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https://mathoverflow.net/questions/450591 | 1 | In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the
Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V\_n)\_n$ of continuous and compact supported functions, which pointwise almost everywhere approximate $V$.
I do know some res... | https://mathoverflow.net/users/508379 | Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support | Let $V\in L^\infty$ and let $V\_\varepsilon=V\*\varphi\_\varepsilon$ be a standard approximation by convolution. Since $V\in L^1$ on any bounded set, it follows that $V\_\varepsilon\to V$ in $L^1$ on bounded sets. Since a convergence in $L^1$ implies a.e. convergence on a subsequence, for every bounded set you find a s... | 7 | https://mathoverflow.net/users/121665 | 450593 | 181,260 |
https://mathoverflow.net/questions/450592 | 2 | Let $$\chi\_S(x,y)=\begin{cases}1&\text{ if }0< x<y< 1\\0&\text{ else }\end{cases}$$
be the indicator function of the simplex $S=\{(x,y)\in (0,1)^2:x<y\}$. I am interested in an explicit partition of unity. That is, I am looking for continuous positive functions $f\_n:\mathbb R^2\to \mathbb R$ so that $\sum\_{n=1}^\i... | https://mathoverflow.net/users/479223 | Partition of unity of simplex | It is easy to construct an explicit continuous (or even smooth) partition of unity over $\mathbb R^2$. Map this partition of unity to the continuous (or smooth) partition of the indicator $\chi\_Q$ of the open square $Q:=(-1,1)^2$ via the bi-smooth map $\mathbb R^2\ni(x,y)\mapsto\frac2\pi\,(\arctan x,\arctan y)\in Q$. ... | 3 | https://mathoverflow.net/users/36721 | 450595 | 181,261 |
https://mathoverflow.net/questions/450585 | 4 | For natural $n$, let
\begin{equation}
p\_n:=2^{1-n}\sum\_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum\_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/2\rfloor$, $l:=\lfloor n/2\rfloor$, and $a\equiv b$ means that $a-b$ is even.
>
> Can this expression for $p\_n$ be... | https://mathoverflow.net/users/36721 | On a double sum involving binomial coefficients | The formula for $p\_n$ was derived to be a probability relating two independent transformed binomials. In particular,
$$p\_n = \mathbb{P}(|U\_n|<|V\_n|)$$
where
$$ U\_n = a\_1 + \dots a\_k \hspace{20pt} V\_n = b\_1 + \dots + b\_l $$
and $k = \lfloor (n+1)/2 \rfloor, l = \lfloor n/2 \rfloor$.
When $n=2m$ then $k = l =... | 5 | https://mathoverflow.net/users/507348 | 450598 | 181,262 |
https://mathoverflow.net/questions/450600 | 9 | How to prove that there can't exist a countable set $\{A\_1,A\_2,\dots\}\subset \mathcal{L}(\mathbb{R})$ (where $\mathcal{L}(\mathbb{R})$ denotes the family of all Lebesgue measurable sets) such that $\sigma(\{A\_1,A\_2,\dots\})=\mathcal{L}(\mathbb{R})$?
(I know that the cardinality of $\mathcal{L}$ is equal to the c... | https://mathoverflow.net/users/508383 | How to prove that the Lebesgue $\sigma$-Algebra is not countably generated? | Let me mount the kind of cardinality argument to which you allude.
You had asked for a proof that the $\sigma$-algebra of Lebesgue measurable sets is not countably generated. But in fact, a much stronger claim is true:
**Theorem.** The $\sigma$-algebra of all Lebesgue measurable sets is not generated by any family ... | 15 | https://mathoverflow.net/users/1946 | 450601 | 181,263 |
https://mathoverflow.net/questions/450553 | 7 | * Let $a(n)$ be [A204262](https://oeis.org/A204262) i.e. [permanent](https://en.wikipedia.org/wiki/Permanent_(mathematics)) of the matrix $n\times n$ with elements $\min(i,j)$.
* Let
$$
f\_{n,\ell}(x)=g\_{n,\ell}(x)+f\_{n,\ell-1}(\ell)-g\_{n,\ell}(\ell), \\
g\_{n,\ell}(x)=\int (n-\ell)^2 f\_{n-1,\ell}(x)\,dx, \\
f\_{n,... | https://mathoverflow.net/users/231922 | Remarkable recursions for the A204262 | I can show the first identity $R(n,0) = f\_{n+1,n+1}(0)$, as a consequence of the more general identity
$$ R(n,q) = \frac{1}{(q+1)!} f\_{n+q+1,n}(n+1)\tag{1}\label{1}$$
for $n,q \geq 0$. Indeed, note that $g\_{n+1,n+1} \equiv 0$ and thus $f\_{n+1,n+1}(0) = f\_{n+1,n}(n+1)$, so $R(n,0) = f\_{n+1,n+1}(0)$ is basically ... | 10 | https://mathoverflow.net/users/766 | 450607 | 181,264 |
https://mathoverflow.net/questions/132618 | 7 | Ogg characterized the finitely many N such that $X\_0(N)\_{\mathbb{Q}}$ is hyperelliptic, and Poonen proved in "Gonality of modular curves in characteristic p" that for large enough N, $X\_0(N)\_{\mathbb{F}\_p}$ is not hyperelliptic.
**Question**: Are there any N such that $X\_0(N)\_{\mathbb{Q}}$ is not hyperelliptic... | https://mathoverflow.net/users/2 | Hyperelliptic modular curves in characteristic p | I already gave the currently accepted answer to this question around 10 years ago. And the answer can be summarised as "Yes, here is an example". However, after 10 years I wanted to come back to this question and answer it again, but now in the opposite direction. Namely, as it turns out the example that I found above ... | 5 | https://mathoverflow.net/users/23501 | 450617 | 181,270 |
https://mathoverflow.net/questions/450627 | 5 | This might be well-known to experts. I was just teaching a course where we went through some parts of Quillen's theorem computing the higher algebraic K-theory of finite fields. Denote by $\mathbb F\_q$ the field with $q$ elements.
>
> **Theorem (Quillen)** Let $q=p^k$ for some prime $p$ and $k \in \mathbb N\_{\geq... | https://mathoverflow.net/users/8176 | Galois action on algebraic K-theory of finite fields | $K(\overline{\mathbb{F}\_p})$ is, after completion at any prime $\ell \neq p$, equivalent to $ku^\wedge\_\ell$. It's true that the equivalence relies on a non-canonical embedding, but the conclusion that the homotopy groups of the $\ell$-completion are a polynomial ring, is independent of the concrete isomorphism. Sinc... | 9 | https://mathoverflow.net/users/39747 | 450628 | 181,271 |
https://mathoverflow.net/questions/450573 | 2 | There is a well know theorem by Coven and Hedlund, in [Sequences with minimal block growth](https://link.springer.com/article/10.1007/BF01762232), stating that the complexity function of an aperiodic sequence\configuration $\omega\in \mathcal{A}^{\mathbb{Z}}$ is at least linear, where the complexity function $c\_\omega... | https://mathoverflow.net/users/143153 | Morse-Hedlund\Coven-Hedlund theorem for non-Abelian groups | Your motivation is about aperiodicity implying high complexity, but you ask whether aperiodicity implies an upper bound on complexity, which is a little strange. Probably there is a typo, and $O(g(r))$ should be $\Omega(g(r))$?
In any case, I'll give a simple generalization of the Morse-Hedlund theorem to all groups.... | 3 | https://mathoverflow.net/users/123634 | 450636 | 181,276 |
https://mathoverflow.net/questions/450623 | 3 | Consider for $i=1,\ldots, N\ge2$
$$X^i\_t=x\_i+W^i\_t,\quad \forall t\ge 0,$$
where $x\_1,\ldots, x\_N\in (0,\infty)$ and $W^1,\ldots, W^N$ are independent Brownian motions. Denote by $\tau\_i$ the first hitting time of $X^i$ at zero, i.e.
$$\tau\_i:=\inf\big\{t\ge 0: X^i\_t\le 0 \big\}.$$
How to prove (rigorou... | https://mathoverflow.net/users/493556 | Can independent Brownian motions hit zero at the same time? | Your question is asking whether two Brownian motions can both first hit zero simultaneously. In fact we can say something stronger; for $N$ independent Brownian motions, the set of times where each Brownian motion hits 0 are pairwise disjoint.
When $N=2$, this is equivalent to asking whether a standard two-dimensiona... | 9 | https://mathoverflow.net/users/507348 | 450641 | 181,277 |
https://mathoverflow.net/questions/27716 | 40 | Fan Chung and Ron Graham's book *Erdos on Graphs: His Legacy of Unsolved Problems* (A. K. Peters, 1998) collects together all of Erdos's open problems in graph theory that they could find into a single volume, complete with bounties where applicable. Of course Erdos posed many other open problems in combinatorics and n... | https://mathoverflow.net/users/3106 | Does there exist a comprehensive compilation of Erdos's open problems? | Recently, Thomas Bloom created a website dedicated exactly to this:
<https://www.erdosproblems.com/>
It currently lists 214 problems, both open and closed. They are all tagged and some problems carry additional information. The list still evolves and if a problem is missing you might want to contact Thomas directly... | 7 | https://mathoverflow.net/users/31469 | 450643 | 181,278 |
https://mathoverflow.net/questions/450606 | 2 | Sounds like a trivial question, but could not find any answer other than the fact that there are many ways to define it. My problem is this: I look at different elevation maps,
1. some with sharp mountains,
2. some rather flat, or
3. with steep but consistent slopes with little gradient variations.
For simplicity, ... | https://mathoverflow.net/users/140356 | What are the best definitions for smoothness of a 2D curve (real-valued function)? | $S(f,[a,b])=\int\_a^b f''(x)^2\,dx$ is, not a measure of smoothness, but rather a measure of nonlinearity (or, better, of non-affinity or, one might say, of "sharp-turn'ness") of a function $f$. Indeed, $S(f,[a,b])$ takes its smallest value $0$ if and only if $f$ is affine on $[a,b]$. (Of course, here I use the term "m... | 2 | https://mathoverflow.net/users/36721 | 450645 | 181,279 |
https://mathoverflow.net/questions/450584 | 3 | A *grating* is a notion in algebraic topology from the 1940 introduced by Alexander. Cartan extended it as follows.
A grating (carapace in french) is defined by a topological space $X$, a module (or a differential ring) $A$ and for each $x \in X$, a surjective morphism $\varphi\_x$ from $A$ to some quotient $A\_x$ su... | https://mathoverflow.net/users/6129 | Sheaves and gratings | I think you may have misunderstood what Cartan was doing. In modern language he defined functors $\Gamma$ from sheaves to gratings and $\mathcal{F}$ from gratings to sheaves but he did not claim that these are mutually inverse equivalences of categories (even allowing for the apparent anachronism). He did write down a ... | 2 | https://mathoverflow.net/users/345 | 450646 | 181,280 |
https://mathoverflow.net/questions/450605 | 0 | Given a centrally symmetric convex body $K$ in the plane (with smooth boundary), it is easy to see that there exists a norm function $g:\mathbb{R}^2\to \mathbb{R}\_{\geq 0}$ for which $K$ is the unit ball.
For the polar body $K^{\circ}$ we also have a norm function $h:\mathbb{R}^2\to \mathbb{R}\_{\geq 0}$ for which $... | https://mathoverflow.net/users/334733 | Norm functions induced by convex bodies | I think you need to assume that $K$ has a smooth boundary and is strictly convex to ensure that $g$ and $h$ are differentiable outside $0$.
