parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/408437 | 9 | Consider the following statement (in $\mathsf{ZF}+\text{AC}\_\omega (\mathbb{R})$):
>
> There exists $(\varphi\_\alpha)\_{\alpha\in\omega\_1}$ with $\varphi\_\alpha : \alpha \rightarrow \mathbb{N}^\mathbb{N}$ injective and $\text{ran}(\varphi\_\alpha)$ is closed and $\text{rank}\_{CB}(\text{ran}(\varphi\_\alpha)) =... | https://mathoverflow.net/users/141146 | How much choice is necessary to prove this statement? | Your statement is equivalent to the assertion that there is a function choosing an enumeration of every countable ordinal. From an enumeration of $\alpha,$ you can easily inject it into a countable set of isolated points in Baire space. For the other direction, if $\varphi\_{\alpha}$ injects $\alpha$ into Baire space, ... | 11 | https://mathoverflow.net/users/109573 | 408442 | 167,257 |
https://mathoverflow.net/questions/406858 | 0 | I need to solve the following equation for the matrix $P \in\mathbb{R}^{r\times d}:$
$$
((PAP^\top)^{-1} P S P^\top (PAP^\top)^{-1})^2 = I\_r,
$$
where $S$ is a symmetric $d\times d$ matrix, $A$ is a PSD $d\times d$ matrix, and $I\_r$ is the identity matrix of dimension $r$.
Is there any easy way to solve this equa... | https://mathoverflow.net/users/156139 | Product of matrices equal identity | $\DeclareMathOperator\diag{diag}\DeclareMathOperator\rank{rank}$Here is how we can construct solutions in $P$.
Necessarily, $r\leq d$, $\rank(P)=r$, $\rank(S)=r\_1\geq r$.
There is $Q$ invertible s.t. $QAQ^T=I\_d$, $QSQ^T=\diag((\lambda\_i)\_{i\leq p\_1},(-\mu\_j)\_{j\leq q\_1},0\_{s})$, where $\lambda\_i,\mu\_j>0$... | 0 | https://mathoverflow.net/users/9091 | 408446 | 167,258 |
https://mathoverflow.net/questions/408458 | 9 | Recall that if we have a metric space $X$ then we can consider the set of its nonempty compact subsets and equip this with a metric called the Hausdorff distance. Denote the resulting metric space $\mathcal{H}(X)$. We of course get an inclusion $X \hookrightarrow \mathcal{H}(X)$. Are any of the homotopy-theoretic prope... | https://mathoverflow.net/users/461492 | Homotopy type of the Hausdorff metric | In J. Andres, M. Väth, *Calculation of Lefschetz and Nielsen Numbers in Hyperspaces for Fractals and Dynamical Systems*, Proc. Amer. Math. Soc. **135** (2007), 479-487, it was shown (esssentially, the result was already implicitly shown in D.W. Curtis, *Hyperspaces of noncompact metric spaces*, Compositio Math **40** (... | 8 | https://mathoverflow.net/users/165275 | 408464 | 167,263 |
https://mathoverflow.net/questions/408406 | 5 | Consider the following well-known statement:
>
> Let $C$ be a small category, $E$ a cocomplete category, and $F\colon C\to E$ a functor. Then there is a (up to isomorphism) unique cocontinuous functor $F'\colon \mathbf{Set}^{C^\mathrm{op}}\to E$ such that $F'\circ y \cong F$, where $y\colon C\to \mathbf{Set}^{C^\ma... | https://mathoverflow.net/users/442261 | Size issue in exhibiting the free cocompletion as a left adjoint | As Denis-Charles says in the comments, the best way to handle this is to replace the presheaf category $\mathbf{Set}^{C^{\mathrm{op}}}$ by the full subcategory $\hat{C}$ of small presheaves. By definition, a presheaf is **small** if it satisfies any of these equivalent conditions:
* it is a small colimit of represent... | 14 | https://mathoverflow.net/users/586 | 408476 | 167,265 |
https://mathoverflow.net/questions/408473 | 7 | Let $f: A \to B\ $ be an abelian group homomorphism. Are there abelian groups $G,\ H,\ K$ such that $K \subseteq H \subseteq G$ and the map
$$\pi \circ i: H \to G/K$$
which is the composition of projection and inclusion is isomorphic to $f$? By isomorphic to $f\ $I mean there exist isomorphisms $\tau: H \to A\ $ and $\... | https://mathoverflow.net/users/166613 | Existence of abelian group extension relative to group homomorphism | The factorization always exists, but in general is not unique.
Let me identify $A$ with $H$. Clearly, $K$ can be identified with $\ker(f)$. Let $f(A)$ be the image of $A$ in $B$. Then $A$ is an abelian extension of $f(A)$ by $K$. If I understand correctly, you are asking whether given an inclusion $f(A)\hookrightarro... | 9 | https://mathoverflow.net/users/6668 | 408487 | 167,271 |
https://mathoverflow.net/questions/408495 | 3 | Let $\mathcal{F}$ be a coherent sheaf on a projective manifold $X$. It is well known that one can construct a resolution of $\mathcal{F}$ by holomorphic vector bundles (locally free sheaves).
Are two such resolutions homotopic? Any reference would be much appreciated.
| https://mathoverflow.net/users/102114 | When are two resolutions of a coherent sheaf homotopic | If this were true, then any short exact sequence of vector bundles would split. Indeed, if $0 \to \mathscr E\_1 \to \mathscr E\_2 \to \mathscr E\_3 \to 0$ is a short exact sequence of vector bundles, then both
\begin{align\*}
K^\bullet = \cdots \to 0 \to \mathscr E\_1 \to \mathscr E\_2 \to 0 \to \cdots
\end{align\*}
an... | 7 | https://mathoverflow.net/users/82179 | 408501 | 167,276 |
https://mathoverflow.net/questions/408441 | 1 | $\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=1}{Y=2} & \ddots & & \vdots \\
\vdots & & \ddots & \vdots \\
\cPr{X=1}{Y=J} &\cdots & \cdots & \cPr{X=J}{Y=J}\end{bmatrix},
$$
w... | https://mathoverflow.net/users/91969 | Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible | Indeed, the condition $\det M\ne0$ can be expressed as a certain non-independence condition, as follows.
For $i$ and $j$ in $[J]:=\{1,\dots,J\}$, let
\begin{equation\*}
p\_{i|j}:=P(X=i|Y=j),
\end{equation\*}
so that $M=[p\_{i|j}]\_{i,j\in[J]}$.
Suppose that $\det M=0$. Then for some real $c\_1,\dots,c\_J$ not a... | 2 | https://mathoverflow.net/users/36721 | 408503 | 167,277 |
https://mathoverflow.net/questions/408471 | 2 | Let $G$ be a subcubic graph.
Suppose that $G$ has an edge coloring $\varphi$ using colors from $\{1,2,3,4,5\}$ such that
* each edge is colored with a set of two elements from $\{1,2,3,4,5\}$ (e.g., $\varphi(e)=\{1,2\}$ for some edge $e$),
* if $e\_1$ and $e\_2$ are adjacent, then $\lvert\varphi(e\_1)\cap \varphi(e... | https://mathoverflow.net/users/375270 | The equivalence of a kind of 2-fold edge coloring and the 2-distance vertex coloring for subcubic graphs | The answer is **False**. Let $G$ be the [Möbius ladder](https://en.wikipedia.org/wiki/M%C3%B6bius_ladder) on 12 vertices with every edge subdivided, or in SageMath code,
```
>G=Graph('K?AEF@oM?w@o') #Mobius ladder on 12 vertices
>G.subdivide_edges(G.edges(),1)
```
$G$ is a subcubic graph.
```
>max(G.degree())... | 2 | https://mathoverflow.net/users/125498 | 408507 | 167,278 |
https://mathoverflow.net/questions/407964 | 25 | Every undergraduate in mathematics learns about proofs by *mathematical induction*. Moreover, every undergraduate taking a course in theoretical computer science or logic learns about *inductive definitions* (of sets, i.e., types if you will), such as the inductively defined set of first-order formulas for a given sign... | https://mathoverflow.net/users/442261 | Coinduction for all? | This is a question that I've puzzled about myself, and I don't pretend to have The Answer. But here's one thought that I've found illuminating. Let's start by comparing the behavior of induction and coinduction in a programming language that has them both.
Inductive datatypes are static. They store information in a p... | 19 | https://mathoverflow.net/users/49 | 408509 | 167,279 |
https://mathoverflow.net/questions/405658 | 2 | I am trying to prove (or disprove) the following assertion:
Consider a probability triple $(X,\mathcal{B},\mu)$, $X$ separable Banach space (complete), $\mathcal{B}$ the Borel $\sigma-$algebra and $\mu$ a countably additive probability measure there on. Let $Y$ be a different separable Banach space.
Given a Baire 1... | https://mathoverflow.net/users/401135 | Baire 1 function equivalence in measure | Let $F$ be a closed subset of $[0,1]$ (say) with empty interior and $\mu(F)>0$ (where $\mu$ is Lebesgue measure), for example given by a fat Cantor set. Clearly, $1\_F$ is Baire class $1$. I claim that $1\_F$ cannot be almost everywhere equal to an almost everywhere continuous function.
