converge... | 3 | https://mathoverflow.net/users/127690 | 357864 | 150,847 |
https://mathoverflow.net/questions/357828 | 19 | A stealth missile $M$ is launched from space station. You, at another space station far away, are trusted with the mission of intercepting $M$ using a single cruise missile $C$ at your disposal .
You know the target missile is traveling in straight line at constant speed $v\_m$. You also know the precise location an... | https://mathoverflow.net/users/75935 | Intercept the missile | There is no hope. Timothy Budd already explained why staying on the sphere $S\_t$ of possible locations of $M$ will not work; so this is to explain why leaving it will not help. What is enough to show is the following **claim**: at large times, if $t\_1<t\_2$ and $C\_{t\_{1,2}}$ are at $\epsilon$-neighborhoods of $S\_{... | 8 | https://mathoverflow.net/users/56624 | 357867 | 150,849 |
https://mathoverflow.net/questions/357868 | 2 | Regarding the matrix inner product based on singular values, Lewis (1995) *"The convex analysis of unitarily invariant matrix functions"* states the result by von Neumann that $\langle X,Y \rangle \leq \langle \sigma\_X ,\sigma\_Y \rangle$. Does anyone know any easy proof or reference for it. I couldn't understand the ... | https://mathoverflow.net/users/156509 | Bound for matrix inner product based on singular values | For a "pedagogical" proof, see [A Note on von Neumann's Trace Inequality](http://page.math.tu-berlin.de/~grigo/10_Neumann_trace.pdf) by Rolf Dieter Grigorieff.
>
> It has been remarked in the literature that *"unexpectedly, finding a
> decent proof of this seemingly simple result turns out to be anything
> but tr... | 1 | https://mathoverflow.net/users/11260 | 357873 | 150,851 |
https://mathoverflow.net/questions/357865 | 1 | $\DeclareMathOperator\U{U}\DeclareMathOperator\GL{GL}$Let $E/F$ be a quadratic extension of local fields and $G=U(V)$ a unitary group associated to hermitian space $V$ over $E/F$. We fix a minimal parabolic subgroup $P\_0$ of $G$ and call $P=NM$ a standard parabolic subgroup of $G$ if it contains $P\_0$. Write $M=\GL\_... | https://mathoverflow.net/users/29422 | Weyl group actions on standard parabolic subgroups of classical groups | Not quite, since the Weyl group of the quasi-split unitary group is not the symmetric group. See page 1272 of Goldberg, "R-Groups and Elliptic Representations of Unitary Groups," <http://www.jointmathematicsmeetings.org/proc/1995-123-04/S0002-9939-1995-1224616-6/S0002-9939-1995-1224616-6.pdf>
| 1 | https://mathoverflow.net/users/64244 | 357875 | 150,852 |
https://mathoverflow.net/questions/357283 | 7 | So-called **$\Delta$-generated** spaces are topological spaces in which paths "determine" the topology of the space. In particular, $X$ is $\Delta$-generated if a set $U\subseteq X$ is open (resp. closed) if and only if $\alpha^{-1}(U)$ is open (resp. closed) in $[0,1]$ for all paths, i.e. continuous functions, $\alpha... | https://mathoverflow.net/users/5801 | Paths through convergent sequences in $\Delta$-generated spaces | The answer to the question is negative. To construct a counterexample, choose a maximal almost disjoint infinite family $\mathcal A$ of infinite subsets of $\omega$.
Endow $\mathcal A$ with the discrete topology and consider the product $[0,1]\times \mathcal A$. For every subset $A\subseteq \omega$, let $$2^{-A}=\{0\... | 3 | https://mathoverflow.net/users/61536 | 357885 | 150,858 |
https://mathoverflow.net/questions/357901 | 15 | I am looking for a reference to the following simple statement; it must be classical. (It is easy to proof, but I want to have a reference.)
>
> A metric space $X$ that corresponds to a Riemannian manifold $(M,g)$ completely determines the underlying smooth manifold $M$ and the metric tensor $g$.
>
>
>
| https://mathoverflow.net/users/1441 | Riemannian manifold as a metric space | Isn't this the [Myers-Steenrod theorem](https://en.wikipedia.org/wiki/Myers%E2%80%93Steenrod_theorem)? "If $(M,g)$ and $(N,h)$ are connected Riemannian manifolds and $f:(M,d\_g)\to(N,d\_h)$ is an isometry, then $f:(M,g)\to(N,h)$ is a smooth isometry"
| 20 | https://mathoverflow.net/users/156492 | 357902 | 150,862 |
https://mathoverflow.net/questions/357889 | 3 | In Brezis's Functional Analysis book through chapters 3-4, I've seen the $\sigma(L^\infty,L^1)$ topology on $L^\infty$ but did not see (so far) any application of it in differential equations. Is there an example of a differential equation where this topology might be helpful in analyzing the $L^\infty$ norm of the sol... | https://mathoverflow.net/users/105925 | Usefulness of the $\sigma(L^\infty,L^1)$ topology in the context of differential equations | This topology is ubiquitous in modern analysis of PDEs, typically in conjunction with the [Banach-Alaoglu theorem](https://en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem). (Bounded sets in $L^\infty$ are relatively compact for the predual $\sigma(L^\infty,L^1)$ / weak-$\ast$ topology.)