Anyway, I do not think that the result is true. Assume that $K$ is the unit ball associated to the $\ell^p$-norm with $p>1$. Then $K^\*$ is the unit ball associated to the $\ell^... | 2 | https://mathoverflow.net/users/169474 | 450650 | 181,281 |
https://mathoverflow.net/questions/293563 | 3 | Can one use lattice basis reduction algorithms, such as LLL over (low-rank) module lattices over rings of number fields of degree greater than 1? Is there any work on lattice reductions over Euclidean rings (for example, using BKZ, lattice enumeraiton, etc)?
| https://mathoverflow.net/users/24541 | Lattice basis reduction over rings of number fields | Apparently, this was studied in the following work: <https://eprint.iacr.org/2019/1035>
| 0 | https://mathoverflow.net/users/24541 | 450653 | 181,282 |
https://mathoverflow.net/questions/450657 | 6 | If I'm not mistaken, the question I put on the title used to be on this site, but I'm not being able to find it at all. I'm therefore reposting it so that someone can either give me the old link or help me answer this question anyway.
If $\tau$ is a topology on the real numbers, then saying that $(\mathbb{R}, \tau, +... | https://mathoverflow.net/users/494553 | Topologies that turn the real numbers into a compact Hausdorff topological group | A compact abelian group $A$ is the same as the Pontryagin dual of a discrete abelian group $B$. The group $A$ is divisible ($nA=A$ for all $n\ge 1$) if and if $B$ is torsion-free, and $A$ is torsion-free iff $B$ is divisible. So $A$ is divisible torsion-free iff $B$ is divisible torsion-free, i.e., $B\simeq\mathbf{Q}^{... | 6 | https://mathoverflow.net/users/14094 | 450659 | 181,284 |
https://mathoverflow.net/questions/450570 | 4 | I believe that a catenoid supports a parametrization $\sigma : U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ that forms a conjugate system (i.e., $\sigma\_{uv} \in\mathrm{span}(\sigma\_u, \sigma\_v)$) with the additional property that the coordinate curves are geodesics. I tried the following computational approach ... | https://mathoverflow.net/users/109420 | Building a geodesic conjugate parameterization on catenoid | I thought about your problem and realized that there is no coordinate parametrization of the catenoid with the properties that you want.
Here is my argument: First, note that, in the given $uv$-parametrization, the first and second fundamental forms of the immersion are
$$
I= \cosh(u)^2 (\mathrm{d}u^2 + \mathrm{d}v^2... | 8 | https://mathoverflow.net/users/13972 | 450668 | 181,288 |
https://mathoverflow.net/questions/323192 | 1 | A finite reduced Laver-like algebra is a finite algebra $(X,\*,1)$ that satisfies the identities $1\*x=x,x\*1=1,x\*(y\*z)=(x\*y)\*(x\*z)$ and where there is a natural number $n$ and a function $\mathrm{crit}:X\rightarrow n+1$ where
1. $\mathrm{crit}(x)=n$ if and only if $x=1$,
2. $\mathrm{crit}(x\*y)=\mathrm{crit}(y)... | https://mathoverflow.net/users/22277 | Is every critically subsimple Laver-like algebra a quotient of a critically simple Laver-like algebra on the same number of generators? | No. A critically subsimple Laver-like algebra is not necessarily a quotient of a critically simple Laver-like algebra with the same number of generators and more critical points. Our strategy for producing counterexamples is to exhibit finite reduced Laver-like algebras $X$ generated by $(x\_a)\_{a\in A}$ such that the... | 1 | https://mathoverflow.net/users/22277 | 450670 | 181,289 |
https://mathoverflow.net/questions/448764 | 3 | Let $G$ be a split reductive group over a nonarchimedean local field $F$ (I'm particularly interested in the case of $\operatorname{GSp}\_{2n}$).
---
>
> Given a parahoric subgroup $K \subset G(F)$, and a parabolic subgroup
> $P$, is there a "nice" group-theoretic description of the orbits of
> $K$ on the flag ... | https://mathoverflow.net/users/2481 | Orbit of a parahoric subgroup on a flag variety | Let $G$ a reductive group over a nonarchimedean local field $F$.
Let $P\_0$ be a minimal parabolic subgroup of $G$ and $A$ a maximal split torus contained in $P\_0$.
The normalizer $N\_G(A)(F)$ acts on the apartment of $A$ via the extended affine Weyl group,
which contains the affine Weyl group $W\_\mathrm{aff}$ with f... | 3 | https://mathoverflow.net/users/507750 | 450683 | 181,293 |
https://mathoverflow.net/questions/450680 | 4 | Let $(X,\succeq)$ be a poset. I have the following two questions:
1. Is it true that there exists a finite complemented distributive lattice (a Boolean lattice) $(S, \succeq^\*)$ such that $X\subseteq S$ and $\succeq^\*$ agrees with $\succeq$ over $X$? If not, are there conditions on $(X,\succeq)$ that guarantee this... | https://mathoverflow.net/users/98626 | Is every finite poset a subset of a finite complemented distributive lattice? | As Sam mentioned in the comments, the answer to question 1 is yes. One can map every condition $p$ in the partial order to the lower cone, the set $S\_p=\{q\in P\mid q\leq p\}$ of conditions below $p$. It is clear that $q\leq p\iff S\_q\subset S\_p$, and so this maps $P$ into the powerset algebra, which is a Boolean al... | 6 | https://mathoverflow.net/users/1946 | 450696 | 181,297 |
https://mathoverflow.net/questions/450554 | 1 | Suppose that $u$ is $C^1$ in $[0,\pi/2]$ and $u(0)=u’(\pi/2)=0$. I want to derive the following Poincaré inequality
$$
\int\_0^{\pi/2} u(x)^2\,dx \leq \int\_0^{\pi/2} u’(x)^2\,dx.
$$
Since $u(0)=0$, we have that $u(x) = \int\_0^x u’(s)\,ds$. Hence
\begin{align}
u^2(x) & \leq \biggl( \int\_0^x u’(s)\,dx\biggr)^2 \\
&... | https://mathoverflow.net/users/480661 | Poincaré inequality in dimension one with mixed boundary condition | We want to show that
$$\int\_0^{\pi/2} u(x)^2\,dx \le \int\_0^{\pi/2} u'(x)^2\,dx \tag{1}\label{1}$$
for all $u\in C^1[0,\pi/2]$ such that $u(0)=0$ and $u'(\pi/2)=0$.
Let us show that **\eqref{1} holds even without any condition on $u'(\pi/2)$**.
We have the [Poincaré inequality](https://en.wikipedia.org/wiki/Poinc... | 2 | https://mathoverflow.net/users/36721 | 450698 | 181,298 |
https://mathoverflow.net/questions/450676 | 0 | I am confused on [the Lemma 5.7 (Lipschitz maximal inequality) here](https://web.math.princeton.edu/%7Ervan/APC550.pdf). Let me first restate the definition and the lemma:
**Def**
$\{X\_t\}\_{t\in T}$ is called Lipschitz for metric $d$ on $T$ if there exists a random variable $C$ such that
$$|X\_t-X\_s|\leq Cd(t,s),\... | https://mathoverflow.net/users/494410 | Lipschitz maximal inequality for random process | $\newcommand\ep\epsilon\newcommand\si\sigma\newcommand\td{\tilde d}$You wrote: "I am a bit confused on the applicability of this lemma on random process that its $E[\sup X\_t]$ is negative."
One way to use this result, whether $E\sup\_{t\in T} X\_t$ is negative or not, is as follows. We have
>
> $$E\sup\_{t\in T}... | 1 | https://mathoverflow.net/users/36721 | 450703 | 181,300 |
https://mathoverflow.net/questions/450679 | 3 | For an $n \times n$ matrix $M$, the $\infty\to 1$ and [cut](https://en.wikipedia.org/wiki/Matrix_norm#Cut_norms) norms are given by
$$\|M\|\_{\infty \to 1} := \max\limits\_{x, y \in \{\pm 1\}^n} \sum\limits\_{i, j} m\_{i, j} x\_i y\_j, \qquad \|M\|\_{\square} := \max\limits\_{A, B} \left|\sum\limits\_{i \in A, j \in ... | https://mathoverflow.net/users/127521 | How to turn $\{-1, 0, 1\}$-valued optimization problem into integer program? | Given $n \times n$ matrix $M$, you want to find $A,B,C \subset \{1,\dots,n\}$ to maximize
$$\sum\limits\_{i \in A, j \in C} m\_{ij} - \sum\limits\_{i \in B, j \in C} m\_{ij}$$
subject to $A < B < C$ and $|A|=|B|=|C|$.
Introduce binary decision variables $a\_i, b\_i, c\_i \in \{0,1\}$ to indicate whether $i\in A, i\in... | 2 | https://mathoverflow.net/users/141766 | 450718 | 181,305 |
https://mathoverflow.net/questions/450523 | 8 | My question is motivated by [this one](https://mathoverflow.net/q/316985), but within real matrices instead of complex ones.
>
> ${\bf Sym}\_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\langle A,B\rangle={\rm Tr}(AB)$, this is a Euclidean space. Does there exis... | https://mathoverflow.net/users/8799 | Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices | Here is an example with $n=4$. (ADDED BELOW: An example for any power of $2$)
I identify $\mathbb R^4$ with the quaternions, and describe $10$ subspaces such that any two of the resulting orthogonal projections satisfy the required condition $4Tr(\pi\pi')=2d+2d'-4$. One of the ten is the zero-dimensional subspace, an... | 8 | https://mathoverflow.net/users/6666 | 450721 | 181,306 |
https://mathoverflow.net/questions/450615 | 4 | Is there a smooth embedding of $S^2$ into some Euclidean space that is equivariant with respect to a linear representation of $PSL(2,\mathbb C)$?
A counterexample to a more general question can be found [here.](https://mathoverflow.net/questions/149867/is-there-an-analogue-of-mostow-palais-equivariant-embedding-theorem... | https://mathoverflow.net/users/1573 | An analogue of Mostow-Palais equivariant embedding theorem for the group of conformal automorphisms of the 2-sphere | The answer is 'no'. In fact, a stronger statement is true: If $V$ is a finite dimensional vector space and $G\subset\mathrm{GL}(V)$ is a (connected) non-compact simple Lie group, then the only bounded orbits of $G$ are fixed points. *A fortiori*, the only compact $G$-orbits are fixed points.
To see this, it's enough ... | 4 | https://mathoverflow.net/users/13972 | 450738 | 181,311 |
https://mathoverflow.net/questions/450699 | 2 | How to prove that the following expression is positive for $u\ge 0$ and $q\in(0,\pi/3)$. $$\frac{2 \sqrt{2} ((2+u) (1+\cos(q)))^{3/4}}{3^{3/4}}-\frac{2^{3/4} (1+\sec(q))}{\sqrt{3} \sec(q)^{3/4}}-\left(1+u+\sqrt{-1+(1+u)^2} \sqrt{1-\frac{4 \sin(q)^2}{3}}\right)^{3/4}$$
| https://mathoverflow.net/users/504719 | An inequality involving 2 variables | Letting $c:=\cos q$, we have $c\in[1/2,1]$. Writing $\sec q=1/c$ and $\sin^2q=1-c^2$, and multiplying the big expression by $3^{3/4}c^{1/4}$, we see that the inequality in question can be rewritten as
\begin{equation\*}
f(c,u)\overset{\text{(?)}}>0, \tag{10}\label{10}
\end{equation\*}
where $(c,u)\in[1/2,1]\times[0,\i... | 1 | https://mathoverflow.net/users/36721 | 450740 | 181,312 |
https://mathoverflow.net/questions/450756 | 4 | Let $G$ be a group and $\pi$ be a finite-dimensional (not necessarily unitary) representation of $G$ on a complex Hilbert space $H$. We shall say that $\pi$ is completely reducible if there exists a decomposition of $H$ into *orthogonal* irreducible sub-representations of $\pi$.