Indeed, assume that $f = 1\_F$... | 3 | https://mathoverflow.net/users/17064 | 408515 | 167,281 |
https://mathoverflow.net/questions/408512 | 7 | Let $a(n)$ be [A214973](https://oeis.org/A214973), number of terms in greedy representation of $n$ using Fibonacci and Lucas numbers.
Let $b(n)$ be [A329320](https://oeis.org/A329320), sequence which arises from attempts to simplify computing of [A329319](https://oeis.org/A329319). Here
$$b(n)=b\left(\left\lfloor\fra... | https://mathoverflow.net/users/231922 | One conjecture by sequencedb.net | Sure. Start with the Zeckendorf representation of $n$, $$ n = \sum\_{ i} F\_{j\_i} $$ where $F\_j$ are the Fibonacci numbers and $j\_i \geq j\_{i-1} + 2$.
Then the fibbinary number associated to $n$ is $\sum\_i 2^{j\_i}$ and $c(n)$ is obtained from this by removing a zero in front of each $1$, i.e. $$c(n) = \sum\_i 2... | 20 | https://mathoverflow.net/users/18060 | 408516 | 167,282 |
https://mathoverflow.net/questions/408520 | 2 | Suppose that $\Omega=[0,1]^2$. I will say that a real valued function $u$ on $\Omega$ satisfies periodic boundary values if
$$u(x,0)=u(x,1), \;u(0,y)=u(1,y),\;\;\;\text{ for all }x,y\in[0,1].$$
Now let $f\in L^2(\Omega)$, and suppose that I want to look for a solution of the following problem
$$ \begin{cases}-\Delta ... | https://mathoverflow.net/users/345667 | Elliptic equation on square with periodic boundary values for the solution and it's partial derivatives | You are using the wrong space, that's all. The correct space is
$$
X=\{u \in H^1\_{\textrm{loc}}(\mathbb R^2): u(\cdot+n)=u(\cdot) \quad \forall n=(n\_1,n\_2)\in \mathbb Z^2\}.
$$
Then you apply Lax-Milgram and you are good to go. Look at
Asymptotic Analysis for Periodic Structures, by Bensoussan, Lions, Papanicolaou (... | 1 | https://mathoverflow.net/users/40120 | 408522 | 167,285 |
https://mathoverflow.net/questions/408524 | 1 | I am considering the minimizing movement scheme related to the gradient of entropy functional in 2-Wasserstein space. The problem is to minimize the following functional for each fixed $\eta$ which is a probability density w.r.t. $(\mathbb{R}^d,Leb)$ with finite second moments: $$\int\rho\log\rho
dx+W\_2^2(\rho,\eta)... | https://mathoverflow.net/users/174600 | How to prove the limit of minimizing sequence of measures is again absolutely continuous(w.r.t. Lebesgue) in the minimizing movement scheme? | $\newcommand{\ep}{\varepsilon}\newcommand\R{\mathbb R}$Yes, the minimizer $\mu$ is absolutely continuous (w.r. to the Lebesgue measure $|\cdot|$).
Indeed, you showed that
\begin{equation\*}
F(\rho\_n)\to m:=\inf\_\rho F(\rho) \tag{-1}
\end{equation\*}
and
\begin{equation\*}
\mu\_{\rho\_n}\to\mu \tag{-0.5}
\end{equa... | 2 | https://mathoverflow.net/users/36721 | 408533 | 167,289 |
https://mathoverflow.net/questions/408508 | 12 | In algebraic topology, relative (co)homology is very useful. For example, we have a long exact sequence which is often helpful for lots of calculations.
In algebraic geometry, we have local cohomology, which is basically the same thing and has the same long exact sequence. However, while the commutative algebra commu... | https://mathoverflow.net/users/131975 | Why doesn't local cohomology seem to be used as much in algebraic geometry? | I don't agree with the premise of this question. Local cohomology (per se and not in the wider context of the six functor formalisms) and its consequences are still very much used in algebraic geometry. If you just glance at SGA2, you will find that the following results are proved using local cohomology:
$\bullet$ L... | 20 | https://mathoverflow.net/users/37214 | 408534 | 167,290 |
https://mathoverflow.net/questions/408555 | 4 | $\DeclareMathOperator\GL{GL}$Let $(\pi,V)$ be an infinite dimensional irreducible admissible representation of $\GL\_2(\mathbb{Q}\_p)$. Let us fix an element $v\_0\in V$ and define a vector space
$$V\_{v\_0} := \left\{\pi\begin{pmatrix}a& \\ &1\end{pmatrix}v\_0,\;\text{s.t.}\;a\in \GL\_1(\mathbb{Q}\_p)\right\}.$$
We ... | https://mathoverflow.net/users/173538 | Schur lemma and Whittaker functions | $\DeclareMathOperator\GL{GL}$Let me try to clarify. The formula for the Whittaker functional in Theorem 4.6.5 of "[Automorphic forms and representations](https://doi.org/10.1017/CBO9780511609572)" of Bump states that
$$W\left(\pi\begin{pmatrix}p^k& \\ &1\end{pmatrix}v\_0\right) = W(v\_0)\frac{\alpha\_1^{k+1}-\alpha\_2^... | 9 | https://mathoverflow.net/users/62154 | 408564 | 167,300 |
https://mathoverflow.net/questions/408545 | 2 | Let $P(z)$ be a complex polynomial of degree $n.$ I am working on the class of polynomials assiociated to $P(z)$ such that their moduli are identical with that of $P(z)$ on the imaginary axis.
For example if $Q(z)$ is a polynomial obtained by the replacement of coefficients of $P(z)$ by their complex conjugates and $... | https://mathoverflow.net/users/128472 | A polynomial having same modulus that of an associated polynomial on the imaginary axis | The set of all polynomials associated with a given one is described
as follows:
Let the given polynomial be
$$P(z)=c(z-z\_1)\ldots(z-z\_n).$$
Then any associated polynomial is of the form
$$Q(z)=\lambda c(z-\sigma\_1(z\_1))\ldots(z-\sigma\_n(z\_n)),$$
where each $\sigma\_j(z)=z$ or $-\overline{z}$, and $|\lambda|=1$.
S... | 7 | https://mathoverflow.net/users/25510 | 408570 | 167,303 |
https://mathoverflow.net/questions/408542 | 3 | Let $\mathcal P$ be the set of probability densities on $[0,1]$ with mean $1/2$, i.e. $p\in \mathcal P$ iff
$$\int\_0^1 p(x)dx=1,\quad \int\_0^1 xp(x)dx=\frac{1}{2}\quad \mbox{and}\quad p(x)\ge 0, ~~\forall x\in [0,1].$$
How to solve the minimization problem below ?
$$\min\_{p\in\mathcal P}~ \left\{V(p) ~:=~ \int... | https://mathoverflow.net/users/nan | Minimization of an entropy type functional | As in the comment by leo monsaingeon, let
$$p\_\*(x):=e^{h(x)}/c,$$
where $h(x):=-x\ln x-(1-x)\ln(1-x)$ and $c:=\int\_0^1 e^{h(x)}\,dx$, so that $p$ is a pdf on $(0,1)$ with mean $1/2$, and
$$V(p)=\int\_0^1(p(x)\ln p(x)-h(x)p(x))\,dx.$$
For any pdf $q$ on $(0,1)$ with $V(q)<\infty$, the directional derivative of $V$ ... | 1 | https://mathoverflow.net/users/36721 | 408571 | 167,304 |
https://mathoverflow.net/questions/408553 | 1 | I happen to have the heat kernel on the two-dimensional hyperbolic space and I need to take partial derivatives in order to check that it satisfies the heat equation as expected. The problem is I can not apply the Leibniz formula because I get zero in the denominator. The function is
$$P\_2(x,t)=\frac{\sqrt{2}e^{-t/4}}... | https://mathoverflow.net/users/411616 | Partial derivative of the heat kernel | A partial integration can remove the singularity:
$$P\_2(x,t)=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int\_x^\infty\frac{se^{-s^2/4t}ds}{\sqrt{\cosh s -\cosh x }}=$$
$$\qquad =\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int\_x^\infty\frac{2\sqrt{s-x}\,s e^{-s^2/4t}}{\sqrt{\cosh s -\cosh x}}\left(\frac{d}{ds}\sqrt{s-x}\righ... | 3 | https://mathoverflow.net/users/11260 | 408574 | 167,305 |
https://mathoverflow.net/questions/408595 | 3 | Given two non-negative Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$, that are finite on compact sets, such that $\nu\ll\mu$, it is well known that
$$\frac{d\nu}{d\mu}(x)= \lim\_{\epsilon\to 0} \frac{\nu(B\_\epsilon(x))}{\mu(B\_\epsilon(x))}$$
holds $\mu$ a.e., where $B\_\epsilon(x)$ is the open ball of radius $\ep... | https://mathoverflow.net/users/131718 | Is the ball ratio theorem for Radon–Nikodým derivative known for general metric spaces? | Arbitrary metric spaces, no.
