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https://mathoverflow.net/questions/385560 | 0 | Given $10 \times 10$ matrices $A$ and $B$, I would like to find $10 \times 10$ matrix $X$ such that
$$A = X B X^T \tag{1}$$
$$B = X A X^T \tag{2}$$
How can I solve the issue? if there is a way to solve only equation (1) or (2) that is ok also.
If anyone can already solve this and show me the way it's fine too.
... | https://mathoverflow.net/users/175154 | Solving two quadratic matrix equations | I calculated the determinant of your two matrices:
I find $\det A=7.95808\cdot 10^{-13} \, i$ while $\det B=-4.54747\cdot 10^{-13} + 2.27374\cdot 10^{-13} i$. If your equations (1) and (2) hold, then $\det A=(\det C)^2\det B$ and $\det B=(\det C)^2\det A$, so $(\det A)/(\det B)=(\det B)/(\det A)$, which fails.
Henc... | 0 | https://mathoverflow.net/users/11260 | 385571 | 160,282 |
https://mathoverflow.net/questions/385546 | 48 | Inspired by the question [here](https://mathoverflow.net/questions/385129/), I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI.
A general comment is that sheaf-theoretic methods do not a priori produce "material set theo... | https://mathoverflow.net/users/6074 | Sheaf-theoretic approach to forcing | Yes, this is a model of ETCSR. Unfortunately, I don't know of a proof of this in the literature, which is in general sadly lacking as regards replacement/collection axioms in topos theory. But here's a sketch.
As Zhen says, the filterquotient construction preserves finitary properties such as Booleanness and the axio... | 28 | https://mathoverflow.net/users/49 | 385573 | 160,283 |
https://mathoverflow.net/questions/385431 | 2 | I am looking for references which might study the following problem in non-trivial cases:
Let $X\subset\mathbb{P}^n\_{\mathbb{C}}$ be a general hypersurface of degree $d$ and consider $G:=G(k+1,n+1)$ the Grassmannian of $k$-planes $\mathbb{P}^k\subset\mathbb{P}^n$. We get an induced family $\pi:Y\to G$ with fibre $Y\... | https://mathoverflow.net/users/386 | Loci of singular plane sections of a generic hypersurface | Consider the flag variety $F = Fl(k,k+1;V)$ and let
$$
U\_k \subset U\_{k+1} \subset V \otimes \mathcal{O}
$$
be the tautological flag of subbundles. We have an exact sequence
$$
0 \to L \to U\_{k+1}^\vee \to U\_k^\vee \to 0,
$$
where $L$ is a line bundle (in fact, $L \cong \det(U\_{k+1}^\vee) \otimes \det(U\_k)$). It ... | 2 | https://mathoverflow.net/users/4428 | 385589 | 160,285 |
https://mathoverflow.net/questions/385576 | 4 | Consider the following system:
$$
\begin{cases}
x\_1 + 3 x\_3 = 4a, \\
f(x\_1) + 3 f(x\_3) = 8 f(a), \\
f'(x\_1) = 3 f'(x\_3).
\end{cases}
$$
I want to find all functions (or at least learn some properties that hold for all of them) $f : [0,1] \to [0,1]$ that are continuous, differentiable on $[0,1]$, monotonically d... | https://mathoverflow.net/users/101533 | Solve differential system of equations | Assume such an $f$ exists. For each $a\in(0,1)$ fix some solution $x\_1(a),x\_3(a)$ of the system. The first equality can be restated as
$$ \frac{1}{4}x\_1(a) + \frac{3}{4}x\_3(a) = a.$$
In other words, $a$ is a convex combination of $x\_1(a),x\_3(a)$. Consider a sequence $a\_k \searrow 0$. As $x\_1(a\_k),x\_3(a\_k) \g... | 5 | https://mathoverflow.net/users/85570 | 385591 | 160,286 |
https://mathoverflow.net/questions/385169 | 1 | Working in first order logic with equality and membership $``\sf FOL(=,\in)"$
Let $\phi x$ be a formula in which only $x$ occur free, and never bound.
Let $\pi\_i x \vec{z}$ be the formula $\forall y (y \in x \leftrightarrow \psi\_i y\vec{z})$ where $\psi\_i y \vec{z}$ is a formula in which only symbols $``y,z\_1,.... | https://mathoverflow.net/users/95347 | Is there a known counter-example to this rule? | Take $\psi(y)$ to be $(y=y)\land \exists u\, \forall v\, \neg(v\in u)$, so that $\psi$ expresses the statement that there is some $\in$-minimal object. Thus, $\pi(x)$ is just false if there are no $\in$-minimal objects, but if there are $\in$-minimal objects then it says $x$ has all objects as elements.
Let $\phi(x)$... | 2 | https://mathoverflow.net/users/3199 | 385610 | 160,293 |
https://mathoverflow.net/questions/344124 | 0 | Consider the uniform distribution $\lambda$ on $[0,1]$, and a point measure $\rho$ with density $\frac{1}{2} (\delta\_{x\_1} + \delta\_{x\_2})$, where we have $0\le x\_1 \le x\_2 < 1/2$.
If our cost is just the distance $c(x,y) = | x - y|$, it seems reasonably clear that the optimal transport map from $\lambda$ to $\... | https://mathoverflow.net/users/120706 | A problem with the dual form of semi-discrete optimal transport | Just like to share for anyone who ends up here - my problem was a lack of strict convexity in the cost function when represented as $c(x,y) = h(x-y)$. The function $h(x-y) = |x - y|$ is not strictly convex, so according to theory that can be found, for example in [Gangbo and McCann](https://projecteuclid.org/euclid.act... | 0 | https://mathoverflow.net/users/120706 | 385615 | 160,296 |
https://mathoverflow.net/questions/385597 | 5 | Let $K$ be a number field with $[K:\mathbb{Q}]=n$ with $n \geq 2$ and let $\mathcal{O}\_K$ be its ring of integers. Suppose that $\alpha\_1, \cdots, \alpha\_n \in \mathcal{O}\_K$ are distinct algebraic integers such that $N\_{K/\mathbb{Q}}(\alpha\_j) = a$ for some fixed rational integer $|a| > 1$ and the principal idea... | https://mathoverflow.net/users/10898 | Linear independence of algebraic integers of equal norm | We adapt an idea from a now-deleted answer by **Kenny Lau**
to construct examples for any $n>2$ with the $\alpha\_j$ all contained in
2-dimensional space. Let $a$ be prime, and choose distinct integers
$x\_1,\ldots,x\_n$ that remain different mod $a$ for which
$$
P(x) := \left[\prod\_{j=1}^n (x-x\_j)\right] - a
$$
is i... | 5 | https://mathoverflow.net/users/14830 | 385616 | 160,297 |
https://mathoverflow.net/questions/330279 | 10 | My question is of local nature.
Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative.
Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)$$ is a CR manifold. Let me assume that $M$ is the simplest possible kind of CR manifold, namely that it is folia... | https://mathoverflow.net/users/5690 | Complex manifold with boundary | Perhaps Giuseppe Della Sala's paper might be useful here: <https://www.ams.org/journals/proc/2011-139-07/S0002-9939-2010-10746-3/home.html>
It precisely deals with the equivalence of smooth Levi-flats. There are examples in the paper
| 2 | https://mathoverflow.net/users/2783 | 385619 | 160,299 |
https://mathoverflow.net/questions/385581 | 5 | [Turán's theorem](https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem) says the following.
>
> Take any natural $n$ and $r$. Suppose that
> \begin{equation\*}
> |G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0}
> \end{equation\*}
> where $|G|$ is the number of edges of an (undirected) graph $G$ with $n$ vertices. Then $... | https://mathoverflow.net/users/36721 | Randomized version of Turán's theorem | No. Consider the following distribution: Let $M$ be an integer, say $M= \frac{n}{\log n}$. Then for each vertex $v \in [n]$ assign an integer $m(v)$ where $m(v)$ is chosen according to the uniform distribution on $\{0,1,\ldots, M-1\}$, and where the $m(v)$s; $v \in [n]$; are mutually independent. Then for each pair $u$... | 3 | https://mathoverflow.net/users/122188 | 385629 | 160,302 |
https://mathoverflow.net/questions/385202 | 0 | One of the famous problem in SDP is the matrix norm minimization (see S. Boyd, *Convex Optimization*, p. 170).
Consider:
\begin{equation}\label{eq:Lasse}
\begin{aligned}
&\min\_{\mathbf{x}}
& & \|A(x)-M\|\_2 \\
& & & A(x)=-A(x)^T
\end{aligned}
\end{equation}
Here
1. $x\in \mathbb{R}^n$
2. $A(x)=x\_1A\_1+\cdots... | https://mathoverflow.net/users/93600 | Matrix norm minimization and matrix inner product | [The comments to this question of mine](https://mathoverflow.net/questions/231727/add-a-multiple-of-i-to-a-matrix-to-minimize-its-operator-norm#comment573211_231727) show that, in general, the operator-norm and Frobenius-norm minima are distinct for this problem.
Let me summarize the argument here. Let $M=diag(\alpha... | 1 | https://mathoverflow.net/users/1898 | 385640 | 160,306 |
https://mathoverflow.net/questions/385628 | 10 |
>
> Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently of $n$?
>
>
>
First, I thought as obvious that the translation length of $a^nb$ has to tend to $+ \infty$ as ... | https://mathoverflow.net/users/122026 | Translation lengths in CAT(0) spaces | Consider the following two transformations of $\mathbb{R}^3$: $a:(x,y,z)\mapsto (x+1,y,z)$ and $b:(x,y,z)\mapsto (-x,-y,z+1)$. The translation axis for $a^nb$ is the line $x=n/2$, $y=0$ and $a^nb$ translates this line by a distance of 1, independent of $n$.
| 14 | https://mathoverflow.net/users/124004 | 385651 | 160,311 |
https://mathoverflow.net/questions/385652 | 1 | I have two lognormally distributed random variables $Y\_i=e^{X\_i}$ where $X\_i \sim \mathcal{N}\big(\mu\_i, \: \sigma\_i^2 \big)$ for $i=1,2$, and $X\_1$ and $X\_2$ are correlated by $\rho\_{12}$. Now, Let $Z=\alpha Y\_1 - \beta Y\_2$.
***Question 1***
Is $Z$ lognormally distributed?
***Question2***
When $\mu\... | https://mathoverflow.net/users/175345 | Confidence interval for the difference of lognormally distributed random variables | Question 1: $Z$ will not be lognormally distributed in general. E.g., if $\beta>0$ or $\alpha<0$, then $P(Z<0)>0$, and hence $Z$ is not lognormally distributed.