Here is one (not so) spec... | 5 | https://mathoverflow.net/users/33741 | 357907 | 150,863 |
https://mathoverflow.net/questions/357384 | 4 | Let $x\_0<x\_1<\ldots<x\_n$ and $f\_0,f\_1,\ldots,f\_n$ be real numbers and
$$s\_i=(f\_i-f\_{i-1})/(x\_i-x\_{i-1}),~~~c\_i=(s\_{i+1}-s\_i)/(x\_{i+1}-x\_{i-1}).$$
If $f$ is a convex function defined on $[x\_0,x\_n]$ with $f(x\_i)=f\_i$ for $i=0,\ldots,n$ then all $c\_i$ are nonnegative. Conversely, this condition guar... | https://mathoverflow.net/users/56920 | algorithm for convex $C^2$ interpolation | If $c\_i$'s are all positive, there are infinitely many such convex $C^2$ functions. As I have pointed out in my comment above, nonnegativity of $c\_i$'s is insufficient to guarantee the existence of a $C^2$ function. One very simple construction via Bezier curve is as follows.
Draw a straight line through each poin... | 2 | https://mathoverflow.net/users/32660 | 357909 | 150,864 |
https://mathoverflow.net/questions/357800 | 0 | Let $X\in\{0,1\}^2$ have mean $\mu=\left[\begin{smallmatrix}p\_1\\p\_2\end{smallmatrix}\right]$ and $\Pr[X\_1 = X\_2 = 1] = p\le \min\{p\_1,p\_2\}$.
(Note we must have $1-p\_1-p\_2+p\ge 0$ for the distribution to be well defined.)
We can then compute the covariance matrix $\Sigma = E[(X-\mu)(X-\mu)^T] = \left[\begin{sm... | https://mathoverflow.net/users/5429 | Bounding $E[\|\Sigma^{-1/2}(X-\mu)\|_2^3]$ for 2-dimensional Bernoulli | Your inequality does not hold in general:
You don't need to compute to compute $\Sigma^{-1/2}$, because $\|\Sigma^{-1/2} x\|\_2^2=x^T\Sigma^{-1}x$ for all $x\in\mathbb R^2$. Using this simple observation with e.g. $p=0$, $p\_1=1/2$, $p\_2=1/2-\epsilon$, and $\epsilon\downarrow0$, we find
$$\gamma
=\frac{\left(1-\eps... | 1 | https://mathoverflow.net/users/36721 | 357912 | 150,865 |
https://mathoverflow.net/questions/357572 | 2 | Recall that a topological ring $A$ is Tate if there is an open subring $A\_0$ such that the induced topology on $A\_0$ is t-adic for some $t \in A\_0$ that becomes a unit in $A.$ One can, given a Tate ring define a notion of powerbounded elements, denote the ring of powerbounded elements by $A^\circ.$ One says that the... | https://mathoverflow.net/users/145417 | Reduced complete Tate ring which is not uniform? | While I do not have an idea as to how easy or difficult the following is compared to other examples, here is *an* example:
Let $A\_0$ be the subring of $\mathbb{Z}\_p[[T]]$ consisting of all the power series $\sum\_i a\_iT^i$ satisfying the condition
$$v\_p(a\_i)\geq \sqrt{i}$$
(here $v\_p$ denotes the $p$-adic valua... | 3 | https://mathoverflow.net/users/60903 | 357919 | 150,869 |
https://mathoverflow.net/questions/357879 | 7 | There is a supermartingale convergence theorem which is often cited in texts which use Stochastic Approximation Theory and Reinforcement Learning, in particular the famous book "Neuro-dynamic Programming" the theorem is:
"Let $Y\_t, X\_t, Z\_t, t = 1,2,3,....$ be three sequences of random variables and let $\mathcal{... | https://mathoverflow.net/users/123273 | Proof of extended supermartingale convergence theorem | Here's one approach.
First notice that
$$
R\_t: = Y\_{t} + \sum\_{i=1}^{t-1} X\_i - \sum\_{i=1}^{t-1} Z\_i
$$
is a supermartingale, since
$$
R\_{t+1} - R\_t = Y\_{t+1} - Y\_t + X\_t - Z\_t,
$$
giving
$$
E(R\_{t+1} - R\_t | \mathcal{F}\_t) = E (Y\_{t+1} | \mathcal{F}\_t) - Y\_t + X\_t - Z\_t
$$
which is less than ... | 5 | https://mathoverflow.net/users/5784 | 357922 | 150,871 |
https://mathoverflow.net/questions/219938 | 27 | Is there a known example of a ring endomorphism $f: \mathbb{Z}[x\_1, \ldots, x\_n] \to \mathbb{Z}[x\_1, \ldots, x\_n]$ such that $f \circ f = f$ but whose image is not isomorphic to a polynomial ring?