A classic text discussing such "derivation" is:
*Hayes, C. A.; Pauc, C. Y.*, Derivation and martingales, Ergebnisse der Mathematik und ihrer Grenzgebiete. 49. Berlin-Heidelberg-New York: Springer-Verlag. VII, 203 p. (1970). [ZBL0192.40604](https://zbmath.org/?q=an:0192.40604).
I think... | 3 | https://mathoverflow.net/users/454 | 408606 | 167,313 |
https://mathoverflow.net/questions/401232 | 4 | Boutot's theorem says that if $X$ is a variety over a field of characteristic 0 with rational singularities, and if $G$ is a reductive group acting on $X$, then the quotient $X/G$ has rational singularities as well.
Is it known whether an analog of this result is true for log terminal singularities?
Namely suppose ... | https://mathoverflow.net/users/334560 | Does the quotient of a variety with log terminal singularities also have log terminal singularities? | Now I can give you a definite answer. In general, the quotient of a klt singularity by a reductive group is not klt, because for instance, the canonical divisor of the quotient may not be $\mathbb{Q}$-Cartier.
However, one can define a broader notion: klt type. A singularity $(X;x)$ is said to be of klt type if there... | 3 | https://mathoverflow.net/users/37338 | 408611 | 167,316 |
https://mathoverflow.net/questions/408387 | 3 | $\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}$I am investigating the interplay between freeness criteria and Ext vanishing. A nice example is a vast literature around the Auslander-Reiten conjecture (ARC) (in the loca... | https://mathoverflow.net/users/115603 | Vanishing of $\operatorname{Ext}_R(\operatorname{Tr} M,N)$ and freeness criteria | I am not sure if this is the kind of thing you are interested in, but let me at least state the easiest to prove criteria that I know for free-ness of $M$ in terms of vanishing of certain $\text{Ext}\_R^i(\text{Tr}M, -)$ .
Proposition: If $M$ is a finitely generated module over a Noetherian local ring $R$ such that $... | 2 | https://mathoverflow.net/users/135253 | 408615 | 167,318 |
https://mathoverflow.net/questions/408601 | 40 | As all analytic number theorists know, iterated logarithms ($\log x$, $\log \log x$, $\log \log \log x$, etc.) are prevalent in analytic number theory. One can give countless examples of this phenomenon. My question is, can someone give an intuitive account for why this is so? Specifics regarding any of the famous theo... | https://mathoverflow.net/users/17218 | Iterated logarithms in analytic number theory | There are two main sources of repeated logs. (These sources can be further refined into natural subcategories, but I'll only mention a couple of those subcategories.) Those two main sources are:
**Type 1**: Repeated logs occur because that is just the truth of the matter.
One of my favorite examples is a 2008 theor... | 36 | https://mathoverflow.net/users/3199 | 408620 | 167,320 |
https://mathoverflow.net/questions/408614 | 7 | Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let $\pi: X \to \text{Spec}(R)$ be minimal resolution of singularities with exceptional divisor $E=\pi^{-1}(\mathfrak m)$. ... | https://mathoverflow.net/users/386496 | $2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve | The answer is *yes*, at least over $\mathbb{C}$, since $2$-dimensional (cyclic) quotient singularities are *taut* (*starr*, in German), namely, they are uniquely characterized, up to biholomorphisms, by their resolution graph.
In other words, every $2$-dimensional normal singularity, having the same resolution graph ... | 3 | https://mathoverflow.net/users/7460 | 408634 | 167,326 |
https://mathoverflow.net/questions/408514 | 2 | I'm searching for some software or open source project which is able to prove propositions of predicate logic of first order in the way of natural deduction introduced for example in the book of Lemmon (beginning logic) and can put the results in LaTeX code.
(You can find a short introduction [here](https://plato.sta... | https://mathoverflow.net/users/462601 | Software to prove statements in the way of natural deduction (tabular form introduced by Lemmon) | Here are two tools for teaching natural deduction proofs that were developed in London in the 1990s.
**Jape**
Developed by
[Bernard Suffrin](http://www.cs.ox.ac.uk/bernard.sufrin/personal/), at Worcester College Oxford, and
[Richard Bornat](http://www.eis.mdx.ac.uk/staffpages/r_bornat/), formerly at Queen Mary and ... | 2 | https://mathoverflow.net/users/2733 | 408638 | 167,328 |
https://mathoverflow.net/questions/408623 | 3 | Let $\{\phi(n)\}\_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
1. $\phi(0)=0$ and $|\phi(n)|\leq \frac{C\_1}{|n|}$ for all $n\neq 0$ and $C\_1>0$ is independent of $n.$
2. $|\phi(n+1)-\phi(n)|\leq \frac{C\_2}{n^2}$ for all $n\neq 0$ and $C\_2>0$ is independent of $n.$
3. $\sum\_{-N... | https://mathoverflow.net/users/136860 | Discrete singular integrals | $\newcommand{\Z}{\mathbb{Z}}\newcommand{\ep}{\epsilon}$Let $a\_n:=\phi(n)$. Then
\begin{equation}
K(x)=\sum\_{n\in\Z}a\_n 1(n-1/2\le x<n+1/2).
\end{equation}
So, $K(x)=a\_0=0$ if $1/2\le x<1/2$. So, for $\ep\in(0,1/2)$,
\begin{equation}
I\_\ep:=\int\_{1/\ep<|x|<\ep}K(x)\,dx=\int\_{|x|<\ep}K(x)\,dx
=\sum\_{n\in\Z}a\... | 1 | https://mathoverflow.net/users/36721 | 408650 | 167,332 |
https://mathoverflow.net/questions/408646 | 3 | **Summary:** I have (I think) a generalisation of Hölder's inequality for positive reals that I can neither prove nor find references to. Pointers would be appreciated or, indeed, a counterexample. Thank you.
The original [enquiry at math stackexchange](https://math.stackexchange.com/questions/4305820) met with no re... | https://mathoverflow.net/users/461993 | Array power-means / generalisation of Hölder inequality | This inequality is known. Indeed, [inequality (1.1)](https://projecteuclid.org/journals/annals-of-probability/volume-19/issue-1/Hypercontraction-Methods-in-Moment-Inequalities-for-Series-of-Independent-Random/10.1214/aop/1176990550.full) by Kwapień and Szulga states that
$$\Big(\int\_S\Big(\int\_T|f(s,t)|^q\mu(dt)\Big)... | 3 | https://mathoverflow.net/users/36721 | 408657 | 167,337 |
https://mathoverflow.net/questions/408644 | 5 | $\require{AMScd}$I am currently thinking about [(strict) henselisations](https://stacks.math.columbia.edu/tag/0BSK) but I don't know too much literature about the topic. So I am wondering if there is a natural way to restrict maps between strict henselisations to henselisations:
Let $A$, $B$ be local rings with an in... | https://mathoverflow.net/users/103737 | Restricting maps between strict henselisations | No. Let $A= k[x]\_{(x)}$, the localization of the ring of polynomials in one variable at the maximal ideal $(x)$, and $B = k(x)$.
Assume (for simplicity) that the characteristic of $k$ is not $2$.
Then there exists $y \in A^h$ satisfying $y^2 = 1+x$, as that polynomial splits into distinct linear factors modulo $x$... | 4 | https://mathoverflow.net/users/18060 | 408661 | 167,339 |
https://mathoverflow.net/questions/408666 | 2 | Suppose I am working over a field $\mathbb{F}$ and have $n$ points in the point-value representation $(x\_0,x\_1,\cdots,x\_{n-1})$. What is the fastest way to do polynomial interpolation and convert this to the coefficient form, that is, obtain the coefficients of polynomial $f()$ such that $f(i)=x\_i, i \in \{0,\cdots... | https://mathoverflow.net/users/437976 | Fastest Implementation of polynomial interpolation? | Polynomial interpolation can be done via multiplying a Vandermonde matrix (or its inverse) by your coefficient/evaluation vector --- it is a change of basis on the vector space of polynomials of bounded degree. If this matrix has special structure (such as when the evaluation points are roots of unity --- here the matr... | 3 | https://mathoverflow.net/users/101207 | 408667 | 167,342 |
https://mathoverflow.net/questions/408680 | 2 | I have $A$ an associative algebra and $B$ at least an alternative algebra. Is there a sufficient condition on $A$ or $B$ to have $A \otimes B$ an alternative algebra?
| https://mathoverflow.net/users/83165 | Alternativity on $A \otimes B$ | The monograph “Alternative Loop Rings” by Goodaire, Jespers and Polcino Milies (North Holland Mathematics Studies **184**, 1996) contains, in chapter I (“Alternative Rings”), §5 (“Tensor Products”), the following proposition (5.13; I'm changing the notation to match yours):
>
> Let $B$ be an alternative algebra ove... | 4 | https://mathoverflow.net/users/17064 | 408689 | 167,352 |
https://mathoverflow.net/questions/408702 | 4 | Let $\mathfrak{S}\_\mathbb{N}$ be the symmetric group of all positive integers. Let $\ell^\infty(\mathbb{N})^\*$ be the dual space of $\ell^\infty(\mathbb{N})$ equipped with weak\*-topology. There is a natural group action of $\mathfrak{S}\_\mathbb{N}$ on $\ell^\infty(\mathbb{N})^\*$, that is to define
$$\sigma(u)\big(... | https://mathoverflow.net/users/140578 | The symmetric group of positive integers acting on $\ell^\infty(\mathbb{N})^*$ | No, it's not continuous.