>
> **Question 1.** Suppose $(\pi\_1... | https://mathoverflow.net/users/165204 | When are finite-dimensional representations on Hilbert spaces completely reducible? | I don't know about Q1 off the top of my head, but I think that for Q2 you are unlikely to get a good answer. Of course I may have a different view from you as to what "mild" adjectives are reasonable to impose.
The reason I say this is that in your definition of completely reducible you are requiring the summands of ... | 7 | https://mathoverflow.net/users/763 | 450761 | 181,314 |
https://mathoverflow.net/questions/450759 | 0 | In a Banach space X, given a norm one bounded linear functional $f$ and $c\in \mathbb{C}\backslash \{0\}$, define $H = \big\{ x\in X \,\vert\, f(x) = c\big\}$ and $\inf H$ = $\inf\_{h\in H} \|h\|$.
1. Is there a necessary condition for $\inf H = \vert\, c\,\vert$?
| https://mathoverflow.net/users/151332 | Infimum of norms of elements in a hyperplane | I am assuming your linear functional means "complex linear", then $\inf H = |c|$ will always be true if $\lVert f \rVert = 1$.
On the one hand, since $\lVert f \rVert = 1$, we can choose a sequence $\{x\_n\}$ of norm $1$ elements in $X$ such that $|f(x\_n)| > 0$ and $|f(x\_n)| \rightarrow \lVert f \rVert = 1$. Replac... | 1 | https://mathoverflow.net/users/166298 | 450765 | 181,317 |
https://mathoverflow.net/questions/450766 | 2 | Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.
In this case, I am (hopefully) trying to find an explicit formula for the Green function for the Laplacian $-\Delta$.
That is, what w... | https://mathoverflow.net/users/56524 | Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? | Well it depends on what you mean by "explicit". Let $(\varphi\_k)\_k \subset L^2(\mathbb{S}^1)$ be the eigenfunctions of the Laplacian on $\mathbb{S}^1$, these have an explicit form that comes by solving the relative ODE, and let $(\lambda\_k)\_k$ be the relative eigenvalues. Then $G\_{\mathbb{S}^1}(p,q)=\sum\_{k\ge 1}... | 5 | https://mathoverflow.net/users/508380 | 450768 | 181,318 |
https://mathoverflow.net/questions/450706 | 18 | $\DeclareMathOperator\Sha{Sha}$I calculated the Tate–Shafarevich group $\Sha(E/K)[2]$ of the elliptic curve $E:y^2=x^3+17x$ over $K=\Bbb{Q}(\sqrt{-37})$.
I calculated that by hand and I reached the conclusion that $\Sha(E/K)[2]$ has order $1$ or $4$.
Magma code
```
K<b>:=QuadraticField(-37);
A:=EllipticCurve([K!... | https://mathoverflow.net/users/144623 | Discrepancy in Magma's calculation and Sage's of elliptic curve? | Well spotted. This is a problem. After quite a bit of fiddling I found that the error is in Denis Simon's script used by Sage.
In fact, when executed with higher values of the parameters so that the script finds the rational points, it also prints the correct information.
The command
```
K.<t> = QuadraticField(-37... | 10 | https://mathoverflow.net/users/5015 | 450776 | 181,321 |
https://mathoverflow.net/questions/450619 | 5 | Let $P\_{n}(x)$ the $n-th$ [Legendre polynomial](https://en.wikipedia.org/wiki/Legendre_polynomials). It is well-knonw that $$\int\_{-1}^1 P\_n(x) P\_m(x) P\_h(x) \, dx=2\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right)^{2}\tag{1}$$ where $\left(\begin{array}{ccc}
n & m & h\\
0 & 0 & 0
\end{array}\right... | https://mathoverflow.net/users/68301 | Generalized Wigner 3-j symbol and Legendre functions | **Summary:** this is a topic of active research [1,2,3,4]; the answer to the general case is not known; answers exist for $a\in\mathbb{C}$, $b,c,\in\mathbb{N}$, and for $a,b,c\in\mathbb{C}$ with $b=c$.
---
**Case 1:** Consider first the case that one of the three parameters $a,b,c$ is complex, while the other two... | 4 | https://mathoverflow.net/users/11260 | 450782 | 181,322 |
https://mathoverflow.net/questions/450640 | 4 | Let $\Omega$ be a measurable space equipped with a $\sigma$-ideal $\mathcal{N}$ (though of as the "null sets"). Define the compact Hausdorff space
$$ \tilde{\Omega} := \mathrm{Spec}(L^\infty(\Omega)), $$
where $\mathrm{Spec}$ denotes the Gelfand spectrum. In [this question](https://mathoverflow.net/questions/99091/spec... | https://mathoverflow.net/users/103549 | Representing measurable map to compact space as a continuous map | I think I have found a precise formulation and proof of the cited statement.
Let's suppose for simplicity that $(\Omega, \Sigma, \mathbb{P})$ is a probability space. Define the set of $K$-valued random variables up to almost sure equality (with $K$ some compact space),
$$ \mathrm{RV}(\Omega, K) := \{X: \Omega \to K \... | 0 | https://mathoverflow.net/users/103549 | 450783 | 181,323 |
https://mathoverflow.net/questions/450742 | 4 | Let $p$ be an odd prime. What's the condition on $q$ for
$$
p^{1+2r}\cdot\operatorname{Sp}(2r,p)\leqslant \operatorname{GU}(p^r,q)\;?
$$ I did some computation and seemed that $q\equiv -1$(mod $p$) does give the embedding. I feel that there is some work already done about it, am I right? Or it is an obvious question an... | https://mathoverflow.net/users/488802 | Condition on $q$ for inclusion $p^{1+2r}\cdot\operatorname{Sp}(2r,p)\leqslant \operatorname{GU}(p^r,q)$ | Too long for a comment. Your conjecture is correct: $q\equiv-1\pmod p$ is a necessary and sufficient condition. This follows directly from the two lemmas below.
It's not a silly question, but it can be answered with some standard tools of representation theory, and a couple of facts about $Sp\_{2r}(p)$: It is perfect... | 7 | https://mathoverflow.net/users/99221 | 450784 | 181,324 |
https://mathoverflow.net/questions/450781 | 4 | In an infinite-dimensional Banah space $(X, \|\cdot\|)$ with a countable Schauder basis $\{x\_n\}$, define:
$$
F\_r: \operatorname{Span}(\{x\_n\}) \rightarrow \operatorname{Span}(\{x\_n\}), \hspace{0.3cm} \sum\_{i\leq N} \lambda\_i x\_i \mapsto \sum\_{i\leq N} \operatorname{Re}(\lambda\_i) x\_i
$$
and:
$$
F\_i: \... | https://mathoverflow.net/users/151332 | The real and the imaginary part of a vector | $\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\C}{\mathbb C}$No, in general the map $F\_r$
\begin{equation\*}
\sum\_1^n w\_j b\_j\mapsto \sum\_1^n \Re(w\_j) b\_j \tag{10}\label{10}
\end{equation\*}
from the span of the $b\_j$'s into itself cannot be continuously extended to the entire $X$, where ... | 4 | https://mathoverflow.net/users/36721 | 450789 | 181,325 |
https://mathoverflow.net/questions/450791 | 12 | Essential undecidability of PA says every complete extension of PA includes a non r.e. set of new "axioms," all undecidable in PA. In that sense lots of sentences of PA are undecidable in PA. And it is trivial to design a measure on, say sentences of PA with Godel number below $n$, where the set of decidable sentence... | https://mathoverflow.net/users/38783 | Is there a useful measure of density of decidable sentences in PA? | Asymptotic density seems a very natural measure. The density of a set of sentences is the limit as $n\to\infty$ of the proportion of those sentences of length at most $n$ amongst all sentences of length at most $n$.
(One should use a formalism that has only finitely many sentences of a given length—for example, one c... | 12 | https://mathoverflow.net/users/1946 | 450792 | 181,326 |
https://mathoverflow.net/questions/450804 | 6 | Disclaimer: I'm a relative beginner in this area. I'm trying to prove that if one has a commutative ring $R$ and a prime number $p$, then there is an exact sequence of the form
$$\DeclareMathOperator\THH{THH}
0 \rightarrow \pi\_n(\THH(R)\_{h\mathbb{T}})/p\pi\_n(\THH(R)\_{h\mathbb{T}}) \rightarrow \pi\_n(\THH(R; \mathbb... | https://mathoverflow.net/users/508592 | An exact sequence involving THH | Let's extract a clear question (about spectra in general) from your question, and then answer it. Let $E$ be any spectrum.
There is the degree $p$ map $p:S\to S$ from the sphere spectrum to itself. Smashing with $E$, this gives a map $E\to E$; we also call this map $p$. It multiplies elements of $\pi\_n(E)$ by $p$. D... | 8 | https://mathoverflow.net/users/6666 | 450809 | 181,333 |
https://mathoverflow.net/questions/450530 | 3 | Consider the numbers $$a\_n=\frac{1}{n+1}\sum\_{k=0}^{n}\frac{2^{k-1}\binom{n+1}{k}B\_k}{2^{s+k-1}-1}, \ n\geq0,$$ where $s\neq1;0;-1;-2;-3;...$ is a fixed real number, and the $B\_k$ are the Bernoulli numbers ($B\_1=-1/2$).
I am trying to show that the power series $\sum\_{n=0}^{+\infty} a\_n x^n$ converges. Numerical... | https://mathoverflow.net/users/109569 | Convergence of a power series | Let $s > 1$. Using Faulhaber's formula and some well known properties of the Bernoulli numbers, we have $$a\_n = \frac{1}{n+1} \sum\_{k = 0}^n 2^{-s} \left(\sum\_{j = 0}^{+\infty} 2^{-(s+k-1) j}\right) \binom{n+1}{k} B\_k = \frac{2^{-s}}{n+1} \sum\_{j = 0}^{+\infty} 2^{-(s-1) j} \sum\_{k = 0}^n (2^j)^{-k} \binom{n+1}{k... | 2 | https://mathoverflow.net/users/508605 | 450816 | 181,334 |
https://mathoverflow.net/questions/450817 | -2 | I want to coin a notion of "strong provability", to be defined as:
$S$ is strongly provable in $T$ if and only if there is a Gödel code of its proof in $T$ that is strictly smaller than any Gödel code of a proof of its negation in $T$
Formally:
$ S \text { is strongly provable in } T \iff \\ \exists x: \operatorn... | https://mathoverflow.net/users/95347 | Can we have consistent theories stating opposing provability statements that are non-standardly coded? | This idea in play here is due to Rosser and is the main idea behind the [Gödel-Rosser theorem](https://en.wikipedia.org/wiki/Rosser%27s_trick).