My own interest in this has to do with better understanding the (categorical) Cauchy completions of Lawvere theories... | https://mathoverflow.net/users/2926 | What are retracts of polynomial rings? | Existence of such an example follows from the same result of Asanuma that is crucial for Gupta's work, see the article
Teruo Asanuma, "Polynomial fibre rings of algebras over noetherian rings", Inventiones mathematicae 87 (1987), 101–127 ([DOI link](https://doi.org/10.1007/BF01389155)).
It follows from Corollary ... | 13 | https://mathoverflow.net/users/1306 | 357926 | 150,872 |
https://mathoverflow.net/questions/357138 | 3 | What can be said about the solutions of, resp. solving, the following system polynomial equations, in which the $x\_i$ and $y\_j$ are the variables and $c\_{i,j},\,d\_i\in\mathbb{R}$ are constants:
$$\begin{matrix}
c\_{0,1}x\_1+\,\cdots+\,c\_{0,n}x\_n&=&d\_0\\
c\_{1,1}x\_1y\_1+\,\cdots+\,c\_{1,n}x\_ny\_n&=&d\_1\\
c\... | https://mathoverflow.net/users/31310 | Solvability of a system of polynomial equations | For the special case where all $c\_{i,j}$'s are equal to 1 and $m=2n-1$, take a look at Ramanujam's paper: <http://ramanujan.sirinudi.org/Volumes/published/ram03.pdf>. Needless to say, it is an ingenious method. The steps are the following:
(1) The key idea is to recognize that the coefficients (w.r.t $\theta$, upto ... | 3 | https://mathoverflow.net/users/155380 | 357941 | 150,877 |
https://mathoverflow.net/questions/357841 | 6 | For two $k$-partitions $X,Y\in k^\omega$ of $\omega$
(seen as functions $\omega\rightarrow k$),
we say $X,Y$ are *almost disjoint*
iff $X^{-1}(i)\cap Y^{-1}(i)$ is finite
for all $i<k$.
Question: Does there exist a set
$Q\subseteq 3^\omega\times (2^\omega)^r$
such that:
1. for every $X\in 3^\omega$, there exists... | https://mathoverflow.net/users/74918 | Complexity of a combinatorial constraint | The answer is yes for $r=3$. Take an ultrafilter $\mathcal{U}$ on $\omega$. Define a function $f \colon 3^\omega \rightarrow 3$ such that $f(X):=i$ iff $X^{-1}(i) \in \mathcal{U}$. Note that if $X\_1$ and $X\_2$ are almost disjoint, then $f(X\_1) \neq f(X\_2)$.
Let $Q:=\{(X,Y) \in 3^\omega \times (2^\omega)^3 \colon ... | 3 | https://mathoverflow.net/users/134910 | 357942 | 150,878 |
https://mathoverflow.net/questions/357931 | 0 | First of all i denote $\{1,2,3,...,m\}$ by $[m]$
Let there be a collection of sets $\alpha=\{A\_{1},A\_{2},...,A\_{m}\}$ such $\bigcup\_{i\in[m]}A\_{i}\subseteq [n]$
Consider any function $f:\mathcal{P}([n])\rightarrow \mathbb{C}$
It is well known that $$f(\bigcup\_{i\in[m]}A\_{i})=\sum\_{i=1}^{m}(-1)^{i-1}\sum\_{... | https://mathoverflow.net/users/156535 | Express inclusion-exclusion principle in terms of matrix operations | As darij grinberg noted, the inclusion-exclusion principle holds only if $f$ is additive over $[n]$ or, equivalently, is a (signed) measure over $[n]$, and then
$$f(A)=\int x\_A\,df$$
for any $A\subseteq[n]$.
Now, to get the inclusion-exclusion principle from properties of indicators $x\_A$ of sets $A$, write
\beg... | 0 | https://mathoverflow.net/users/36721 | 357947 | 150,880 |
https://mathoverflow.net/questions/357929 | 4 | I know that the largest eigenvalue of a graph is bounded between the minimal and maximal row sum of the matrix. If I have a $0-1$ symetric matrix (an adjacency matrix) and I know $k$ of the rows have at least $k$ zeros in them (more specifically, I know $k$ is the maximum size of an all zeros principal sub-matrix of th... | https://mathoverflow.net/users/156518 | Relation of row sums to largest eigenvalue | You are asking for relationships between the maximum independent set and the eigenvalues. If you search with those terms you will find several. For example, [Haemers proved](https://core.ac.uk/download/pdf/82168366.pdf) that the maximum size of an independent set is bounded above by $$ n\frac{-\lambda\_1\lambda\_n}{\de... | 2 | https://mathoverflow.net/users/9025 | 357948 | 150,881 |
https://mathoverflow.net/questions/357950 | 0 | Let $(E,\mathcal{A},\mathbb{P})$ be a probability space $\{X\_n\}$ be a sequence of random variable, such that:
$$
(1)~.~~~\sup\_n\mathbb E (|X\_n|)<\infty\Rightarrow
$$
$$
(2)~.~~~\dfrac{M\_j}{2}<\int\_{j-1<|X\_n|\leq j}{|X\_{n}(t)|d\mathbb{P}(t)}\leq M\_j+\dfrac{1}{j^2} \qquad\forall n\geq 1 \text{ et }1\leq j\leq ... | https://mathoverflow.net/users/156512 | $\sum_{n=1}^{\infty}{\frac{1}{n^{1+\epsilon}}\mathbb{E}\big((|X_n|\mathbb{1}_{|X_n|\leq n})^{1+\epsilon}\big)}<\infty,~~\forall\epsilon>0 $ | Let $N\_j:=M\_j+1/j^2$, so that
$$\sum\_1^\infty N\_j<\infty,$$
and for $j\le n$
$$E|X\_n|1\_{j-1<|X\_n|\le j}\le N\_j$$
and hence
$$(j-1)P(j-1<|X\_n|\le j)\le E|X\_n|1\_{j-1<|X\_n|\le j}\le N\_j,$$
so that for $j=2,\dots,n$
$$P(j-1<|X\_n|\le j)\le \frac{N\_j}{j-1}\ll\frac{N\_j}j,$$
where $a\ll b$ means $a=O(b)$. Sin... | 3 | https://mathoverflow.net/users/36721 | 357954 | 150,884 |
https://mathoverflow.net/questions/357776 | 2 | What are examples of expander family of 3-transitive Schreier graphs?
