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https://mathoverflow.net/questions/381177 | 2 | I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits\_{\mathbb{R}^k}}\_{\sum\_{i=1}^k y\_i=x} e^{-N^2r(\sum\_{i=1}^k y\_i^2-\frac{1}{k}x^2)} dy\_1\dots dy\_k,$$
where this notation means that I want to integrate over $\mathbb{R}^k$ restricted to the plane where $\sum\_{i=1}^k y... | https://mathoverflow.net/users/171313 | Change of variables in a Gaussian integral in matrix form | $\newcommand\R{\mathbb R}\newcommand\1{\mathbf1}$When you say "I want to integrate over $\mathbb{R}^k$ restricted to the plane where $\sum\_{i=1}^{k}y\_i=x$", you have to specify the measure over the plane over which you want to integrate.
It appears you want this measure to be induced by the Lebesgue measure on $\R^... | 5 | https://mathoverflow.net/users/36721 | 381215 | 158,685 |
https://mathoverflow.net/questions/381205 | 1 | It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is **not** a Hopf subalgebra?
| https://mathoverflow.net/users/153228 | Example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra | The Hopf algebra $H = k[x^{\pm 1}]$ with $\Delta(x) = x \otimes x$ is the group algebra of $\mathbb{Z}$, the free group on one generator. Its subalgebra $k[x]$ is the "monoid algebra" for the submonoid $\mathbb{N} \subset \mathbb{Z}$, the free monoid on one generator. $k[x] \subset k[x^{\pm 1}]$ is not a Hopf subalgebr... | 4 | https://mathoverflow.net/users/78 | 381216 | 158,686 |
https://mathoverflow.net/questions/381222 | 4 | Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0\_C$ of the Picard scheme of $C$ is isomorphic to the Jacobian $J\_C$, so in particular the $n$-torsion of the class group of $C$, ${\rm Cl}(C)[n]={... | https://mathoverflow.net/users/50351 | p-torsion in the Picard group of a regular projective curve | Take $p=3$ and $C\subset \mathbb{P}^2$ with equation $y^2 z=x^3 - t z^3$ where $t\in K$ is not a cube. Then $C$ is regular but, putting $L:=K(t^{1/3})$, $C\_L$ is isomorphic to the usual cuspidal cubic (explicitly, the equation becomes $y^2 z=(x - t^{1/3} z)^3$).
Thus, putting $J:=\mathrm{Pic}^0\_{C/K}$, it follows t... | 11 | https://mathoverflow.net/users/7666 | 381226 | 158,688 |
https://mathoverflow.net/questions/381224 | 6 | Let $V$ be a $\mathbb{Z}\_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}\_2$-graded Lie algebra (aka [super Lie algebra](https://en.wikipedia.org/wiki/Lie_superalgebra)). The free super Lie algebra is also graded by the number of generators (the generators $V$ sit in degree 1, and th... | https://mathoverflow.net/users/2622 | Lie powers of a graded vector space and Klyachko's theorem | Let me work over $\mathbb{C}$ for simplicity. We have
$$L(V) \cong \bigoplus\_{n\ge0} V^{\otimes n} \otimes\_{S\_n} \text{Lie}(n),$$
where $\text{Lie}(n)$ is the $n^{th}$ space of the Lie operad, with specific $S\_n$ actions on $\text{Lie}(n)$ and $V^{\otimes n}$ to be precised below; this is, to be clear, true for... | 8 | https://mathoverflow.net/users/290 | 381232 | 158,689 |
https://mathoverflow.net/questions/359882 | 11 | *This series is divergent; therefore, we may be able to do something with it.* -- Oliver Heaviside
[Edit (1/14/21) from the answer by Count Iblis to a recent MO-Q on math vids: An enthusiastic intro is that to the set of lectures by Carl Bender "[Perturbation and Asymptotic Series](https://www.youtube.com/playlist?li... | https://mathoverflow.net/users/12178 | Resources for divergent / asymptotic series | As far as on-line-available things go, I've attempted to modernize some arguments and give examples of asymptotics of integrals (both Watson's Lemma and easy Laplace/saddle-point examples), as well as asymptotics for ordinary differential equations, both regular and certain irregular singular points. On-line, as well a... | 2 | https://mathoverflow.net/users/15629 | 381233 | 158,690 |
https://mathoverflow.net/questions/381039 | 10 | Let $(W,S)$ be a Coxeter group, $I\subseteq S$ a subset of simple reflections, and $W\_I \subseteq W$ the corresponding parabolic subgroup (we could also assume $|W\_I|<\infty$, if needed).
Let also $t\_1,t\_2\in W$ be two reflections (i.e. elements in $W$ conjugated to some $s\_1,s\_2 \in S$ respectively) such tha... | https://mathoverflow.net/users/95513 | Product of two reflections lying in a parabolic subgroup of a Coxeter group | $\DeclareMathOperator\Im{Im}\DeclareMathOperator\Fix{Fix}$As [suggested](https://mathoverflow.net/questions/381039/product-of-two-reflections-lying-in-a-parabolic-subgroup-of-a-coxeter-group/381234#comment968750_381039) by Sam, I am posting this as an answer.
As mentioned in the comments, the above question can be so... | 7 | https://mathoverflow.net/users/26751 | 381234 | 158,691 |
https://mathoverflow.net/questions/381241 | 1 | I have the following recursive sequence:
$Z\_k = Z\_{k-1} - AA^TZ\_{k-1}xx^T$ where $Z\_k \in \mathbb R^{n \times d}, A \in \mathbb R^{n \times d}, d > n, rank(A) = n, x \in \mathbb R^{d \times 1}$
$A$ is a constant matrix, $x$ is a constant vector. Theoretically, this sequence of matrices $Z\_k$ is entirely determ... | https://mathoverflow.net/users/169760 | Recurrence relation in matrices | Let $B=AA^T$ and $X=xx^T$. Note that
$X^k=\|x\|^{2k-1}X$.
\begin{align\*}
Z\_1& = Z\_{0} - BZ\_{0}X \\
%
Z\_2& = Z\_{1} - BZ\_{1}X
%
= Z\_{0} - BZ\_{0}X -B\big[Z\_{0} - BZ\_{0}X\big]X
%
= Z\_{0} - 2BZ\_{0}X + B^2Z\_{0}X^2 \\
%
Z\_3& = Z\_{2} - BZ\_{2}X
%
= Z\_{1} - BZ\_{1}X -B\big[Z\_{1} - BZ\_{1}X\big]X
%... | 5 | https://mathoverflow.net/users/170118 | 381243 | 158,693 |
https://mathoverflow.net/questions/381254 | 6 | I am interested in the category-theoretic description of trees (and operads?) and have started a course of study that will allow me to engage with these two ([1](https://arxiv.org/pdf/0807.2874.pdf), [2](http://mat.uab.cat/%7Ekock/cat/polynomial.pdf)) manuscripts of Joachim Kock.
An essential prerequisite to the earl... | https://mathoverflow.net/users/172326 | How do we describe the right adjoint? | It's worth thinking about the simplest example, namely when $A$ is a single point. In this case, $g^\*$ is the product functor $-\times B$, and its right adjoint is the set of sections: it sends $f\colon X\to B$ to the set $\Gamma(B,X)$ of sections $s\colon B\to X$ of $f$. This is vaguely similar behaviour to how the i... | 7 | https://mathoverflow.net/users/4177 | 381255 | 158,695 |
https://mathoverflow.net/questions/381168 | 1 | Consider the bounded linear operator $M\_a$ defined by $M\_au(x)=\frac{1}{\sqrt{a}}u\left(\frac{x}{a}\right)$, for $a>1$. On $L^2(\mathbb{R})$, it is easy to see that this is a unitary operator and that (either directly or by an application of Stone's theorem to the continuous one-parameter group) it has spectrum the e... | https://mathoverflow.net/users/142740 | Spectrum decomposition of the scaling operator on weighted spaces | It is continuous spectrum, since the adjoint has no eigenvalues. In fact $M\_a^\*v(y)=\sqrt {a} v(ay) e^{(1-a^2)y^2}$ (duality with respct to the measure $e^{-y^2} dy$) and assuming $M\_a^\* v=\lambda v$ we get the functional equation
$$\sqrt av(ay)e^{(1-a^2)y^2}=\lambda v(y) \quad {\rm or}
\quad v(ay)e^{-a^2y^2}=\mu v... | 1 | https://mathoverflow.net/users/150653 | 381264 | 158,698 |
https://mathoverflow.net/questions/381251 | 2 | Reading a book about Ramsey theory this is the first example of a compact (semitopological) semigroup, which is a nonempty semigroup S with compact Hausdorff topology for which $x \mapsto x\*s$ is a
continuous map for all $s$ in $S$
>
> If $X$ is a compact Hausdorff space, then the Tychonov cube $X^X$ is a compact ... | https://mathoverflow.net/users/153785 | The Tychonov cube $X^X$ of a compact space $X$ is a compact semigroup with the composition operation | To check that $f \to f \circ g$ is continuous in $f$ as a map $X^X \to X^X$ for a fixed $g \in X^X$: take a net $f\_i \to f$ ($i \in I$, some directed set) in $X^X$ converging to $f \in X^X$. This means exactly that $$\forall x \in X: f\_i(x) \to f(x)\tag{1}$$ in $X$. So in particular for any $x \in X$, $(f\_i \circ g)... | 3 | https://mathoverflow.net/users/2060 | 381272 | 158,699 |
https://mathoverflow.net/questions/381122 | 4 | Given an infinite field $k$, consider a quiver $\Gamma$ with one vertex and two arrows $x,y$ and define $R=k\Gamma/(x,y)^2.$ This is a three-dimensional $k$-algebra.
Now consider the additive group of 2 by 2 matrices over $R$ denoted by $M\_2(R)$. To define the multiplication, consider matrices $$X\_t=\begin{bmatrix}... | https://mathoverflow.net/users/131868 | Using computer algebra to check if a family of algebras are pair-wise non-isomorphic | There's a practical computer-algebra way to determine whether
(a) they're all isomorphic up to finitely many exceptions, or
(b) there "(bounded finite)-to-one" non-isomorphic, i.e., there exists $n$ such that for each $t$ the set of $s$ such that $A\_s\simeq A\_t$ has cardinal $\le n$.
See [this question](https:/... | 1 | https://mathoverflow.net/users/14094 | 381276 | 158,700 |
https://mathoverflow.net/questions/381268 | 1 | Let $X$ and $Y$ be random variables on the same probability space. The $\infty$-Wasserstein distance between $X$ and $Y$ is defined as
$$d\_{\infty}(X, Y) = \inf \|X\_1 - Y\_1\|\_{L\_{\infty}},$$
where the infimum is over all random variables $X\_1$ and $Y\_1$ with same distribution as $X$ and $Y$, respectively.
... | https://mathoverflow.net/users/78173 | Definition of Wasserstein distance through cumulative distribution | $\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $d:=d\_\infty(X,Y)$, $\|\cdot\|:=\|\cdot\|\_{L\_\infty}$,
$$E:=\{\ep>0\colon F\_X(t-\ep)\le F\_Y(t)\le F\_X(t+\ep) \text{ for all } t \in \R\},$$
$$D:=\inf E.$$
We need to show that $d=D$.