Specifically, Rosser proposes to consider the sentence $\rho$ asserting that for every proof of $\rho$ in PA, there is a smaller proof of $\neg\rho$. In your terminology, $\r... | 7 | https://mathoverflow.net/users/1946 | 450818 | 181,335 |
https://mathoverflow.net/questions/450810 | 2 | Let $F$ be a p-adic field. A character on $F^\times$ is defined as a *continuous* group homomorphism $F^\times\longrightarrow\mathbb{C}^\times$. But is there any way to construct a non-continuous group homomorphism $F^\times\longrightarrow\mathbb{C}^\times$? Is there even any such homomorphism?
| https://mathoverflow.net/users/32746 | Non-continuous group homomorphism from p-adic field to C* | Since $\mathbf C^\times$ is a divisible group, Zorn’s lemma tells us for every abelian group $G$ and subgroup $H$ that each group homomorphism $H\to \mathbf C^\times$ extends (somehow) to a group homomorphism $G\to\mathbf C^\times$.
Use $G = F^\times$ and $H = \langle u\rangle$, where $u$ is in $\mathcal O\_F^\times$... | 7 | https://mathoverflow.net/users/3272 | 450822 | 181,336 |
https://mathoverflow.net/questions/450787 | 13 | From [Wikipedia](https://en.wikipedia.org/wiki/Second-order_arithmetic#Projective_determinacy) (I couldn't find the original source):
>
> $\text{ZFC} + \{\text{there are $n$ Woodin cardinals: $n$ is a natural number}\}$ is conservative over $\text{Z}\_2$ with projective determinacy.
>
>
>
(where projective det... | https://mathoverflow.net/users/65915 | How much determinacy do you need for second order arithmetic to be as strong as ZFC? | Because ZFC proves soundness of $\text{Z}\_2$, no consistent finite extension of $\text{Z}\_2$ proves all second order arithmetic statements that are provable in ZFC (for example, the statement "the conjunction of the new axioms implies their own consistency with $\text{Z}\_2$"). Moreover, for each fixed $n$, $Σ^1\_n$ ... | 11 | https://mathoverflow.net/users/113213 | 450839 | 181,342 |
https://mathoverflow.net/questions/450838 | 2 | The question is in the title.
Equation $\sum\_{i=1}^n x\_i^3 = 0$ has no non-trivial integer solutions for $n=3$. For $n=4$, there are known descriptions of all integer/rational solutions, see
Choudhry, Ajai. "On equal sums of cubes." The Rocky Mountain journal of mathematics (1998): 1251-1257.
My question is abo... | https://mathoverflow.net/users/89064 | Describe all integer/rational solutions to $x^3+y^3+z^3+t^3+s^3=0$ | This is not possible in any meaningful way.
In fact the variety you describe defines a smooth cubic threefold $X$ in $\mathbb{P}^4$. By a famous theorem of Clemens and Griffiths these are not even rational varieties over $\mathbb{C}$. This means that these are no way to parametrise the $\mathbb{C}$-points, let alone ... | 19 | https://mathoverflow.net/users/5101 | 450843 | 181,344 |
https://mathoverflow.net/questions/450850 | 1 | O. Baues and F. Gruewald in their paper "AUTOMORPHISM GROUPS OF POLYCYCLIC-BY-FINITE GROUPS AND ARITHMETIC GROUPS" have stated that a polycyclic-by-finite group has a unique maximal finite normal subgroup (see page 216, Section 1.2). I was wondering if somone could give a proof for that? Does this result still hold for... | https://mathoverflow.net/users/114476 | Does a finitely presented amenable group contain a unique maximal finite normal subgroup? | In a group $G$, the polyfinite radical $W(G)$ is the union of all finite normal subgroups. So, $G$ has a maximal finite normal subgroup iff $W(G)$ is finite.
You're asking whether $W(G)$ is finite for every finitely presented elementary amenable group. The answer is negative. There are indeed finitely presented solva... | 4 | https://mathoverflow.net/users/14094 | 450852 | 181,348 |
https://mathoverflow.net/questions/450842 | 3 | The following comes from Definition 2 in Pavel Pudlak, "A new proof of the congruence lattice representation theorem," *Algebra Universalis* **6** (1976), 269-275.
Let $X$ be a set. Let $F$ be a family of functions from $X$ to itself containing the identity map and closed under composition.
Define a binary relation... | https://mathoverflow.net/users/51389 | Why Is Pudlak's relation on the family of one- or two-element subsets of a set transitive? | I don’t see what the problem is supposed to be; you just compose the functions witnessing the two domination relations in the obvious way:
Fix $c=u\_0,\dots,u\_n=d$ and $f\_1,\dots,f\_n\in F$ such that $\{f\_i(a),f\_i(b)\}=\{u\_{i-1},u\_i\}$ for each $i=1,\dots,n$.
Fix $e=v\_0,\dots,v\_m=f$ and $g\_1,\dots,g\_m\in ... | 3 | https://mathoverflow.net/users/12705 | 450863 | 181,350 |
https://mathoverflow.net/questions/450796 | 1 | Let $X\_{t}=\sum\_{i=1}^n(1+s\cdot w)\sin(t\_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $s$ such that $X\_t$ is strictly positive with high probability? i.e. when $n\rightarrow\infty$,
$$P(\inf X\_t>0)\r... | https://mathoverflow.net/users/494410 | the infimum of a random process | We have $X\_t=Y\sum\_{i=1}^n\sin(t\_i)$, where $Y:=1+sw$,
$t=(t-1,\dots,t\_n)\in T^n:=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim N(0,1)$, and $s$ is a real number.
So,
$$\inf\_{t\in T^n}X\_t=\min\_{t\in T^n}X\_t=-n|Y|;$$
this minimum is attained at $t=(-\frac\pi2,\dots,-\frac\pi2)$ if $Y\ge0$ and at $t=(\frac\pi2,\dots,\fr... | 0 | https://mathoverflow.net/users/36721 | 450867 | 181,351 |
https://mathoverflow.net/questions/450860 | 1 | Let $S$ be a finite semigroup and $K$ a field of characteristic 0 (we can assume the complex numbers for simplicity).
>
> Question: Is there a characterization when the center of the semigroup algebra $KS$ is one-dimensional and a basis is given by the idendity of $KS$ (meaning $KS$ has an identity)?
>
>
>
If ... | https://mathoverflow.net/users/61949 | Semigroup algebras with one dimensional center | Let me answer what goes on with bands. I'll assume the band is a monoid for simplicity. I think everything works so long as the band is unital but this can get messier to describe. Details of what I'm saying can be found in [Margolis, S., & Steinberg, B. (2012). Quivers of monoids with basic algebras. Compositio Mathem... | 1 | https://mathoverflow.net/users/15934 | 450871 | 181,354 |
https://mathoverflow.net/questions/450788 | 3 | Let $f(x)\in\mathbb{Z}[x]$ be a polynomial of degree $d$ and naive height (maximum of the absolute values of the coefficients) at most $H$. Is there anything known about the number of prime factors of the discriminant of $f$? Particularly upper bounds?
[This](https://mathoverflow.net/questions/11880/what-primes-divid... | https://mathoverflow.net/users/477216 | Number of prime factors of a polynomial discriminant | The discriminant is $O(H^{2(d-1)})$ so its number of prime factors is bounded by $$ (1 + o(1)) \frac{ \log (H^{2(d-1)})}{\log \log (H^{2(d-1)})} = (2(d-1) + o(1)) \frac{ \log H}{\log \log H}$$ by the asymptotics for primorials (a consequence of the prime number theorem).
This is not too far from optimal (in the worst... | 3 | https://mathoverflow.net/users/18060 | 450874 | 181,355 |
https://mathoverflow.net/questions/450858 | 8 |
>
> **Question:**
>
>
> what is the geometric interpretation of the subject of machine learning and/or deep learning?
>
>
>
Being "forced" to have a closer look at the subject, I have the impression that it all boils down to approximately reconstruct the characteristic function that decides about set-membershi... | https://mathoverflow.net/users/31310 | Geometric formulation of the subject of machine learning | There are several geometric aspects of machine learning.
1. You can think of the goal of ML as function approximation. Your question mentions reconstructing a characteristic function, but people often look at approximating other types of functions as well ("regression" as opposed to "classification"). The geometry he... | 10 | https://mathoverflow.net/users/1227 | 450877 | 181,357 |
https://mathoverflow.net/questions/449817 | 1 | Let $A,B,C$ be Banach spaces and $m\,:\,A\times B\to C$ be a bilinear map such that $\|m(a,b)\|\leq \textrm{const}\,\|a\|\|b\|$. We denote by $\mathcal{S}(\mathbb{R}^d)$ be the standard space of Schwartz test functions and we denote by $\mathcal{S}(\mathbb{R}^d,B)$ the space of Schwartz test functions valued in $B$.
... | https://mathoverflow.net/users/47256 | Banach space valued distributions and test functions | Yes.
The tensor product $\mathcal S(\mathbb R^d)\otimes B$ can be identified with a subspace of $\mathcal S(\mathbb R^d,B)$ by denoting by $g\otimes b$ the function $x\mapsto g(x)b$ for all $g\in\mathcal S(\mathbb R^d)$ and $b\in B$ and this is the space you're first defining $g\mapsto\langle F,g\rangle\_m$ on.
The... | 2 | https://mathoverflow.net/users/508539 | 450881 | 181,359 |
https://mathoverflow.net/questions/450873 | 1 | Let $X\_{t}=\sum\_{i=1}^n(1+s\cdot w\_i)t\_i\sin(t\_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise.
How to find the condition on $s$ such that $X\_t$ is strictly positive with high probability? i.e. when $n\rightarrow... | https://mathoverflow.net/users/494410 | Lower bounding the infimum of a random process | This will not hold under any condition on $s$.
Indeed, by the continuity of $X\_t$ in $t$,
$$\inf\_{t\in T} X\_t=\inf\_{t\in[-\pi/2,\pi/2]^n} X\_t
\le X\_{\vec 0}=0.$$
So, for any real $s$,
$$P(\inf X\_t>0)=0\not\to1.$$
| 0 | https://mathoverflow.net/users/36721 | 450883 | 181,360 |
https://mathoverflow.net/questions/450880 | 0 | Assume that $u~ \colon \mathbb{R}\_+ \to [-M,M]$ is a bounded continuously differentiable function such that $u(0) = 0$ and $$u(t) \leq \int\_0^t \lambda(s)~u(s)~\mathrm{d}s + C \label{1}\tag{1}$$ where $\lambda ~ \colon \mathbb{R}\_+ \to [-\infty,-\alpha]$ is a strictly negative function with $\lambda(t) \leq -\alpha ... | https://mathoverflow.net/users/163454 | On the validity of a certain Grönwall-type inequality | No. A counterexample is $C=3$, $\lambda(t)=-1$, and $u(t)=\sin t$ for all $t$.
| 2 | https://mathoverflow.net/users/36721 | 450884 | 181,361 |
https://mathoverflow.net/questions/450888 | 3 | Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $ K $ having positive characteristic.
I am interested in determining for which fields $f(x)$ remains irreducible, base... | https://mathoverflow.net/users/215016 | irreducibility of the polynomial $ x^4 +1 $ | Among $-1$, $2$, and $-2$, there are $0$, $1$ or $3$ squares. This determines the irreducible factorization of $X^4+1$.