Meaning for an action that is 3-transitive.
It is better to have an option for randomization. We know that choosing 2 elements at random in a simple Lie group leads to expander family of Cayley graphs.
Is the same thing true for example, in ca... | https://mathoverflow.net/users/142777 | Examples of 3-transitive expander family of Schreier graphs | Well, it was solved.
It is a bit long for here but given $G=SL(2,p)$, $Cay(G, S) $ expander,
we can see that $Sc:=Sch[G,P^1(F\_p),S]$ is also an expander, by comparing their mixing time and find it is less than that of the cayley graph.
The action is 3-transitive because the action of $GL$ is.
The idea is that... | 0 | https://mathoverflow.net/users/142777 | 357978 | 150,892 |
https://mathoverflow.net/questions/357961 | 6 | From reading the literature of the 1970s heyday of locally convex spaces, it seems that it was an important open question whether there is an infra-Pták (i.e. $B\_r$-complete) space that is not Pták (i.e. $B$-complete). Is this still open?
| https://mathoverflow.net/users/99234 | Infra-Pták space that is not Pták | Valdivia constructed counterexamples in his paper [Br-Complete Spaces which are not B-Complete](https://eudml.org/doc/173399).
| 4 | https://mathoverflow.net/users/99234 | 357980 | 150,893 |
https://mathoverflow.net/questions/351834 | 2 | Suppose the convex polytope is the set of feasible solutions $\mathbf{x}\in\mathbb{R}^n$ for the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}\,,\; \mathbf{A}\in\mathbb{R}^{m\times n}$ subject to some (linear) constraints $\mathbf{0}\leq\mathbf{x}\leq\mathbf{v}$; how is it possible to select an interior point of said ... | https://mathoverflow.net/users/151189 | Interior point of a convex polytope | Finding all vertices of the polytope would have the same complexity as the Vertex Enumeration Problem. I do not think that's a practical approach.
The polytope is the intersection of the box $0\leq x \leq v$, and the hyperplanes $Ax=b$. A point in the interior of the box, which also lies on the hyperplanes is what i... | 3 | https://mathoverflow.net/users/155380 | 357991 | 150,896 |
https://mathoverflow.net/questions/357984 | 8 | Let $k\geq 2$. Consider the following norm of exponenetial sum:
$$
I(N,p,k)=\int\_0^1\int\_0^1 \left|\sum\_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy.
$$
Bourgain mentioned on Page 118 of
<https://math.mit.edu/classes/18.158/bourgain-restriction.pdf>
that $I(N,6,2)\gtrsim N^3\log N$, where he referenced the foll... | https://mathoverflow.net/users/111012 | Lower bound on exponential sums | There are a few things to clear up.
The first is that, on the page in the Bourgain paper you mention, he actually proves the lower bound $I(N,6,2)\gg N^3\log N$ from the fact that
$$ \left\lvert\sum\_{n=0}^N e(nx+n^2y)\right\rvert \gg N/q^{1/2}$$
whenever $\lvert x-b/q\rvert \ll 1/N$ and $\lvert y-a/q\rvert \ll ... | 8 | https://mathoverflow.net/users/385 | 358004 | 150,900 |
https://mathoverflow.net/questions/303860 | 9 | Has anyone seen the ring $\Lambda[x\_0, x\_1, x\_2, \ldots]/(x\_i x\_j - (i+1) x\_0 x\_{i+j})$ in some natural context?