Take any real $c>d$. Then for some random variables (r.v.'s) $X\_1$ and $Y\_1... | 3 | https://mathoverflow.net/users/36721 | 381278 | 158,701 |
https://mathoverflow.net/questions/381248 | 0 | On page 2 of *"Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm"* by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at [arXiv](https://arxiv.org/abs/1312.6536), they claim the following on the bottom right of the page:
>
> "...in the stationary case the intensity of ... | https://mathoverflow.net/users/172324 | Question about the intensity of a cox process, Diggle–Moraga–Rowlingson–Taylor (2013) | The [intensity function](https://mathworld.wolfram.com/IntensityFunction.html) of a [point process](https://en.wikipedia.org/wiki/Point_process#Definition) $(N\_B)$ over $\mathbb R^d$ is defined as the density of the intensity measure $\mu$ of $(N\_B)$ relative to the Lebesgue measure over $\mathbb R^d$ (in your case, ... | 2 | https://mathoverflow.net/users/36721 | 381283 | 158,702 |
https://mathoverflow.net/questions/380865 | 2 | A classic reference on cluster expansions in mathematical physics (specially statistical mechanics) is [these lecture notes by professor Brydges](http://www.math.ubc.ca/%7Edb5d/Seminars/les_houches_84.pdf) for a les Houches course in 1984 on the mentioned topic. However, these lectures took place over 30 years ago and ... | https://mathoverflow.net/users/150264 | What's the current state of cluster expansions? | An impressive result of the method of cluster expansion is to derive the van der Waals equation of state for gases. Indeed, two-body interactions give the corrections to the ideal gas law. So it can be used for practical calculations!
But as Abdelmalek has remarked, the method is often used in mathematical physics to... | 5 | https://mathoverflow.net/users/172359 | 381285 | 158,703 |
https://mathoverflow.net/questions/343521 | 6 | Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, and I can ask that a sheaf be constant after pulling back to the orbits of this action (i.e. rays in the fibers of $E$).... | https://mathoverflow.net/users/87785 | How to understand the Fourier-Sato transform and microlocalization functors? | I know comes from far in the future, and you've probably figured it out by now, but I wanted to add to Vivek's comment.
Along a complex hypersurface $\{f=0\}$ that is non-singular at point $p$, there are identifications $\left [ \mu\_{\{f=0\}}(F) \right ]\_{(p;d\_p f)} = (\varphi\_f F)\_p = R\Gamma\_{\{Re(f) \geq 0\}... | 2 | https://mathoverflow.net/users/11906 | 381293 | 158,706 |
https://mathoverflow.net/questions/381290 | 0 | Suppose $W\_p(\mu\_n,\mu)\to 0$ and $d(E(\mu\_n),E(\mu))<r\_n$. Here, $W\_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu\_n,\mu$ are probability measures on some metric space (say, $\mathbb{R}^n$, but this isn't crucial). Can anything be concluded regarding the convergence rate of ... | https://mathoverflow.net/users/171444 | Arbitrarily bad rates of convergence in Wasserstein metric | Since you are talking about "[c]onvergence of the first moments", I will be assuming that by $E(\mu)$ you mean the mean of the probability measure $\mu$.
Anyhow, the convergence of the first moments does not help at all. For instance, let $\mu$ be the distribution of the constant-zero random variable (r.v.), that is,... | 0 | https://mathoverflow.net/users/36721 | 381300 | 158,711 |
https://mathoverflow.net/questions/381279 | 1 | What is the relationship between the spaces $X\_1\triangleq \mathscr{P}(C([0,1],\mathbb{R}))$ and $X\_2\triangleq C([0,1],\mathscr{P}(\mathbb{R}))$; where $\mathscr{P}(\cdot)$ denotes the Borel probability measures on a space and it is equipped with the total-variation topology. Specifically, I wonder, is $X\_1$ contin... | https://mathoverflow.net/users/36886 | Embeddings of spaces of probability measures | This is an extended version of my comment above. It is not an answer, or at least: not a positive answer. (Perhaps it is sort of a negative answer?)
---
There is a (sort of natural) candidate for an embedding $\Phi$ from $X\_2$ into $X\_1$, defined as follows. Take a family of measures $\mu = (\mu\_t, t \in [0,1]... | 1 | https://mathoverflow.net/users/108637 | 381303 | 158,712 |
https://mathoverflow.net/questions/381309 | 0 | Let $T\in B(\mathcal{H} \otimes \mathcal{H})$ where $\mathcal{H}$ is a Hilbert space. We can define operators
$$T\_{[12]}= T \otimes 1;\quad T\_{[23]}= 1 \otimes T$$
and if $\Sigma: \mathcal{H} \otimes \mathcal{H} \to \mathcal{H} \otimes \mathcal{H}$ is the "flip" map, then we can define
$$T\_{[13]}= \Sigma\_{[23]}T\_{... | https://mathoverflow.net/users/nan | $(ST)_{[13]}= S_{[13]}T_{[13]}$ for $S,T \in B(\mathcal{H}\otimes \mathcal{H}).$ | You can probably do it the way that you suggested, but my instinct is to try and exploit the idea that the leg notation is really just about relabelling of the factors in the triple tensor product of Hilbert spaces, and "anything you do spatially" will be weak-star continuous at the level of B(triple tensor product).
... | 2 | https://mathoverflow.net/users/763 | 381320 | 158,720 |
https://mathoverflow.net/questions/381194 | 1 | $\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
>
> Let $f\_n \rightharpoonup f$ in $W^{1,2}(\M,\N) $ with $Jf\_n > 0$ a.e., and suppose that the volume $V(\{x \in \M \, | \, Jf\_n... | https://mathoverflow.net/users/46290 | Does weak continuity of Jacobians hold for non nondegenerate maps? | There is a counterexample, however there might be ways to avoid it.
Take $\mathcal{M} = \mathcal{N} =\mathbb{S}^2$, but now consider sequence of maps that cover the sphere twice, where you shrink the preimage of one of them to a point. Specifically consider using the stereographic projection as you did, consider $g\_... | 2 | https://mathoverflow.net/users/51695 | 381326 | 158,722 |
https://mathoverflow.net/questions/381319 | 1 | **TL; DR.**
In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing **extra symmetries**. Explicit examples come from compact groups, and I wonder if there are more such examples.
**Formal description**
To be more precise, let $R$ be an associative ... | https://mathoverflow.net/users/124549 | Making use of extra symmetries; more examples? | Your setup is much more specific than necessary. All you need is two rings $R, S$ with commuting actions on an abelian group $M$ (which is therefore an $(R, S^{op})$-bimodule) such that $M$ is semisimple as an $S$-module. Then, as an $S$-module, $M$ decomposes canonically into a direct sum of isotypic components, and s... | 3 | https://mathoverflow.net/users/290 | 381327 | 158,723 |
https://mathoverflow.net/questions/381257 | 4 | This is a [cross-post from MSE](https://math.stackexchange.com/questions/3968628/polynomial-implicitization-with-alternating-signed-powers) where it did not receive a response.
For integer $n\geq 2$, consider a parameterization of the coordinates $(x\_1, x\_2, ..., x\_{n})$ in terms of the parameters $(s\_{1},s\_{2},... | https://mathoverflow.net/users/18526 | A polynomial implicitization | I think many places will call these functions *supersymmetric power sums*. They appear in the work of Kerov, Okounkov and others on the asymptotic representation theory of the symmetric group (via Thoma's simplex etc.). Many of the usual identities on symmetric functions have supersymmetric analogs, including the Newto... | 7 | https://mathoverflow.net/users/2384 | 381335 | 158,725 |
https://mathoverflow.net/questions/381333 | 3 | Let $X\_{\bullet}^+$ be a strictly simplicial algebraic space and for a morphism $\delta:[m]\to[n]$ in $\Delta^+$, let $\delta:X\_n\to X\_m$ also denote the associated map (by abuse of notation). Then one can consider the category $\operatorname{Mod}(\mathcal{O}\_{X\_{\bullet}^+})$ of $\mathcal{O}\_{X\_{\bullet}^+}$-mo... | https://mathoverflow.net/users/112142 | Derived Category of strictly simplicial algebraic space vs. systems of objects in the derived categories | This kind of functor is flat-out never faithful, because it is very easy for two natural transformations to have components that are equivalent in the derived category without being equivalent in a natural way leading to equality in $D$. For extremely simple diagram shapes it can be full and essentially surjective, but... | 3 | https://mathoverflow.net/users/43000 | 381336 | 158,726 |
https://mathoverflow.net/questions/381341 | 5 | Sufficiently powerful theories (Peano arithmetic, ZFC, and so on — this question came from thinking about Coq) can't prove their own consistency. However, are there cases of two theories, $A$ and $B$, where $A$ proves $B$ is consistent and $B$ proves $A$ is consistent? (To make up a potential example, "Peano arithmetic... | https://mathoverflow.net/users/172385 | Are there "typical" formal systems that have mutual consistency proofs? How long a chain of these can we build? | No, this cannot happen, although it's a little bit trickier than one might expect to prove this!
---
First, a miniature result:
>
> Suppose $T,S$ are computably axiomatizable theories in the language of arithmetic, each containing the theory $\mathsf{I\Sigma\_1}$, with $T\vdash Con(S)$ and $S\vdash Con(T)$. T... | 5 | https://mathoverflow.net/users/8133 | 381346 | 158,731 |
https://mathoverflow.net/questions/381120 | 4 | To any space $X$ you can associate its [de Rham space](https://ncatlab.org/nlab/show/de+Rham+space) $X\_{dR}$. Vector bundles on $X\_{dR}$ are the same thing as vector bundles on $X$ with a flat connection.
Can anything like this be said for **meromorphic** connections?
---
For instance, a naive idea is that th... | https://mathoverflow.net/users/119012 | A de Rham space for meromorphic connections? | Here is one way to construct an object $X\_{mdR}$ for $X$ say, a complex variety. Recall that $X\_{dR}$ can be constructed as follows. Let $\hat{X}$ be the formal completion of the diagonal inside $X\times X$. Then $X\_{dR}$ is the quotient of $X$ by the equivalence relation defined by $\hat{X}$.
Let $\eta$ denote th... | 3 | https://mathoverflow.net/users/51424 | 381347 | 158,732 |
https://mathoverflow.net/questions/381218 | 10 | The exact question I am interested in is the following.
Fix a small $\varepsilon\in(0,1)$ and an integer $q\ge 2$ (you may assume that $q$ is prime if it helps though I believe it shouldn't matter too much). For a large prime $P$ and an integer $a\in\mathbb Z$, define $G(a,P)=\{aq^m\mod P: m=0,1,2,\dots\}$ where the ... | https://mathoverflow.net/users/1131 | Does every geometric progression contain a small remainder modulo a large prime? | $\newcommand{\F}{\mathbb F}$
$\newcommand{\eps}{\varepsilon}$
(As reqested by the OP, and to address @Mark Lewko's comments, here is the argument showing that the statement is true for the primes satisfying a certain condition; the missing counterpart is to prove that almost all primes satisfy the condition in question... | 3 | https://mathoverflow.net/users/9924 | 381354 | 158,735 |
https://mathoverflow.net/questions/381357 | 17 | I am wondering if the orthogonal group $O\_n({\bf Q})$ is dense in $O\_n({\bf R})$?