1. If none is a square, it is irreducible;
2. If only $-1=i^2$ is a square, the irreducible factorization is $(X^2-i)(X^2+i)$;
3. If only $-2=u^2$ is a square, the irreducible factorization is $(X^2... | 12 | https://mathoverflow.net/users/14094 | 450900 | 181,366 |
https://mathoverflow.net/questions/450890 | 16 | I asked this on Math Stack Exchange, but it didn't get a single answer. So, I am now asking it here. Let our signature be that of a single binary operation $+$. I define the constant identity to be $x+y=z+w$. The commutative identity is, of course, the well-known identity $x+y=y+x$. I wonder, is there an identity stric... | https://mathoverflow.net/users/43439 | Is there an identity between the commutative identity and the constant identity? | Yes:
$$(x + x) + y = y + x$$
The constant identity implies this because both sides are $+$es. This does not imply the constant identity because it is true about any set with an operation that is commutative, associative, and idempotent (meaning $x + x = x$ for all $x$), the smallest nontrivial example is $\{x,y\}$ wher... | 25 | https://mathoverflow.net/users/508667 | 450905 | 181,369 |
https://mathoverflow.net/questions/450799 | 6 | I need to prove the following statement.
Let $ n, g, m, a ,t$ be integers. Prove that the following statement is true for all $ n \geq g(1+2m)+1 $, $ g\geq 2t $, $ m\geq t $, $ 0\leq a <t $, and $ t>0 $:
$$\sum\_{l=0}^{m}(-1)^l\frac{\binom{m}{l}}{\binom{n/g-l}{m+1}}\left[\frac{\binom{n-2t}{gl-a}}{\binom{n}{gl}}-\frac... | https://mathoverflow.net/users/508578 | A summation involving fraction of binomial coefficients | First let's notice that
\begin{split}
\frac{\binom{m}l}{\binom{n/g-l}{m+1}} &= \binom{n/g}l\binom{n/g-l-m-1}{m-l} \frac{m!(m+1)!\Gamma(n/g-2m)}{\Gamma(n/g+1)} \\
&=(-1)^{m-l}\binom{n/g}l\binom{2m-n/g}{m-l} \frac{m!(m+1)!\Gamma(n/g-2m)}{\Gamma(n/g+1)}
\end{split}
and similarly
$$\frac{\binom{n-2t}{gl-a}}{\binom{n}{gl}} ... | 3 | https://mathoverflow.net/users/7076 | 450906 | 181,370 |
https://mathoverflow.net/questions/450879 | 1 | Let $\texttt{R}$ be an $\texttt{E}$-infinity ring and let $\texttt{M,N}$ be $\texttt{E}$-infinity modules. Under what conditions do we have
$$ \texttt{[M, N] ≅ [M,R] ⊗ N}$$
Under ordinary circumstances (with modules), we can show that a canonical map is an isomorphism using rank or dimension. What is the easiest ap... | https://mathoverflow.net/users/30211 | [M,N]≅ [M,R] ⊗ N for E-infinity modules | See page 70 of EKMM. The map $$F\_R(M,N) \wedge\_R F\_R(M',N') \rightarrow F\_R(M \wedge\_R M', N \wedge\_R N')
$$ is an isomorphism if $M,M'$ are strongly dualizable or if $M$ is strongly dualizable and $N=R$. Being strongly dualizable is equivalent to being a wedge summand of a finite cell $R$ module.
| 3 | https://mathoverflow.net/users/134512 | 450908 | 181,372 |
https://mathoverflow.net/questions/450897 | 3 | For each $n \in \mathbb{N}$, the Hermite function $\psi\_n : \mathbb{R} \to \mathbb{R}$ is a Schwartz function defined by
\begin{equation}
\psi\_n(x):=(-1)^n(2^n n!\sqrt{\pi})^{-1/2} e^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}
\end{equation}
according to the Wikipedia article : <https://en.wikipedia.org/wiki/Hermite_polynomials... | https://mathoverflow.net/users/56524 | $L^\infty$ bound of $x^m \psi_n(x)$ where $\psi_n$ is a Hermite function and $m,n \in \mathbb{N}$ - extension from Cramer's inequality | $\newcommand\ep\varepsilon$According to the 6th display in [this section](https://en.wikipedia.org/wiki/Hermite_polynomials#Asymptotic_expansion), for any fixed $\ep>0$ we have
$$\psi\_n(x)=(2/\pi)^{1/4}(\pi n)^{-1/4}(\sin t)^{-1/2}s\_n(x)$$
if $n\to\infty$, $x=\sqrt{2n+1}\,\cos t$, and $t\in[\ep,\pi-\ep]$, and
$$s\_n(... | 3 | https://mathoverflow.net/users/36721 | 450910 | 181,373 |
https://mathoverflow.net/questions/450928 | 6 | Circumstances: I'm studying Grothendieck's Galois Theory and recently encountered a proposition that discussed the stability of coproducts under pullback. And I found the page [pullback-stable colimit](https://ncatlab.org/nlab/show/pullback-stable+colimit) in nLab.
On the mentioned nLab page, pullback-stability wasn'... | https://mathoverflow.net/users/508703 | Why isn't pullback-stability defined for individual colimits but for colimits with the same shape? | **Pullback-stability *is* sometimes considered for individual colimits, or at least, smaller classes than “all colimits of shape $D$”.** However, it’s most often used in settings where it holds for large classes of colimits, so authors usually define it just at the level of generality they need. In particular, the nlab... | 12 | https://mathoverflow.net/users/2273 | 450938 | 181,383 |
https://mathoverflow.net/questions/450948 | 10 | Context: I've just started reading Tate's thesis. In it, we start with a local field $k.$ The aim of the section is to describe the structure of the character groups of $k^+$ (the additive group) and $k^\*$(the multiplicative group). But for some reason when looking at the character group for $k^+$, we are looking only... | https://mathoverflow.net/users/508721 | Why are we defining character groups differently for additive and multiplicative group in Tate's thesis? | I think the basic reason for the apparently different definitions boils down to the different topologies of $k^{\*}$ versus $k^{+}$.
For simplicity of discussion, let's consider the case that $k= \mathbb{Q}\_p$. If $\psi: \mathbb{Q}\_p^{+}$ is a continuous additive character, then it must have image in the unit circl... | 19 | https://mathoverflow.net/users/2627 | 450953 | 181,391 |
https://mathoverflow.net/questions/450966 | 4 | Fix natural numbers $d,N$ and a polynomial $\Delta \in \mathbb{C}[x\_1,\ldots,x\_d]$. Let $S\_{d,N}$ be the set of field extensions $K/ \mathbb{C}(x\_1,\ldots,x\_d)$ such that
1. The degree $[K: \mathbb{C}(x\_1,\ldots,x\_d)]$ is bounded by $N$.
2. The discriminant of $K/ \mathbb{C}(x\_1,\ldots,x\_d)$ is $\Delta$.
3. ... | https://mathoverflow.net/users/4690 | Finiteness of number of extensions with bounded degree and discriminant | Yes, and I don't think you need 3. Let $D\subset \mathbb{C}^d$ be the locus where $\Delta $ vanishes. You are looking at étale covers of a fixed degree $n\ (\leq N)$ of $\mathbb{C}^d\smallsetminus D$, up to birational isomorphism. Such covers are classified by an action of $\pi \_1(\mathbb{C}^d\smallsetminus D)$ on a s... | 8 | https://mathoverflow.net/users/40297 | 450972 | 181,395 |
https://mathoverflow.net/questions/450974 | 13 | Let $A$ be a $N \times N$ symmetric positive semi-definite matrix with $N \geq 2$. Let $D$ be a diagonal matrix of dimension $N$. I would like to measure how much $A$ "is far" from $D$, i.e. I am trying to find a way to quantify how $A$ differs from a diagonal matrix. More generally, the aim is to come up with a measur... | https://mathoverflow.net/users/508747 | Measuring the "distance" of a matrix from a diagonal matrix | **Q:** *Is there a measure that captures the overall "level of orthogonality" of two matrices $A$ and $B$.*
You can collect the $N^2$ elements of $A$ and $B$ into a pair of vectors $a$, $b$, and then take the inner product $(a,b)=\sum\_{ij}\bar{A}\_{ij}B\_{ij}={\rm tr}\,A^\ast B$. This would generalize the usual meas... | 15 | https://mathoverflow.net/users/11260 | 450976 | 181,396 |
https://mathoverflow.net/questions/445334 | 5 | **Background on heights**
Consider $P = [a: b] \in \mathbb{P}^1(\mathbb{Q})$, where $a,b$ are coprime integers. We define the naive (multiplicative) height as
$$H(P) = \max \{|a|, |b|\}$$
We can change the coordinates of $\mathbb{P}^1$, which would induce a different height function. For example, consider the autom... | https://mathoverflow.net/users/479911 | How should multiplicative height on projective space interact with automorphisms? | The naive height is not at all intrinsic. It is just a convenient choice to work with for notational simplicity. If one is doing things properly one should be counting rational points of bounded height with respect to every choice of adelic metric on the line bundle $\mathcal{O}(1)$. For different choices of adelic met... | 1 | https://mathoverflow.net/users/5101 | 450993 | 181,399 |
https://mathoverflow.net/questions/450973 | 6 | This is probably a $y=f(x)$ question, but I searched several times on the MathOverflow without success so I decided to explicitly ask for the help of other members: please feel free to ask me to remove the question if inappropriate.
Nearly a month or so, in a MathOverflow answer (I guess so, but at this point I shou... | https://mathoverflow.net/users/113756 | On a fast high precision numerical analysis C library | Since you speak about mathematical proofs, probably you don't want an arbitrary-precision library, but a verified computation library based on interval arithmetic.
Maybe [Arb](https://arblib.org/)? Or [boost-interval](https://www.boost.org/doc/libs/1_82_0/libs/numeric/interval/doc/interval.htm)?
And maybe the post ... | 13 | https://mathoverflow.net/users/1898 | 450995 | 181,400 |
https://mathoverflow.net/questions/450992 | 0 | I am a bit puzzled about some probable implication in a paper.
Let $\varphi:\mathbb{P}^1\_K\to \mathbb{P}^1\_K$ be a rational map, where $K$ is a number field and let $\alpha\in K$ be such that $\varphi(\alpha)\neq\infty$. Then is it true that $\alpha$ is an exceptional point (backward orbit of $\alpha$ under $\varp... | https://mathoverflow.net/users/481562 | A question about the backward orbit and forward orbit of a rational map in 1-dimensional projective space | Let $\alpha\_0=\alpha$, and for each $n\ge1$, choose a point $$\alpha\_n\in\phi^{-1}(\alpha\_{n-1})\subset \mathbb P^1(\overline K).$$ (The map $\phi$ is surjective on $\mathbb P^1(\overline K)$.) The points $\alpha\_n$ are in the backward orbit of $\alpha$, which you have assumed is finite. Hence I can find some $m>n\... | 3 | https://mathoverflow.net/users/11926 | 450998 | 181,401 |
https://mathoverflow.net/questions/451007 | 8 | Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$, and let $c:[\omega]^\omega\to\{0,1\}$ be a function. We say that $a\in [\omega]^\omega$ is **monochromatic** with respect to $c$ if the restriction $c|\_{{\cal P}(a)\cap [\omega]^\omega}$ of $c$ to the infinite subsets of $a$ is constant. More... | https://mathoverflow.net/users/8628 | Non-Ramsey functions $c:[\omega]^\omega\to\{0,1\}$ and the Axiom of Choice | The keyword for searching is "infinite exponent partition relation."