Indeed, fix a non-principal ultrafilter $U$ supported by the set of even numbers, and define $m\in\ell^\infty(\mathbf{N})^\*$ by $m(f)=\lim\_{n\to U}f(n)$.
Now define $\tau\_n$ as the transposition $(2n,2n+1)$ and $s\_n=\prod\_{k\ge n}\tau\_k$. Then $s\_n$ tends to the identity map for the ... | 3 | https://mathoverflow.net/users/14094 | 408706 | 167,357 |
https://mathoverflow.net/questions/408705 | 3 | *Note: This is a concrete case of the following question: [Are almost all measure-preserving flows on compact manifolds ergodic?](https://mathoverflow.net/questions/408676/are-almost-all-measure-preserving-flows-on-compact-manifolds-ergodic.)*
Let $M$ be a Riemannian manifold with its natural Riemannian measure, and ... | https://mathoverflow.net/users/173490 | Are $C^1$ vector fields generating an ergodic flow $C^0$ dense? | You can't always approximate by ergodic flows, because ergodic flows might not even exist.
For example, on $S^2$ the Poincare-Bendixson theorem rules out ergodic flows, but there are many measure-preserving flows.
| 3 | https://mathoverflow.net/users/1227 | 408707 | 167,358 |
https://mathoverflow.net/questions/408704 | 2 | Let $\Gamma: \mathcal{Y} \twoheadrightarrow \mathcal{X}$ be an upper hemicontinuous correspondence with non-empty values and a closed graph. A continuous selection function in general does not exist, but is it always possible to construct a finite number of continuous functions, $f\_i: \mathcal{Y} \to \mathcal{X}, i=\{... | https://mathoverflow.net/users/466713 | Collection of continuous selections for upper hemicontinuous map | $\newcommand{\Ga}{\Gamma}\newcommand{\N}{\mathbb N}$Not in general.
E.g., let $Y:=\mathcal Y:=[0,1]$, $X:=\mathcal X:=[-1,1]$, $\Ga(0):=[-1,1]$, and $\Ga(y):=\{g(y)\}$ for $y\in(0,1]$, where $g(y):=\sin\frac1y$.
Then $\Ga\colon Y\twoheadrightarrow X$ is an [upper hemicontinuous](https://en.wikipedia.org/wiki/Hemico... | 1 | https://mathoverflow.net/users/36721 | 408713 | 167,360 |
https://mathoverflow.net/questions/408597 | 0 | In a [2014 article](https://link.springer.com/article/10.1007%2FJHEP04%282014%29186) by Chapman, Hoyos and Oz, the authors study non-equilibrium fluid dynamics and describe a method for deriving [Kubo formulas](https://en.wikipedia.org/wiki/Kubo_formula) for thermal transport coefficients of superfluids (the method rel... | https://mathoverflow.net/users/119114 | Generalising results on superfluid Kubo formulas | Not quite sure what you are asking, but Shukla and Kovtun have provided all Kubo formulas for non-dissipative transport coefficients [here](https://link.springer.com/article/10.1007%2FJHEP10%282018%29007)
| 3 | https://mathoverflow.net/users/466792 | 408715 | 167,362 |
https://mathoverflow.net/questions/408663 | 5 | Let $\mathbb{G}$ be a compact quantum group with function algebra $(C(\mathbb{G}), \Delta)$ (in the sense of Woronowicz). Let $X \in M(B\_0(H) \otimes C(\mathbb{G}))$ be a (possibly infinite-dimensional) representation of the quantum group $\mathbb{G}$, and let $K$ be an $X$-invariant subspace of $H$, i.e. if $p\in B(H... | https://mathoverflow.net/users/216007 | Subrepresentations of C*-algebraic compact quantum groups | The following was my original answer, dealing with the case where $X$ is unitary.
It is a nontrivial fact that the orthogonal complement of an invariant subspace is again an invariant subspace. Thus, the projection $p$ in the question will automatically satisfy the stronger property $(p \otimes 1)X = X(p \otimes 1)$.... | 4 | https://mathoverflow.net/users/159170 | 408728 | 167,365 |
https://mathoverflow.net/questions/408716 | 4 | Let $\mathrm{Bundle}$ be the category whose objects are [smooth vector fiber bundles](https://en.wikipedia.org/wiki/Fiber_bundle) over $\mathbb{R}$, and morphisms are fiberwise smooth linear map (that is, the base is not assumed to be fixed).
Let $Base \colon \mathrm{Bundle} \to \mathrm{Diff}$ be the functor returnin... | https://mathoverflow.net/users/148161 | Classification of functorial smooth vector fiber bundles | This is called [natural bundle](https://ncatlab.org/nlab/show/natural+bundle). Apparently, all known information is in [Kolár, Slovák, Michor: Natural operations in differential geometry](https://www.mat.univie.ac.at/%7Emichor/kmsbookh.pdf) (recommended by Stefan Waldmann).
From the description, it is also the best c... | 2 | https://mathoverflow.net/users/148161 | 408731 | 167,367 |
https://mathoverflow.net/questions/408741 | 6 | By Kuratowski's theorem, every nonplanar graph contains a (topological) minor of $K\_5$ or $K\_{3,3}$.
But I observed that every time I construct a $4$-connected nonplanar graph, it always contains not only a $K\_{3,3}$-minor but also a $K\_5$-minor.
Moreover, although I tried many times, I cannot construct a $4$... | https://mathoverflow.net/users/384338 | Does every $4$-connected nonplanar graph contain a $K_5$-minor? | Yes, this is true and follows from [Wagner's theorem](https://en.wikipedia.org/wiki/Wagner%27s_theorem). Wagner's theorem asserts that every graph with no $K\_5$ minor can be built from $0$-, $1$-, $2$-, and $3$-sums from planar graphs and a fixed $8$ vertex non-planar graph called the [Wagner graph](https://en.wikiped... | 10 | https://mathoverflow.net/users/2233 | 408744 | 167,371 |
https://mathoverflow.net/questions/408747 | 1 | Let $f(z)$ be a holomorphic function in the angle $A=\{0<\arg z<\frac{\pi}2\}$, continuous in $\bar A$, satisfying $|f(z)|\le M$ on $\partial A$ and satysfying the following growth condition:
$$
|f(x+i y)|\le Ce^{y^2}\quad\hbox{in $A$}. \label{1}\tag{$\ast$}
$$
**Question**. Does it follow that $f$ is bounded in $A$?
... | https://mathoverflow.net/users/14551 | Phragmén–Lindelöf principle for the critical exponent | For a small $\alpha > 0$, write $$\beta = \frac{\sin^2 \alpha}{\sin(2 \alpha)} = \frac{\tan \alpha}{2} ,$$ and define $$g(z) = f(z) \exp(i \beta z^2).$$ Then $$|g(z)| \leqslant |f(z)| \leqslant C \exp(|z|^2)$$ for $z \in A$, $$|g(i r)| = |f(i r)| \leqslant M$$ for $r > 0$, and $$\begin{aligned}|g(r e^{i \alpha})| & = |... | 1 | https://mathoverflow.net/users/108637 | 408767 | 167,383 |
https://mathoverflow.net/questions/408669 | 6 | Let $\kappa$ and $\lambda$ be cardinals. A thin $(\kappa,\lambda)$-list is a function $L:[\lambda]^{<\kappa}\longrightarrow [\lambda]^{<\kappa}$ such that for all $x\in[\lambda]^{<\kappa}$, $L(x)\subseteq x$ and $\{L(y)\;|\;y\subseteq x\}$ has cardinality $<\kappa$.
Say that $(\kappa,\lambda)$-STP holds iff whenever ... | https://mathoverflow.net/users/138274 | Are the following two "tree properties" equivalent? | The $(\kappa,\lambda)$-STP and $(\kappa,\lambda)$-SSTP are equivalent for any uncountable cardinals $\kappa\leq\lambda$: Let $\mu<\kappa$ and $(L\_\gamma)\_{\gamma<\mu}$ a sequence of thin $(\kappa,\lambda)$-lists. We can then "amalgamate" these lists into one list $L$ as follows:
Let $h:\lambda\times\mu\rightarrow\lam... | 4 | https://mathoverflow.net/users/125703 | 408769 | 167,385 |
https://mathoverflow.net/questions/408687 | 2 | Let $A$ and $B$ be $C^\*$-algebras. Given $f \in B^\*$, we can form the right slice map
$$\iota \otimes f: A \otimes B \to A: a \otimes b \mapsto af(b)$$
which extends uniquely to a bounded linear map
$$\iota \otimes f: M(A \otimes B) \to M(A)$$
that is strictly continuous on the unit ball.