Question 2: If $\mu\_Z$ is known, then it does not make sense to use a confidence interval for $\mu\_Z$.
If $\mu\_Z$ and $\sigma\_Z$ are both unknown, then... | 1 | https://mathoverflow.net/users/36721 | 385654 | 160,312 |
https://mathoverflow.net/questions/385270 | 4 | Let $g(x)$ be a polynomial with integral coefficients.
For $r\geq 1$, We define the sequence $a\_{g}$ for some polynomial $g(x)$ as follows:
$\clubsuit)a\_{g}(1)=g(x)$
$\clubsuit)a\_{g}(r)=g(a\_{g}(r-1))$ for $r\geq 2$
Now we are given a polynomial $f(x)$ of $\deg(f)\geq 2$ such that if $x=0$, $a\_{f}(r)\righta... | https://mathoverflow.net/users/160943 | Is it true that sum of reciprocal of primes $p$ such that $p|a_{f}(p)$ converges? | Have you seen [Silverman](https://arxiv.org/pdf/0707.1505.pdf) (Section 4)? The sum that he estimates there is the dynamical analogue of the relevant sums that appear when studying the analogue of $\gcd(n,f^n(0))$ (as in Kim for EDS, Sanna and myself for Lucas sequences... I am guessing this based on your post history)... | 2 | https://mathoverflow.net/users/nan | 385668 | 160,314 |
https://mathoverflow.net/questions/385659 | 0 | In Itô calculus, it is easy to construct an associativity rule. Namely, if $B\_t$ is a Brownian motion and $M\_t = \int\_0^t X\_s dB\_s$ for suitable $X\_t$, then we have the following associativity rule: $Z\_t = \int\_0^t Y\_s dM\_s = \int\_0^t Y\_s X\_s dB\_s$. Such a rule can be derived by defining a martingale calc... | https://mathoverflow.net/users/149959 | Associativity rule for integration against fractional Brownian motion | For $H > 1/2$ and assuming that both $X$ and $Y$ have trajectories that are almost surely $\alpha$-Hölder continuous for some $\alpha > 1/2$, there is only one sensible definition of the stochastic integral (Riemann-Stieltjes) and associativity holds since it does so for smooth functions and the integral is stable unde... | 3 | https://mathoverflow.net/users/38566 | 385672 | 160,316 |
https://mathoverflow.net/questions/385669 | 2 | Is there an analytic entire function $f:\mathbb{C} \rightarrow \mathbb{C}$ such that
1. $f(z)=\overline{f(\overline{z})},$
2. For every $\varepsilon>0$ there is a $\delta >0$ such that if $\textrm{Im }z > \delta,$ then $\|f(z)-i\|<\varepsilon,$
3. $f(0)=0?$
(Note that the tangent function satisfies the three proper... | https://mathoverflow.net/users/32470 | Entire reflection symmetric function which is near $i$ when $\textrm{Im }z$ is big | The construction is as follows. Property 1 will follow from the fact that $f$ maps the real line to itself, which is easier to verify. First consider the building block,
$$g(z)=\int\_\gamma\frac{e^{e^\zeta}}{\zeta-z}d\zeta.$$
where $\gamma$ consists of two parallel rays $\{ z=\pm(\pi/2+\epsilon)i+t:t\geq 0\}$
and a seg... | 3 | https://mathoverflow.net/users/25510 | 385678 | 160,321 |
https://mathoverflow.net/questions/385683 | 1 | The dimension of any irreducible $\frak{sl}\_n$-representation $V$ is clearly equal to the dimension of its dual representation $V^\*$. Does the dimension of an irreducible $\frak{sl}\_n$-representation determine it uniquely up to its dual? What happens for the $B,C$, and $D$ series?
| https://mathoverflow.net/users/153228 | Dimensions of $\frak{sl}_n$-representations | Recall that the dimension of the $\mathfrak{sl}\_n$ representation indexed by partition $\lambda$ is the number of semistandard Young tableaux of shape $\lambda$ with entries in $\{1,2,\ldots,n\}$.
You can check that there are 15 SSYTs of shape $(3,1)$ with entries in $\{1,2,3\}$, and similarly 15 SSYTs of shape $(4)... | 13 | https://mathoverflow.net/users/25028 | 385685 | 160,322 |
https://mathoverflow.net/questions/385585 | 3 | $\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then
\begin{equation}
\om(G)\ge\sum\_{i\in[n]}\frac1{n-d\_i},\tag{1}
\end{equation}
where $d\_i$ is the degree of vertex $i$ in $G$. Inequality (1) is the... | https://mathoverflow.net/users/36721 | Randomized version of Turán's theorem II | For completeness, I'll repeat here the construction as the other thread. Let $M$ be an integer say $M=\frac{1}{p}$. Then for each $v \in [n]$, let $m(v)$ be an integer chosen uniformly from $\{0,1,\ldots, M-1\}$, and then $u$ and $v$ form an edge iff $m(u)\not = m(v)$.
Then all of 0.--3. as above are satisfied, and $... | 1 | https://mathoverflow.net/users/122188 | 385692 | 160,325 |
https://mathoverflow.net/questions/385647 | 6 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). We say that $I\subseteq V$ is an *independent set* if $e\not\subseteq I$ for all $e\in E$.
We say that $H$ is *tameable* if every independent set is contained in a maximal independent set. Every graph [is tameable](https://mathoverflow.net/a/3... | https://mathoverflow.net/users/8628 | Tameable hypergraphs | No. For a counterexample let $H=(\omega,E)$ where $E=\{e\_n:n\in\omega\}$ and $e\_n=[n,\omega)=\{x\in\omega:x\ge n\}$.
If a subset $E'\subseteq E$ is finite then $(\omega,E')$ is tameable; every vertex cover contains a finite vertex cover which contains a minimal vertex cover, and (equivalently) every independent set... | 3 | https://mathoverflow.net/users/43266 | 385693 | 160,326 |
https://mathoverflow.net/questions/385648 | 5 | Can someone point me to a reference where the notion of "Lie crossed module" appeared for the first time?
I see many papers "recall" the definition of the Lie crossed module but, I do not see any mention of a "first-time" reference.
The definition of Lie crossed module I am referring to is mentioned as Definition 1... | https://mathoverflow.net/users/118688 | First time appearance of Lie crossed module (crossed module of Lie groups) in literature | Crossed modules of Lie algebras are defined by Kassel and Loday in Definition A.1 of [Extensions centrales d’algèbres de Lie](https://doi.org/10.5802/aif.896) (published 1982).
Crossed modules of Lie groups are defined by Mackenzie in Definition 1.5 of [Classification of principal bundles and lie groupoids with presc... | 5 | https://mathoverflow.net/users/402 | 385694 | 160,327 |
https://mathoverflow.net/questions/385695 | 28 | (Crossposted on math stack exchange: <https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation>)
It was recently brought to my attention that there is a striking similarity between the [Class Number Formula](https://en.wikipedia.org/wiki/Class_number_formula) and [... | https://mathoverflow.net/users/92433 | Relation between Schanuel's theorem and class number equation | **A spirit.** A general approach that may interest you is the following: counting laws can be obtained by studying suitable generating functions, *via* Tauberian arguments: the rightmost pole (resp. residue) of the generating function gives the growth order (resp. leading constant) in the counting law.
In the case of... | 21 | https://mathoverflow.net/users/43737 | 385699 | 160,328 |
https://mathoverflow.net/questions/385702 | 1 | Given a vector $v = (v\_1, \ldots, v\_n) \in \mathbb{R}^n$, we can associate a rational linear subspace with this vector: assume $\{1, v\_i \text{ for }i \in I\}$ is a linear basis of $\{1, v\_1, \ldots, v\_n\}$ over $\mathbb{Q}$: there exist positive integer $m$ and integer $n\_{j,i}, i \in \{0\} \cup I$, such that fo... | https://mathoverflow.net/users/129960 | Rational linear subspace corresponding to an irrational vector | Invariant description (which yields your basis-independence claim) is the following: *$V$ is the space of sequences $(x\_1,\ldots,x\_n)$ such that $\sum m\_ix\_i=0$ whenever $\sum m\_iv\_i$ is rational.*
To prove it, note that $\sum m\_i v\_i$ is rational if and only if $\sum m\_i [v\_i]=0$, where $[v\_i]$ is a class... | 3 | https://mathoverflow.net/users/4312 | 385705 | 160,329 |
https://mathoverflow.net/questions/385655 | 2 | Suppose that you have $A, B$ two unital $C^\*$ algebras and let $A \ast B$ the reduced free product (I think that it is the reduced amalgamated product over the common $\*$-subalgebra $\mathbb{C} 1$) and let $A\hat{\*} B$ be the full free product.
The questions are:
1. Is it true that $A\oplus B = \frac{A\hat\ast B... | https://mathoverflow.net/users/173852 | Uniqueness of the direct sum of $C^*$ algebras as quotient of free products | This question initially confused me, as "of course" the direct sum of $C^\ast$-algebras is simply the vector space direct sum, with the max norm. However, this is not the [direct product of rings](https://en.wikipedia.org/wiki/Direct_sum#Direct_sum_of_rings) exactly because of the unital issue: the natural maps $A\righ... | 1 | https://mathoverflow.net/users/406 | 385718 | 160,332 |
https://mathoverflow.net/questions/385715 | 2 | This is from Hartshrone exercise 6.6 part (a).
Let $A$ be a regular local ring and $M$ be a finitely generated $A$-module, prove the following
>
> $M$ is projective $\iff$ $\operatorname{Ext}^{i}(M,A)=\{0\}$ for all $i>0$
>
>
>
The hint is to use the following
**Proposition (6.11 A)**
If $A$ is a regular l... | https://mathoverflow.net/users/129919 | Characterization of projective modules in terms of Ext groups | Along the lines of Hartshorne:
by (1) for all finitely generated $\mathrm{N}$ we have $\mathrm{Ext^i(M,N)}=0$ ($i>\mathrm{dim(A)}$).