It is easily checked for $n = 2$ but I think that there is a general principle concerning compact algebraic groups underneath.
| https://mathoverflow.net/users/6129 | Is $O_n({\bf Q})$ dense in $O_n({\bf R})$? | There's an easy argument based on the Cayley transform: If $a$ is a skew-symmetric $n$-by-$n$ real matrix, then $I\_n+a$ is invertible (since $(I\_n-a)(I\_n+a)=I\_n-a^2$ is a positive definite symmetric matrix and hence invertible), and
$$
A = (I\_n-a)(I\_n+a)^{-1}
$$
is orthgonal (i.e., $AA^T = I\_n$). Note that $(I\_... | 40 | https://mathoverflow.net/users/13972 | 381365 | 158,738 |
https://mathoverflow.net/questions/381329 | 2 | Let $A$ be a $C^\*$-algebra (not necessarily unital). Let $X$ be a compact Hausdorff space. We can consider the minimal $C^\*$-tensor product $A \otimes C(X)$. On this space, we can consider the slice map
$$\text{id}\_A\otimes \text{ev}\_x: A \otimes C(X) \to A: a \otimes f \mapsto f(x)a$$
Suppose that $(\text{id}\_A... | https://mathoverflow.net/users/nan | If $(\text{id}_A\otimes \text{ev}_x)(z)= 0$ for all $x \in X$. Do we have $z=0$? | The answer is "yes".
For $A\otimes C(X)$ we have the standard identification with $C(X,A)$ the space of (bounded) continuous maps $X\rightarrow A$ with the sup norm. Here $a\otimes f$ is identified with the function $x\mapsto f(x)a$. Then $(\operatorname{id}\_A\otimes \operatorname{ev}\_x)$ is identified with the map... | 2 | https://mathoverflow.net/users/406 | 381377 | 158,742 |
https://mathoverflow.net/questions/381385 | 0 | The problems
1. 'Given $0<a<b$ and a prime $p<a$ is there an integer $\ell\in[a,b]$ such that $p|\ell$?'
2. 'Given $0<a<b$ and an integer $q\not\in[a,b]$ is there an integer $\ell\in[a,b]$ such that $q$ is coprime to $\ell$?'
are in $\mathsf{NP}$.
Are these problems in $\mathsf{BPP}$ or at least in $\mathsf{P/pol... | https://mathoverflow.net/users/10035 | What are the complexity classes of these problems about divisibility and coprimality? | Both problems are provably in P. For the first one, this is immediate - using division with remainder (which is polynomial time), write $a=qp+r,b=q'p+r'$ with $0\leq r,r'<p$. Then there is $\ell$ divisible by $p$ in this interval iff $r=0$ or $q'>q$. For the record, this doesn't depend on $p$ being prime.
For the sec... | 3 | https://mathoverflow.net/users/30186 | 381389 | 158,746 |
https://mathoverflow.net/questions/381390 | 0 | The question asks what is known about integer solutions $(\mathbf{x}, \mathbf{y}) = ((x\_1, x\_2, x\_3), (y\_1, y\_2, y\_3))$ to the equation
$$\displaystyle x\_1^{r-1} y\_1^r + x\_2^{r-1} y\_2^r = x\_3^{r-1} y\_3^r$$
where $r \geq 4$ is an integer, $x\_1 x\_2 x\_3 y\_1 y\_2 y\_3 \ne 0$, and $\gcd(x\_1 y\_1, x\_2 y... | https://mathoverflow.net/users/10898 | Coprime integer solutions to $x_1^{r-1} y_1^r + x_2^{r-1} y_2^r = x_3^{r-1} y_3^r$ | Assuming the abc conjecture, there are only finitely many solutions with $r\geq 5$. Indeed, more generally, abc conjecture implies there are only finitely many sums of the form $a+b=c$ in which $a,b,c$ are coprime and all prime exponents in them are at least $4$. Indeed, for any such triple we have
$$rad(abc)\leq (abc)... | 4 | https://mathoverflow.net/users/30186 | 381397 | 158,750 |
https://mathoverflow.net/questions/381402 | 4 | I am trying to prove that the following equation is equal to zero.
$$
0=
\sum\_{j=J+1}^N \Big(j (1-q)+ (j-J) (q N-j) \Big) \cdot q^{j} (1-q)^{N -j} \binom{N}{j} \label{zero1}$$
Where
$J,N \in \mathbb{Z}^+$ and $J<N$ and
$0<q<1$ is a probability.
[Numerical simulations (see link)](https://drive.google.com/file/d/1E... | https://mathoverflow.net/users/171932 | How to prove that these partial binomial sums are zero? | Write $p=1-q$, $qN-j=(N-j)q-jp$, the $j$-th summand is $$(j(1-j+J)p+(j-J)(N-j)q)q^jp^{N-j} {N\choose j}.$$
Expand the brackets and consider it as a homogeneous polynomial in $p$ and $q$ of degree $N+1$. The coefficient of $q^{j+1}p^{N-j}$ equals
$$
(j-J)(N-j){N\choose j}+(j+1)(J-j){N\choose j+1}=0.
$$
| 7 | https://mathoverflow.net/users/4312 | 381403 | 158,752 |
https://mathoverflow.net/questions/381400 | 2 | Recall that a topological space is called Frechet-Urysohn if the operations of closure and sequential closure coincide.
Let $X$ be a locally compact Hausdorff space. It is known that $C(X)$ is not Frechet-Urysohn with respect to the compact-open topology. Indeed, it was proven in the paper *Hernández, Mazón - On the ... | https://mathoverflow.net/users/53155 | Is $C_0(X) $ Frechet-Urysohn with respect to the compact-open topology? | In general, $C\_0(X)$ need not be Frechet-Urysohn.
For a counterexample, let $X = \omega\_1$ be the first uncountable ordinal with its order topology. Let $S \subset \omega\_1$ be the set of successor ordinals in $\omega\_1$. For $\alpha \in S$, let $1\_\alpha : \omega\_1 \to \mathbb{R}$ be the function which is $1$ ... | 3 | https://mathoverflow.net/users/4832 | 381404 | 158,753 |
https://mathoverflow.net/questions/376840 | 2 | Let $\mathcal{C}$ and $\mathcal{D}$ be two categories and $F$ and $G: \mathcal{C}\to \mathcal{D}$ be two functors. Suppose $F$ and $G$ have right adjoints $F^{\wedge}$ and $G^{\wedge}: \mathcal{D}\to \mathcal{C}$.
Now let $T:F\Rightarrow G$ be a natural transformation.
>
>
> >
> > My question is: does $T$ induc... | https://mathoverflow.net/users/24965 | Does a natural transformation of functors induce a natural transformation between their right adjoints? | I should have made my comment an answer earlier. The answer is yes:
Let $\eta: 1 \to F^\wedge F$ denote the unit of the adjunction and $\varepsilon: G G^\wedge \to 1$ the counit. Then form the composite
$$G^\wedge \stackrel{\eta G^\wedge}{\to} F^\wedge F G^\wedge \stackrel{F^\wedge TG^\wedge}{\to} F^\wedge G G^\wed... | 6 | https://mathoverflow.net/users/2926 | 381406 | 158,754 |
https://mathoverflow.net/questions/381405 | 0 | Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ which is convex and satisfies $\frac{\partial{f}^2 }{\partial{x}\partial{y}} \leq 0$. The last condition is equivalent to the inequality $f(x\_1,y\_1) + f(x\_2,y\_2) \geq f(\min\left(x\_1,x\_2\right), \min\left(y\_1,y\_2\right)) + f(\... | https://mathoverflow.net/users/43628 | Planar function inequality on parallelograms | E.g., let $f(x,y):=(x-y)^2+x^2+y^2$ for $(x,y):=R=[0,1]^2$, $v\_1:=(1,1)$, $v\_2:=(0,t)$, $v\_3:=(0,0)$, and $v\_4:=(1,1-t)$, where $t\in(0,1/2)$. Then $f$ is convex, $\frac{\partial{f}^2 }{\partial{x}\partial{y}} \le 0$, $v\_1v\_2v\_3v\_4$ is a parallelogram inscribed into $R$, but $f(v\_2) + f(v\_4)\not\ge f(v\_1) + ... | 2 | https://mathoverflow.net/users/36721 | 381409 | 158,755 |
https://mathoverflow.net/questions/380126 | 14 | For $1 \leq r \leq n$, let $\mathcal{B}^n\_r$ denote the polytope of all real matrices
$$ \pi = \begin{pmatrix} \pi\_{1,1} & \pi\_{1,2} & \cdots & \pi\_{1,n} \\
\pi\_{2,1} & \ddots & \cdots & \pi\_{2,n} \\
\vdots & \ddots & \ddots & \vdots \\
\pi\_{n,1} & \cdots & \cdots & \pi\_{n,n} \end{pmatrix} \in \mathbb{R}^{n\ti... | https://mathoverflow.net/users/25028 | Ehrhart period collapse for $123\ldots k$-avoiding Birkhoff polytope? | The Ehrhart function $L(\mathcal B^n\_r;t)$ is an honest polynomial. We will show this by following Per's suggestion and proving that it coincides with the Ehrhart function of a certain Gelfand-Tsetlin polytope. The following two steps can probably be combined into one, but I thought it was natural to think of it this ... | 4 | https://mathoverflow.net/users/2384 | 381410 | 158,756 |
https://mathoverflow.net/questions/381339 | 9 | Under what conditions on a $C^\*$ algebra $A$ we have the following inequality:
$$x^\*a^\*ax+a^\*x^\*xa\leq x^\*x+a^\*x^\*ax+x^\*a^\*xa\;\;\; \forall x,a\in A$$
The second identity which I am looking for is the following:
Does the following inequality imply that the algebra is commutative:
$$xx^\*\leq k x^\*x\;... | https://mathoverflow.net/users/36688 | Two inequalities in $C^*$ algebras | The second condition also implies that $A$ is commutative. If $A$ is not commutative then it has an irreducible representation on some Hilbert space $H$ of dimension at least $2$. Find unit vectors $v,w \in H$ with $\langle v, w\rangle = 0$. By Kadison transitivity there exists $x \in A$ with $xv = 0$ and $xw = v$. The... | 7 | https://mathoverflow.net/users/23141 | 381418 | 158,759 |
https://mathoverflow.net/questions/380357 | 14 | The well-known theorem of Dirichlet on primes in arithmetic progression states that given coprime natural numbers $a\le q$, there are infinitely many prime numbers congruent to $a\pmod q$. The standard proof is via *analytic* number theory; however, the special case $a=1, q=2$, is a celebrated theorem of Euclid, whose ... | https://mathoverflow.net/users/166628 | Euclid-style proof of Dirichlet’s theorem on primes in certain arithmetic progression | A proof of the construction of a polynomial, in English, is in the paper of Murty and Thain, *Primes in Certain Arithmetic Progressions* (Funct. Approx. Comment. Math. **35** (2006) pp. 249-259, doi:[10.7169/facm/1229442627](https://doi.org/10.7169/facm/1229442627)). See Section 2, which builds up to the Euclid-style p... | 14 | https://mathoverflow.net/users/3272 | 381423 | 158,761 |
https://mathoverflow.net/questions/381361 | 19 | I was recently reminded about [this old question](https://math.stackexchange.com/q/629357/88262) on math.stackexchange.
Let $\operatorname{Mod}R$ be the category of (right) modules for a ring $R$. The questioner mistakenly thought that the Freyd-Mitchell embedding theorem implied that for every ring $R$ there was ano... | https://mathoverflow.net/users/22989 | Vopěnka's principle and contravariant full embeddings between module categories | Assuming (M) (= there is only a set of measurable cardinals), the category $\bf{Vec}$ of vector spaces
(over every field) has a small dense subcategory. This is an old result of Isbell (see also
<https://arxiv.org/pdf/1812.10649.pdf>). Hence $\bf{Vec}$$^{\text{op}}$ is boundable, i.e., it can be fully embedded
to a cat... | 11 | https://mathoverflow.net/users/73388 | 381433 | 158,763 |
https://mathoverflow.net/questions/381437 | 4 | Let $S,S'$ be schemes, let $\pi : S' \to S$ be a morphism which is faithfully flat and locally of finite presentation, set $S'' := S' \times\_{S} S'$ and $S''' := S' \times\_{S} S' \times\_{S} S'$ with projections $p\_{1},p\_{2} : S'' \to S'$ and $p\_{12},p\_{13},p\_{23} : S''' \to S''$. Let $\mathcal{E}'$ be a vector ... | https://mathoverflow.net/users/15505 | A noneffective descent datum: isomorphism not satisfying the cocycle condition | This already fails for line bundles on smooth projective curves: let $X$ be 'the' pointless conic over $\mathbb{R}$, given by the closed subscheme of $\mathbb{P}^2\_{\mathbb{R}}$ cut out by $X^2+Y^2+Z^2 = 0$.