Here $\Lambda[x\_0, x\_1, x\_2, \ldots]$ is the (graded-)commutative algebra (either over the integers or the integers localized at 2) freely generated by elements $x\_0,x\_1,x\_2,\ldots$ of odd hom... | https://mathoverflow.net/users/6872 | Curious anti-commutative ring | I noticed this now, and I want to remark that the underlying abelian group can in fact be described very precisely. To do that, note that:
(1) the defining relations easily imply that the abelian group of elements of degree $d\ge 2$ in this algebra is certainly generated by $x\_0^{d-1}x\_k$, $k\ge 0$, and
(2) as di... | 6 | https://mathoverflow.net/users/1306 | 358007 | 150,901 |
https://mathoverflow.net/questions/358003 | 9 | A word is cubefree if it cannot be written as $xyyyz$ where $y$ has positive length.
Let $h$ be the morphism from $\{0,1,2,3,4\}^\*$ to $\{0,1\}^\*$ given for words of length 1 as follows ($a\to h(a)$):
$$0\to 001001010011$$
$$1\to 001001101011$$
$$2\to 001010011011$$
$$3\to 001101001011$$
$$4\to 010011001011$$
and e... | https://mathoverflow.net/users/4600 | A cubefree-preserving morphism from 5 to 2? | An $\infty$ to 2 (hence 5 to 2) cube-free morphism was constructed by Bean-Ehrenfeucht-McNulty. The fact that your morphism is cube-free follows from their theorem. See Theorem 2.4.1 of my book "Combinatorial Algebra:syntax and semantics".
| 10 | https://mathoverflow.net/users/nan | 358011 | 150,903 |
https://mathoverflow.net/questions/357997 | 3 | Suppose $\mathfrak{M}$ is a left proper celluar model category and $S$ is a set of cofibrations in $\mathfrak{M}$. What are the generating trivial cofibrations of $L\_S\mathfrak{M}$? Are they $J\cup S$, where $J$ is the set of generating trivial cofibrations of $\mathfrak{M}$?
| https://mathoverflow.net/users/149491 | Generating trivial cofibrations of Bousfield localization | No, definitely not! They are much more complicated to characterize. You need to take horns on the set of morphisms you just wrote down. This is all detailed carefully in Hirschhorn's book, summarized [here](https://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories). Search in there for "generating acycli... | 2 | https://mathoverflow.net/users/11540 | 358013 | 150,904 |
https://mathoverflow.net/questions/358014 | -1 | Let $G$ be a compact Lie group.
>
> Is each conjugacy class a closed subset of $G$?
> Define the conjugacy equivalent relation $g\sim h$ if $g$ is conjugate to $h$.Is $G/\sim$ a Haussdoef space with the quotient topology?Is a there a natural manifold structure on it? If the answer is "yes", is there a natural Riem... | https://mathoverflow.net/users/36688 | A manifold or Riemannian structure on the space of all conjugacy classes of a compact Lie group | After you choose a maximal torus $T$, the space of conjugacy classes is identified with with the quotient $T / W(T)$ of $T$ by the Weyl group, see <https://en.wikipedia.org/wiki/Maximal_torus#Weyl_group>. The quotient is not a manifold in general (if $G$ is simply connected, then it can be identified with a Weyl chambe... | 4 | https://mathoverflow.net/users/17047 | 358018 | 150,905 |
https://mathoverflow.net/questions/357650 | 6 | Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), embedded in $S$, and such that the (weighted) shortest path metric in $G$ is quasi-isometric to $m$ (this means that ther... | https://mathoverflow.net/users/45855 | Quasi-isometric embedding of graphs in non-compact riemannian surfaces | This follows from the usual "economic covering" method: By Zorn (but alternatively, you can easily do it with your bare hands without using the choice axiom), $S$ admits a maximal family $(x\_i)$ of 1-separated points (meaning that the distance between $x\_i$ and $x\_j$ is at least 1 for $i\neq j$). Then, the balls $B\... | 3 | https://mathoverflow.net/users/105095 | 358025 | 150,906 |
https://mathoverflow.net/questions/357982 | 2 | Let $U$ be a bounded domain in the Euclidean space with sufficiently smooth boundary. Let $\{f\_i\}$ be a orthonormal basis of $H^1\_0(U)$ satisfying $-\Delta f\_i = \lambda\_i f\_i$ where $\lambda\_i \leq \lambda\_{i+1}$.
For fixed $N\in\mathbb N$ let $V\_N$ be the subspace of $H^1\_0(U)$ spanned by $\{f\_1, \dots ,... | https://mathoverflow.net/users/56524 | Approximating functions in $H^1_0(U) \cap H^2(U)$ via $H^1$ norm and $L^2$ projection | Yes, this is possible and actually true, up to a slight shift in the index and a missing constant:
$$
\|\nabla u-\nabla \pi\_N(u)\|^2\_{L^2}\leq \frac{d}{\lambda\_{N+1}}\| D^2 u\|^2\_{L^2}.
$$
Here $d$ is the dimension of $U\subset \mathbb R^d$.
**Proof**
Let me first remind a classical fact: Since the $f\_i$'s are o... | 5 | https://mathoverflow.net/users/33741 | 358026 | 150,907 |
https://mathoverflow.net/questions/358020 | 21 | Consider the following curious statement:
>
> $(S)$ $\;$ Let $X$ be a non-empty set and let $f:X \to X$ be fixpoint-free (that is $f(x) \neq x$ for all $x\in X$). Then there are subsets $X\_1, X\_2, X\_3 \subseteq X$ with $X\_1\cup X\_2\cup X\_3 = X$ and $$X\_i \cap f(X\_i) = \emptyset$$ for $i \in \{1,2,3\}$.