The following article, for example, has many interesting results concerning the infinite exponent partition relation $\omega\to(\omega)^\omega$, which asserts that every coloring function has your Ramsey property, and its connection with weak forms ... | 8 | https://mathoverflow.net/users/1946 | 451019 | 181,408 |
https://mathoverflow.net/questions/450997 | 1 | Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations.
Is the action of $G$ always amenable?
(Amenable action, in the sense of Zimmer, which is equivalent to the action being orbit-equivalent to some action of the integers on $(... | https://mathoverflow.net/users/173694 | Nonamenable p.m.p. action on a standard probability space | A measure preserving action of a non-amenable group on a finite measure space is always **non-amenable**, if this is what you meant.
It is pretty obvious if one uses the most natural definition of an amenable action in terms of asymptotically equivariant families of probability measures. This definition (due to Renau... | 2 | https://mathoverflow.net/users/8588 | 451021 | 181,409 |
https://mathoverflow.net/questions/451018 | 1 | Let $\Theta(x,t)$ be the Jacobi-Theta function:
\begin{equation}
\Theta(x,t):=1+\sum\_{n=1}^\infty e^{-\pi n^2 t} \cos(2\pi n x)
\end{equation}
Usually, the heat equation with the periodic boundary conditions is solved by the "spatial" convolution with $\Theta(x,t)$. However, I wonder what would happen if we perform ... | https://mathoverflow.net/users/56524 | Convolution with the Jacobi Theta-function on "both the space and time variables" - still jointly smooth? | $\newcommand\Th\Theta\newcommand{\Z}{\mathbb Z}$The answer here is no.
E.g., for $(x,t)\in[0,1]\times[0,\infty)$ let
\begin{equation}
f(x,t):=2x\,1(x<1/2)+(2-2x)\,1(x\ge1/2).
\end{equation}
Then
\begin{equation}
(\Th\*f)(x,t)=\frac t4-\frac2{\pi^3}\sum\_{n=1}^\infty a\_n\cos2n\pi x
=\frac t4-\frac{c\_t}{\pi^3}\,g... | 2 | https://mathoverflow.net/users/36721 | 451024 | 181,411 |
https://mathoverflow.net/questions/450488 | 2 | Moufang identities
$$x(y⋅xz)=(xy⋅x)z,$$
$$(zx⋅y)x=z(x⋅yx),$$
$$xy⋅zx=x(yz⋅x)$$
are identities deeply related with alternativity (since setting $z=1$ one recovers left and right alternativity), while a Moufang plane is a strictly geometrical concept that encodes the fact that the group of automorphisms that fixes every ... | https://mathoverflow.net/users/83165 | Moufang identities and Moufang plane | This is discussed and proved in detail in *Hall, Marshall jun.*, The theory of groups, New York: The Macmillan Company. xiii, 434 p. (1959). [ZBL0084.02202](https://zbmath.org/?q=an:0084.02202), specifically in chapter 20 "Group Theory and Projective Planes", and in there in section 20.5 "Moufang and Desarguesian Plane... | 4 | https://mathoverflow.net/users/8338 | 451025 | 181,412 |
https://mathoverflow.net/questions/450990 | 7 | Let $f:\mathbb{C} \to \mathbb{D}$ be a functor of 2-categories and let $\operatorname{Fun}(\mathbb{C},\operatorname{Cat})^{\operatorname{lax}}$ denote the 2-category of functors and lax natural transformations.
Is it true in general that restriction along $f$,
$f^\*:\operatorname{Fun}(\mathbb{D},\operatorname{Cat})... | https://mathoverflow.net/users/117760 | Preservation of lax limits in categories of functors and lax natural transformations | I believe this functor preserves all *oplax* limits, and more generally all "$l$-rigged limits" in the sense of [Enhanced 2-categories and limits for lax morphisms](https://arxiv.org/abs/1104.2111) by Steve Lack and myself. (I don't think it's reasonable to ask it to preserve *lax* limits, since these 2-categories don'... | 6 | https://mathoverflow.net/users/49 | 451034 | 181,416 |
https://mathoverflow.net/questions/451049 | 1 | Let $B$ be a standard Brownian motion. Its characteristic exponent (or Fourier transform) is easily calculated to be
$$ \mathbb E [e^{ixB\_t}] = e^{-\frac 12 x^2 t}. $$
Now I want to apply a Girsanov transformation with density $\exp(\int\_0^t A\_s d B\_s - \int\_0^t A\_s^2 ds)$ for some predictable process $A\_s$ sati... | https://mathoverflow.net/users/166168 | Characteristic exponent after Girsanov transformation | By Doob-Dynkin we have that $A\_t=f(t,\{B\_s\}\_{0\leq s\leq t})$.
If $\mu\_0$ is the law of Brownian motion then under the measure $\mu=\exp\left(\int\_0^T A\_sdB\_s-1/2\int\_0^T A\_s^2 ds\right)\mu\_0$ we have that the law of $B$ is the law of the solution to the (possibly path dependent) SDE $dX\_t=f(t, \{X\_s\}\_... | 4 | https://mathoverflow.net/users/479223 | 451056 | 181,419 |
https://mathoverflow.net/questions/451039 | 2 | This posting is related to the [answer](https://mathoverflow.net/a/450985/95347) to this question.
Lets extend the language of $\sf PA$ with a monadic symbol "$\vdash$", add to the formula formation rules, the rule:
* if $(\phi)$ is a sentence in the language of $\sf PA$, then $(\vdash \phi)$ is a formula.
Now ad... | https://mathoverflow.net/users/95347 | Do these two provability theories over PA differ in consistency strength? | No, both of these theories are equiconsistent with PA.
The reason is that given any model of PA, we may expand it to a model in the language with $\vdash$ by interpreting $\vdash\psi$ as true in the model exactly when PA proves $\psi$. That is, we interpret $\vdash\psi$ inside the model as being the same as provabili... | 6 | https://mathoverflow.net/users/1946 | 451057 | 181,420 |
https://mathoverflow.net/questions/451055 | 0 | This problem comes from this commutative algebra problem
>
> Let $R$ be a commutative ring with identity, $I$ is a finite generated
> ideal of $R$ such that $I^2=I$, then exists $e\in R$ such that $I=Re$.
>
>
>
This problem can be proved using the Nakayama lemma, but the condition for finite generated is neces... | https://mathoverflow.net/users/476484 | Can every idempotent ideal be generated by an idempotent? | I have a counterexample from analysis. Let $R= C[0,1]$, the ring of real-valued continuous functions on the unit interval, and let $$I = \{ f \in C[0,1] : f(0) = 0 \}.$$
Then $I$ cannot be generated by any idempotent $p$, for if it were $p$ would have to satisfy $p(0) =0$ and $p(x) = 1 \ (x \neq 0)$, contradicting cont... | 4 | https://mathoverflow.net/users/507821 | 451066 | 181,423 |
https://mathoverflow.net/questions/451069 | 1 | Let $G(t, x) := \frac{1}{\sqrt{4 \pi t}} \exp\left( -\frac{x^2}{4 t }\right)$ for all $(t, x) \in (0, T) \times \mathbb{R}$ be the fundamental solution to the heat equation $\partial\_tu = \partial\_{xx}u$.
Evans PDE book (not verbatim but its clear from the context) states that
$$
\lim\_{\delta \downarrow 0} \int\_{... | https://mathoverflow.net/users/500621 | Does convolution with heat kernel converge to pointwise evaluation? | What you want cannot be true. Elements of $L^2((0,T); L^2(\mathbb{R}))\cap L^\infty((0,T),L^\infty(\mathbb{R}))$ can be discontinuous everywhere. Without doing any "time integration" you cannot get the "pointwise in time" limit.
For an example: let $k$ be a fixed, non-trivial, $C^\infty\_0(\mathbb{R})$ function. And ... | 3 | https://mathoverflow.net/users/3948 | 451074 | 181,427 |
https://mathoverflow.net/questions/451047 | 1 | Consider a family of stochastic processes $dX^h\_t=(g(X^h\_t)+h(s))\,dt+dW\_t$ and a functional $I\_f:h(s) \rightarrow E[f(X\_t^h)] $. I would like to compute the kernel of the derivative of this functional with respect to $h$ using the Girsanov formula or Malliavin calculus. For example, Gisranov's formula gives
\begi... | https://mathoverflow.net/users/508809 | Linear response for SDE | For the original question of computing the kernel of the derivative of this functional with respect to h, I am more leaning on just regular Frechet derivative (if h is some deterministic function) and using the Girsanov formula to study the difference $I\_{f}(h+\epsilon v)-I\_{f}(h)$. For just the exponentials we have
... | 1 | https://mathoverflow.net/users/99863 | 451081 | 181,428 |
https://mathoverflow.net/questions/451075 | 0 | Let us consider the sequence space $c\_0$ with the equivalent norm
$$\Vert x \Vert^2 = \max\_{i\ge1} \vert x^i \vert^2 + \sum\_{i=2}^{\infty} 2^{-i+1} \vert x^i \vert^2 $$
for $x=(x^1,x^2,\ldots)\in c\_0$.
Let us take two sequences $(x\_{2n})$ and $(x\_{2n+1})$ in $c\_0$ such that $x\_{2n} \xrightarrow{w} x$ and $x\_... | https://mathoverflow.net/users/494605 | Renorming on a separable Banach space | The answer is no. E.g., let $x\_{2n}=x:=(\frac12,\frac12,\frac13,\frac14,\ldots)$ and $x\_{2n+1}=y:=(0,\frac12,\frac13,\frac14,\ldots)$ (for all $n$). Then (i) $x\_{2n}=x\to x$ and $x\_{2n+1}=y\to y$ in any sense and (ii) $\|x\_{2n}+x\_{2n+1}\|=2\|x\_{2n}\|=2\|x\_{2n+1}\|$ and hence $I\_n=0$ for all $n$. However, $x\ne... | 0 | https://mathoverflow.net/users/36721 | 451089 | 181,429 |
https://mathoverflow.net/questions/451092 | 7 | Let $FA\_\kappa (\mathbb{P})$ be the claim that for every family $\mathscr{D}$ of dense sets in the poset $\mathbb{P}$ with $\vert \mathscr{D} \vert = \kappa $ there is a filter $G$ such that for all $D \in \mathscr{D} $, we have $G \cap D \neq \emptyset$. So basically what I am asking for is a forcing axiom for a sing... | https://mathoverflow.net/users/495743 | Forcing axiom for a single poset | There are several immediate things to say.
Some instances are outright provable in ZFC:
* $\newcommand\FA{\text{FA}}\FA(\omega,\newcommand{\P}{\mathbb{P}}\P)$ is a theorem for every poset $\P$.
* $\FA(\kappa,\P)$ holds whenever $\P$ is ${<}\kappa$-closed, and indeed, merely ${\prec}\kappa$-strategically closed is s... | 9 | https://mathoverflow.net/users/1946 | 451093 | 181,432 |
https://mathoverflow.net/questions/450951 | 5 | For Grassmannians, the Schubert cells can be indexed by certain Young Tableaux, whose partition determines the dimensions of intersections of the chosen subspace with the standard complete flag. For example, up to change in convention, in $Gr(2,4)$, the subspace $V = \langle e\_2,e\_4\rangle$ may belong to $X\_{(1,0)}$... | https://mathoverflow.net/users/131046 | geometric meaning to pairs of SYT indexing for the basis of cohomology ring of full flag variety | Since, surprisingly, there are still no answers or even comments, let me note that the answer to the last question is well known to be "yes": the Schubert cell containing a flag $(E\_1,\dots,E\_m)$ is indeed determined by the values $\dim(E\_i\cap F\_j)$.