Assume $X \in M(A \otimes ... | https://mathoverflow.net/users/216007 | $(\iota \otimes f)(X) = 0$ for all $f \in B^*$ implies $X=0$ | From comments, it seems that the OP is using the "abstract" definition of multipliers (compare below). A good reference is indeed the appendix of [arXiv:funct-an/9707009](https://arxiv.org/abs/funct-an/9707009). Let's use some remarks from there (bottom of page 38) to show that $\iota\otimes f:A\otimes B\rightarrow A$ ... | 5 | https://mathoverflow.net/users/406 | 408776 | 167,387 |
https://mathoverflow.net/questions/408740 | 3 | Let $n\geq2$. Are there sets $A, B \subseteq \mathbb{N}$ such that $|A|=|B|=n$ and all numbers in $A+B$ are primes?
A well-known conjecture is that there are infinite set $A$ and finite set $|B|=n$ such that all numbers in $A+B$ are primes.
Are there infinite sets $A, B \subseteq \mathbb{N}$ such that all numbers in ... | https://mathoverflow.net/users/345221 | Are there infinite sets $A$ and $B$ such that all numbers in $A+B$ are primes? | Yes, conditional on the Hardy-Littlewood prime tuples conjecture.
Let $A$ and $B$ be two finite sets such that $A +B$ consists of primes, and such that for all primes $p$, there are residue classes $x\_p$ and $y\_p$ mod $p$ with $x\_p+y\_p \neq 0 \mod p$ such that $A$ does not contain any numbers congruent to $x\_p$ ... | 11 | https://mathoverflow.net/users/18060 | 408782 | 167,389 |
https://mathoverflow.net/questions/408743 | 5 | Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial sheaves on $\mathsf{Man}$),
and let $\mathcal S$ denote the $\infty$-category of $\infty$-groupoids (the usual model catego... | https://mathoverflow.net/users/5690 | Geometric realisation of smooth $\infty$-stacks | The case when $M$ is a smooth manifold follows from the [smooth Oka principle](https://ncatlab.org/nlab/show/shape+via+cohesive+path+%E2%88%9E-groupoid#ConsequenceSmoothOkaPrinciple).
See there for an expository account of the argument and references to additional sources.
Indeed, the left side of (\*) is $$\def\Hom{... | 6 | https://mathoverflow.net/users/402 | 408790 | 167,392 |
https://mathoverflow.net/questions/408725 | 4 | I was working on finding a series expression for a function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(x^y) = f(x)^{f(y)}$ along the way for construction of such a function I came across a limit that I don't seem to have any tools to evaluate:
If we use the usual notation for tetration as $^{n}x$ then I am i... | https://mathoverflow.net/users/46536 | Finding closed forms/related constants to a limit involving tetration | There is an implementation around (Pari/GP; in the [tetration-forum](https://math.eretrandre.org/tetrationforum/showthread.php?tid=1017)) which claims to have a Kneser-implementation. It is a bit difficult to handle, so I'll show here a simpler version (essentially polynomial) of a tetration-function which seems to app... | 3 | https://mathoverflow.net/users/7710 | 408796 | 167,395 |
https://mathoverflow.net/questions/408797 | 2 | When can you reconstruct the power series of a function by taking the limits of the coefficients of the polynomials that interpolate its values at $0,1,2,\dots,j$?
More precisely:
Let $f\colon\mathbb{R}\to\mathbb{R}$. For all nonnegative integers $j$, let $p\_j$ be the unique polynomial of degree $j$ with real coef... | https://mathoverflow.net/users/132459 | Power series whose coefficients are limits of coefficients of polynomial interpolations | In the paper, A note on convergence of Newton interpolating polynomials,
by D. Dimitrov and J. Philipps, Journal of Computational and Applied Mathematics
Volume 51, Issue 1, 30 May 1994, Pages 127-130, the following simple criterion is mentioned: $f$ is entire of exponential type less than $\log 2$,
then the sequence o... | 3 | https://mathoverflow.net/users/25510 | 408802 | 167,397 |
https://mathoverflow.net/questions/408818 | 2 | Conway constructed a field of characteristic 2 whose elements are the finite Nim values, indexed by the natural numbers. What is known about nontrivial automorphisms of this field? Do any of them correspond to arithmetically tractable bijections from the set of natural numbers to itself?
| https://mathoverflow.net/users/3621 | Does Conway’s field of finite nim values have arithmetically tractable isomorphisms? | The field of natural numbers under nim operations is precisely the quadratic closure $\mathbb{F}\_{2^{2^\infty}}$ of $\mathbb{F}\_2$, viꝫ. the inductive limit (“union”) of the subfields $\mathbb{F}\_{2^{2^d}}$ given by the nim multiplication on the integers $0,\ldots,2^{2^d}-1$.
The Galois group $\operatorname{Gal}(\... | 4 | https://mathoverflow.net/users/17064 | 408826 | 167,404 |
https://mathoverflow.net/questions/408463 | 7 | Can we upper bound the convergence rate of
$$\max\_{\textbf{v}: \left\Vert \textbf{v}\right\Vert\_2=1} \left\{ \left\Vert \textbf{T}^n \textbf{v}\right\Vert^2\_2 - \left\Vert \textbf{T}^{n+1} \textbf{v}\right\Vert^2\_2 \right\}~,$$
where $\textbf{T}\in \mathbb{R}^{d \times d}$ is a contraction operator
($\left\Vert\tex... | https://mathoverflow.net/users/100796 | Bounding the decrease after applying a contraction operator $n$ vs $n+1$ times | You can easily get a slightly cruder bound $d/n$ (or, if you want, $r/n$) as follows.
Let $A\_n=(T^\*)^nT^n$ and $B\_n=A\_n-A\_{n+1}$. Then $(B\_nv,v)=\|T^nv\|^2-\|T^{n+1}v\|^2\ge 0$, so $B\_n$ is positive definite. Also $B\_{n+1}=T^\*B\_nT$, so, since $T$ is a contraction, $Tr B\_{n+1}\le Tr B\_n$ (this is obvious i... | 5 | https://mathoverflow.net/users/1131 | 408834 | 167,405 |
https://mathoverflow.net/questions/408839 | 6 | Let $T$ be a compact Hausdorff extremally disconnected set (so $T$ is a compact Hausdorff space, such that the closure of each open subset is again open). Let $S \subseteq T$ be a closed subset.
**Question:** Is $S$ extremally disconnected?
For me, this looks like a very natural question about extremally disconnect... | https://mathoverflow.net/users/148992 | Is a closed subset of an extremally disconnected set again extremally disconnected? | No, the Stone-Cech compactification $\beta\mathbf{N}$ of $\mathbf{N}$ is extremally disconnected, but not the Stone-Cech boundary $\beta\mathbf{N}\smallsetminus\mathbf{N}$.
To see this, it is enough to find an increasing sequence $(F\_n)$ of clopen subsets with no supremum (=least upper bound) in the Boolean algebra ... | 10 | https://mathoverflow.net/users/14094 | 408844 | 167,408 |
https://mathoverflow.net/questions/408810 | 5 | Consider the equation $\bar{\partial} f=g$ on the complex plane. We may assume $g$ is compactly supported, but we need the case that $g$ is only assumed to be continuous. Is there a solution to this equation? (I mean classical solution.) If yes, is it the solution given by Cauchy integral formula?
1. If $g$ is $C^1$,... | https://mathoverflow.net/users/27205 | Inhomogeneous Cauchy–Riemann equation on complex plane with continous right hand side | The integral operator
$$Ph(z)=-\frac{1}{\pi}\int\int h(\zeta)\left(\frac{1}{\zeta-z}-\frac{1}{z}\right)dxdy$$
acts on $L^p$, $p>2$, and the result satiafies $(Ph)\_{\overline{z}}=h$
in the sense of distributions. For continuous $h$, this equation may not
have a classical $C^1$ solution.
Edit. The following example wa... | 5 | https://mathoverflow.net/users/25510 | 408847 | 167,409 |
https://mathoverflow.net/questions/408846 | 3 | Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some positive integer $k$.
Are the "simple and broad conditions" guaranteeing that exist a class $C^k$ Monge map $T:\mathbb{R... | https://mathoverflow.net/users/36886 | Regularity of transport map | This is an open question. The difficulty is that Monge's cost function is very degenerate, so doesn't satisfy the assumptions of the standard regularity theory of optimal transport.