>
>... | https://mathoverflow.net/users/8628 | Does the "three-set-lemma" imply the Axiom of Choice? | The three-set lemma is listed as form 285 in Howard and Rubin's "Consequences of the axiom of choice". According to their book, the earliest appearance seems to be a problem in a 1963 issue of the American Mathematical Monthly (problem 5077).
As mentioned already in the comments by Emil Jerabek, this form of choice ... | 33 | https://mathoverflow.net/users/12976 | 358030 | 150,908 |
https://mathoverflow.net/questions/357956 | 5 | Before introducing *block spin transformations* in chapter four of [Random Walks, Critical Phenomena and Triviality in Quantum Field Theory](https://link.springer.com/book/10.1007/978-3-662-02866-7), the authors state the following:
>
> "In this chapter we sketch a specific method for constructing scaling
> (conti... | https://mathoverflow.net/users/150264 | Renormalization group strategies | In statistical mechanics one is mostly interested in some fixed probability measure for some spin configurations on the infinite volume lattice $\mathbb{Z}^d$. The two main problems related to such a measure are P1) the construction of the infinite volume limit and P2) the study of the long distance behavior of correla... | 3 | https://mathoverflow.net/users/7410 | 358032 | 150,909 |
https://mathoverflow.net/questions/358052 | 2 | Let $\mathbb T\_\theta^2$ be quantum tori generated by two unitary operators $u,v$. can $u,v$ be finite dimensional?
| https://mathoverflow.net/users/136860 | A question on quantum tori | Let's fix conventions, so that $vu = e^{2\pi i \theta}uv$. It follows from a result of [Slawny](https://projecteuclid.org/euclid.cmp/1103857742) that $C(\mathbb{T}^2\_\theta)$ is simple iff $\theta$ is irrational, so that $C(\mathbb{T}^2\_\theta)$ has a finite-dimensional representation only if $\theta$ is rational.
... | 4 | https://mathoverflow.net/users/6999 | 358058 | 150,918 |
https://mathoverflow.net/questions/357568 | 5 | Let A be an Artin algebra and assume all modules are basic, then a classical result says that tilting modules $T$ are in bijection with complete cotorsion pairs $(T^{\perp}, \check{ add(T)})$ (with some additional assumptions on the cotorsion pairs) , where $T^{\perp}= \{ X | Ext\_A^i(T,X)=0 $ for all $i >0 \}$ and $\c... | https://mathoverflow.net/users/61949 | On tilting and cotilting modules | We have $^\perp(T^\perp) = \check{\operatorname{add}}\, T$, and $(\check{\operatorname{add}}\, T)^\perp = T^\perp$, and $(^\perp U)^\perp = \hat{\operatorname{add}}\, U$, and $^\perp(\hat{\operatorname{add}}\, U) = {^\perp U}$. Condition 1. and 2. are equivalent to
1. $\hat{\operatorname{add}}\, U \subseteq T^\perp$... | 3 | https://mathoverflow.net/users/130741 | 358076 | 150,927 |
https://mathoverflow.net/questions/358069 | 1 | Let $R$ be a Dedekind domain. Let $A$ and $B$ be two finitely generated domains over $R$. Assume that for every maximal ideal $\mathfrak{p}\subset R$ the $R\_{\mathfrak{p}}$-algebras $A\_{\mathfrak{p}}$ and $B\_{\mathfrak{p}}$ are isomorphic. Are $A$ and $B$ isomorphic?
| https://mathoverflow.net/users/156612 | Locally isomorphic algebras over a Dedekind domain | **Counterexample.** Let $R$ be a Dedekind domain with $\operatorname{Cl}(R) \neq 0$. Let $I \subseteq R$ be an ideal that is not principal (in algebraic geometry language, let $\mathscr L$ be a nontrivial line bundle), and let $J = R$ be the trivial ideal (let $\mathcal O$ be the trivial line bundle). Then $I\_{\mathfr... | 3 | https://mathoverflow.net/users/82179 | 358077 | 150,928 |
https://mathoverflow.net/questions/358057 | 13 | It is not too difficult to show that if $X$ is an infinite set, then there exists a two-element subset of the group $\operatorname{Sym}(X)$ with trivial centralizer iff $\lvert X\rvert \leq \lvert\mathbb{R}\rvert$.
My question is if this is true if we replace $\operatorname{Sym}(X)$ with $\operatorname{End}(X)$.
... | https://mathoverflow.net/users/156473 | For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both? | The answer is no: **for every set $X$ there exists a pair in the monoid $X^X$ of self-maps of $X$, with centralizer reduced to $\{\mathrm{id}\}$**.