It is smooth, projective, geometrically integral over $\mathbb{R}$, and $X(\mathbb{R})= \emptyset$.
In your not... | 8 | https://mathoverflow.net/users/110362 | 381444 | 158,767 |
https://mathoverflow.net/questions/381429 | 0 | Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ satisfying
* $f \geq 0$, $f(0,0) = 0$,
* $\frac{\partial{f}}{\partial{x}} \geq 0$, $\frac{\partial{f}}{\partial{y}} \leq 0$,
* $\frac{\partial^2{f} }{\partial{x}\partial{y}} \leq 0$.
The last condition is equivalent to the inequal... | https://mathoverflow.net/users/43628 | Extending functional inequality from rectangles to parallelograms | Take for instance $f(x,y)=x^2$, that satisfies the assumptions. Since one has $f(x,y)+f(1-x,1-y)= 1-2x+2x^2 < 1=f(0,0)+f(1,1)$ for all $0<x<1$, it verifies the stated inequality for no parallelogram (inscribed in $R$, with vertices in $(0,0)$ and $(1,1)$) with non-vertical edges.
| 1 | https://mathoverflow.net/users/6101 | 381448 | 158,768 |
https://mathoverflow.net/questions/381334 | 7 | This question has [also been posted on MSE](https://math.stackexchange.com/questions/3986936/characterization-of-disks-of-constant-curvature-and-whose-boundaries-have-consta), but maybe here is the right place to post it.
Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ ... | https://mathoverflow.net/users/85934 | Constant Gaussian curvature disks | This is an addendum to the proofs by Anton and Deane, completing the missing part of the argument.
Lemma. Let $D$ be the closed unit disk and $f: D\to S^2$ an immersion such that $f(\partial D)$ is a circle $C$ in $S^2$. Then $f$ is 1-1.
Proof. Let $J: S^2\to S^2$ denote the reflection in $C$. Double $D$ cross its ... | 5 | https://mathoverflow.net/users/39654 | 381449 | 158,769 |
https://mathoverflow.net/questions/381456 | 63 | By density of primes, I mean the proportion of integers between $1$ and $x$ which are prime. The prime number theorem says that this is asymptotically $1/\log(x)$.
I want something much weaker, namely that the proportion just goes to zero, at whatever rate. And I want the easiest proof possible.
The simplest proof ... | https://mathoverflow.net/users/126543 | What is the simplest proof that the density of primes goes to zero? | I'm summarising the discussion in GH from MO's answer as a separate answer for clarity.
The fact that the primes have (natural) density zero can be deduced from a (seemingly) more general statement:
>
> **Theorem** Let $1 < n\_1 < n\_2 < \dots$ be a sequence of natural numbers that are pairwise coprime. Then this... | 64 | https://mathoverflow.net/users/766 | 381462 | 158,772 |
https://mathoverflow.net/questions/381419 | 1 | Take a polynomial $f(x)$ of even degree $n$ of the form—$${n \choose {n/2}}x^{n/2}(1-x)^{n/2} k,$$where $k>1$ is the $(n/2)$th Bernstein coefficient of the polynomial. (With these properties, $f$ peaks at the point 1/2 in the interval [0, 1] and is nonnegative everywhere in [0, 1].)
Suppose $f(1/2) \in (0, 1)$, and s... | https://mathoverflow.net/users/171320 | On the degree elevation needed to bring Bernstein coefficients to [0, 1] | We have $$ f(x) = {n \choose {n/2}}x^{n/2}(1-x)^{n/2} k = 2^n f(1/2) x^{n/2}(1-x)^{n/2} = 2^n f(1/2) x^{n/2} (1-x)^{n/2} ( x+ (1-x))^r = \sum\_{j=0}^r 2^n f(1/2) \binom{r}{j} x^{n/2+j}(1-x)^{n/2 + r-j} $$
so the $n/2+j$th [Bernstein coefficient](https://en.wikipedia.org/wiki/Bernstein_polynomial) is $$ \frac{ 2^n f(1... | 1 | https://mathoverflow.net/users/18060 | 381465 | 158,773 |
https://mathoverflow.net/questions/381466 | 1 | Consider a general $4\times 4$ matrix:
$$
X:=\left(
\begin{array}{cccc}
X\_0 & X\_1 & X\_2 & X\_3 \\
X\_4 & X\_5 & X\_6 & X\_7 \\
X\_8 & X\_9 & X\_{10} & X\_{11} \\
X\_{12} & X\_{13} & X\_{14} & X\_{15}
\end{array}
\right)
$$
and let $Y\_k\subset\mathbb{P}^{15}$ the variety of matrices of rank equal to $k$. What is... | https://mathoverflow.net/users/nan | Picard groups of determinantal varieties | There is a natural morphism
$$
Y\_k \to Gr(k,4) \times Gr(k,4)
$$
associating to a matrix its image and coimage. Moreover, this morphism factors as the composition
$$
Y\_k \hookrightarrow \mathbb{P}\_{Gr(k,4) \times Gr(k,4)}(U\_1^\vee \otimes U\_2) \to Gr(k,4) \times Gr(k,4),
$$
where $U\_i$ are the tautological vector... | 4 | https://mathoverflow.net/users/4428 | 381469 | 158,775 |
https://mathoverflow.net/questions/381478 | 5 | In §1.6 of Atiyah's *K-theory*, he defines the notion of a $G$-(vector)-bundle, which is a sort of "equivariant vector bundle" with respect to a finite group action. More specifically, let $G$ be a finite group and $X$ a compact Hausdorff space. A $G$-bundle (not in the principal sense) is a vector bundle $\pi:E\righta... | https://mathoverflow.net/users/147463 | Locally trivializing a G vector bundle? | To get an idea of what is going on, suppose the base space $X$ was a single orbit of $G$, say $X = G/H$, with $H$ a subgroup. A $G$-equivariant vector bundle over $G/H$ will be isomorphic to one of the form $G \times\_H V \rightarrow G/H$, where $V$ is a representation of $H$. This will only be isomorphic to $G/H \time... | 4 | https://mathoverflow.net/users/102519 | 381481 | 158,780 |
https://mathoverflow.net/questions/381484 | 7 | I fix $C$ a symmetric monoidal model category (with a cofibrant unit if it helps). I'm assuming that it is closed, or at least that the tensor product commutes to colimits in each variable.
If $X$ is a monoid in $C$, and $A \to B$ is a map in $C$, I'm calling free extension of $X$ by $A \to B$ (along some map $A \to ... | https://mathoverflow.net/users/22131 | Free extension of algebra for an operad | This result is true and is due to Spitzweck, Berger–Moerdijk, Fresse, and Elmendorf–Mandell.
A complete set of references can be found around Proposition 5.7
in the paper <https://arxiv.org/abs/1410.5675>.
| 5 | https://mathoverflow.net/users/402 | 381489 | 158,782 |
https://mathoverflow.net/questions/381314 | 13 | In Paul Cohen's original 1963 paper on forcing, *[The independence of the Continuum Hypothesis](https://doi.org/10.1073/pnas.50.6.1143)*, published in PNAS, he gives his general proof sketch of how he intends to create a model of ZFC that doesn't support CH:
1. Start with a countable model of ZFC.
2. Within the model... | https://mathoverflow.net/users/24611 | What did Paul Cohen mean by saying that generic sets of natural numbers have "no asymptotic density?" | Perhaps it's worth collecting the comments into a community wiki answer.
When Cohen was trying to prove the independence of the continuum hypothesis from ZFC, at some point he realized that what one needed to do was to take a countable transitive model $M$ of ZFC and "adjoin" a suitable subset of $\mathbb N$ to it, m... | 11 | https://mathoverflow.net/users/3106 | 381495 | 158,786 |
https://mathoverflow.net/questions/381447 | 7 | In (single-variable) complex analysis, given analytic functions $f$ and $g$ with no common zeros, one can find analytic functions $u$ and $v$ such that $uf+vg=1$. I’d like to know if the same holds in several variables; as a simple case, specifically,
>
> Let $f,g\colon\mathbb{D}^2\to\mathbb{C}$ be analytic (in the... | https://mathoverflow.net/users/166628 | Bezout’s identity for analytic functions of several variables | Make an open cover $D^2=\cup\_j(U\_j\cup V\_j)$, for example, by polydisks
such that $f$ has zeros only in $U\_j$ and $g$ has no zeros in $U\_j$. This is possible since zeros of $f$ and $g$ are disjoint.
Solve the [1st Cousin problem](https://en.wikipedia.org/wiki/Cousin_problems) with Cousin data $-1/(fg)$ in $U\_j$... | 8 | https://mathoverflow.net/users/25510 | 381499 | 158,787 |
https://mathoverflow.net/questions/379301 | 1 | Suppose that $A$ is an $\omega$-REA set (so $A^{[n]}$ is r.e. in the prior columns). It is a well-known result that if each column of $A$ is computable then $A \leq\_T \emptyset^2$ ($\emptyset^2$ can recover the computable index for each column from indices of prior columns and thus compute $A$).
A suggestive way to ... | https://mathoverflow.net/users/23648 | Computable in $\omega$-REA degree but not double jump of finitely many columns | Ok, so unfortunately the answer is no. I've attached a proof that builds a $\omega$-REA set such that $A^{[\leq n]}$ is low for all $n$ yet $A$ computes $\emptyset'' \oplus X$ where $X$ is a natural non-$\Delta^0\_3$ set [1] r.e. in $\emptyset''$. The proof is a bit too long to attach as an image so here's a [link](htt... | 0 | https://mathoverflow.net/users/23648 | 381507 | 158,791 |
https://mathoverflow.net/questions/381475 | 2 | Suppose $A, B, C\in\mathbb{R}^{n\times n}$ are all symmetric positive definite matrices, and they satisfy the inequality $A \succeq B + C$. Assume also that all of the three matrices are bounded, i.e., $A\_{\min}I \preceq A \preceq A\_{\max}I$, $B\_{\min}I \preceq B \preceq B\_{\max}I$ and $C\_{\min}I \preceq C \preceq... | https://mathoverflow.net/users/172027 | Matrix inversion inequality | Well, this is true. We consequently have the following:
**Lemma 1.** If $X\succeq I$, then $X^{-1}\preceq I$.
**Proof.** Write $X$ in the diagonal basis.
**Lemma 2.** If $X\succeq Y\succ 0$ then $X^{-1}\preceq Y^{-1}$.
**Proof.** We have $X=Y+Z=Y^{1/2}(I+Y^{-1/2}ZY^{-1/2})Y^{1/2}$ for $Z\succeq 0$, then $$X^{-1... | 2 | https://mathoverflow.net/users/4312 | 381508 | 158,792 |
https://mathoverflow.net/questions/381435 | 4 | Let $E$ be a vector space over a field $K$. If $u \in \mathscr L(E)$ is an endomorphism of $E$, can it be written as a linear combination of projections (i.e. endomorphisms $p$ of $E$ such that $p \circ p = p$)?