Since $\mathrm{N}$ is finitely generated, we may find an exact sequence of the form
$$0\rightarrow\mathrm{K}\rightarrow\mathrm{A}^{\oplus r}\rightarrow\mathrm{N}\rightarrow 0.$$
Takin... | 3 | https://mathoverflow.net/users/104669 | 385719 | 160,333 |
https://mathoverflow.net/questions/352803 | 3 | $\let\op=\operatorname$In $\op{Set}$, we have an $(\op{Epi},\op{Mono})$ [orthogonal factorization system](https://ncatlab.org/nlab/show/orthogonal+factorization+system). Strikingly, if we reverse the roles, we get the no-less-important $(\op{Mono},\op{Epi})$ [*weak* factorization system](https://ncatlab.org/nlab/show/w... | https://mathoverflow.net/users/2362 | What do you call a map of spaces which is weakly left orthogonal to all $n$-connected maps? | This question was answered in the comments $\mathcal L\_n$ comprises those maps which are retracts of relative $\leq n+1$-dimensional relative cell complexes.
Tom Goodwillie explains in the comments a cohomological characterization of $\mathcal L\_n$ for sufficiently large $n$.
| 1 | https://mathoverflow.net/users/2362 | 385750 | 160,339 |
https://mathoverflow.net/questions/385745 | 5 | This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int\_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above equation can be augmented either with Dirichlet boundary condition $u=g$ on $\partial\Omega$ or with Neumann boundary ... | https://mathoverflow.net/users/112207 | Seeking for references on some PDEs | Some time ago I've addressed a similar problem in [this Q&A](https://mathoverflow.net/questions/322568/laplace-equation-with-integral-source-terms/325079?r=SearchResults&s=7%7C4.4618#325079), so I feel I can offer something useful regarding the posed question.
>
> **Question 1:** What is the name and application of... | 5 | https://mathoverflow.net/users/113756 | 385753 | 160,340 |
https://mathoverflow.net/questions/385151 | 1 | Let $D = \mathbb{R^+} \times (\mathbb{R}\backslash \{0\})$
Let $\mu(dt \times dx)$ be a $\sigma$-finite measure on the Borel $\sigma$-algebra $\sigma(D)$.
Let $M(dt \times dx)$ be the Poisson random measure with intensity measure $\mu$, i.e. for $B \in \sigma(D)$, $M(B)$ is a Poisson random variable with intensit... | https://mathoverflow.net/users/130369 | Poisson point process in polar coordinates | This kind of thing is studied at length in *Random Measures, Point Processes, and Stochastic Geometry* by Baccelli, Blaszczyszyn, and Karray. The book is made freely available in pdf form by the authors, and you can find it by searching for the title.
**Theorem.** Given two locally compact second-countable Hausdorff ... | 1 | https://mathoverflow.net/users/24840 | 385758 | 160,343 |
https://mathoverflow.net/questions/385765 | 2 | **1)** How can we prove that the logistic sequence
$$x\_{n+1}=rx\_n(1-x\_n),\ x\_1=a\in (0,1)$$
converges to $\frac{r-1}{r}$, for $r\in [1,3]$?
**2)** Also I wonder how can we prove that the sequence $(x\_n)\_{n\in\mathbb{N}^\*}$ has two limit points (the fixed points $\dfrac{r^2+r+\sqrt{(r-3)(r+1)}}{2r^2}$ and $... | https://mathoverflow.net/users/61629 | Logistic sequence convergence | A boundary point of the immediate basin of attraction of an attracting fixed point for a continuous function is either a non-attracting periodic point of period $2$, or a non-attracting fixed point, or a point mapped to a non-attracting fixed point. If $r \in [1,3]$ there are no real points of period $2$, the only othe... | 1 | https://mathoverflow.net/users/13650 | 385768 | 160,348 |
https://mathoverflow.net/questions/385725 | 3 | If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}\_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\Omega^{i}(M)\times\Omega^{n-i}\_{X/S}\longrightarrow\Omega^{n}\_{X/S}$.
I am currently reading Deligne and Illusie's pap... | https://mathoverflow.net/users/70751 | Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism | First of all, Deligne-Illusie worked with relative Frobenius $F = F\_{X/S}: X \to X'$, so the target of the composition should be $\Omega\_{X'/S}^p$ instead of $\Omega\_{X/S}^p$.
Your question about why the composition
$$F\_\*\Omega\_{X/S}^p \to H^pF\_\*\Omega\_{X/S}^\bullet \to \Omega\_{X'/S}^p$$
is the trace map is... | 4 | https://mathoverflow.net/users/14037 | 385771 | 160,349 |
https://mathoverflow.net/questions/385785 | 0 | I need reference where they talk about to prove that spheres of dimension n≥2 don't admit symmetric flat connections.
| https://mathoverflow.net/users/175211 | Does spheres of dimension n≥2 admit symmetric flat connections? | Manifolds which admit symmetric flat connections are known as affine manifolds. A standard result on these spaces is that the fundamental group of a compact affine manifold must be infinite.
For a reference, see Corollary 1.14 of the following lectures.
<https://arxiv.org/abs/1802.03624>
| 1 | https://mathoverflow.net/users/125275 | 385787 | 160,354 |
https://mathoverflow.net/questions/385661 | 1 | Suppose that $D(A)$ is the derived category of of a ring A. Let $b\in D(A)$ be a compact object and $B$ the localizing subcategory generated by b (having arbitrary coproduct).
Does the inclusion functor $ D(A)\leftarrow B: i$ have a left adjoint ?
Let $W: D(A)\rightarrow B$ the right adjoint to the inclusion functor.... | https://mathoverflow.net/users/136909 | localizing subcategories of a nice triangulated category | As noted in the comments, there is no reason for the functor to have a left adjoint in general, as the inclusion will not preserve limits.
For the other two questions, the inclusion functor is fully-faithful, which occurs if and only if the unit map $\text{id} \to W \circ i$ is an equivalence. Finally, $A \in D(A)$ i... | 4 | https://mathoverflow.net/users/16785 | 385790 | 160,356 |
https://mathoverflow.net/questions/385792 | 4 | Let $R$ be a commutative ring and let $\mathrm{Mod}\_R$ be the category of (left) $R$-modules.
>
> Question: Is it true that the categories $\mathcal{Z}(\mathrm{Mod}\_R)$ and $\mathrm{Mod}\_R$ are equivalent?
>
>
>
I have read the previous claim in a couple of places but without any proof or reference, and I c... | https://mathoverflow.net/users/167503 | Drinfeld center of $\mathrm{Mod}_R$ | Let $(X,\Phi)$ be an object of the Drinfeld center.
We'd like to prove that $(X,\Phi)$ is isomorphic to $(X,$ standard symmetry isomorphism $)$, which would prove the equivalence as you say we already have fully faithfulness.
Compose the isomorphism $\Phi$ with the inverse of the standard symmetry isomorphism, to g... | 6 | https://mathoverflow.net/users/102343 | 385795 | 160,357 |
https://mathoverflow.net/questions/385796 | 7 | Let $T$ be an invertible positive operator and $S$ be another positive operator on a complex Hilbert space.
We then study
$$ \Vert (T+S)^{-1/2}T(T+S)^{-1/2}\Vert$$
I would assume that this norm is bounded by one.
But I fail to see how one could actually show this? Cause the definition of the square root using the f... | https://mathoverflow.net/users/150549 | Is this operator bounded? | Denote $Q=(T+S)^{-1/2}T(T+S)^{-1/2}$. The inequality $\|Q\|\leqslant 1$ is equivalent to $\langle Qx,x\rangle\leqslant \langle x,x\rangle$ for all vectors $x$. Denote $(T+S)^{-1/2}x=y$, we get $$\langle Qx,x\rangle=\langle (T+S)^{-1/2}Ty,x\rangle=\langle Ty,(T+S)^{-1/2}x\rangle=\langle Ty,y\rangle\leqslant \langle (T+S... | 13 | https://mathoverflow.net/users/4312 | 385799 | 160,358 |
https://mathoverflow.net/questions/368619 | 11 | I came across Villani's paper titled *"Hypocoercive diffusion operators"* and couldn't figure out a computation that is skipped in that paper. Specifically, consider the following transformed Fokker-Planck equation, where $h(t,x,v)$ is the unknown, $(x,v) \in \mathbb{R}^n \times \mathbb{R}^n$, $V(x)$ is some potential ... | https://mathoverflow.net/users/163454 | Modified energy method for transformed Fokker-Planck equation (tricky integration by parts…) | There is a worked-out proof in page 10 and following of [Hérau's lecture notes.](https://hal.archives-ouvertes.fr/hal-01616979/document) The detailed steps are for $n=1$, $V=0$, but I assume once that is understood, the more general case would follow smoothly.
As a short-hand notation we write $ \|\partial\_x h \|^2=... | 4 | https://mathoverflow.net/users/11260 | 385802 | 160,360 |
https://mathoverflow.net/questions/385810 | 2 | In trying to solve another the problem posed in the question <https://www.mathoverflow.net/q/385777/78539>, I'm led to consider the following problem.
Let $\mu\_\gamma$ be the Marchenko-Pastur distribution with parameter $\gamma \in (0,1)$. Note that $\mu\_\gamma$ is supported on $[t\_-,t\_+]$, where $t\_{\pm} = (1\p... | https://mathoverflow.net/users/78539 | For fixed $\lambda \ge 0$, Integrate the function $f_\lambda(x):=x/(x + \lambda)^2$ w.r.t. Marchenko-Pastur density | $$I(\lambda)=\int\_{t\_-}^{t\_+}\frac{\sqrt{\left(t\_+-t\right) \left(t-t\_-\right)}}{2 \pi {\gamma} (t+\lambda)^2}\,dt
=\frac{-\sqrt{ {\gamma}^2+2 {\gamma} ( {\lambda}-1)+( {\lambda}+1)^2}+ {\gamma}+ {\lambda}+1}{2 {\gamma} \sqrt{ {\gamma}^2+2 {\gamma} ( {\lambda}-1)+( {\lambda}+1)^2}}.$$
| 3 | https://mathoverflow.net/users/11260 | 385815 | 160,363 |
https://mathoverflow.net/questions/385818 | 2 | Let $q$ be a power of a prime and $S \subseteq \mathrm P^2 \mathbf F^q$ such that
$$ \forall g \in \operatorname{PGL}(3,q), gS \cap S \neq \emptyset.$$
Can it be that $\vert S \vert < 1+q$ ?