Specifically, if $\sigma$ is the corresponding permutation, th... | 4 | https://mathoverflow.net/users/19864 | 451096 | 181,433 |
https://mathoverflow.net/questions/450980 | 3 | I am having trouble in understanding Theorem 8.2 of [Adams's book](https://press.uchicago.edu/ucp/books/book/chicago/S/bo21302708.html) and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this case will be
$$\varprojlim\_{a}{}^{p}E^{q}(X\_... | https://mathoverflow.net/users/160383 | Spectral sequence in Adam's book, Theorem 8.2 | This is not a complete answer, but it is too long for a comment. I will assume you are interested in the spectral sequences in Section 8 of Adams, not the Atiyah-Hirzebruch spectral sequences of Section 7. There are two different spectral sequences there, depending on whether you start with a CW complex filtered by its... | 6 | https://mathoverflow.net/users/4194 | 451100 | 181,435 |
https://mathoverflow.net/questions/451094 | 8 | Skolem wrote an interesting paper in 1960 about the use of a three-valued logic to create a set theory with true comprehension that solves Russell's paradox. That paper is [A set theory based on a certain 3-valued logic](https://doi.org/10.7146/math.scand.a-10600).
He basically uses what most today would call the str... | https://mathoverflow.net/users/24611 | Using three-valued logic for set theory with singletons and equality | One way to adopt three-valued truth in set theory is with paraconsistent set theory, which typically uses Kleene three-valued logic, but with the understanding that truth value 1/2 is regarded as "both true and false," and so in particular, both 1/2 and 1 are designated as instances of truth (and this logic is then cal... | 9 | https://mathoverflow.net/users/1946 | 451103 | 181,436 |
https://mathoverflow.net/questions/451085 | 4 | Suppose I work over $\mathbb{C}$. Is it known for which locally convex topological vector spaces $V$, we have $\text{Ext}^i(V, \mathbb{C})=\{0\}$ for all $i>0$, working with the type of Ext groups that Wengenroth uses. For instance, is it true when $V$ is Frechet and nuclear?
| https://mathoverflow.net/users/3396 | Ext groups of locally convex topological vector spaces | The Hahn-Banach theorem implies that $\mathbb C$ is an injective object in the category of complex locally convex topological vector spaces. Therefore, Ext$^i(V,\mathbb C)=0$ for every locally convex space $V$.
| 8 | https://mathoverflow.net/users/21051 | 451122 | 181,442 |
https://mathoverflow.net/questions/451123 | 0 | This question seem a bit elementary, but I find it more subtle than its looks. So, I post the question here.
Let $f(x,t) : [0,2\pi] \times [0,1] \to \mathbb{C}$ be a function such that $f(0,t)=f(2\pi,t)$ for all $t \in [0,1]$. Also, further suppose that $f(x,t)$ is $C^\infty$ in $x$ for each $t \in [0,1]$ and jointly... | https://mathoverflow.net/users/56524 | If $f(x,t)=\sum_{n \in \mathbb{Z}} a_n(t) e^{in x}$ is $C^\infty$ in $x$ and all $a_n(t)$ continuous, $x$ derivatives of $f$ are continuous in $t$? | I am going to intepret the question as: given an $f$ satisfying the conditions in the question, must $\partial\_x f$ be continuous as a function on $[0,2\pi]\times [0,1]$?
@Echo's suggestion seems to give a concise, explicit counterexample. However, I thought it might be worth spelling out a line of thought that can ... | 5 | https://mathoverflow.net/users/763 | 451136 | 181,446 |
https://mathoverflow.net/questions/451139 | 5 | I was looking for a complex extension of the surreals and then I found $\operatorname{No}[i]$, what are its properties and how do I express $x+yi$ in the $a | b$ notation?
| https://mathoverflow.net/users/508896 | What are the properties of $\operatorname{No}[i]$? | The surreal complex field $\text{No}[i]$, known as the [surcomplex field](https://en.wikipedia.org/wiki/Surreal_number#Surcomplex_numbers), is a proper-class-sized set-saturated algebraically closed field. It is universal for all fields of characteristic 0. Indeed, under global choice, every class field structure in ch... | 5 | https://mathoverflow.net/users/1946 | 451143 | 181,449 |
https://mathoverflow.net/questions/451149 | 5 | The axiom $W\_\kappa$, for $\kappa$ a cardinal, is the statement that for all sets $X$, either $|X|\leq\kappa$ (that is, there is an injection $X\to\kappa$) or $\kappa\leq|X|$. Is there literature on the dual notion $W^\*\_\kappa$ that for all $X$, $|X|\leq^\*\kappa$ (that is, there is a surjection $\kappa\to X$, or $X... | https://mathoverflow.net/users/478588 | References for the axiom of surjective comparability | Not too much that I'm aware of. It comes up in the case of $\omega$, since that would imply that even if Dedekind finite sets exist, they can at least be mapped onto $\omega$ (equivalently, "the power set of any infinite set is Dedekind infinite").
In the general case, this isn't as helpful, and there's not too much ... | 5 | https://mathoverflow.net/users/7206 | 451150 | 181,453 |
https://mathoverflow.net/questions/451110 | 1 | I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question).
As an example consider the Laplacian operator. The inverse of the Laplacian is given by
$$(-\Delta)^{-1} u(x) = C \int\_{\mathbb{R}^n} u(x-... | https://mathoverflow.net/users/498931 | Reference request: inverse of differential operators | There are many possible answers to this question, depending on the type of the differential operator, the domain of the functions, and whether you want to impose any additional conditions such as specifying boundary data.
For example, if you assume that the differential operator has constant coefficients, then the [M... | 3 | https://mathoverflow.net/users/613 | 451157 | 181,455 |
https://mathoverflow.net/questions/451014 | 5 | Let $F:\mathbf{\Delta}\to\mathcal{S}\_{\leqslant n-1}$ be a cosimplicial object in the $\infty$-category of $(n-1)$-truncated spaces. Is it always a right Kan extension of its restriction along $\mathbf{\Delta}^{\leqslant n}\hookrightarrow\mathbf{\Delta}$? (Then one might call $F$ to be $n$-coskeletal.)
I want to use... | https://mathoverflow.net/users/42571 | Connectedness of truncated version of cosimplicial indexing category | I don't believe the first claim is true, but I can give a somewhat formal argument for the connectivity of $\Delta\_{\le n} \times\_{\Delta} \Delta\_{/m}$.
Let us write $u$ for the inclusion $\Delta\_{\le n} \to \Delta$. The crucial result follows the following special case of Appendix A of <https://arxiv.org/abs/2207.... | 2 | https://mathoverflow.net/users/76636 | 451158 | 181,456 |
https://mathoverflow.net/questions/451153 | 3 | $\quad$Let $\mathcal{Sch}/\mathbb C$ denote the category of schemes over $\mathbb C$. For an arbitrary $X\in\mathcal{Ob}(\mathcal{Sch}/\mathbb C)$, Deligne in his [Article](http://publications.ias.edu/sites/default/files/Number19.pdf) defined a polarizable Hodge complex by replacing $X$ by a smooth simplicial scheme $\... | https://mathoverflow.net/users/508433 | Geometric Interpretation of absolute Hodge cohomology | In fairness, some of the credit for the general construction of absolute cohomology should go to Beilinson (see his *Notes on Absolute cohomology*).
I think you'll find the homology version, that you are asking about, discussed in Janssen's article, *Deligne homology, Hodge D-conjecture, and motives*.
| 4 | https://mathoverflow.net/users/4144 | 451159 | 181,457 |
https://mathoverflow.net/questions/451161 | 4 | Let $j: K\to K′$ be a categorical equivalence of simplicial sets. By [HTT, Remark 2.1.4.11], we have a Quillen equivalence (with the covariant model structures)
$$
j\_!:\mathsf{sSet}\_{/K}\rightleftarrows\mathsf{sSet}\_{/K'}:j^\*.
$$
On the other hand, we have the functor $\mathbf{\Delta}\to\mathsf{sSet}, [n]\mapsto\De... | https://mathoverflow.net/users/42571 | Categorical equivalences vs. categories of simplices | No it is not. But there is a replacement for this. There is another model of $\infty$-categories: marked simplicial sets. If $X$ is a simplicial set with a set of marked $1$-simplices $S$, the fibrant replacement of $(X,S)$ is a model of the localization by $S$ of the $\infty$-category corresponding to $X$ (as a mere s... | 13 | https://mathoverflow.net/users/1017 | 451164 | 181,459 |
https://mathoverflow.net/questions/451079 | 3 | **Background:** If $\mathcal{H}$ is a Hilbert space and $U:\mathcal{H}\rightarrow\mathcal{H}$ is a unitary operator, then for each $f \in \mathcal{H}$ the sequence $(\langle U^nf,f\rangle)\_{n = 1}^\infty$ is positive definite, so it can be viewed as the Fourier coefficients of some measure $\mu\_f$ on the torus $\math... | https://mathoverflow.net/users/508836 | Spectral disjointness of unitary representations of Type I groups and orthogonality | A friend provided an answer to my question in a private communication. I have added some details and will present it below in my own words.
**Additional notation:** Below I use $\mu\_{\pi}$ to refer to the measure $\mu$ from my initial post with respect to a particular representation $\pi$. I also use $[\mu]$ to deno... | 0 | https://mathoverflow.net/users/508836 | 451172 | 181,460 |
https://mathoverflow.net/questions/451004 | 5 | Let $a>0$ and consider the operator
$$Tf(t,x)= \int\_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$
When $a=2$, the function $Tf$ solves the Cauchy problem associated with the homogeneous Schrödinger equation:
$$i \, \partial\_t u+i \, \Delta u=0,\qquad u(0)=f.$$
A classic... | https://mathoverflow.net/users/116555 | Maximal operator estimates for the Schrödinger equation | $\newcommand\dotcup{\mathbin{\dot\cup}}$I shared your confusion until a year ago, when Markus Haase explained to me that \eqref{2} can actually be rewritten as assertion (ii) in the following theorem, which I find much clearer.
**Theorem.** Let E be a Banach space, let $(\Omega,\mu)$ be a measure space, let $p \in [1... | 3 | https://mathoverflow.net/users/102946 | 451174 | 181,461 |
https://mathoverflow.net/questions/450895 | 2 | Suppose $X\_1,\ldots,X\_n\sim \text{i.i.d.} \operatorname N(\mu,\sigma^2).$ Consider these functions of the foregoing random variables:
* Any weighted average of $X\_1,\ldots,X\_n,$ with constant (i.e. non-random) weights not depending on $\mu$ or $\sigma^2.$
* $\operatorname{median}\{X\_1,\ldots,X\_n\}.$ (If $n$ is ... | https://mathoverflow.net/users/6316 | Classification of all unbiased estimators of the mean of a normal population | $\newcommand{\R}{\mathbb R}\newcommand{\si}{\sigma} $At least in your setting, statistics (also called estimators) are random variables of the form $T(X)$, where $T$ is a Borel-measurable function from $\R^n$ to $\R$ and $X:=(X\_1,\ldots,X\_n)$.