If the dimension of $E$ is finite, this is true. Indeed, one can check that if $(E\_{i,j})\_{1 \leq i,j \l... | https://mathoverflow.net/users/80602 | Is every endomorphism a linear combination of projections? | The answer is yes: see [Goerge Lowther's answer](https://math.stackexchange.com/a/887623/35400) at Pierre-Yves Gaillard's MathSE linked question: every endomorphism is a $K$-linear combination of $\le 9$ idempotents.
In infinite dimension, the argument even shows that every endomorphism is a $\mathbf{Z}$-linear combi... | 1 | https://mathoverflow.net/users/14094 | 381513 | 158,795 |
https://mathoverflow.net/questions/381486 | 0 | I am interested in understanding the solutions $\phi$ of the following integral equation: $$0=\int\_0^1 \int\_0^1 \phi(x,y)(x-y)\thinspace dx\thinspace dy.$$ Equivalently, I am interested in understanding the kernel of the linear function $F:C^k(\mathbb{R}^2)\to\mathbb{R}$ where $F(\phi)=\int\_0^1 \int\_0^1 \phi(x,y)(x... | https://mathoverflow.net/users/170545 | Solution set of integral equation/ Kernel of linear operator | I think it's easiest to work in the Hilbert space setting for this problem, i.e., to consider $F$ is a functional on the space $L^2([0,1]^2)$, where $[0,1]^2$ is endowed with the Lebesgue measure.
Let $1 \in L^2([0,1])$ denote the constant function with value $1$, and let $h \in L^2([0,1])$ be given by $h(x) = x$ for... | 4 | https://mathoverflow.net/users/102946 | 381516 | 158,798 |
https://mathoverflow.net/questions/380191 | 2 | Remember a degree $\mathbb{d}$ is the $n$-lub of $\mathbb{c}\_j$ in the Turing degrees if it is the least element (not merely a minimal element) set of $\mathbb{c}^{(n)}$ such that $\mathbb{c}$ computes every $\mathbb{c}\_j$. It is non-trivial if it's not the $n$-th jump of a finite join of the degrees $\mathbb{c}\_n$.... | https://mathoverflow.net/users/23648 | Arithmetic non-trivial 2-l.u.b | Yes, it turns out that there is such an arithmetic (indeed 3-REA) non-trivial 2-lub. In fact, $\emptyset'''$ is such a degree. Consider the construction I give [here](https://invariant.org/papers/double%20jump%20hypo.pdf) in answer to this [question](https://mathoverflow.net/questions/379301/computable-in-omega-rea-deg... | 0 | https://mathoverflow.net/users/23648 | 381518 | 158,800 |
https://mathoverflow.net/questions/381269 | 5 | I hope this post is on topic as a reference request.
I have seen somewhere the idea of (and saw it written just like this):
$$\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }.$$
I am writing a note and want to use this idea and am just wondering is there a good reference where such matters are d... | https://mathoverflow.net/users/35482 | Reference request for Deterministic $\subset$ Random $\subset$ Quantum | The way I interpret your question is: deterministic = pure state on an abelian C${}^\*$-algebra, random = arbitrary state on an abelian C${}^\*$-algebra, quantum = pure state on an arbitrary C${}^\*$-algebra. There's one further level of generality, arbitrary state on an arbitrary C${}^\*$-algebra, which gives you stat... | 4 | https://mathoverflow.net/users/23141 | 381526 | 158,802 |
https://mathoverflow.net/questions/381527 | 26 | Apologies if the answer is trivial, this is far from my domain.
In order to define the field of [Hahn series](https://en.wikipedia.org/wiki/Hahn_series#Summable_families_2), one needs the following fact: if $A,B$ are two well-ordered subsets of $\mathbb{R}$ (or any ordered group — with the induced order of course), the... | https://mathoverflow.net/users/40297 | The sum of two well-ordered subsets is well-ordered | Ramsey theory! Suppose $A + B$ is not well-ordered. Then there is a strictly decreasing sequence $a\_1 + b\_1 > a\_2 + b\_2 > \cdots$. Observe that for any $i < j$, either $a\_i > a\_j$ or $b\_i > b\_j$ (or both). Make a graph with vertex set $\mathbb{N}$ by putting an edge between $i$ and $j$ if $a\_i > a\_j$, for any... | 44 | https://mathoverflow.net/users/23141 | 381528 | 158,803 |
https://mathoverflow.net/questions/381531 | 7 | I'm trying to understand the footnote to Example 5.3 in [Wiegand - Sheaf cohomology of locally compact totally disconnected spaces](https://www.ams.org/journals/proc/1969-020-02/S0002-9939-1969-0253324-8/) which is about constructing a locally compact Hausdorff and totally disconnected space whose sheaf cohomology with... | https://mathoverflow.net/users/15934 | A set theoretic question arising from trying to understand a sheaf cohomology question | This is not onto for any uncountably infinite $T$, even one much smaller than the power set (if the continuum hypothesis is false).
Fix $X \in 2^{ S \times T}$ such that the induced map $h \colon T \to 2^S$ where $h(t) = \{ s \mid (s,t) \in X\}$ (i.e. taking vertical fibers of $X$) has uncountable image. (For example... | 8 | https://mathoverflow.net/users/18060 | 381534 | 158,805 |
https://mathoverflow.net/questions/381535 | 0 | Let $\varepsilon<1/1000$ and $n$ is big enough. Is it true that every graph with at most $\varepsilon n^2$ edges has independence number bigger than $\varepsilon n$?
| https://mathoverflow.net/users/100359 | Independence number of graph with $\varepsilon n^2$ edges | No. Will's example is better, but you may also proceed probabilistically.
A random graph $G(n,\varepsilon)$ (every two vertices are joined with probability $\varepsilon$) has at most $\varepsilon n^2$ edges with probability almost 1 and does not have such huge independent sets, also with probability almost 1 (the pro... | 3 | https://mathoverflow.net/users/4312 | 381537 | 158,807 |
https://mathoverflow.net/questions/381532 | 12 | Let $M$ be a connected compact Riemann surface. Let $f, g$ be two nonconstant meromorphic functions. Why is there a two-variable complex polynomial $F(x,y)$ that vanishes for $(x, y)=(f, g)$, (in other words $F(f,g)=0$)?
| https://mathoverflow.net/users/166540 | Two non constant meromorphic functions over a connected compact Riemann surface, could not be algebraically independent | Let $F$ be a polynomial of degree at most $n$.
For a point $x$ where $f$ has a pole of order $a$ and $g$ has a pole of order $b$, $F(f,g)$ has a pole of order at most $n\max(a,b)$. Locally near $x$, we can write $F(f,g)$ as a Laurent series $$c\_{ -N} z^{-N} + c\_{1-N} z^{1-N} + \dots + c\_{-1} z^{-1}+ c\_0 + c\_1 z ... | 18 | https://mathoverflow.net/users/18060 | 381541 | 158,810 |
https://mathoverflow.net/questions/381492 | 2 | On page 4 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at [arXiv](https://arxiv.org/abs/1312.6536), they claim the following on the top right of the page:
>
> $$ℓ^\*(Λ,X)=\prod\_{i=1}^nΛ(x\_i)\{\int\_A Λ(x)dx... | https://mathoverflow.net/users/172324 | General form for likelihood of Cox process, from Diggle–Moraga–Rowlingson–Taylor | Write the inhomogeneous [Poisson point process](https://en.wikipedia.org/wiki/Point_process#Representation) on a Borel subset $A$ of $\mathbb R^d$ as
$$\sum\_{i=1}^N\delta\_{X\_i},$$
where $\delta\_a$ is the Dirac probability measure at point $a$, $N$ is the random variable whose value is the number of points that appe... | 2 | https://mathoverflow.net/users/36721 | 381558 | 158,816 |
https://mathoverflow.net/questions/381542 | 6 | Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square lattice points. However,
1. Given any positive integer *n*, can we always find a sufficiently large circle drawn on the... | https://mathoverflow.net/users/142600 | On circles and ellipses drawn on an infinite planar square lattice | (1-2) Yes.
For each integer $n > 0$ the circle $x^2 + y^2 = 13^{n-1}$ passes through
exactly $4n$ lattice points, namely those with
$$
z := x+iy = \zeta (3+2i)^a (3-2i)^b
$$
with $a,b$ nonnegative integers such that $a+b=n-1$, and
$\zeta \in \{1, i, -1, -i\}$.
Given $(a,b)$, exactly one of the four choices of $\zeta$ m... | 13 | https://mathoverflow.net/users/14830 | 381560 | 158,817 |
https://mathoverflow.net/questions/381506 | 1 | Let $\{\mathbb{P}\_{\theta}\}\_{\theta}$ be an exponential family of probability measures, all with finite mean. Under what conditions is the parameterization map $\theta\mapsto \mathbb{P}\_{\theta}$ continuous with respect to the Wasserstein-$1$ distance?
| https://mathoverflow.net/users/36886 | Parameterization of exponential family | $\newcommand\om\omega\newcommand\Om\Omega\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $(P\_t)\_{t\in T}$ be an exponential family over a separable complete metric space $(X,d)$, where $T$ is an open subset of $\mathbb R^k$ and $t=(t\_1,\dots,t\_k)$ is a [natural parameter](https://en.wikipedia.org/wiki/Exponentia... | 1 | https://mathoverflow.net/users/36721 | 381567 | 158,820 |
https://mathoverflow.net/questions/380425 | 5 | Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the module of Kähler differentials). Is $f$ necessarily in $k$?
| https://mathoverflow.net/users/4181 | Kähler differentials on an Artinian local ring | I finally remembered the example (though not the reference). Take $f=x^2y^2+x^5+y^5\in R=\mathbb{C}[[x,y]]$. Then $f\_x,f\_y$ form a regular sequence in $R$ and thus $R/I$ where $I=(f\_x,f\_y)$ is an Artin local ring. One checks $f\not\in I$. Thus, $df=0\in\Omega^1\_{R/I}$, but $f\neq 0$ in $R/I$.
| 2 | https://mathoverflow.net/users/9502 | 381583 | 158,825 |
https://mathoverflow.net/questions/381566 | 6 | I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.
I just wanted to ask if there is a notion of fractional derivative that is linear and satisfy the following property $D^u((f)^n) = \alpha D^u(f)f^{(n-1)}$ where $... | https://mathoverflow.net/users/nan | A question on fractional derivatives | There are basically no interesting solutions to this equation beyond first and zeroth order operators, even if one only imposes the stated constraint for $n=2$.
First, we can [depolarise](https://en.wikipedia.org/wiki/Polarization_identity) the hypothesis
$$ D^u(f^2) = \alpha\_2 D^u(f) f \quad (1)$$
by replacing $f$ ... | 16 | https://mathoverflow.net/users/766 | 381587 | 158,828 |
https://mathoverflow.net/questions/381580 | 7 | In some of his writings, Paul Cohen gave an informal, motivational discussion about the word *generic* (as it is used in forcing). While very suggestive, the discussion leaves the meaning of the word ambiguous, and could lead someone to guess that it is related to probability theory. Indeed, there was a [recent MO ques... | https://mathoverflow.net/users/3106 | Independence result where probabilistic intuition predicts the wrong answer? | If the Borel-Cantelli lemma counts as probabilistic intuition, then here's an example. Think of the real $x$ that you adjoin to a ground model as a sequence of $0$'s and $1$'s, and let $f(n)$ be the length of the $n$-th run of consecutive $1$'s in $x$. If the bits in $x$ were chosen by independent flips of a fair coin ... | 10 | https://mathoverflow.net/users/6794 | 381588 | 158,829 |
https://mathoverflow.net/questions/381544 | 3 | I work on a problem in my research. I have a graph, $G$, with $2n$ vertices. It has one connected component of order $2n-1$ and an isolated vertex. $\lambda\_1\geq \lambda\_2\geq \ldots \geq \lambda\_{2n}$ are the eigenvalues of $G$. I have some bounds for them.
$$2n-3\leq \lambda\_1<2n-2,\\
0\leq \lambda\_2\leq 1,\\
... | https://mathoverflow.net/users/152342 | How to find non-isomorphic graphs with specific orders? | So here is a family of graphs that satisfies your requirements.... is this the only family?