(I first left my original "groupwise" answer because it's easier and because it has other follow-up questions. It's now deleted and copied as an [answer to another questi... | 12 | https://mathoverflow.net/users/14094 | 358078 | 150,929 |
https://mathoverflow.net/questions/358086 | 2 | Given a finite set of points $(x\_1, y\_1), (x\_2, y\_2), \ldots, (x\_n, y\_n)$ in the plane, *Linear Regression* tells us how to find the straight line "$y=a+bx$" best approximating the given points, in the sense that the quantity
$$
E(a, b)= \sum\_{i=1}^n\big (ax\_i+b-y\_i\big )^2
$$
is as small as possible. Howe... | https://mathoverflow.net/users/97532 | Non-parametric regression and curvature | Yes, that is the cubic smoothing spline with $\lambda$ (multiplier on the integral, which controls the amount of smoothing) ) = 1. See <https://en.wikipedia.org/wiki/Smoothing_spline#Cubic_spline_definition> .
| 2 | https://mathoverflow.net/users/75420 | 358088 | 150,935 |
https://mathoverflow.net/questions/358096 | 7 | I hope to understand the connection between statistics and information theory in a deep philosophical sense.
I suppose the best place to start would be David MacKay's Information Theory, Inference, and Learning Algorithms, but I was curious what else there may be. Are there any other good books out there like this? W... | https://mathoverflow.net/users/156621 | Books to develop a unified view of statistics and information theory? | The booklength tutorial by Shannon award winner Imre Csiszár and Paul Shields is freely available online [here](https://www.nowpublishers.com/article/Details/CIT-004):
*Information Theory and Statistics: A Tutorial*
I. Csiszár, Rényi Institute of Mathematics, Hungarian Academy of Sciences
There is also an [article]... | 3 | https://mathoverflow.net/users/17773 | 358101 | 150,938 |
https://mathoverflow.net/questions/357899 | 6 | For odd primes, we have an equalizer diagram for the $K(1)$- local sphere given by
$$L\_{K(1)}S \rightarrow K{{ \xrightarrow{\Psi^g}}\atop{\xrightarrow[i\_K ] {}}} K$$
where $g$ is a topological generator of $\mathbb{Z}\_p^{\times}$ and $\Psi^g$ denotes the Adams operation. Now we can apply the functor $Mod(-): \textrm... | https://mathoverflow.net/users/70889 | Descent for $K(1)$-local spectra | It's not quite true: need to require a $p$-adic continuity condition for the $\Psi^g$-semilinear automorphism of the $K(1)$-local $K$-module. You can see <https://arxiv.org/pdf/2001.11622.pdf> Proposition 3.10 for a slight variant which also works at the prime 2.
| 7 | https://mathoverflow.net/users/3931 | 358104 | 150,939 |
https://mathoverflow.net/questions/358103 | 7 | I am reading about the KZ equations in Kassel's Quantum groups. In definition XIX.3.1 (page 455) he defines the differential system $(KZ\_n)$ as
$$
dw = \frac{h}{2\pi\sqrt{-1}} \sum\_{1 \leq i <j \leq n} \frac{t\_{ij}}{z\_i-z\_j}(dz\_i- dz\_j)w $$
where the $t\_{ij}$ are some element of the universal enveloping alge... | https://mathoverflow.net/users/155559 | What kind of object are the solutions of the Knizhnik-Zamolodchikov Equations | There are really 3 levels at which you can define this equation:
1. for functions
$$G:Y\_n \longrightarrow V^{\otimes n}$$ where $V$ is a f.d. module and $h$ is a complex number. You can then take a formal expansion around $h=0$ and regard $G$ as being valued in $V^{\otimes n}[[h]]$.
2. for functions
$$G:Y\_n \longri... | 5 | https://mathoverflow.net/users/13552 | 358115 | 150,943 |
https://mathoverflow.net/questions/340269 | 6 | It is known that the field of multisymmetric rational functions (over a field of characteristic $0$), that is,
rational function in variables $x\_{11}, \ldots, x\_{1m}, \ldots, x\_{n1}, \ldots, x\_{nm}$ that are invariant under the $S\_n$ action $g.x\_{ij}=x\_{g(i)j}$ is rational of transcendent degree $nm$. In the la... | https://mathoverflow.net/users/134624 | Transcendent basis for the field of multisymmetric functions | The answer is yes.
Indeed, Remark 7 of the paper
Paul Görlach, Cordian Riener, Tillmann Weißer. Deciding positivity of multisymmetric polynomials. Journal of Symbolic Computation 74 (2016), 603-616.
(<https://www.sciencedirect.com/science/article/abs/pii/S074771711500098X>, see also <https://arxiv.org/abs/1409.270... | 0 | https://mathoverflow.net/users/1306 | 358118 | 150,945 |
https://mathoverflow.net/questions/357645 | 2 | For any set $X$ let $[X]^2 = \big\{\{a,b\}: a\neq b\in X\big\}$. We say that a graph $G$ is *self-complementary* if $G\cong \bar{G}$ where $\bar{G} = (V, [V]^2\setminus E)$.