An estimator $T(X)$ of $\mu$ is called unbiased (for $\mu$) if
\begin{eq... | 2 | https://mathoverflow.net/users/36721 | 451188 | 181,463 |
https://mathoverflow.net/questions/451186 | 6 | For context, I heard that the Atiyah-Segal formalism of quantum field theory is equivalent to the traditional Wightman approach (at least within the scope of certain theories). However, I could not find a precise proof of this, only physical arguments.
| https://mathoverflow.net/users/164131 | Is Segal's notion of conformal field theory a quantum field theory in the sense of Wightman axioms? | My understanding is that Segal invented his formalism (which was then adapted by Atiyah) by thinking about the same thing Wightman was thinking about: formalising the theory of local operators. In hindsight, the theory Segal was groping towards was essentially the theory of [factorization algebra](https://www.cambridge... | 9 | https://mathoverflow.net/users/78 | 451189 | 181,464 |
https://mathoverflow.net/questions/451086 | 5 | $\DeclareMathOperator\SO{SO}$Recently I came across this old [MSE post](https://math.stackexchange.com/questions/4267430/measuring-distance-in-son) or [this paper (w.o. proof)](https://epubs.siam.org/doi/10.1137/S0895479801383877) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant ... | https://mathoverflow.net/users/491352 | Geodesic distance on $\mathrm{SO}(n)$ | The formula holds with a minus sign on the RHS and only for small enough distances. Here's a short proof: by invariance you may set $A = I\_n$. Then for $B$ close enough to $I\_n$, the unit-speed minimising geodesic from $I\_n$ to $B$ is a one-parameter subgroup $\gamma(s) = \exp(s \Xi)$, for some $\Xi \in \mathfrak{so... | 4 | https://mathoverflow.net/users/14708 | 451196 | 181,466 |
https://mathoverflow.net/questions/451191 | 4 | Question:
Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B\_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive on vertices and also transitive on directed edges?
Context:
We often understand a group by its actions. For a fi... | https://mathoverflow.net/users/387190 | Action of braid groups on regular trees | In the article *A group theoretic criterion for property FA*, Culler and Vogtmann notice that
>
> for $n \geq 5$, the braid group $B\_n$ has property $A \mathbb{R}$,
>
>
>
meaning that every *non-trivial* (i.e. without a global fixed point) action of $B\_n$ on a tree has an invariant bi-infinite geodesic. As a... | 12 | https://mathoverflow.net/users/122026 | 451204 | 181,467 |
https://mathoverflow.net/questions/451201 | 0 | Let $R=k[x\_1,...,x\_n]$ be a polynomial ring over a field. Let $f$ be a homogeneous polynomial and $I=(f\_1,...,f\_m)$ a homogeneous ideal.
With Macaulay2, one can compute the Groebner basis of $I$ when $k=\mathbb{Q}$ or when $k=\mathbb{F}\_p$. This helps one decide whether $f$ is in $I$ or not. This is of course ju... | https://mathoverflow.net/users/103381 | Ideal membership and change of fields | The ideal membership question is completely algorithmic, and indeed you can solve is using the reduced Gröbner basis of your ideal (for any ordering of your choice).
The reduced Gröbner basis of $(f\_1,\ldots,f\_m)$ is computed via the Buchberger algorithm, and it is clear from the way the algorithm works, that everyth... | 2 | https://mathoverflow.net/users/1306 | 451205 | 181,468 |
https://mathoverflow.net/questions/451202 | 4 | Consider the number of integer partitions of $n$, usually denoted by $p(n)$ and generated by
$$\sum\_{n\geq0}p(n)x^n=\prod\_{k\geq1}\frac1{1-x^k}.$$
I have encountered an interesting enumeration.
Take all partitions of $n+2$ which do not use $1$ as a part. Then, count all different integers appearing in every such ... | https://mathoverflow.net/users/66131 | Number of partitions of $n$ and number of different integers in 1-avoiding partitions | Note that $u(n)=|\Theta(n+2)|$ where $\Theta(k)$ is the set of partitions of $k$ without 1's with one part being labelled (i.e., multisets of integers greater than 1 summing up to $k$ with one labelled element, like $\{{\bf 6}\},\{{\bf 4},2\},\{{\bf 2},4\},\{{\bf 3},3\}, \{{\bf 2},2,2\}$ for $k=6$).
Bijection from pa... | 7 | https://mathoverflow.net/users/4312 | 451212 | 181,474 |
https://mathoverflow.net/questions/451220 | 1 | Let $f(t) : [0,\infty) \to \mathbb{R}$ be an $L^1\_\text{loc}$ function.
Then, I wonder if the following series
\begin{equation}
\sum\_{n=1}^\infty e^{-n^2 T} \int\_0^T e^{n^2 t} \lvert f(t)\rvert
\, dt
\end{equation}
converges, where $T \in (0, \infty)$ is fixed.
This seems related to Dawson integral. Could anyon... | https://mathoverflow.net/users/56524 | Does $\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt$ converge for $L^1_\text{loc}$ $f : [0,\infty) \to \mathbb{R}$? | The updated version is indeed correct. I will show that the sum is finite for almost all $T\in [1, 2]$ but it should be applicable for all $T\in (0, \infty)$.
The reason why the result can fail for a particular $T$ is that our function $f$ can resemble too closely $\delta$-function at the point $T$ so that we will be... | 5 | https://mathoverflow.net/users/104330 | 451228 | 181,479 |
https://mathoverflow.net/questions/451219 | 1 | I'm looking for a reference and a proof of the following version (or eventually a more general version) of the Central Limit Theorem for complex random variables.
**Theorem.** *Let $Z\_1, Z\_2, \dots, Z\_n$ be independent identically distributed complex Goodman (see Note 1) random variables with mean $\mu$ and comple... | https://mathoverflow.net/users/503895 | Reference request and clarification for Central Limit Theorem for complex random variables | On the "Furthermore":
The central limit theorem for vectors involves the variance-covariance matrix. Let $Z = X+iY$ be a complex random variable (with mean $0$ for simplicity). If $X,Y$ have variance-covariance matrix:
$$\begin{pmatrix} \mathbf E[X^2] & \mathbf E[XY] \\ \mathbf E[XY] & \mathbf E [Y^2] \end{pmatrix}... | 3 | https://mathoverflow.net/users/18060 | 451229 | 181,480 |
https://mathoverflow.net/questions/451242 | 7 | A try to capture the informal notion of "this sentence is not provable in less than $n$-many steps" where $n$ is a concrete natural.
Use the diagonal lemma to coin the following sentences, per each concrete natural $n$:
$\zeta\_n \iff \ulcorner \zeta\_n \urcorner \text { is not provable in }< S\_n(0)\text{-many ste... | https://mathoverflow.net/users/95347 | Can this provide an example of incompleteness under the assumption of mere consistency? | No, quite the opposite in fact. Namely, if we have PA as the background theory, or some other minimal theory of arithmetic, and this theory is consistent, then all the sentences $\zeta\_n$ are true!
Let me assume that each $\zeta\_n$ asserts of itself that it has no proof in PA (or some other fixed sufficient backgro... | 11 | https://mathoverflow.net/users/1946 | 451243 | 181,485 |
https://mathoverflow.net/questions/451252 | 18 | Let $G = \langle A \mid R \rangle$ be a finitely presented group, given by a finite presentation. If $G$ is abelian, then we can verify this fact: simply verify the fact that $[a, b] = 1$ for all generators $a, b \in A$ (this can be done by enumerating all words equal to $1$ in the group). That is, the property "being ... | https://mathoverflow.net/users/120914 | Is solvability semi-decidable? | The Adian–Rabin theorem, although it is not usually stated like this, says that Markov properties are not co-semi-decidable, thus « not being metabelian » or « not being solvable » are not semi-decidable.
However, whether « being solvable » or « being solvable of derived length $\le n$ » is semi-decidable is an open pr... | 17 | https://mathoverflow.net/users/153080 | 451265 | 181,491 |
https://mathoverflow.net/questions/451260 | 2 | Let $\omega$ denote the first infinite ordinal (also known as the set of natural numbers). We call a set $E\subseteq {\cal P}(\omega)$ a *deck* if for all $n\in \omega$, the set $E$ contains exactly one member of cardinality $n$. (More formally, $E$ is a deck if for all $n\in \omega$ we have $|\{e\in E: |e| = n\}| = 1$... | https://mathoverflow.net/users/8628 | Possible chromatic numbers of a hypergraph on $\omega$ with a deck of edges | The answer is plainly negative for $k=1$ and positive for $k=2$. The answer is negative for $k\gt2$ because $(\omega,E)$ is $2$-colorable whenever the edge-set $E$ is a "deck," i.e., contains just one edge of size $n$ for each integer $n\ge2$. To see this consider a random $2$-coloring of $\omega$; the probability that... | 3 | https://mathoverflow.net/users/43266 | 451276 | 181,494 |
https://mathoverflow.net/questions/451287 | 3 | Let $\mathcal g\_1$ denote the usual Godel sentence defined as: $$ \mathcal g\_1 \iff \neg\exists x:\operatorname {Proof}\_T(x, \ulcorner \mathcal g\_1 \urcorner)$$
Lets suppose that $\sf T$ is consistent (metatheoretically), effectively generated, extends $\sf PA $, and complete.
Accordingly, $T \vdash \neg \mathc... | https://mathoverflow.net/users/95347 | Can we use remote provability to prove the first incompleteness theorem sans $\omega$-consistency? | It doesn't contradict PA to have such a descending sequence. For example, you could instead simply keep substracting one from $g^{\*1}$, and this would be provably descending, but there is no contradiction in that — it doesn't mean somehow that at some stage, substracting one gives a larger number.
The point is that ... | 9 | https://mathoverflow.net/users/1946 | 451293 | 181,499 |
https://mathoverflow.net/questions/451297 | 1 | Suppose $X$ is a singular quasi-projective curve over the complex numbers, and $X'$ is a good nonsingular compactification of a resolution of singularities $Y\to X$. Let $D$ be the complement of $Y$ in $X'$.
>
> Can one compute $H^\*(X,\mathbf{Q})$ in terms of $H^\*(X',\mathbf{Q})$ and $H^\*(D,\mathbf{Q})$ and perh... | https://mathoverflow.net/users/nan | Cohomology of singular curves | You don't need all this machinery in this case (unless your goal was to understand the machinery). You have to exact sequences
$$0\to W\_0\to H^1(X) \to H^1(Y)\to 0$$
$$0 \to H^1(X')\to H^1(Y)\to \oplus\_{p\in D} \mathbb{Q}(-1)\to H^2(Y)\to 0$$
The last sequence (which is Gysin) can be shortened (non canonically) to
$$... | 3 | https://mathoverflow.net/users/4144 | 451300 | 181,500 |
https://mathoverflow.net/questions/451302 | 2 | Let $\mathcal{R}$ be a $1$-category. Assume that one has a pushout of representable $1$-presheaves $\mathrm{y} A \cup\_{\mathrm{y} B} \mathrm{y} C$ in $\mathsf{PSh}(\mathcal{R})$. Under which conditions does then the inclusion $\mathsf{PSh}(\mathcal{R}) \to \mathcal{P}(\mathcal{R})$ into $\infty$-preheaves preserve thi... | https://mathoverflow.net/users/145805 | Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts? | The inclusion from discrete presheaves to spaces doesn't preserve all pushouts, even when $\mathcal R$ is a point.
But it does preserves some pushouts. The usual condition is to check that at least one of $yB \to yA$ or $yB \to yC$ is a levelwise monomorphism of sets. Then because the Kan-Quillen model structure is l... | 6 | https://mathoverflow.net/users/2362 | 451307 | 181,503 |
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