1. Let $X\_1$ be the graph consisting of $n-1$ disjoint copies of $K\_2$. Then the spectrum of $X\_1$ is $$\underbrace{1,1,\ldots,1}\_{n-1},\underbrace{-1,-1,\ldots,-1}\_{n-1}$$
2. Next let $X\_2$ be the complement of $X\_1.$ ... | 7 | https://mathoverflow.net/users/1492 | 381596 | 158,834 |
https://mathoverflow.net/questions/381598 | 13 | The question is inspired by [an answer](https://mathoverflow.net/a/262700/41291) to [The concept of Duality](https://mathoverflow.net/q/73711/41291)
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of the manifold.
Spanier-Whitehead... | https://mathoverflow.net/users/41291 | Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles? | Let $X$ be a finite complex. Then the functor
$$\lim\_X:\operatorname{Fun}(X,\operatorname{Sp})\to \operatorname{Sp}$$
sending a local system of spectra $E$ to its limit preserves all colimits. Indeed it preserves all finite colimits by stability, and it preserves all filtered colimits by the finiteness of $X$. Therefo... | 15 | https://mathoverflow.net/users/43054 | 381603 | 158,836 |
https://mathoverflow.net/questions/381608 | 2 | I'm currently reading the paper [Well-posedness for the Zakharov system with the periodic boundary condition](https://projecteuclid.org/download/pdf_1/euclid.die/1367241476) by Takaoka. In the proof of Lemma 2.3 about the integral $I\_1$ one needs to establish the estimate
$$\int\_{-\infty}^{\infty} \frac{d\tau'}{\la... | https://mathoverflow.net/users/146998 | Estimate on an integral involving the Japanese bracket | I assume that you mean $a>1/2$, for $a=1/2$ the integral diverges.
Denote $\tau=\tau'+\alpha n^2$ and use the relations $\langle x\rangle\asymp \max(1,|x|)$, $(x+y)^a\asymp \max(x^a,y^a)$ for $x,y>0$ (here $A\asymp B$ means $c\_1 b\leqslant a\leqslant c\_2 b$ for universal constants $c\_1,c\_2$). Your integral become... | 4 | https://mathoverflow.net/users/4312 | 381612 | 158,838 |
https://mathoverflow.net/questions/381601 | 2 | Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak continuous.
Under what additional conditions can we guarantee that $\operatorname{span}(f(E))$ is a *finite-dimensional* s... | https://mathoverflow.net/users/36886 | Control on dimension of image | Just to give an example on how weird this can become: take $E = \mathbb{R}$ and $F = \ell^2$ with standard Hilbert basis $e\_0 e\_1, e\_2, \ldots$. Then take a smooth bump function $\chi \in C^\infty(\mathbb{R})$ with support in the unit interval and, say $\chi(1/2) = 1$ to make things non-trivial.
Define the highly ... | 5 | https://mathoverflow.net/users/12482 | 381613 | 158,839 |
https://mathoverflow.net/questions/381605 | 5 | Let $f\colon \mathbb{P}^2\dashrightarrow\mathbb{P}^2$ be a rational map ($\mathrm{deg}(f)$ may be high), let $\Gamma\subset\mathbb{P}^2\times\mathbb{P}^2$ be the closure of the graph. Let $x\in\mathbb{P}^2$ be a point in the source, then the fiber $\Gamma|\_x$ can be viewed as a union of rational curves in the target $... | https://mathoverflow.net/users/nan | Bounding degree of rational curves in the exceptional locus | The answer is no. Let us consider the case where $f$ is birational. In this case, for each point $x\in \mathbb{P}^2$ such that $f$ is not defined, the intersection of $\Gamma$ with $x\times \mathbb{P}^2$ contains all curves of $\mathbb{P}^2$ that are contracted by the inverse $f^{-1}$. The degree of these curves can be... | 6 | https://mathoverflow.net/users/23758 | 381624 | 158,845 |
https://mathoverflow.net/questions/381618 | 8 | Using the Serre-Swan's theorem, one can do vector bundle theory on noncommutative manifold $(A,H,D)$, by replacing vector bundle by finitely generated projectve module $M$. For the construction of tangent bundle, one can use derivation, but I am not sure how to define derivation on the module. The idea seems to be defi... | https://mathoverflow.net/users/172458 | tangent bundle on noncommutative manifold | Noncommutative Riemannian (spin) geometry via spectral triples is grounded in an approach to noncommutative differential calculus that privileges the cotangent bundle over the tangent bundle: given a spectral triple $(\mathcal{A},H,D)$, you have an $\mathcal{A}$-bimodule
$$
\Omega^1\_D := \operatorname{Span}\{ a \cdot... | 9 | https://mathoverflow.net/users/6999 | 381628 | 158,847 |
https://mathoverflow.net/questions/381616 | 5 | Take $S^n$ and consider the union $Z$ of $k\_1$ circles, $k\_2$ 2-dimensional spheres, ..., $k\_{n-2}$ $(n−2)$-dimensional spheres, embedded in $S^n$ in an unknotted way, with no mutual intersection and no mutual linking.
I want to understand the complement $S^n \setminus Z$.
To that end, I calculated its homology ... | https://mathoverflow.net/users/109520 | Complement to a union of spheres in a sphere | Your answer is correct, if "unknotted" is equivalent to: being homeomorphic to a union of standard sphere complements within disjoint convex balls. Philosophically, you could imagine giving a proof by saying that such a space is a connected sum of sphere complements, and understand enough about them to prove it inducti... | 3 | https://mathoverflow.net/users/360 | 381641 | 158,850 |
https://mathoverflow.net/questions/381626 | 2 | Consider the contour integral
>
> $\frac{1}{2\pi i}\oint\_\gamma\chi(z)\frac{dz}{z}\,,$
>
>
>
where $\gamma$ is a (not necessarily simple) closed curve lying in $\mathbb{C}\setminus{0}$ and $\chi\colon\mathbb{C}\to\mathbb{R}\_{\ge 0}$ is a continuous function. My question is
>
> Are there special/generic h... | https://mathoverflow.net/users/166628 | Twisted winding number | The integral over the curve can be reduced to the integral over the region bounded by the curve using Green's formula:
$$
\frac{1}{2\pi i}\int\_\gamma \chi(z)\frac{dz}{z}=\frac{1}{\pi}\int\_U\bar{\partial}\left(\frac{\chi(z)}{z}\right)d\text{Area}(z)=\frac{1}{\pi}\int\_U\left(\frac{\bar{\partial}\chi(z)}{z}\right)d\t... | 1 | https://mathoverflow.net/users/56624 | 381643 | 158,851 |
https://mathoverflow.net/questions/381640 | 2 | Does there exist a sequence of decreasing continuous functions $(f\_n)\_{n\in\mathbb{N}}$ satisfying the following two conditions?
* For every $n\in\mathbb{N}$, $\lim\_{x\to\infty}f\_n(x)=0$;
* For any other decreasing continuous function $g$ tending to zero at infinity, there exists $n\in\mathbb{N}$ so that $\frac{g... | https://mathoverflow.net/users/142808 | Sequence of tending to zero functions that majorizes any other tending to zero function | Let $C\_d$ be the space of all decreasing continuous functions tending to 0 at infinity, equipped with the sup metric $d\_\infty$. Note this is a complete metric space.
Suppose such a sequence $f\_n$ did exist. Then for every $g \in C\_d$ there would exist $n$ such that $g/f\_n$ is decreasing to 0; since $g/f\_n$ is ... | 11 | https://mathoverflow.net/users/4832 | 381647 | 158,852 |
https://mathoverflow.net/questions/381632 | 5 |
>
> Given a prime $p$ and an integer $n\ge p$, what is the smallest possible degree of a polynomial $Q\in\mathbb F\_p[x\_1,\dotsc, x\_n]$ such that $Q$ vanishes on every vector $x\in\{0,1\}^n$ of weight $w(x)=p$, but $Q(0)\ne 0$? (Here the *weight* of a vector is the number of its nonzero coordinates.)
>
>
>
I a... | https://mathoverflow.net/users/9924 | Polynomials vanishing on prescribed layers | For both modified questions, the answer is $n+1-p$. Since every polynomial that vanishes on vectors of weight $\geq p$ vanishes on vectors of nonzero weight divisible by $p$, other than $0$, it suffices to prove the upper bound for polynomials vanishing on vectors of weight $\geq p$ and the lower bound for vectors of n... | 4 | https://mathoverflow.net/users/18060 | 381661 | 158,856 |
https://mathoverflow.net/questions/360913 | 2 | Let $n\geq3$ be a fixed positive integer, consider the parameter space $|\mathcal{O}\_{\mathbb{P}^{n}}(d)|$ of degree $d$ hypersurfaces ($d\geq n+1$) in $\mathbb{P}^{n}$. Let $Z\_{unir}\subset |\mathcal{O}\_{\mathbb{P}^{n}}(d)|$ be the subspace parameterizing uniruled hypersurfaces, do we know some lower bounds on $\ma... | https://mathoverflow.net/users/nan | Uniruled locus in the parameter space of hypersurfaces | One can find such bounds in [Corollary 4.2, "Rational curves on complete intersections in positive characteristic" by E. Riedl and M. Woolf]: The space of multidegree $\underline{d}$ complete intersections in $\mathbb{P}^n$ containing a rational curve has codimension at least $\sum d\_i −2n +2$. The space of uniruled h... | 1 | https://mathoverflow.net/users/nan | 381672 | 158,860 |
https://mathoverflow.net/questions/381533 | 2 | I was primarily interested in the following question. Let $n\geq 3$, and let $X\subset \mathbb{P}^n$ be a degree $d$ hypersurface. Assume that its singularity locus $S$ (with reduced structure) is irreducible and smooth of dimension $k$. (As pointed out by @abx , this should not be a strict inclusion)
Is it true that
$... | https://mathoverflow.net/users/98788 | On the dimension of the dual variety of a singular hypersurface | No. Consider the cubic surface $X$ defined by the equation $$x\_0^2x\_2+x\_1^2x\_3=0.$$
Then $X$ is singular along the line $x\_0=x\_1=0.$ Then $X^{\vee}$ is a hypersurface (actually isomorphic to $X$) and $S^{\vee}\cong \mathbb P^1$ (consisting of the hyperplanes containing the line). So
$$\dim S^{\vee}=1<2=\dim X... | 1 | https://mathoverflow.net/users/74322 | 381673 | 158,861 |
https://mathoverflow.net/questions/381550 | 3 | What is the best known bound for the Mertens function along arithmetic progressions? More specifically, what is the best bound known for
$$\sum\_{n<x}\mu(kn)$$
as $k,x\to\infty$. [This](https://arxiv.org/pdf/1406.7326.pdf) paper of Lynelle Ye gives a very complete solution, but only under the RH so it is not much u... | https://mathoverflow.net/users/159298 | Best known bounds for $\left|\sum_{n<x}\mu(nk)\right|$ (Reference request) | We use the bound given by my previous answer: $$\sum\_{n<x}\mu(kn)\ll\_A \frac{kx}{(\log (kx))^A}.$$ Now we improve this bound by using that the sum $$\sum\_{d \in \mathbb N\atop d\mid n, d\mid k}\mu(d)$$ is $1$ or $0$ according to if $\gcd(n,k)=1$ or not. We get $$\sum\_{n<x}\mu(kn)=\mu(k)\sum\_{n<x\atop\gcd(k,n)=1}\m... | 3 | https://mathoverflow.net/users/9232 | 381675 | 158,862 |
https://mathoverflow.net/questions/381670 | 21 | In 2019, Shitov has shown a counterexample ([Ann. Math, **190**(2) (2019) pp. 663-667](https://doi.org/10.4007/annals.2019.190.2.6)) to [Hedetniemi’s conjecture](https://en.wikipedia.org/wiki/Hedetniemi%27s_conjecture),
$$\chi(G \times H)=\min(\chi(G),\chi(H))$$
where $\chi(G)$ is the [chromatic number](https://en.wi... | https://mathoverflow.net/users/63938 | Smallest known counterexamples to Hedetniemi’s conjecture | Yes, Xuding Zhu did this in *Relatively small counterexamples to Hedetniemi's conjecture* (J. Comb. Theory B **146** (2021) pp. 141-150, doi:[10.1016/j.jctb.2020.09.005](https://doi.org/10.1016/j.jctb.2020.09.005), arXiv:[2004.09028](https://arxiv.org/abs/2004.09028)) where the sizes of the graphs are $3403$ and $10501... | 25 | https://mathoverflow.net/users/2384 | 381678 | 158,863 |
https://mathoverflow.net/questions/381479 | 3 | Let $f : Y \to X$ be a morphism of projective manifolds, with $(X,D)$ log smooth. Consider the exact sequence which defines the relative (logarithm) cotangent bundle (where $\Delta := f^{\ast} D$):
$$0 \to f^{\ast}\Omega\_X^1(\log D) \to \Omega\_Y^1(\log \Delta) \to \Omega\_{Y/X}^1(\log \Delta) \to 0.$$
Let $\maths... | https://mathoverflow.net/users/172177 | Exact sequence of relative differential forms | For $0 \to A \to B \to C \to 0$ an exact sequence of modules, we obtain a filtration on $\wedge^d B$ where $F^k / F^{k+1}$ is $\wedge^k A \otimes \wedge^{d-k} C$.