(I asked a version of this question few years ago [here](https://math.stackexchange.com/questions/2073541/sets-smaller-than-a-l... | https://mathoverflow.net/users/102887 | Smallest subset in $P^2 \mathbf F_q$ which cannot be disjointed from itself by a homography | Assume that $|S|\leqslant q$. Choose a random projective map $g$. The probability that $gs=t$ for fixed $s,t\in S$ equals $1/|\mathrm P^2 \mathbf F^q\|=1/(q^2+q+1)$, so the sum over all pairs $(s,t)\in S\times S$ is strictly less than 1 and there exists $g$ such that $gs\ne t$ for all pairs.
| 3 | https://mathoverflow.net/users/4312 | 385821 | 160,365 |
https://mathoverflow.net/questions/385824 | 2 | Let $\alpha \neq 1.$
**If $X,Y$ are two independent random variable such that $U=X+Y$ and $V=X+\alpha Y$ are independent, then $X$ and $Y$ are normally distributed.**
In term of characteristic functions this means that $$\forall x, y \in \mathbb{R}, \phi\_X(x+y)\phi\_Y(x+\alpha y)=\phi\_X(x)\phi\_Y(x)\phi\_X(y)\phi... | https://mathoverflow.net/users/172528 | Functional equations and normal distribution | Let $a:=\alpha$.
If $a\ne0$, then $X$ and $Y$ are normal by the [Darmois--Skitovich theorem](https://en.wikipedia.org/wiki/Darmois%E2%80%93Skitovich_theorem).
If $a=0$ and the distribution of $X$ is nondegenerate, then $U=X+Y$ and $V=X$ cannot be independent.
If $a=0$ and the distribution of $X$ is degenerate, th... | 2 | https://mathoverflow.net/users/36721 | 385828 | 160,367 |
https://mathoverflow.net/questions/385830 | 6 | This is [cross-posted from `math.se`](https://math.stackexchange.com/questions/4043136/putting-sheaves-to-work-for-algebraic-topology) after receiving points and no answers. I apologise if this question is too basic for MathOverflow.
I'm refreshing my memory of covering space theory, and this time around, I know some... | https://mathoverflow.net/users/123769 | Putting sheaves to work for algebraic topology? | For sufficiently nice topological spaces $X$ (e.g., locally connected for the last two categories to be equivalent, and semilocally simply connected and locally path-connected for all three to be equivalent), the following three categories are equivalent:
* Functors from the fundamental groupoid of $X$ to the categor... | 11 | https://mathoverflow.net/users/402 | 385833 | 160,369 |
https://mathoverflow.net/questions/385829 | 9 | Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I wasn't able to find a discussion of this question in the literature on topological groups and the Hilbert 5th problem.
... | https://mathoverflow.net/users/23935 | Are locally compact, Hausdorff, locally path-connected topological groups locally Euclidean? | Under the additional assumption of finite topological dimension pointed by YCor in the comments to the OP, the answer is *yes*, see e.g. Theorem 10, pp. 120 of the paper of K. Whittington, *Local connectedness in topological groups*, [Topology and its Applications **180** (2015) 111-123](http://dx.doi.org/10.1016/j.top... | 9 | https://mathoverflow.net/users/11211 | 385839 | 160,371 |
https://mathoverflow.net/questions/385846 | 8 | Is every Hausdorff, locally compact group that does not contain any non-trivial compact group, finitely dimensional?
| https://mathoverflow.net/users/110389 | About locally compact groups without compact subgroups | Yes, it's even a Lie group whose unit component is a semidirect product $R\rtimes S^n$, where $R$ is a simply connected solvable Lie group and $S$ is the universal covering of $\mathrm{SL}\_2(\mathbf{R})$.
Indeed, by van Dantzig, every locally compact group $G$ has an open subgroup $U$ such that $U/U^\circ$ is compac... | 12 | https://mathoverflow.net/users/14094 | 385848 | 160,374 |
https://mathoverflow.net/questions/385583 | 6 | Let $\mathbb{C}$ be the field of the complex numbers. Let $R=\mathbb{C}[x]$, $T=\mathbb{C}\langle x\rangle$ be the ring of entire series with convergence radius at least $1$, and let $S=\mathbb{C}\langle\langle x\rangle\rangle$ be the ring of entire series with infinite convergence radius. We have $R\subset S \subset T... | https://mathoverflow.net/users/66686 | If $M\otimes_S T$ is an $A$-module, is $M$ an $A$-module? | I may have a counter-example. Suppose $A$ is $\mathbb{C}[x,t]/(t^2=x+1)$, and let $M$ be $\mathbb{C}\langle \langle x\rangle\rangle $ as an $S$-module. Then, $M\otimes\_S T=\mathbb{C}\langle x\rangle $ is given an $A$-module structure by defining the action of $t$ as the multiplication by $\sqrt{x+1}$ where
$$\sqrt{x+1... | 2 | https://mathoverflow.net/users/66686 | 385855 | 160,376 |
https://mathoverflow.net/questions/385868 | 4 | Let $X$ be a smooth, plane projective curve of degree $6$ and genus $10$ (over complex numbers).
**Question :** Is it possible that there exists a special divisor $\Delta$ of degree $10$ on $X$ such that it has exactly $5$ independent sections?
**Observations :** $(1)$ From Clifford's Theorem, we have $h^0(\mathcal... | https://mathoverflow.net/users/156533 | Special divisors on smooth plane curves | Such a divisor cannot exist. Let $H$ be the divisor of a line. By the base-point free pencil trick, we have an exact sequence
$$0\rightarrow H^0(\Delta -H)\rightarrow H^0(\Delta)^2\rightarrow H^0(\Delta +H)\,;$$since $\deg (\Delta +H)=16$, we have $h^0(\Delta +H)\leq 8$, hence $h^0(\Delta -H)\geq 2$.
Then $D:=\Delta -H... | 6 | https://mathoverflow.net/users/40297 | 385874 | 160,381 |
https://mathoverflow.net/questions/385822 | 3 | For any $p$-dic field $K$, we have an equivalence of categories
$$D\_{st}:Rep\_{\mathbb{Q}\_p}^{st}(G\_K)\rightarrow MF\_K^{ad}(\varphi,N),\quad V\mapsto (B\_{st}\otimes\_{\mathbb{Q}\_p} V)^{G\_K}$$
with quasi-inverse $V\_st$ determined by
$$V\_{st}(D)=(B\_{st}\otimes\_{K\_0} D)^{\varphi=1,N=0}\cap Fil^0(B\_{dR}\otimes... | https://mathoverflow.net/users/152554 | Restriction of $(\varphi, N)$-modules | Don't confuse $(\phi, N)$-modules (which are finite-dimensional vector spaces over $\mathbf{Q}\_p$ with various extra structures) with $(\phi, \Gamma)$-modules (which are modules over a much bigger and more complicated ring, but see all Galois reps, not just semistable ones). De Shalit and Porat are working on the $(\p... | 1 | https://mathoverflow.net/users/2481 | 385885 | 160,383 |
https://mathoverflow.net/questions/384575 | 6 | I would like to know if anyone has studied the following ``Hadamard product" of binary (or ternary) matroids. (There is a notion of Hadamard product of matroids studied e.g. [here](https://arxiv.org/abs/2003.10529) but I think that one is different.)
Let $M,N$ be simple binary matroids of rank $r$ and $s$, respective... | https://mathoverflow.net/users/150898 | A Hadamard product of binary (or ternary) matroids | I hope that below is the proof of Conjecture (for $n\geqslant 2$, for $n=1$ it is false by trivial reasons), but please check carefully.
If $M$, $N$ are matroids on the ground set $E$ which are represented over a field $\mathbb{F}$: $M=\{x\_i:i\in E\}$, $N=\{y\_i:i\in E\}$ ($x\_i$ and $y\_i$ are vectors in correspond... | 1 | https://mathoverflow.net/users/4312 | 385889 | 160,385 |
https://mathoverflow.net/questions/385873 | 8 | I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
More generally, let $F$ be a field of characteristic 0, $L/F$ a quadratic extension, and $H$ be an $L/F$-Hermitian fo... | https://mathoverflow.net/users/4149 | Parabolics and simple roots for a special unitary group: reference request | I have edited in some remarks from the comments ([1](https://mathoverflow.net/questions/385873/parabolics-and-simple-roots-for-a-special-unitary-group-reference-request/385903#comment983612_385903) [2](https://mathoverflow.net/questions/385873/parabolics-and-simple-roots-for-a-special-unitary-group-reference-request/38... | 4 | https://mathoverflow.net/users/2383 | 385903 | 160,391 |
https://mathoverflow.net/questions/385901 | 8 | *This was previously [asked and bountied](https://math.stackexchange.com/questions/2326484/is-zfcv-l-consistently-omega-complete) on MSE:*
For brevity, let $T$ be $\mathsf{ZFC+V=L}$.
Say that an extension of $\mathsf{ZFC}$ is *$\omega$-complete* iff it has exactly one $\omega$-model up to elementary equivalence. Wh... | https://mathoverflow.net/users/8133 | Is $\mathsf{ZFC+V=L}$ consistently $\omega$-complete? | Claim: $T+$"$T$ is $\omega$-complete" is inconsistent. For suppose it's consistent and now work in a model $V$ of this theory. Let $T^+$ be the resulting completion of $T$ (i.e. the unique theory of the $\omega$-models of $T$ in the sense of $V$). Then note that $T^+$ is a $\Delta^1\_1$ real, so $T^+\in L\_{\omega\_1^{... | 13 | https://mathoverflow.net/users/160347 | 385907 | 160,393 |
https://mathoverflow.net/questions/385814 | 4 | We are trying to write a snippet of Magma code to clarify the steps in the simplified procedure of applying $3$-, $6$-, $12$-descent and hopefully resolve the missing generator of the following $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ elliptic curve
```
[1,0,1,-7378946443270262192188413585705099138466525... | https://mathoverflow.net/users/95511 | 3-, 6-, 12-descent for Z2xZ6 elliptic curves | The snippet below worked for me. The numerical values of $b\_6$ and $b\_{12}$ are the lowest powers of $2$ to produce rational points on the picked coverings.
```
SetSeed(1);
SetClassGroupBounds("GRH");
E := EllipticCurve([1,0,0,-990429690240960203866170,343844266388187332499781887080604900]);
P1 := E (Proc. AMS, 1993) it is shown by Dow, Frankiewicz and Zbierski that in the $\aleph\_2$-Cohen model every compact zero-dimensional $F$-space of weight at most $\mathfrak{c}$ is embeddable on $\omega^\*$.