The first difficulty is that the solutions to this problem need not be unique. However, the transport will occur along disjoint line seg... | 4 | https://mathoverflow.net/users/125275 | 408855 | 167,411 |
https://mathoverflow.net/questions/408861 | 16 | Let $f$ be a meromorphic function on $\mathbb{C}$ which is algebraic over the field of rational functions $\mathbb{C}(z)$ (i.e. satisfies a nontrivial equation $\sum a\_i(z)f(z)^{i}=0$, with $a\_i(z)\in \mathbb{C}(z)$). Is $f$ actually rational?
| https://mathoverflow.net/users/40297 | Meromorphic function on $\mathbb{C}$ algebraic over $\mathbb{C}(z)$ | The following argument is based on Christian Remling's proof (given in a [comment](https://mathoverflow.net/questions/408861/meromorphic-function-on-mathbbc-algebraic-over-mathbbcz#comment1048802_408861)), but is more elementary. Let us examine the behavior of $f(1/z)$ as $z\to 0$. The function $f(1/z)$ is algebraic ov... | 15 | https://mathoverflow.net/users/11919 | 408869 | 167,415 |
https://mathoverflow.net/questions/408865 | 1 | Consider the vector field $V:\mathbb{R}^4\rightarrow\mathbb{R}^4$, defined by
\begin{equation}
V(x,v,M\_0,M\_1)=(v,\kappa^{-1}(\beta M\_0-v-kx),-M\_0+v M\_1,-M\_1+1-vM\_0),
\end{equation}
such that $\beta,\kappa,k$ are constants. The only equilibrium point occurs at $P^\*=(0,0,0,1)$ and the Jacobian matrix of $V$ at... | https://mathoverflow.net/users/167822 | Finding the eigenvalues and eigenvectors of Jacobian at equilibrium point of nonlinear ODEs | The characteristic polynomial is
$$P(\lambda) = (\lambda+1)\left(\lambda^3 + \frac{\kappa+1}{\kappa} \lambda^2 + \frac{k-\beta+1}{\kappa} \lambda + \frac{k}{\kappa}\right) $$
Since $\kappa \lambda^3 + (\kappa+1) \lambda^2 + (k-\beta+1) \lambda + k$ is irreducible over the rationals, there's no further factorization pos... | 1 | https://mathoverflow.net/users/13650 | 408873 | 167,417 |
https://mathoverflow.net/questions/408830 | 2 | Let us take a relative category $(\mathcal{C},\mathcal{W})$, and consider its hammock localization $L\_H \mathcal{C}$. It seems to me that for every two objects $X,Y \in \mathcal{C}$ the mapping simplicial set $L\_H \mathcal{C}(X,Y)$ has the (strict) right lifting property against all inclusions $\partial \Delta^n \hoo... | https://mathoverflow.net/users/134438 | Are hammock localizations locally truncated? | Your calculation is correct. For every two objects $X, Y \in \mathcal{C}$, the hom space $L^H\mathcal{C}(X,Y)$ has the right lifting property against $\partial \Delta^n \to \Delta^n$ for $n \geq 3$.
First note that the nerve of any category has the strict right lifting property against $\partial \Delta^n \to \Delta^n... | 2 | https://mathoverflow.net/users/184 | 408887 | 167,420 |
https://mathoverflow.net/questions/408872 | 2 | I have a question that is clearly not research level, but it's confusing me so I will ask anyway.
There must be some little logic flaw I am missing. Take $\Omega$ a bounded smooth domain in $\mathbb R^N$ and assume $ \lambda\_k$ is the $k$ eigenvalue of $ -\Delta$ in $H^1\_0(\Omega)$.
Let $v$ denote a smooth solut... | https://mathoverflow.net/users/66623 | Simple elliptic pde problem | If you apply the maximum principle, at a point $p$ where the function $v$ reaches its minimum, you get $-\lambda^2 v(p) \geq \lambda^2$ so $v(p) \leq -1$. In particular, the function $u$ is not globally defined as it has to go to $-\infty$ at least at $p$.
| 5 | https://mathoverflow.net/users/24271 | 408890 | 167,422 |
https://mathoverflow.net/questions/408819 | 1 | In one of Soundararajan's papers, he claims without proof that it is a standard exercise to show that the number $N(X)$ of positive square-free integers $d \equiv 1 \; \bmod \; 8$ less than $X$, with at least two primes factors all of which is congruent to $\pm1 \; \bmod \; 8$ is asymptotically bounded by $\frac{X}{\sq... | https://mathoverflow.net/users/167999 | Asymptotic lower bound for the number of square free with at least two prime factors | Let us introduce a multiplicative function supported on squarefrees by $\alpha(p)=\mathbf{1}\_{p \equiv \pm 1 \bmod 8}$.
We are interested in
$$\sum\_{\substack{n \le x\\ n \equiv 1 \bmod 8\\ n \text{ has }\ge 2 \text{ prime factors}}} \alpha(n) \gg \frac{x}{\sqrt{\log x}}.$$
The primes have density $1/\log x = o(1/\sq... | 4 | https://mathoverflow.net/users/31469 | 408918 | 167,430 |
https://mathoverflow.net/questions/408917 | 2 | Let $m \in \mathbb{R}$.
Let $f(n)$ be [A007814](https://oeis.org/A007814), exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Let $g(n)$ be [A129760](https://oeis.org/A129760), bitwise $\operatorname{AND}$ of binary representation o... | https://mathoverflow.net/users/231922 | Modulo $2$ binomial transform of $m^n$ | If $n\in \mathbb N$ has binary expansion $n=\sum\_{k\in S}2^k$ for a finite subset $S\subset \mathbb N$, then
$$ \sum\_{j=0}^n\Big[\big( {n \atop j}\big)\text{mod}\, 2 \Big]x^jy^{n-j}=\prod\_{k\in S} (x^{2^k}+y^{2^k})$$
More generally, but not needed here (see below): for sums of $r$ indeterminates $x\_i$, the binary... | 3 | https://mathoverflow.net/users/6101 | 408920 | 167,431 |
https://mathoverflow.net/questions/408900 | 4 | Let $ G $ be a Lie group, $ H $ a closed subgroup, and $ G/H $ compact. Under what conditions do we have that
$$
G/H \cong K/(K\cap H)
$$
where $ K $ is a maximal compact subgroup of $ G $? Obviously the result is trivial if $ G $ is compact.
If $ G=\mathbb{R} $ and $ H=\mathbb{Z} $ this is not true. I guess that is ... | https://mathoverflow.net/users/387190 | Does the maximal compact subgroup always act transitively on a compact homogeneous space? | (Comment converted to answer per request:)
The “surprising result” about simply connected homogeneous spaces in your (currently) last paragraph is Montgomery's Theorem ([1950](//ams.org/mathscinet-getitem?mr=0037311)): generally (in your notation) if $G/H$ is compact and $H$ closed ***connected***, then $K$ is transi... | 4 | https://mathoverflow.net/users/19276 | 408937 | 167,436 |
https://mathoverflow.net/questions/408901 | 1 | Let $\{\phi(n)\}\_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
1. $\phi(0)=0$ and $|\phi(n)|\leq \frac{C\_1}{|n|}$ for all $n\neq 0$ and $C\_1>0$ is independent of $n.$
2. $|\phi(n+1)-\phi(n)|\leq \frac{C\_2}{n^2}$ for all $n\neq 0$ and $C\_2>0$ is independent of $n.$
3. $\sum\_{-N... | https://mathoverflow.net/users/136860 | Extending a discrete singular kernel | $\newcommand\R{\mathbb R}\newcommand{\Z}{\mathbb{Z}}\newcommand{\ep}{\epsilon}\newcommand{\fl}[1]{\lfloor#1\rfloor}$The answer is yes to each of your two questions.
Let $a\_n:=\phi(n)$. Then
\begin{equation\*}
K(x)=\sum\_{n\in\Z}a\_n R(x-n).
\end{equation\*}
Note that for all $j\in\Z$ we have $K(j)=a\_j$ and $K$ li... | 1 | https://mathoverflow.net/users/36721 | 408941 | 167,437 |
https://mathoverflow.net/questions/408700 | 9 | There are the complex p-adic numbers.
But what is the p-adic analogue of the Cayley–Dickson construction?
Or more important: What is the p-adic analogue of the octonions?
It would be nice if the (unit)-multiplication table of such a p-adic analogue corresponds to the projective plane over the finite field with p elemen... | https://mathoverflow.net/users/466686 | p-adic analogue of octonions | Defining and classifying the octonion algebras (composition algebras of dimension $8$) over fields $k$, or, in more sophisticated terms, computing the Galois cohomology set $H^1(k, G\_2)$, is the topic of the book *Octonions, Jordan Algebras and Exceptional Groups* by T. A. Springer & F. D. Veldkamp (2000, appropriatel... | 12 | https://mathoverflow.net/users/17064 | 408946 | 167,439 |
https://mathoverflow.net/questions/408966 | 0 | Consider a Brownian motion $B$ and let $f(r)=\sqrt{2r \ln(\ln(r))}.$
Is it true that $\lim\_{r \to +\infty}\frac{1}{f(r)}(B\_r-B\_{\left \lfloor{f(r)}\right \rfloor})= 0$ a.s. ?
If so, how to prove it? Otherwise what counter-example do you suggest ?
| https://mathoverflow.net/users/172528 | $\lim_{r \to +\infty}\frac{1}{\sqrt{2r \ln(\ln(r))}}(B_r-B_{\left \lfloor{\sqrt{2r \ln(\ln(r))}}\right \rfloor})= 0$ a.s.? | no, the second term is basically $B\_t/t$ which (proabably) does go to 0, but the first is governed by the law of the iterated logarithm and does not. In fact, you know that limsup of that term is 1.
| 1 | https://mathoverflow.net/users/143907 | 408971 | 167,444 |
https://mathoverflow.net/questions/408763 | 6 | Let $E$ be a locally convex topological vector space over $\mathbb{R}$. The projectivization $PE$ is the quotient of $E\backslash\{0\_{E}\}$ with respect to the equivalence relation $e\sim f$ if $e=\lambda f$.
>
> Is $PE$ a Tychonoff (i.e. completely regular Hausdorff) space?