To see this, recall that $\wedge^d B$ is defined as the quotient of $B\otimes \dots \otimes B$ by relations of the form $x \otimes y = - y \otimes x$. We d... | 4 | https://mathoverflow.net/users/18060 | 381684 | 158,864 |
https://mathoverflow.net/questions/373997 | 11 | Suppose $\kappa\leq\mu$ are infinite cardinals. Let us agree to call a family $\mathcal{P}\subseteq[\mu]^{<\mu}$ a countably generating family for $[\mu]^\kappa$ if every member of $[\mu]^\kappa$ can be written as a union of countably many elements of $\mathcal{P}$. Note that we can extend this in the obvious way to st... | https://mathoverflow.net/users/18128 | Can this result in cardinal arithmetic be established without using pcf theory? | There **IS** an easy proof of this, but I just had to reframe the way I was thinking of the problem. The cardinals in question (and many of their relatives) turn out to be $2^{\mu}$ if $\mu$ is strong limit because we can do suitable coding:
Suppose $\mu$ is a singular strong limit cardinal of cofinality $\kappa$. Si... | 6 | https://mathoverflow.net/users/18128 | 381687 | 158,865 |
https://mathoverflow.net/questions/381676 | 10 | It is well known that (working over ZF) AC implies that every [fibration](https://ncatlab.org/nlab/show/fibration) $p:\mathcal{E}\to\mathcal{B}$ can be equipped with a [cleavage](https://ncatlab.org/nlab/show/cleavage) by choosing, for each arrow $u:I\to p(X)$ in the base category whose codomain is in the image of the ... | https://mathoverflow.net/users/92164 | Is AC equivalent over ZF to 'every fibration can be equipped with a cleavage'? | Yes, in fact Grothendieck fibration between groupoids are enough.
Let $p:Y \to X$ be any surjection.
We construct the following groupoid $G$. Its set of objects is $X \amalg Y$. Its morphisms corresponds to the equivalence relation such that two elements of $y$ are equivalent if they have the same image by $p$ and ... | 16 | https://mathoverflow.net/users/22131 | 381690 | 158,866 |
https://mathoverflow.net/questions/381607 | 16 | Let $G$ be a finitely presented group of cohomological dimension $n$.
Apart from the unresolved ambiguity pertaining to the Eilenberg--Ganea conjecture, it is known that we can find an $n$-dimensional model of the classifying space $BG = K(G,1)$.
It is also not hard to see that $BG$ needs to have cells in every dim... | https://mathoverflow.net/users/14233 | How many cells needed to build the classifying space $BG$? | A group $G$ is of type $\mathcal{F}\_n$ if it has a $K(G,1)$ with finite $n$-skeleton. Let $F\_2$ be the free group on $2$ generators. Consider the kernel of the map $F\_2\times F\_2 \times \cdots \times F\_2 \to \mathbb{Z}$, sending each generator to $1$. Then the kernel is of type $\mathcal{F}\_{n-1}$ but not of type... | 7 | https://mathoverflow.net/users/1345 | 381692 | 158,868 |
https://mathoverflow.net/questions/381590 | 2 | 1-planar graphs are those can be drawn in the plane so that there is at most one crossing per edge. We know that the maximum number of edges of an $n$-vertex 1-planar graph is at most $4n-8$, and the maximum number of edges of a bipartite $n$-vertex 1-planar graph is at most $3n-8$. I remember that there is also a resu... | https://mathoverflow.net/users/148974 | The density of a tripartite 1-planar graph | I find the following source. The known bound is 3.5n-7 (Theorem 4.8, pp. 57).
<https://www.springer.com/gp/book/9789811565328>
Beyond Planar Graphs
Communications of NII Shonan Meetings
Editors: Hong, Seok-Hee, Tokuyama, Takeshi (Eds.)
| 1 | https://mathoverflow.net/users/148974 | 381693 | 158,869 |
https://mathoverflow.net/questions/381380 | 9 | **Question:** Are the class numbers of $\mathbb{Q}(\cos(\frac{2\pi}m))$ $O(m^n)$ for some fixed $n$?
Evidences (e.g. a recent paper) showing that the question above is open are also OK.
**Remark:** If such $n$ exists, then $n\geq2$. By the paper [unit groups and class numbers of real cyclic octic fields](https://pd... | https://mathoverflow.net/users/125498 | Are the class numbers of $\mathbb{Q}(\cos(2\pi / m))$ $O(m^n)$ for some fixed $n$? | The answer is **no**.
By Proposition 2 of Gary Cornell and Michael I. Rosen 's paper [The -rank of the real class group of cyclotomic fields](http://www.numdam.org/article/CM_1984__53_2_133_0.pdf) (paraphased):
>
> Let $L/\mathbb Q$ be an abelian -extension of the rationals with Galois group
> $G$. Assume the ine... | 4 | https://mathoverflow.net/users/125498 | 381700 | 158,873 |
https://mathoverflow.net/questions/381637 | 10 | Thanks to a result of Herman and Vaserstein in [[3](https://doi.org/10.1007/BF01388839)], Rieffel's notion of stable rank [[4](https://doi.org/10.1112/plms/s3-46.2.301)] coincides with the Bass stable rank [[1](http://www.numdam.org/item/PMIHES_1964__22__5_0/?source=BSMF_1962__90__323_0)] for every $C^\ast$-algebra $A$... | https://mathoverflow.net/users/35324 | Stable rank one and corners of $C^\ast$-algebras | For separable $C^\ast$-algebras this phenomenon cannot happen, since any corner of a separable stable rank one $C^\ast$-algebra has stable rank one. More generally, this holds for $C^\ast$-algebras $A$ such that all its two-sided closed ideals are $\sigma$-unital.
In fact, suppose $A$ is separable and has stable rank... | 4 | https://mathoverflow.net/users/126109 | 381703 | 158,876 |
https://mathoverflow.net/questions/381525 | 8 | In order theory, an antichain (Sperner family/clutter) is a subset of a partially-ordered set, with the property that no two elements are comparable with each other. A maximal antichain is the antichain which is not properly contained in another antichain. Let's take the power set of $\{1,2,\ldots, n\}$ as our partiall... | https://mathoverflow.net/users/129960 | Verification of a maximal antichain | *Remark.* Originally I claimed this to be a full solution, but that was false, as shown by Emil in the comments. However, this argument proves the following weaker version.
I can prove that it is co-NP-complete to decide for an input family $A$ whether there is a set $S$ that is unrelated to all sets in $A$. I'll cal... | 2 | https://mathoverflow.net/users/955 | 381704 | 158,877 |
https://mathoverflow.net/questions/380455 | 2 | Let $\mathcal{P}\_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and *weak* topology. Let $X\_t$ be a strong solution to the SDE with initial conditions
$$
dX\_t = \mu(t,X\_t)dt + \sigma(t,X\_t) dW\_t, \mbox{ } X\_0=x
$$
for some Lipschitz-continuous functions $... | https://mathoverflow.net/users/36886 | Weak continuity of law | You can write $$X\_t-X\_t'=x-x'+\int\_0^t(\mu(s,X\_s)-\mu(s,X\_s'))ds+\int\_0^t(\sigma(s,X\_s)-\sigma(s,X\_s'))dW\_s $$
Then
$$ \mathbb{E}(\|X\_t-X\_t'\|^2) \leq 3 \left(|x-x'|^2+a^2t\int\_0^t \mathbb{E}(\|X\_s-X\_s'\|^2)ds + b^2\int\_0^t\mathbb{E}(\|X\_s-X\_s'\|^2) ds \right)$$ with $a,b$ the lipschitz constant of $\m... | 2 | https://mathoverflow.net/users/99045 | 381733 | 158,886 |
https://mathoverflow.net/questions/381667 | 0 | For a given $q\in (0,1]$, define the $l\_q$-ball as
$$\mathbb{B}\_q(R\_q)\mathrel{:=}\left\{\theta\in\mathbb{R}^d\,\middle\vert\,\sum\_{j=1}^d \lvert\theta\_j\rvert^q\leq R\_q \right\}. $$
For a given integer $s\in\{1,2,\dotsc,d\}$, the best $s$-term approximation to a vector $\theta^\*\in\mathbb{R}^d$ is defined as
$$... | https://mathoverflow.net/users/163923 | Properties of $l_q$-balls | WLOG, let $\theta^\*=(\theta^\*\_1,...,\theta^\*\_d)$ with $|\theta^\*\_1|\geq |\theta^\*\_2| \geq\cdots\geq |\theta^\*\_d|$. Then we have
$$\|\Pi\_s(\theta^\*)-\theta^\*\|\_2^2 = \sum\_{j=s+1}^d |\theta^\*\_j|^2 \leq |\theta^\*\_s|^{2-q} \sum\_{j=s+1}^d |\theta^\*\_j|^q = \left(\frac{1}{s} \sum\_{i=1}^s |\theta^\*\_s|... | 1 | https://mathoverflow.net/users/172610 | 381735 | 158,888 |
https://mathoverflow.net/questions/369729 | 9 | Serre's conjecture says that given any odd, irreducible, continuous representation $\rho:G\_{\mathbb{Q}}\rightarrow GL\_2(\overline{\mathbb{F}\_p})$ there is some eigenform $f$ of weight $k(\rho)$, level $N(\rho)$, and nebentype $\epsilon(\rho)$, such that $\rho$ is isomorphic to the mod $p$ representation $\bar \rho\_... | https://mathoverflow.net/users/124710 | Is the weight in Serre's conjecture "minimal"? | I believe a modular form as you describe indeed cannot exist. I think it's easier to think about these issues if they're translated into the representation-theoretic language of Serre weights. Associated to $\overline{\rho}$ is a set of Serre weights, i.e., of irreducible mod $3$ representations of $\mathrm{GL}(2,\math... | 4 | https://mathoverflow.net/users/379 | 381736 | 158,889 |
https://mathoverflow.net/questions/381740 | 2 | Is there a good integration by parts formula to compute
$$\int\_{0}^\infty f \ H (f') dx,$$
where $H$ denotes the [Hilbert transform](https://en.wikipedia.org/wiki/Hilbert_transform) and $f$ is a smooth function?