Answer to 3... | 6 | https://mathoverflow.net/users/5903 | 385980 | 160,415 |
https://mathoverflow.net/questions/385851 | 5 | Some authors define superharmonicity at infinity in the following way. A function $u$ is superharmonic on an open set $V\subset\mathbb{R}^m\cup\{\infty\}$ (one point compactification), containing infinity, if it is superharmonic on $V\setminus\{\infty\}$ in the regular way, and at infinity, $u$ is lower semicontinuous ... | https://mathoverflow.net/users/100746 | Superharmonicity at infinity | 1. None of the definitions implies the other:
* The function $|x|^{2-m}$ is superharmonic (in fact: harmonic) in $\mathbb R^m \cup \{\infty\} \setminus \{0\}$ according to the second definition, but it is not according to the first one.
* Conversely, the function $-1$ is superharmonic (in fact: harmonic) in $\mathb... | 2 | https://mathoverflow.net/users/108637 | 385986 | 160,418 |
https://mathoverflow.net/questions/385984 | 2 | Take six distinct points $p\_1,\dots,p\_6\in\mathbb{P}^1$ and consider the double covering $f:C\rightarrow \mathbb{P}^1$ ramified over $p\_1,\dots,p\_6\in\mathbb{P}^1$. Then $C$ is a smooth curve of genus two.
Can we degenerate $C$ to a singular rational curve or to a union of smooth rational curves by collapsing som... | https://mathoverflow.net/users/nan | Degenerations of hyperelliptic coverings | If $p\_1 = p\_2 \ne p\_3 = p\_4 \ne p\_5 \ne p\_6$ then the normalization of the double cover branched at the divisor $D = \sum\_{i=1}^6 p\_i$ is a smooth irreducible rational curve. If also $p\_5$ and $p\_6$ collide, the normalization of the double cover is the union of two smooth rational curves.
| 6 | https://mathoverflow.net/users/4428 | 385987 | 160,419 |
https://mathoverflow.net/questions/385794 | 1 | Let $G$ be a compact Lie group and let $H$ be a closed subgroup of $G$, with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$.
We denote $G\times\_H \mathfrak{g} / \mathfrak{h}$: the set of orbits $(G \times \mathfrak{g} / \mathfrak{h})/H $ of the right action of $H$ on $ G \times \mathfrak{g} / \mathfrak{h}$ ($H$ acts... | https://mathoverflow.net/users/172459 | Description of $A^\bullet(G/H)$ | By definition, $k$-forms are sections of the bundle $\bigwedge{}^kT^\*(G/H)$, which is the associated bundle $G\times\_H\bigwedge^{k}(\mathfrak{g}/\mathfrak{h})^\*$. You then apply the general formula that the sections of the associated bundle for any $H$-representation $V$ is $(C^{\infty}(G)\otimes V)^H$: the tensor p... | 2 | https://mathoverflow.net/users/66 | 385989 | 160,421 |
https://mathoverflow.net/questions/385993 | 31 | I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation
\begin{equation}\label{eq}
x^{x+1}=(x+1)^x
\end{equation}
Let us define that with the expression "algebraic way" the student really means "the solution $x$ to the equation is an algebraic number".
... | https://mathoverflow.net/users/146431 | How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental? | The number $x$ is transcendental, and your Gelfond-Schneider argument almost works.
Suppose to the contrary that $x$ is algebraic. Then $x+1$ and $x/(x+1)$ are also algebraic, and so the [Gelfond-Schneider theorem](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem) guarantees that $x = (x+1)^{\frac{x}{x... | 48 | https://mathoverflow.net/users/48142 | 385996 | 160,423 |
https://mathoverflow.net/questions/385959 | 8 | Suppose $M$ is a (closed, connected, oriented, smooth) manifold.
If $M$ is aspherical, i.e., if the inversal covering $\tilde{M}$ is contractible, $M$ is a $B\pi\_1(M)$. This is often enforced by geometry, for instance it holds if $M$ admits a metric of non-positive sectional curvature (Cartan--Hadamard).
We deduce... | https://mathoverflow.net/users/14233 | Cohomological dimension bounds on the fundamental group of a manifold | Let $M$ be an orientable manifold. The relation between the cohomological dimension of $\pi\_1(M)$ and that of $M$ comes from the following:
**Proposition:**
$M$ has the homotopy type of a fibration over $B\pi\_1(M)$ with fiber $\tilde M$.
*Proof:*
Let $B\pi\_1(M) = E\pi\_1(M)/\pi\_1(M) $ be a classifying space for... | 7 | https://mathoverflow.net/users/173096 | 386008 | 160,428 |
https://mathoverflow.net/questions/385995 | 3 | Let $\lambda\vdash n$ denote an [integer partition](https://en.wikipedia.org/wiki/Partition_(number_theory)) of $n$ and $\frak{H}\_{\lambda}$ be the multiset of [hook lengths](https://en.wikipedia.org/wiki/Hook_length_formula) of $\lambda$. Further, let $o(\lambda)=\#$ of odd entries and $e(\lambda)=\#$ of even entries... | https://mathoverflow.net/users/66131 | Generating function for parity in hooks | Yes, the generating function is
$$\sum\_{n\geq 0} F\_{n}(q,t)x^n=\prod\_{k\geq 1}\frac{(1-q^{2k}x^{2k})^2}{1-q^kx^k}\cdot \prod \_{k\geq 1}\frac{1}{(1-q^kt^kx^{2k})^2}.$$
This follows from the usual bijection between partitions and 2-cores (corresponding to the first product on the right) and 2-quotients (corresponding... | 5 | https://mathoverflow.net/users/2384 | 386009 | 160,429 |
https://mathoverflow.net/questions/386011 | 44 | When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the circle that induces the identity on the circle. From this lemma, you easily deduce the Brouwer fixed point Theorem for t... | https://mathoverflow.net/users/37214 | "Cute" applications of the étale fundamental group | Using the étale fundamental group one can construct an injective group homomorphism
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \hookrightarrow \operatorname{Out}(\widehat{F\_2})$
which is canonical in the sense that there are no choices involved in its construction (once the algebraic closure is fixed), ... | 38 | https://mathoverflow.net/users/175608 | 386014 | 160,431 |
https://mathoverflow.net/questions/386024 | 1 | Let $A$ be a unital $C^\*$-algebra. Suppose $\rho\_1$ and $\rho\_2$ are two states on $A$. If $\rho\_1=\rho\_2$, we have $\|\rho\_1+i\rho\_2\|=\sqrt{2}$.
If we have $\|\rho\_1+i\rho\_2\|=\sqrt{2}$, can we conclude that $\rho\_1=\rho\_2$?
| https://mathoverflow.net/users/153196 | Calculation of the norm of linear combinitation of two states on a $C^*$-algebra | The answer is YES.
Consider any $a\in A$ with $0\le a\le 1$ and
put $\alpha:=(\rho\_1+i\rho\_2)(a)$ and $\beta:=(\rho\_1+i\rho\_2)(1-a)=(1+i)-\alpha$.
Then $x:=(\bar{\alpha}/|\alpha|)a+(\bar{\beta}/|\beta|)(1-a)$ has norm at most $1$ and so
$$|1+i|\geq|(\rho\_1+i\rho\_2)(x)|=|\alpha|+|(1+i)-\alpha|.$$
Hence $\alpha$ mu... | 8 | https://mathoverflow.net/users/7591 | 386040 | 160,437 |
https://mathoverflow.net/questions/386036 | 0 | Consider the invertible matrices in $\mathbb F\_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z\_{\geq1}$ independent of $n$) generators for the group on the condition if $a$ is a generator $a\neq a^{-1}$ and $a^{-1}$ is a member of the set and is considered... | https://mathoverflow.net/users/10035 | Generators of $SL(n,\mathbb F_2)$? | OK, I'll answer the intended question! ${\rm SL}(2,2) \cong S\_3$ and the answer to the question is no in that case, so assume that $n>2$.
Don Taylor wrote down explicit sets of two generators for the finite classical groups. Search for "D.E. Taylor Pairs of Generators for Matrix Groups". (These are the generators us... | 7 | https://mathoverflow.net/users/35840 | 386041 | 160,438 |
https://mathoverflow.net/questions/386046 | 2 | Let $X$ and $Y$ be independent random symmetric matrices. What can one say about $\mathbb{E} [X Y X Y]$ or $\mathrm{trace} \mathbb{E} [X Y X Y]$ in terms of properties of $X$ and $Y$?
In particular, can we compute such expectations if $X$ is Wishart distributed and $Y$ is inverse-Wishart distributed (i.e. the inverse... | https://mathoverflow.net/users/175634 | Expectation of product of random matrices | You could compute these expectation values from the known marginal distribution of the matrix elements of the Wishart and inverse Wishart ensembles; as a simpler test case here is the answer for a single product $XY$ of a $(p,n)$ Wishart and a $(p,\nu)$ inverse Wishart matrix (with identity scale matrices):
$${\rm tr}\... | 1 | https://mathoverflow.net/users/11260 | 386057 | 160,443 |
https://mathoverflow.net/questions/386026 | 3 | I have functions $A, B, F, S$ that are zero on $(-\infty, 0)$.
And I have successfully represented the below equation as convolution and multiplication:
$\int\_0^t {dt\_1} \int\_0^t {dt\_2} B(t - t\_2)F(t\_2 - t\_1)S(t - t\_1)F(t\_1)$
$=\int\_0^t {dt\_1}F(t\_1) S(t - t\_1)\int\_{t\_1}^t {dt\_2} B(t - t\_2)F(t\_2 - ... | https://mathoverflow.net/users/175619 | Can it be represented by convolution and multiplication | $\newcommand\R{\mathbb R}$
Most likely, the integral
$$I:=\int\_0^t dt\_1 \int\_0^t dt\_2 \int\_0^t dt\_3\, B(t-t\_3)F(t\_3-t\_2)
S(t-t\_2)F(t\_2-t\_1)F(t\_1)$$
cannot be expressed in terms of products and convolutions -- because the finite interval $[0,t]$ is not a group (or even a semigroup), and the restrictions $t\... | 3 | https://mathoverflow.net/users/36721 | 386065 | 160,446 |
https://mathoverflow.net/questions/282663 | 10 | Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost sense (i.e. considering the category of extensions in the sense of "almost mathematics"). I want to know whether this is ... | https://mathoverflow.net/users/7108 | periodic cyclic homology and tilting in the sense of Scholze | Of course, there cannot be a direct relation at the categorical level: After all, one category is $R$-linear while the other is $R^\flat$-linear (I write $R^\flat=R'$ for the tilt, as usual).