Given an infinite cardinal $\kappa$, is there a collection ${\cal C}$ of pairwise non-isomorphic self-complementary graphs on the vertex set $\... | https://mathoverflow.net/users/8628 | Collection of pairwise non-isomorphic infinite self-complementary graphs | As you know, there are $2^\kappa $ nonisomorphic graphs of cardinality $\kappa$ for every infinite cardinal $\kappa$. (In fact there are $2^\kappa$ nonisomorphic *trees* of cardinality $\kappa$, see [this answer](https://mathoverflow.net/questions/281039/mutually-non-isomorphic-connected-graphs-on-kappa-points/358126#3... | 5 | https://mathoverflow.net/users/43266 | 358121 | 150,947 |
https://mathoverflow.net/questions/358133 | 6 | In Ravi Vakil's lecture notes ("Foundations of Algebraic Geometry", Classes 53 and 54) one can find a relative version of Serre duality (Exercise 6.1), namely:
"Suppose $\pi: X\rightarrow Y$ is a flat projective morphism of
locally Noetherian schemes, of relative dimension $n$. Assume all of the geometric fibers
are ... | https://mathoverflow.net/users/nan | Serre duality in families | In this case, Serre duality in families = Grothendieck duality. At the level of generality you are asking, I suggest the paper:
Kleiman, Steven L.: Relative duality for quasicoherent sheaves. *Compositio Math.* **41** (1980), no. 1, 39–60.
<http://www.numdam.org/item/?id=CM_1980__41_1_39_0>
But if you are serious... | 10 | https://mathoverflow.net/users/6348 | 358135 | 150,951 |
https://mathoverflow.net/questions/356429 | 6 | Is there a connected Banach manifold $M$ and a smooth map $f:M \to M$ such that the rank of $Df\_x$ is finite for every $x\in M$ but this rank is not uniformly bounded
| https://mathoverflow.net/users/36688 | A smooth map on a Banach manifold whose pointwise rank is finite but its rank is not globally bounded | Let $\mathbb H$ and $(e\_n)\_{n\geq0}$ be your favourite second countable Hilbert space and orthonormal Hilbert basis. Mine is the space $\ell^2(\mathbb N)$ of square integrable sequences, with $e\_n$ the indicator function of the singleton $\lbrace n\rbrace$. I write, for any $x\in\mathbb H$, $x^n$ for the $n$th coord... | 5 | https://mathoverflow.net/users/129074 | 358150 | 150,955 |
https://mathoverflow.net/questions/357335 | 0 | Does anyone know about a statistical divergence of this type?
\begin{equation}
\text{D}(P||Q) = \frac{1}{2} \left[\text{KL}(M||P) + \text{KL}(M||Q)\right]
\end{equation}
where $M = \frac{1}{2} [P+Q]$.
This divergence is very similar to the Jensen-Shannon Divergence $\text{D}(P||Q) = \text{KL}(P||M) + \text{KL}(Q||M)... | https://mathoverflow.net/users/156139 | Statistical divergence | That is J-divergence, commonly just called 'symmetrised KL divergence.'
* [Kullback, Solomon, and Richard A. Leibler. "On Information and Sufficiency." *The Annals of Mathematical Statistics* 22.1 (1951): 79-86.](https://projecteuclid.org/euclid.aoms/1177729694)
* [Jeffreys, Harold. "An Invariant Form for the Prior P... | 1 | https://mathoverflow.net/users/10668 | 358151 | 150,956 |
https://mathoverflow.net/questions/358149 | 3 | I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf\_\ast$ takes a sheaf $F$ to the sheafification of the cohomology presheaf $V\mapsto \mathrm H^\bullet(f^{-1}V,F)$. When $f$ is the identity, the sheafification is zero fo... | https://mathoverflow.net/users/69037 | Subspace inclusion with non-vanishing higher direct images | You expect the higher direct images of $f: X \to Y$ to be nonzero if for arbitrarily small neighborhoods $y\_0 \in U$ the space $f^{-1}(U)$ has non-vanishing higher cohomology. I.e. $R^i f\_\* \mathbb Z$ vanishes if and only if all of its stalks do.
For instance, think about the inclusion of $\mathbb R^2 - 0$ into $... | 5 | https://mathoverflow.net/users/52918 | 358161 | 150,960 |
https://mathoverflow.net/questions/358107 | 9 | I am currently reading "[Schiffer variations and the generic Torelli theorem for hypersurfaces](https://arxiv.org/abs/2004.09310)" by Voisin, where it is claimed that the subgroup of $\mathrm{SL}\_{2m}$ ($m \geq 3$) which preserves a generic subspace of $\bigwedge^2 \mathbb{C}^{2m}$ of dimension bigger than $3$ must be... | https://mathoverflow.net/users/37214 | Subgroup of $\mathrm{GL}_n$ stabilizing linear subspace skew-symmetric matrices | Here is the worst possible proof of all but one case, namely $m=r=3$ (but this is not included in your question as stated, though I think it is claimed in the paper.)
Let $r$ denote the dimension of a generic subspace $W$ of $\bigwedge^2\mathbb{C}^{2m}$. We assume $3\leq r\leq\frac{1}{2}\operatorname{dim}(\bigwedge^2... | 3 | https://mathoverflow.net/users/51424 | 358164 | 150,962 |
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