| https://mathoverflow.net/users/110835 | Integration by parts with Hilbert transform | Perhaps something like this? (with integration from $-\infty$ to $\infty$ to arrive at a nicely symmetric answer):
$$\int\_{-\infty}^\infty f \ H (f') dx=\frac{1}{\pi}\text{P.V.}\,\int\_{-\infty}^\infty \int\_{-\infty}^\infty \frac{f(x)f'(y)}{x-y}\,dxdy$$
$$\qquad=\frac{1}{\pi}\int\_{-\infty}^\infty \int\_{-\infty}^\in... | 3 | https://mathoverflow.net/users/11260 | 381745 | 158,891 |
https://mathoverflow.net/questions/381706 | 3 | Let $B$ be a quaternion algebra over $\mathbb{Q}$ that is a division algebra over $\mathbb{R}$. The theorem of Hasse-Schilling tells us that the image of the reduced norm of $B^\times$ is $\mathbb{R}^+$.
Let $M$ be a maximal order in $B$. What is known regarding the image of the reduced norm of $M$? To be more specif... | https://mathoverflow.net/users/156310 | Norm of maximal order in quaternion algebra | This is a wonderfully rich question! I'll refer to my book (<http://quatalg.org>).
I'll write $\mathcal{O}$ for your $M$, a definite quaternion order over $\mathbb{Z}$. Let $\mathrm{nrd} \colon \mathcal{O} \to \mathbb{Z}$ be the reduced norm. Then $\mathrm{nrd}$ is a quadratic form on $\mathcal{O}$ (more generally, s... | 5 | https://mathoverflow.net/users/4433 | 381747 | 158,892 |
https://mathoverflow.net/questions/381487 | 1 | I'm reading Demailly's *Complex Analytic and Differential Geometry* In Section I.2.D.4 he uses the following fact: Suppose $u \in \mathcal{D}'(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a distribution such that all of its derivatives are of order zero, i.e.
>
> for every compact $K \subset \Omega$ there is a ... | https://mathoverflow.net/users/123448 | A distribution $u$ such that all of its derivatives are of order zero is smooth | *Step 1.* With no loss of generality we may assume that $u$ is compactly supported (just multiply $u$ by a test function if it is not). Let $B = [-M,M]^n$ be a box that contains the support of $u$.
*Step 2.* Let $v = \partial^{(2,2,\ldots,2)} u$. Since $|\langle v, f\rangle| = |\langle u, \partial^{(2,2,\ldots,2)} f\... | 2 | https://mathoverflow.net/users/108637 | 381757 | 158,894 |
https://mathoverflow.net/questions/381754 | 11 | Let $M$ be the mod-$p$ Moore spectrum where $p \geq 3$ is a (power of) a prime. Then $M$ satisfies the "polynomial equation" $M \wedge M \cong M \oplus \Sigma M$. Is this a general phenomenon, or is it something very special about Moore spectra?
More formally, let $K = K\_0^\oplus(\mathbb S\_{(p)})$ be the direct-sum... | https://mathoverflow.net/users/2362 | Solving polynomial equations in spectra? | Here is a simple argument that would show many finite complexes can not be `integral' in your sense.
If $Sq^{2^k}$ acts nontrivially on $H^\*(X;\mathbb Z/2)$ then $Sq^{2^{k+1}}$ will act nontrivially on $H^\*(X \wedge X;\mathbb Z/2)$. But if $X$ were integral then there would be an upper bound on $k$ such that $Sq^{2... | 17 | https://mathoverflow.net/users/102519 | 381758 | 158,895 |
https://mathoverflow.net/questions/327188 | 5 | Consider the polynomial ring $R=\mathbb Z[x\_1,\ldots,x\_n]$ and an ideal $I\subset R$. Let $<$ be a monomial order, i.e. a total order on the set of monomials in $R$ such that for any monomials $a$, $b$ and $c$ we have $ab<ac$ whenever $b<c$. For such an order the initial ideal $\mathrm{in}\_<(I)$ is the linear space ... | https://mathoverflow.net/users/19864 | Infinitely many initial ideals for non-Artinian monomial orders? | The OP has already figured out the answer since the question was posted. I am writing a slightly more expanded version here because I haven't been able to find this written down somewhere.
Let $\mathcal T\_n$ denote the set of all possible term orders on monomials in the variables $\{x\_1,x\_2,\dots,x\_n\}$. These ar... | 3 | https://mathoverflow.net/users/2384 | 381763 | 158,897 |
https://mathoverflow.net/questions/380134 | 5 | Let $X$ be a metric space. We say that $X$ is *asymptotically geodesic* if for all $\epsilon > 0$, there exists $R > 0$ such that, for all $x,y \in X$, there exists some finite sequence of points $p\_0=x,p\_1,p\_2,...,p\_N = y \in X$ such that each $d(p\_{i-1},p\_i) \le R$ and
$$ \sum\_{i=1}^N d(p\_{i-1},p\_i) \le (1 +... | https://mathoverflow.net/users/169294 | Example of an invariant metric on a nilpotent group which is not asymptotically geodesic | This answer seems to essentially be what @YCor was going for in his comments, but I'll give very explicit examples, since at the level he described the answer, there were some details that needed to be checked that weren't immediately clear to me. (I hope this is an appropriate situation to answer my own question.)
F... | 2 | https://mathoverflow.net/users/169294 | 381766 | 158,898 |
https://mathoverflow.net/questions/381680 | 2 | Let $(X\_n)\_n$ be a sequence of independent random variable, $(u\_n)\_n$ a sequence of positive numbers, such that $$\frac{1}{u\_n}\sum\_{k=1}^nX\_k \Rightarrow X$$
where $X$ is not degenerate.
Prove that either $(u\_n)\_n$ converges or there exist an increasing sequence $(v\_n)\_n$ such that $\lim\_n v\_n =+\infty,... | https://mathoverflow.net/users/172528 | Finding a sequence from weak convergence | **1.** by the symmetrization, we may discuss the sequence $ \{X\_n,n\ge 1\} $ of independent random variables with following representation only:
\begin{equation\*}
X\_k=Y\_k^\prime-Y\_k^{\prime\prime},
\end{equation\*}
where $ \{Y\_k^\prime,Y\_k^{\prime\prime},k\ge 1\} $ is a sequence of independent random variables ... | 2 | https://mathoverflow.net/users/103256 | 381768 | 158,899 |
https://mathoverflow.net/questions/381746 | 1 | Having a white random process $s(t)$, is it possible to write $$s(t)=\sum\_{i=0}^\infty\alpha\_i\phi\_i(t)$$ where the $\alpha\_i$ are random variables and the $\phi\_i$ orthogonal polynomials (Jacobi Polynomials, Legendre Polynomials...)?
| https://mathoverflow.net/users/151969 | Expansion of white noise into infinite series using orthogonal polynomials | Yes, if $\{f\_j\}$ is an orthonormal basis of $L^2[0,1]$ and $Z\_j$ are i.i.d. standard normal variables, then $S(t)=\sum\_j Z\_j f\_j(t)$ is a white noise. Note that this sum does not converge pointwise and should be understood as a distribution. Alternatively,
if $F\_j(t)=\int\_0^t f\_j(r) \,dr$ then $B(t)=\sum\_j Z\... | 1 | https://mathoverflow.net/users/7691 | 381770 | 158,901 |
https://mathoverflow.net/questions/334816 | 3 | The model of perceptron is a linear binary classifier, which is $f(x)=\mathbb{sign}(w^Tx+b)$. $x$ is the datapoint as $w$ as well as $b$ are the parameters.
The cost function of Primal Perceptron is $$\min\limits\_{w,b}-\sum\_{x\_i\in M}{y\_i\left(w^Tx\_i+b\right)}$$.
Where $M$ means the datapoints set where some p... | https://mathoverflow.net/users/142356 | How to inference the dual form of perceptron? | **Disclaimer**: I realize this is a very late response; hopefully, it will be useful to others. Additionally, this is going to be a long answer, as I will try to work from first principles. Don't say I didn't warn you. I will solve for the (slightly more general) case when some errors are allowed as we cannot assume re... | 2 | https://mathoverflow.net/users/171768 | 381780 | 158,907 |
https://mathoverflow.net/questions/381782 | 2 | I'm looking for a simple example in discrete dynamical systems whose periodic points set is not necessary closed.
I've seen some example in websites but they are not that simple and discrete.
Note that :
>
> $(X,f)$ is a Dynamical System if $f:X \to X$ is a homeomorphism and $X$ is a compact metric space.
> \be... | https://mathoverflow.net/users/172647 | Construct a homeomorphism whose periodic points set is not closed | Answering based on the comments of mine and YCor. This answer is also a duplicate of the answer to [the same question on Math.SE](https://math.stackexchange.com/questions/3993791/construct-a-homeomorphism-f-such-that-perf-is-not-closed/3993916#3993916).
Consider the space of infinite strings on a finite alphabet (so ... | 3 | https://mathoverflow.net/users/172450 | 381792 | 158,911 |
https://mathoverflow.net/questions/381788 | 1 | Let $f: X\rightarrow Y$ be proper etale morphism between varieties over the field of complex numbers. Does there exists a finite group $G$ such that $Y$ is the categorical quotient of $X$ under the free action of $G$? Or in other words is every etale cover a principal bundle?
We can consider the group scheme $Aut\_Y(... | https://mathoverflow.net/users/152042 | Is every etale cover a principal bundle? | Not every etale covering is a principal bundle under a group $G$. This is easiest to see using the Galois correspondence for etale coverings: the category of finite etale coverings of $X$ is equivalent to the category of finite continuous $\pi\_1^{et}(X,x)$-sets, for a chosen basepoint $x\in X$. Passed across this form... | 7 | https://mathoverflow.net/users/126183 | 381793 | 158,912 |
https://mathoverflow.net/questions/381799 | 9 | The notion of a $C^\*$-algebra being *nuclear* has many equivalent characterisations. These are considered in the excellent, modern textbook [$C^\*$-Algebras and Finite-Dimensional Approximations](https://www.ams.org/publications/authors/books/postpub/gsm-88) by Brown and Ozawa. They take as definition that the identit... | https://mathoverflow.net/users/406 | Reference request: Brown Ozawa and strong completely positive approximation property? | I am not sure if it is in Brown and Ozawa, but it is in Pisier's recent book "Tensor Products of C\*-algebras and Operator Spaces" as Corollary 10.16. It may also be in his earlier Operator Spaces book, but my copy isn't with me.
| 10 | https://mathoverflow.net/users/34640 | 381812 | 158,917 |
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