On the other hand, what is certainly true is that if $\pi^\flat\in R^\flat$ is some element with $\pi=(\pi^\flat)^\sharp\in R$... | 8 | https://mathoverflow.net/users/6074 | 386070 | 160,447 |
https://mathoverflow.net/questions/386060 | 8 | A colleague and I are working on a problem and part of it comes down to evaluating the residue of a rational function. In particular,
$$
\mathrm{Res} \left( z^{kn-1} \left( az^{m}+1 \right)^{-k}; r \right),
$$
where $a$, $k$, $m$ and $n$ are positive integers satisfying $a \geq 2$ and $0<m<n$ and $r$ is any $m$-th root... | https://mathoverflow.net/users/175660 | residue calculation for rational function | We want to calculate
$$\rho(k,n,m)=\operatorname\*{res}\_{w=1}\left(\frac{w^n}{1-w^m}\right)^k\frac{dw}{w}. $$
If $kn$ is divisible by $m$ then it seems that $\rho(k,n,m)=-\binom{-k}{kn/m-k}/m$. This is because in this case the residues at all $m$'th roots of unity are the same, and the sum of those residues is minus t... | 7 | https://mathoverflow.net/users/10366 | 386074 | 160,449 |
https://mathoverflow.net/questions/385992 | 5 | I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators on said interval.
We can consider the perturbed heat semigroup $T=e^{(\Delta+f)}$ at fixed time $1$. The heat semigro... | https://mathoverflow.net/users/119875 | Backward heat equation and forward perturbed heat equation well posed? | No, this cannot be true if $f$ is just $C^\infty$. Let $u=e^{(\Delta+f)t}u\_0$.
At $t=1$, $u=e^{\Delta+f}u\_0=e^\Delta v\_0$ for some $v\_0$. Then, by well known properties of the heat equation, $u$ is spatially analytic. Moreover, $u\_t=e^{\Delta+f}(\Delta+f)u\_0$. If $u\_0$ is sufficiently smooth, then $(
\Delta+f)u\... | 4 | https://mathoverflow.net/users/12120 | 386077 | 160,451 |
https://mathoverflow.net/questions/385530 | 11 | The *monadic theory of the real line* is the set of all sentences in the [monadic second-order language](https://en.wikipedia.org/wiki/Monadic_second-order_logic) of order which are true in $\mathbb{R}$. In [this 1982 paper](https://www.sciencedirect.com/science/article/pii/0003484382900043), Gurevich and Shelah show t... | https://mathoverflow.net/users/5017 | What is the Turing degree of the monadic theory of the real line? | Gurevich and Shelah showed in [The monadic theory and the “next world”](https://www.semanticscholar.org/paper/The-monadic-theory-and-the-%E2%80%9Cnext-world%E2%80%9D-Gurevich-Shelah/f742479301ee2c2b5917d371e848de4fa82b22de) that the monadic theory of the real line (or even just the Cantor Discontinuum) can compute
-... | 7 | https://mathoverflow.net/users/113213 | 386079 | 160,452 |
https://mathoverflow.net/questions/386052 | 5 | Let $X, Y, Z$ be compact topological manifolds $f: Y \to X, g: Z \to X$ be embeddings of submanifolds meeting transversely and let $W = Y \times\_X Z$:
$$
\begin{array}{ccc}
Y & \to^f & X \\
\uparrow^G & & \uparrow^g\\
W & \to^F & Z \\
\end{array}
$$
My question is: How does one show that the two morphisms $F\_! \... | https://mathoverflow.net/users/2234 | commutativity of restriction and Gysin morphisms in a cartesian square | One way to see this is to use the definition of the Gysin map via Thom isomorphisms. Then (at least in this simplest case where everything is an embedding) the statement reduces to the fact that Thom isomorphisms are natural for pullbacks of bundles.
To explain what I mean, let me give one possible definition of the ... | 3 | https://mathoverflow.net/users/8103 | 386082 | 160,454 |
https://mathoverflow.net/questions/386069 | 4 | I have found the following question to be surprisingly hard:
**Is there a non-zero $f\in L^1(\mathbb R)$ or $f\in L^2(\mathbb R)$ such that
$$
f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere},
$$
where $\hat f$ is the Fourier transform of $f$ and $\cdot$ is the pointwise product?**
I expect the answer to be ... | https://mathoverflow.net/users/58125 | Vanishing of the product of a function and its own Fourier transform | I had to stratch my head a bit to decipher what we wrote 4 years ago but there it is
* Take $\hat{\phi} \in C^\infty\_c(\Bbb{R})$ supported on $(1/2,1)$
* Take $h \in C^\infty(\Bbb{R})$ and $1$-periodic supported on $[0,1/2]+\Bbb{Z}$, let $c\_n=\int\_0^1 h(x)e^{-2i\pi nx}dx$. Then
$$\hat{h}=\sum\_n c\_n \delta(y-n)... | 3 | https://mathoverflow.net/users/84768 | 386096 | 160,461 |
https://mathoverflow.net/questions/385155 | 3 | Let $p \in \mathbb{R}^{n}$ and $p=\lambda\_1 e\_1+...+\lambda\_n e\_n$ where $e\_i$ are standard basis vectors then if I want to find the component along which I can get closest to the point $p$ then it will just be $e\_j$ with $j \in \{1,...,n\}$ such that $\lambda\_j$ satisfies $|\lambda\_j| = max\_{1 \leq i\ \leq n}... | https://mathoverflow.net/users/168019 | The direction that gets me closest to a given point in $\mathbb{R}^n$ | For $p, x\_i\in \mathbb{R}^n$, with $1\le i\le n$, define
$$\theta\_i=\angle(x\_i,\text{span}(\{x\_j \;|\; j\ne i\})),\quad \theta=\min(\{\theta\_i\})$$
$$\beta\_i=\frac{\pi}{2}-\angle(p,x\_i),\quad\beta=\max(\{\beta\_i\})$$
**CLAIM.**
$$\sin(\beta)\ge\frac{sin(\theta)}{\sqrt{n(1+(n-1)\cos(\theta))}}$$
**PROOF.**... | 2 | https://mathoverflow.net/users/2480 | 386102 | 160,465 |
https://mathoverflow.net/questions/386097 | 4 | What on Earth is a homotopy pullback of
$$A \rightarrow B \leftarrow C \ \ \ \ \ ???$$
Here $A,B,C$ are elements of a category ${\mathcal V}$ enriched in topological spaces (any convenient category of topological spaces will do). I understand that it is some kind of a weighted limit. This means that I need to take all ... | https://mathoverflow.net/users/5301 | How to compute Homotopy Pullback |
>
> Are there any explicit ways of calculating it, similar to the methods, working for the bog down homotopy pullbacks and pushout?
>
>
>
Yes, in fact the same formula continues to work in this case.
Consider the (ordinary) pullback
$$A⨯\_B B^{[0,1]} ⨯\_B C.$$
Here $B^{[0,1]}$ denotes the powering of $B$ ove... | 10 | https://mathoverflow.net/users/402 | 386103 | 160,466 |
https://mathoverflow.net/questions/386006 | 7 | Let $x$ be an algebraic number. Must $\arctan(x)/\pi$ have finite irrationality measure? Are there any useful upper bounds?
| https://mathoverflow.net/users/83174 | Upper bounds on the irrationality measure of the arctan of an algebraic number | Let $\alpha=\frac{1+xi}{\sqrt{1+x^2}}$. There are some cases $(\arctan x) /\pi$ is rational. For example, $x=1, \sqrt3$. In these cases, $(\arctan x)/\pi$ has the irrationality measure $1$. These occur precisely when $\alpha$ is a root of unity.
Since $x$ is algebraic, so is $\alpha$. Then $\arctan x = \arg \alpha = ... | 5 | https://mathoverflow.net/users/21090 | 386106 | 160,467 |
https://mathoverflow.net/questions/386104 | 2 | Let $\rho\_p:G\_{\mathbb{Q}\_p} \to \text{Gl}\_n(\mathbb{Q}\_p)$ be semistable representation. In [local to global Galois representation](https://mathoverflow.net/q/41940), it was asked if one can find a geometric global Galois representation $\rho:G\_{\mathbb{Q}}\to \text{Gl}\_n(\mathbb{Q}\_p)$ such that $\rho\vert\_{... | https://mathoverflow.net/users/152554 | Local to global for semistable $G_{\mathbb{Q}_p}$-representations | A basic necessary condition, following the comment you quote, is that the determinant of Frobenius is finite-order. Is this condition sufficient? It might be, sometimes.
If you look instead at $\ell$-adic representations where the classification of representations is much easier, you're going to run into trouble for ... | 3 | https://mathoverflow.net/users/18060 | 386107 | 160,468 |
https://mathoverflow.net/questions/235312 | 17 | Here, $\, j\_U, \, j\_D$ are the canonical elementary embeddings induced by $U,D$ respectively.
I note that it is consistent with the existence of a measurable that the answer be *yes*: it is true in the model $L[D]$ for $D$ a measure on $\kappa$.
| https://mathoverflow.net/users/84846 | If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U(U) = j_D(D)$, is $U=D$? | It is consistent with ZFC that the answer is no, but under the Ultrapower Axiom, the answer is yes, not only for $\kappa$-complete ultrafilters on $\kappa$, but also for arbitrary countably complete ultrafilters.
**First I'll show that in the Kunen-Paris model, there exist distinct normal ultrafilters $U\_0$ and $U\_... | 11 | https://mathoverflow.net/users/102684 | 386116 | 160,471 |
https://mathoverflow.net/questions/386095 | 3 | Let $f:\Sigma \to \Sigma$ be a two side shift map, where $\Sigma=\{1,2,3,4\}^{\mathbb{Z}}$ and let $A:\Sigma \to SL(2,\mathbb{R})$ be a function such that $A((x\_{n}))=A\_{x\_{0}}$. Assume that there are two different Lyapunov exponents $-\lambda$ and $\lambda$, so there there is a Oseledets splitting $\mathbb{R}^{2}=E... | https://mathoverflow.net/users/127839 | An angle between two vectors in Oseledets theorem | Ok, let take $v$ in the bundle at $x$. We may decompose $v=v^{u}+v^{s}\in E^{u}\oplus E^{s}$.
Assume without loss of generality that $\lVert v\rVert=1$.
Applying $A^{i}$, using equivariance and Osceldets' theorem we get $$A^{i}v \approx e^{i\cdot\lambda}\cdot v+e^{-i\cdot\lambda}\cdot v^{s}$$.