>
>
>
As far as I can tell, the th... | https://mathoverflow.net/users/53155 | Is the projectivization of a topological vector space Tychonoff? | The projective space $PE$ of a topological vector space $E$ is Hausdorff but in general is not Tychonoff, not functionally Hausdorff and even not Urysohn (let us recall that a topological space is *Urysohn* if any distinct points have disjoint closed neighborhoods).
As a suitable counterexample, consider the countabl... | 5 | https://mathoverflow.net/users/61536 | 408973 | 167,445 |
https://mathoverflow.net/questions/408979 | 8 | I am interested in smooth nonedegenerate surfaces $X\subset\mathbb{P}^n$, $n\geq 5$, whose secant variety $\sigma(X)$ has dimension $4$. Clearly, the second Veronese embedding of $\mathbb{P}^2$ is such an example. I would be happy about an answer to any of the following
>
> **Questions:** Which other examples exist... | https://mathoverflow.net/users/36563 | Smooth surfaces with defective secant variety | The Veronese surface in $\mathbb{P}^5$ is indeed the only secant defective surface that is not a cone. This is a classical (and non-trivial) result by F. Severi, see p. 6 and Theorem 10.1 in
C. Ciliberto, F. Russo: [Varieties with minimal secant degree and linear systems of maximal dimension on surfaces](http://dx.do... | 10 | https://mathoverflow.net/users/7460 | 408980 | 167,447 |
https://mathoverflow.net/questions/408982 | 3 | **Could you please recommend a reference to or supply a proof of the following identity \eqref{combin-ID-Maclaurin}, or \eqref{first-equiv-form}, or \eqref{combin-ID-Mac-Equiv}, or \eqref{combin-ID-Mac-Reform} for $m\ge2$?**
For non-negative integers $k,n\ge0$, I guess that
\begin{equation}\label{combin-ID-Maclaurin}... | https://mathoverflow.net/users/147732 | Ask for a reference or a proof of a combinatorial identity $\sum_{k=0}^n\binom{2n+1}{2k}\binom {k}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m}$ | We will prove the identity **(A8)** that qifeng618 alluded to. The method is called the Wilf-Zeilberger methodology. To this end, define the two functions (suppressing $x$)
$$F(n,k):=\frac{\binom{2x+1}{2k+1}\binom{x-k}{n-k}(2n+1)}{(2x+1)\binom{x+n}{2n}4^n}
\qquad \text{and} \qquad
G(n,k):=-\frac{F(n,k)\,2k\,(2k+1)}{2(x... | 2 | https://mathoverflow.net/users/66131 | 408996 | 167,453 |
https://mathoverflow.net/questions/408773 | 10 | This question has to do with some experimental phenomenon in groups generated by involutions that I cannot explain.
Let $G$ be a finite, undirected graph, and let $W$ be the corresponding *right-angled Coxeter group*, i.e., the generators of $W$ are the vertices $v \in G$, and we have relations $v^2=1$ for all $v$ an... | https://mathoverflow.net/users/25028 | Reversals of autonomous subsets in right-angled Coxeter groups | Okay, following the ideas from the comments, I do think there is a counterexample.
Let $G=K\_4$, so $W$ is generated by involutions $v\_1,\ldots,v\_4$ subject to no other relations. Consider the Coxeter element $c=v\_1v\_2v\_3v\_4$. We can reverse the autonomous subset $\{v\_1,v\_2\}$ here to get $c'=v\_2v\_1v\_3v\_4... | 2 | https://mathoverflow.net/users/25028 | 409007 | 167,458 |
https://mathoverflow.net/questions/409009 | 4 | Let $\phi$ be an $N$-function, (i.e. $\phi : \mathbb{R}\_{\geq 0} \to \mathbb{R}\_{\geq 0}$ is convex and satisfies $\lim\_{t \to 0} \frac{\phi(t)}{t} = 0, \lim\_{t\to \infty} \frac{\phi(t)}{t} = \infty$).
We can define the associated Luxemburg norm on the appropriate subspace of $\mathbb{R}$-valued random variables ... | https://mathoverflow.net/users/468679 | Weak concentration bounds for averages of independent random variables in Orlicz spaces | In general, the answer is no. Moreover, the answer is no even if
\begin{equation}
\phi(t)=t\ln(1+t). \tag{1}
\end{equation}
Indeed, suppose that $P(Z\_i=0)=1-2p$ and $P(Z\_i=b)=p=P(Z\_i=-b)$ for all $i$, where
\begin{equation\*}
p:=\frac1{2\phi(b)},
\end{equation\*}
$\phi$ is as given by (1),
and $b$ is a large eno... | 3 | https://mathoverflow.net/users/36721 | 409031 | 167,468 |
https://mathoverflow.net/questions/408781 | 4 | Are there examples of
1. a non-zero C\*-algebra which is
2. universally generated by
3. finitely many projections (not all commuting) together with a unit and plus
4. necessarily satisfying some additional relations such that
5. there remain no traces?
In other words, is there a non-zero quotient
$$C^\*(Z/2\*\ldots... | https://mathoverflow.net/users/45494 | Example: traceless C*-algebra universally generated by projections | I'm not sure if the following is exactly what the OP was looking for, but it definitely solves the question. The following lemma implies that $\mathcal O\_2$ is a quotient of $C^\*(\underbrace{(\mathbb Z/2\mathbb Z) \* \dots \* (\mathbb Z/2\mathbb Z)}\_{4\textrm{ times (edited)}} )$.
>
> **Lemma (edited):** Let $A$... | 6 | https://mathoverflow.net/users/126109 | 409043 | 167,469 |
https://mathoverflow.net/questions/406758 | 6 | Let $\{e\_1,\ldots, e\_n\}$ be the standard basis of $\mathbb{C}^n$. Consider the $m$-multilinear form $$v=\sum\_{i=1}^n e\_i^{\otimes m}\in (\mathbb{C}^n)^{\otimes m}$$
and consider the action of $\text{GL}\_n(\mathbb{C})$ on
$(\mathbb{C}^n)^{\otimes m}$.
Question: What is the stabilizer of $v$ in $\text{GL}\_n(\mat... | https://mathoverflow.net/users/41644 | Stabilizers of multilinear forms | It is equal.
Since the ground field is of characteristic zero and $v$ is symmetric one may as well compute the stabilizer of the polynomial $f:=\sum\_{i=1}^nx\_i^m$.
Assume $g\in\mathrm{GL}\_n(\mathbb C)$ stabilizes $f$. Then it also stabilizes the Hessian $$\det\nolimits\_{ij}(\partial\_i\partial\_jf)\in\mathbb C^... | 5 | https://mathoverflow.net/users/89948 | 409061 | 167,474 |
https://mathoverflow.net/questions/409058 | 11 | Define the *divisibility graph* of a set of positive integers as the graph whose vertices are the integers, two of which are joined by an edge if one divides the other.
For all *N*, is it true that integers less than or equal to *N* whose proper divisors have divisibility graph which is planar are more numerous than ... | https://mathoverflow.net/users/60732 | Is the divisibility graph of the proper divisors of n more often planar than not? | No, because almost all numbers have at least $4$ distinct prime factors, making the divisibility graph contain a hypercube and thus be nonplanar.
| 27 | https://mathoverflow.net/users/18060 | 409064 | 167,476 |
https://mathoverflow.net/questions/409080 | 4 | Suppose we are working in the language of a binary operation symbol $\*$. Let $S$ be a set of equations which generate precisely the same equational theory generated by the set containing the commutative and associative equations: $\{ x\*y=y\*x, (x\*y)\*z=x\*(y\*z) \}$. Suppose further that $S$ has at least three eleme... | https://mathoverflow.net/users/43439 | A question regarding equational bases of the theory of the commutative and associative properties | Yes it's true.
Write $S=(s\_i=t\_i)\_{i\in I}$ with $s\_i,t\_i$ in some free magma on $k\_i$ generators, where each of the $k\_i$ variables appears in either $s\_i$ or $t\_i$.
First, since $\mathbf{N}$ with addition satisfies this theory, we see that the length of $s\_i$ and $t\_i$ (number of letters) are the same.... | 5 | https://mathoverflow.net/users/14094 | 409081 | 167,480 |
https://mathoverflow.net/questions/400628 | 0 | This question was inspired by Joel David Hamkins's excellent [question](https://mathoverflow.net/questions/205985/is-there-a-leibnizian-model-with-no-definable-elements-in-a-finite-language) on Leibnizian structures with no definable elements. Let $n$ be a positive integer. Is there an infinite structure in a finite la... | https://mathoverflow.net/users/43439 | An infinite Leibnizian structure in a finite language with precisely $n$ definable elements | *Turning my comment into an answer to move this off the unanswered queue:*
Take a Leibnizian structure with no definable elements and just add $n$ constants to it (and make the structure "otherwise boring" on those elements - e.g. no relation involving one of those elements should hold, and any function involving one... | 2 | https://mathoverflow.net/users/8133 | 409082 | 167,481 |
https://mathoverflow.net/questions/265570 | 6 | This is essentially what Exercise 5.4 in
Boucheron, Lugosi, Massart *Concentration Inequalities* boils down to:
For real $a,b$ and $0<p<1$,
\begin{align\*}
&pa^2\log( \frac{a^2}{b^2+pa^2-pb^2}) +
(1-p)b^2\log(\frac{b^2}{b^2+pa^2-pb^2})
\\
&\le
\frac{p(1-p)(a-b)^2}{1-2p}\log\frac{1-p}p
.