Therefore a unit vector... | 1 | https://mathoverflow.net/users/8857 | 386117 | 160,472 |
https://mathoverflow.net/questions/386110 | 0 | $\DeclareMathOperator\ann{ann}$Let $a$ and $b$ be two non-zero zero divisors of a commutative ring $R$ with 1 such that $\ann(a) \ne \ann(b)$.
is it always possible to find a sequence of non-zero elements $a\_1,\dotsc,a\_k \in R$ such that $a \in \ann(a\_1)$, $a\_1 \in \ann(a\_2)$, …, $a\_{k-1} \in \ann(a\_k)$, and $... | https://mathoverflow.net/users/33047 | Annihilator of an element in a ring | If you allow some $a\_i$ to be $0$, then the answer is obviously yes.
If there are no $0$-divisors in $R$, then the answer is vacuously yes. If there is a $0$-divisor $a$ in $R$, then taking $b = 1$ gives an example where no such sequence exists.
Suppose that $a$ and $b$ both have non-$0$ annihilators. Say $x \ne 0... | 2 | https://mathoverflow.net/users/2383 | 386118 | 160,473 |
https://mathoverflow.net/questions/386111 | 3 | Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $L \in \mathcal C$ be a $\otimes$-invertible object. Then the braiding $L \otimes L \to L \otimes L$ is [simply](https://mathoverflow.net/a/293774/2362) multiplication by $\dim L$, where $\dim L$ is some involution on the unit object $I$.
Thus the un... | https://mathoverflow.net/users/2362 | Is every $\otimes$-invertible object "coherently sym-central"? | Just to confirm Jacob Lurie's comment above (**EDIT:** And the following has been corrected -- a previous version fell for a classic blunder as pointed out by Jacob Lurie below): the group completion of $S$ is $\Omega^\infty \tau\_{\leq 1} \mathbb S$ as an infinite loop space. We can see this using a group completion l... | 3 | https://mathoverflow.net/users/2362 | 386120 | 160,474 |
https://mathoverflow.net/questions/386128 | 1 | We call $E\subseteq {\cal P}(\omega)$ a *Fano-like plane* if
1. for all $x,y\in \omega$ there is $e\in E$ with $\{x,y\}\subseteq e$,
2. whenever $e\_1\neq e\_2\in E$ we have $|e\_1\cap e\_2|=1$, and
3. $|e|>1$ for all $e\in E$.
There are Fano-like planes in which not all edges (members of $E$) have the same cardina... | https://mathoverflow.net/users/8628 | Fano-like planes on $\omega$ | No. Suppose $E\subseteq\mathcal P(\omega)$ is a Fano-like plane. If $e\in E$ and $x\in\omega\setminus e$, then there is a bijection between the edges containing $x$ and the points in $e$. Therefore, given two edges $e\_1,e\_2$ and a point $x\notin e\_1\cup e\_2$, there is a bijection between the points of $e\_1$ and th... | 3 | https://mathoverflow.net/users/43266 | 386131 | 160,478 |
https://mathoverflow.net/questions/14763 | 84 | Let me stress that I am *only* interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader knows what the "1970s version of the local Langlands conjectures" are when writing this question---there are plenty of re... | https://mathoverflow.net/users/1384 | What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field? | Now that our paper [Geometrization of the local Langlands correspondence](https://arxiv.org/abs/2102.13459) with Fargues is finally out (ooufff!!), it may be worth giving an update to Ben-Zvi's answer above. In brief: we give a formulation of Local Langlands over a $p$-adic field $F$ so that it is finally
1. an actua... | 32 | https://mathoverflow.net/users/6074 | 386138 | 160,479 |
https://mathoverflow.net/questions/386073 | 6 | In Lemma 5.4.5.11 of HTT, the proof given relies on Lemma 5.4.5.10. However it seems that Lurie applies Lemma 5.4.5.10, which requires the given simplicial set to be contractable, to an arbitrary $\kappa$-small simplicial set.
This seeming incongruity was pointed out in this [question](https://mathoverflow.net/questi... | https://mathoverflow.net/users/175274 | Lemma 5.4.5.11 of HTT | I think there is a typo in Lemma 5.4.5.11: $K$ is supposed to be $\tau$-small and not $\kappa$-small. Note that if $\tau < \kappa$ and $K$ is $\kappa$-small but not $\tau$-small then the statement of the lemma is simply false: e.g., set $\mathcal{I}=\mathcal{J}=K=\mathbb{N}$ to be the poset of natural numbers (with arr... | 8 | https://mathoverflow.net/users/51164 | 386139 | 160,480 |
https://mathoverflow.net/questions/386013 | 6 | This question has also been posted on MSE, but maybe here is the right place to obtain an answer.
Let $(M^3,g)$ be a compact connected oriented Riemannian $3$-manifold with nonempty boundary. The Hodge-de Rham Theorem says that there is an isomorphism between $H^1\_{dR}(M)$, the first de Rham cohomology group of $M$,... | https://mathoverflow.net/users/85934 | Tangential harmonic $1$-forms are pullbacks of harmonic functions | Any constant map satisfies the requirements of the final question :). More seriously, if you want to find tangential harmonic form of this type, which represents a given $[u]\in [M,S^1]\cong H^1(M,\mathbb{Z})$ in de Rham cohomology, then you can proceed as follows (unless I am missing something):
Pick a smooth map $v... | 2 | https://mathoverflow.net/users/66777 | 386143 | 160,482 |
https://mathoverflow.net/questions/386132 | 13 | Let $K$ be a henselian valuation field with residue field $k$, then the decomposition group surjects onto Galois group of the residue field, with kernel the inertia subgroup, namely we have short exact sequence:$$0\to I\to\mathrm{Gal}\_K\to\mathrm{Gal}\_k\to 0$$
When $K$ is a local field, we can split the sequence by... | https://mathoverflow.net/users/nan | Does $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ always split? | Good question! Let me try to guess what Gabber had in mind there. (Note that he only says "known" (to him), not "well-known"...)
The claim is that the extension splits. Note that to prove this, we are free to replace $K$ by any (algebraic) extension $K'$ whose residue field $k'$ is purely inseparable over $k$. By Zor... | 12 | https://mathoverflow.net/users/6074 | 386147 | 160,483 |
https://mathoverflow.net/questions/320575 | 7 | I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories.
1. In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see how $\mathcal{D}^{/F(x)}$ matches the hypotheses thereof, in that it is not at all obvious to me that it be $\tau$-fil... | https://mathoverflow.net/users/134438 | Stability of accessible $\infty$-categories under some operations | For (2) I suggested a possible solution for this here: [Lemma 5.4.5.11 of HTT](https://mathoverflow.net/questions/386073/lemma-5-4-5-11-of-htt/386139#386139).
For (3) it really appears to be a typo and can be fixed as in the comment of dhy. For (1), as explained by Tim in the comments, there is actually no mathematic... | 3 | https://mathoverflow.net/users/51164 | 386160 | 160,486 |
https://mathoverflow.net/questions/385736 | 0 | We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta t$. How could we prove that the quantities
$$
\begin{align}
Q &= q + {\Delta}t\frac{\partial H}{\partial p}(q,p),\\
P &=... | https://mathoverflow.net/users/172882 | Hamilton equations-Symplectic scheme | I take it that your question is about why the symplectic Euler method is symplectic, while the explicit Euler method is not.
The point is that for a Hamiltonian of the form $H(p,q)=T(p)+V(q)$, the symplectic Euler method can be seen as the composition of the two steps
\begin{align\*}
\tilde{q} &= q\_i+T'(p\_i)\,\Delt... | 2 | https://mathoverflow.net/users/45250 | 386165 | 160,488 |
https://mathoverflow.net/questions/385455 | 5 | For a long time I've been confused about Drinfeld Sokolov/BRST reduction/semiinfinite cohomology for affine Lie algebras. Most treatments define it in what to me feels like a fairly ad-hoc way, by choosing a nilpotent element then applying an elaborate construction. (Of course it's not unmotivated: it generalises the B... | https://mathoverflow.net/users/119012 | Drinfeld Sokolov and the semiinfinite flag variety | Maybe let me try to synthesize my comments into an answer. All of this is contained in Raskin's beautiful paper arxiv.org/abs/1611.04937 on Whittaker categories. Convention: We work here in the derived world, i.e., all our categories are assumed pretriangulated dg (equivalently one can take stable $\infty-$categories).... | 4 | https://mathoverflow.net/users/51424 | 386185 | 160,494 |
https://mathoverflow.net/questions/386184 | 3 | Consider a symmetric Frobenius algebra without unit, that is, a finite-dimensional complex associative algebra $\delta$ with a linear functional $\epsilon$, such that $\epsilon\circ \delta$ is a non-degenerate symmetric bilinear form, and
$$ \epsilon\circ\delta\circ(\delta\otimes \operatorname{id})=\epsilon\circ\delta\... | https://mathoverflow.net/users/115363 | Is the unit in the definition of a symmetric Frobenius algebra necessary? | Denote by $V$ the underlying vector space of your algebra and by $\bullet$ the product. Then the nondegenerate bilinear form $\eta$ identifies $V$ with $V^\*$. I claim that the preimage of the linear form $\varepsilon$ under this identification is the unit. Denote it by 1. Then, for any $a,b \in V$, we have
$$
\eta(1 \... | 2 | https://mathoverflow.net/users/175087 | 386194 | 160,497 |
https://mathoverflow.net/questions/386193 | 2 | Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $r$ containing in $\mathbb{P}^n$. Denote by $I\_X \subset \mathbb{C}[X\_0,...,X\_n]$ the ideal of $X$ defined by some homogeneous polynomials. For $k \gg 0$ and a general choice of $n-r$ homogeneous polynomials $F\_1,...,F\_{n-r} \in I\_X$ of degree $k$,... | https://mathoverflow.net/users/45397 | Bertini type result for complete intersection varieties containg a non-singular variety | The general such intersection is the union $X \cup Y$, where $Y$ is smooth away from $X$. On the other hand, in general $Y$ has singularities in codimension 4. Indeed, if $\mathcal{I}\_X$ is the ideal sheaf of $X$, the polynomials $F\_i$ induce a morphism
$$
\mathcal{O}^{\oplus (n-r)} \to \mathcal{I}\_X(k).
$$
When res... | 4 | https://mathoverflow.net/users/4428 | 386197 | 160,499 |
https://mathoverflow.net/questions/386189 | 6 | Following the computation of the THH (topological Hochschild homology) of $\mathbb{F}\_p$ as outlined in Krause-Nikolaus.
We use the fact that $\mathbb{F}\_p$ is initial $E\_2$ ring with $0=p$ to compute
$$\mathbb{F}\_p \otimes\_{\mathbb{S}} \mathbb{F}\_p \cong \mathbb{F}\_p[{\Omega^2 S^3}]$$
Then,
$$THC(\mathbb{... | https://mathoverflow.net/users/136287 | What is the topological Hochschild cohomology of $\mathbb{F}_p$? | Let me write $HH^S(B) = THH(B) = B \wedge\_{B^e} B$ for topological Hochschild homology, and $HH\_S(B) = F\_{B^e}(B, B)$ for topological Hochschild cohomology, where $B^e = B \wedge\_S B^{op}$. For $B$ commutative the $B^e$-module action on $B$ factors through $\mu : B^e \to B$, so by adjunction we have $F\_{B^e}(B, B)... | 8 | https://mathoverflow.net/users/9684 | 386198 | 160,500 |
https://mathoverflow.net/questions/386068 | 3 | I am still studying Deligne and Illusie's paper (<https://eudml.org/doc/143480>), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}\_{p}$-schemes, then $\operatorname{Lif}(X,\tilde{S})$ is the gerbe of liftings to $\tilde{S}=S(\mathbb{Z}/p^{2})$, a morph... | https://mathoverflow.net/users/70751 | Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence | (1) is elementary deformation theory: if $\alpha\colon \mathcal{O}\_{\tilde U}\to \mathcal{O}\_{\tilde U}$, then $\alpha-{\rm id}\colon \mathcal{O}\_{\tilde U}\to \mathcal{O}\_{\tilde U}$ vanishes modulo $p$ and hence there exists a unique $\delta\colon \mathcal{O}\_U\to \mathcal{O}\_U$ such that $$\alpha(f) = f + p\cd... | 2 | https://mathoverflow.net/users/3847 | 386202 | 160,502 |
https://mathoverflow.net/questions/385908 | 8 | This may be a dumb question, but I ask it here.
In ordinary cohomology, we can construct a Hopf invariant for $f \colon S^{2n-1} \to S^{n}$ by applying $H^{\*}(- \colon \mathbb{F}\_p)$ to the cofibre sequence, so that the Hopf invariant measures the non-triviality of
$$
0 \to H^{\*}(S^{2n}) \to H^\*(C\_f) \to H^{\*}(... | https://mathoverflow.net/users/141140 | Generalization of Hopf invariant | The $E$-based Adams spectral sequence is the homotopy spectral sequence associated to the tower of spectra $\dots \to Y\_2 \to Y\_1 \to Y\_0 = S$ with $Y\_{s+1} \to Y\_s \to E \wedge Y\_s$ a homotopy fiber sequence for each $s\ge0$. The edge homomorphism to filtration $s=0$ detects the Hurewicz image of $\pi\_\*(S)$ in... | 7 | https://mathoverflow.net/users/9684 | 386204 | 160,503 |
https://mathoverflow.net/questions/386208 | 6 | It is well known that if $f(x)$ is a polynomial over $\mathbb Z$ then for every prime $p$ (not dividing the discriminant of $f$ (thanks to KConrad)) the Galois group of that polynomial mod $p$ over $\mathbb{F}\_p$ embeds into the Galois group of $f$ over $\mathbb{Q}$. Where can I find a (easy) proof of this fact?
| https://mathoverflow.net/users/157261 | Galois group of a polynomial modulo $p$ | 1. This result of Dedekind is *not* true for every prime $p$, but only for primes not dividing the discriminant of $f(x)$.
2. There is no “easy” proof for someone who knows only Galois theory (the setting where the result is usually first met). You can find a proof in Jacobson’s Basic Algebra I, attributed to Tate, tha... | 12 | https://mathoverflow.net/users/3272 | 386219 | 160,506 |
https://mathoverflow.net/questions/386224 | -1 | Consider working on a domain $\Omega$ in $ R^N$ and we assume that $r=|x|$ and $ \theta$ is the angle between the $x\_N$ axis and the $ R^{N-1}$ plane. I am looking at functions and domains that depend only on $ r$ and $ \theta$. Is there a name for these coordinates and is there a reference for a bunch of computations... | https://mathoverflow.net/users/66623 | name of coordinates and reference (elliptic pde) | $r$ is called the radius, $\theta$ is called the polar angle.
I'm not sure exactly what kind of computations you are looking for. Most computations in these coordinates are special cases of computations in differential geometry in general coordinates.
For example, the gradient of a function $f=f(r,\theta)$ is
$$ ... | 1 | https://mathoverflow.net/users/144134 | 386229 | 160,509 |
https://mathoverflow.net/questions/386112 | 1 | How to prove
$$ \lVert uv\rVert\_{\dot{B}^{\frac{N}{p}-1}\_{p,1}}\leqslant C \lVert u\rVert\_{\dot{B}^{\frac{N}{p}}\_{p,1}} \lVert v\rVert\_{\dot{B}^{\frac{N}{p}-1}\_{p,1}}$$
when $N\geqslant2 $and$1\leqslant p<2N$.
I know it needs Bony decomposition, but I don’t know how to use the condition $N\geqslant2 $and$1\le... | https://mathoverflow.net/users/175690 | Show a inequality in homogeneous Besov space | I shall write $\mathrm{P}\_k$ for the homogeneous Littlewood-Paley projectors, and the paraproduct decomposition as
$$ uv = u\prec v + u\succ v + u\diamond v $$
Assume $1\le p<2$. Then $p'>p$ and Bernstein applies to estimate the $L^{p'}\_x$ norm in terms of the $L^p\_x$ norm, at a cost of derivatives. The resonant... | 0 | https://mathoverflow.net/users/144134 | 386232 | 160,510 |
https://mathoverflow.net/questions/386238 | 1 | Let $A \in S^{n}\_{+}$ be a positive semi-definite matrix and $D \in S^{n}\_{+}$ a diagonal matrix with all the diagonal entries no smaller than one, i.e., $D\_{ii} \geq 1$ for all $i \leq n$.
I wonder whether the transformation $DAD$ will scale up the eigenvalues? i.e., let $\lambda\_i(M)$ denote the $i$-th largest ... | https://mathoverflow.net/users/97310 | A monotonicity property of eigenvalues | **Yes**.
Write the eigenvalues as $\max\min$ (or as $\min\max$) of Rayleigh quotient. Then use the fact that $\|Dx\|\ge\|x\|$.
| 2 | https://mathoverflow.net/users/8799 | 386240 | 160,513 |
https://mathoverflow.net/questions/385516 | 1 | Suppose one has $N$ iid random walks $X^{(1)}\_t,\ldots,X^{(N)}\_t$ in discrete or continuous time $t$, let us say for example Poisson jump processes, and consider the stochastic process $Y^{(N)}\_t = \text{max}(X^{(1)}\_t,\ldots,X^{(N)}\_t)$. My question is, broadly, what interesting scaling limits this process $Y^{(N... | https://mathoverflow.net/users/76764 | multi-time limit of a maximum of random walks | First, you might as well assume that the mean of your walks is $0$, because that is just a deterministic shift. If not, as you point out, the maximum may be washed out by the deterministic contribution - unless you scale $t$ and $N$ appropriately. In what follows I will assume mean $0$.
The natural decorrelation leng... | 2 | https://mathoverflow.net/users/35520 | 386249 | 160,514 |
https://mathoverflow.net/questions/383789 | 2 | Suppose that ambient space is $\mathbb R^2$, and $\Omega \subset \mathbb{R}^2 $ is a smooth domain, non simply connected domain. To fix ideas,we can assume $$\Omega = \{(x\_1,x\_2) : 1< x\_1^2+x\_2^2 <4\}.$$
Given $p\geq2$, if $V$ is simply connected, given a function $\phi\in W^{1,p}(V;S^1)$ there exists $\theta \in... | https://mathoverflow.net/users/40120 | Defining a map into $S^1$ as an "angle" in a non simply connected domain | Let $\Omega$ be connected. There is a lifting criterion: A map $f:\Omega\rightarrow S^1$ lifts to the universal cover $\mathbb R\rightarrow S^1$ iff $f\_\*(\pi\_1(\Omega))=0$, where $f\_\*$ is the induced map $f\_\*:\pi\_1(\Omega)\rightarrow \pi\_1(S^1)$. Your space $W^{1,p}(\Omega,S^1)$ has different connected compone... | 2 | https://mathoverflow.net/users/12156 | 386252 | 160,515 |
https://mathoverflow.net/questions/378703 | 8 | It is easy to see a forcing of size $\aleph\_1$ is proper if and only if is semiproper. I was wondering when such an equivalency holds between semi-proper and stationary-preserving forcings in $\rm ZFC$? Or consistently in a model where significant fragments of $\rm MM$ fail.
| https://mathoverflow.net/users/38866 | Properness for small forcings | The answer is no, as the following upcoming work of Shelah and Usuba shows:
**Theorem (Shelah-Usuba)**:
The following theories are equiconsistent with ZFC:
ZFC+CH+ “there is an $\omega \_1$-stationary preserving $\sigma$-Baire poset of size $\aleph\_1$
which is not semiproper”.
ZFC+“Martin’s axiom for semiprope... | 3 | https://mathoverflow.net/users/11115 | 386268 | 160,520 |
https://mathoverflow.net/questions/384145 | 17 | I came across this question while making some calculations.
>
> **QUESTION.** Can you find some transformation to "decouple" the double integral as follows?
> $$\int\_0^{\frac{\pi}2}\int\_0^{\frac{\pi}2}\frac{d\alpha\,d\beta}{\sqrt{1-\sin^2\alpha\sin^2\beta}}
> =\frac14\int\_0^{\frac{\pi}2}\frac{d\theta}{\sqrt{\cos... | https://mathoverflow.net/users/66131 | Decoupling a double integral | (Thanks go to Etanche and Jandri)
\begin{align}J&=\int\_0^{\frac{\pi}{2}}\int\_0^{\frac{\pi}{2}} \frac{1}{\sqrt{1-\sin^2(\theta)\sin^2 \varphi}}d\varphi d\theta\\
&\overset{z\left(\varphi\right)=\arcsin\left(\sin(\theta)\sin \varphi\right)}=\int\_0^{\frac{\pi}{2}} \left(\int\_0^ \theta\frac{1}{\sqrt{\sin(\theta-z)\sin... | 10 | https://mathoverflow.net/users/175613 | 386273 | 160,523 |
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