\end{align\*}
This is suppo... | https://mathoverflow.net/users/12518 | Exercise related to log-Sobolev inequalities | By scaling if necessary, we may assume without loss of generality that $a^2p + b^2\bar p = 1$ Substituting $u = a^2p$, we can rewrite the 1-dimensional inequality as
\begin{align\*}
f(u) := u\log \frac{u}{p} + \bar u \log \frac{\bar u}{\bar p} - c(p) (\sqrt{\bar u p} - \sqrt{u \bar p})^2 \le 0.
\end{align\*}
We calcula... | 1 | https://mathoverflow.net/users/20062 | 409091 | 167,485 |
https://mathoverflow.net/questions/408955 | 6 | I’d like to know if a sharp version of Craig’s interpolation theorem for $L\_{\omega\_1 \omega}$ is already known or exists in the literature. By a “sharp” version of this theorem, I mean something like the following statement: if $\phi$ implies $\psi$ then the interpolant $\theta$ is of the same syntactic complexity a... | https://mathoverflow.net/users/42949 | Sharp Craig interpolation theorem for $L_{\omega_1 \omega}$ | It actually isn't true that the complexity of $\theta$ could be meaningfully bounded in the term of complexities of $\varphi$ and $\psi$.
Let us fix an arbitrary recursive ordinal $\alpha$. Below I sketch a construction of infinitary $\Pi\_n$ formulas $\varphi,\psi$, for some finite $n$ such that there are no $\Pi\_\... | 5 | https://mathoverflow.net/users/36385 | 409109 | 167,488 |
https://mathoverflow.net/questions/409118 | 2 | Let $Pr\_{(X,Y)}$ be a probability distribution of a random vector $(X,Y)$. Let $F$ be the cumulative distribution function of $(X,Y)$. Define
$$
\mathcal{A}\equiv \{(x,y): x\leq 2 \text{ and }x-y\leq 3\}
$$
Is there a way to express $Pr\_{(X,Y)}(\mathcal{A})$ as
$$
(\*)\quad \sum\_{k=1}^K F(a\_k, b\_k)\times c\_k
$$
f... | https://mathoverflow.net/users/42412 | Probability measure of trapezoidal area | No, this cannot be done in general. Indeed, let $A:=\mathcal A$. You want to express
$$P((X,Y)\in A)=Ef(X,Y)$$
as
$$\sum\_{k=1}^K c\_k F(a\_k,b\_k)=Eg(X,Y),$$
where
$$f(x,y):=1(x\le2,x-y\le3)$$
and
$$g(x,y):=\sum\_{k=1}^K c\_k 1(x\le a\_k,y\le b\_k).$$
However, for any choice of the numbers $a\_k,b\_k,c\_k$, there will... | 2 | https://mathoverflow.net/users/36721 | 409123 | 167,491 |
https://mathoverflow.net/questions/409122 | 5 | Let $m, n\in \mathbb{N}$ and $|x| < 1$. I look for hints to derive an analytic formula for
$$f\_{m,n}(x) = \sum\_{k \in \mathbb{N}} {n + k \choose k} {m + k \choose k} x^{k}. $$
| https://mathoverflow.net/users/3441 | Generating function of product of binomial coefficients | This is Gauss' hypergeometric function
$F(n+1,m+1,1;x)$. You can then apply the huge theory of hypergeometric functions to derive further expressions. For instance, Euler's transformation formula gives the alternative expression
$$\frac 1{(1-x)^{m+n+1}}\,F(-m,-n,1;x)=\frac 1{(1-x)^{m+n+1}}\sum\_{k=0}^{\min(m,n)}\binom ... | 8 | https://mathoverflow.net/users/10846 | 409124 | 167,492 |
https://mathoverflow.net/questions/409115 | 4 | Let $T$ be an **injective** operator system and $U$ be an arbitrary operator system. Let $\varphi: T \to U$ be a unital completely positive map and $\psi: U \to T$ be a unital completely positive map with $\psi \varphi = \iota\_T$. Is it true that $\varphi$ is a complete isometry?
**Attempt**: I think yes. Here is an... | https://mathoverflow.net/users/216007 | If a completely positive unital map admits a completely positive unital left inverse, it is a complete isometry | Injectivity of $T$ plays no role. The point is that unital completely positive maps are contractive, so if a strict inequality $\|\phi(a)\|< \|a\|$ holds for some $a\in T$, then $\|a\|=\|\psi(\phi(a))\|\leqslant \|\phi(a)\| < \|a\|$, which is a contradiction. This shows that $\phi$ is an isometry and the same argument ... | 9 | https://mathoverflow.net/users/24953 | 409128 | 167,493 |
https://mathoverflow.net/questions/409119 | 8 | In McLarty's *The Rising Sea: Grothendieck on simplicity and generality* I found the following quote:
>
> The same, Grothendieck knew, would work for cases yet unimagined. He notes that Tohoku [Grothendieck 1957] already gave foundations for the cohomology of any topos [Grothendieck 1985–1987, p. P41n.]. That conte... | https://mathoverflow.net/users/469007 | Tohoku and cohomology of toposes | As requested:
By Theorem 1.10.1 in Tohoku, an Grothendieck abelian category has enough injectives. Sheaves of abelian groups on a Grothendieck topos form a Grothendieck abelian category. By Theorem 2.2.2 in Tohoku, one may then take derived functors of global sections.
As mentioned in the comments, though, sheaf co... | 9 | https://mathoverflow.net/users/6936 | 409139 | 167,496 |
https://mathoverflow.net/questions/409073 | 8 | There are two languages endow the theory of coherent sheaves with a six functor formalism (that I "know" of), one being formulated in $\text{ProCoh}(X)$ by Deligne and the other being $D(\mathcal{O}\_{X,\blacksquare})$ of Clausen-Scholze.
I'm curious as to the relation between the two. Every coherent sheaf gives rise t... | https://mathoverflow.net/users/152554 | Relation between ProCoh and solid modules | This is a long comment addressing the functor in question but not the lower shriek. I would like to say that it is close to a fully faithful embedding. For example, we claim that the pro-objects in the category of abelian groups of finite presentation (or equivalent, of finite type) is a full subcategory of solid abeli... | 4 | https://mathoverflow.net/users/176381 | 409142 | 167,497 |
https://mathoverflow.net/questions/409093 | 8 | The classical Calderon-Zygmund decomposition says that if $f\geq 0$ is $L^1$ on a cubes $B$, with average value $\alpha$, then there is a sequence of disjoint cubes $B\_j$, such that the average of $f$ on each $B\_j$ is in between $\alpha$ and $2^n \alpha$, and $f\leq \alpha$ a.e. away from $\bigcup\_j B\_j$.
I am wo... | https://mathoverflow.net/users/130379 | Calderon-Zygmund decomposition on manifolds? | Calderon-Zygmund theory generalises without much difficulty to doubling metric measure spaces (or more generally to "spaces of homogeneous type"). See for instance Chapter 1 of
*Stein, Elias M.*, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murp... | 10 | https://mathoverflow.net/users/766 | 409145 | 167,499 |
https://mathoverflow.net/questions/409079 | 7 | If $R$ is a ring and $M$ an $R$-module, $M$ is *uniserial* if its lattice of submodules is a chain. Over an Artinian $R$, the chain will be finite. From what I understand, deciding when two uniserial modules are isomorphic is an open problem. D'Este, Kaynarca, and Tutuncu point out in the introduction to their paper *I... | https://mathoverflow.net/users/8027 | Rings of finite uniserial type | It is an open problem which Artin algebras have only finitely many uniserial indecomposable modules.
This is stated for example as problem 2 in the open problems section in the book "Representation theory of Artin algebras" by Auslander-Reiten-Smalo.
No good characterisation exists as far as I know or an algorithm to c... | 1 | https://mathoverflow.net/users/61949 | 409148 | 167,501 |
https://mathoverflow.net/questions/408976 | 3 | Can someone please give me an example of a Noetherian normal local domain of dimension two such that there exists a prime ideal $P$ of height one having the property $P^{(n)}$ is not a principal ideal for any $n \geq 1$. Here $P^{(n)}$ is the [symbolic $n$-power](https://en.wikipedia.org/wiki/Symbolic_power_of_an_ideal... | https://mathoverflow.net/users/97962 | Symbolic powers of a prime ideal of height one | Take $E$ an elliptic curve $zy^2 - x(x-z)(x-tz)$ say over $\mathbb{C}$ and choose a point $Q$ of infinite order (or so for instance the divisor $Q - O$ has infinite order in the divisor class group, here $O$ is the point at infinity).
It follows that in the graded ring of dimension $2$,
$$\mathbb{C}[x,y,z]/(zy^2 - x(... | 4 | https://mathoverflow.net/users/3521 | 409155 | 167,505 |
https://mathoverflow.net/questions/408816 | 4 | Let $P$ be a convex simplicial polytope in $\mathbb{R}^n$. Can we find a convex simplicial polytope $P\_0$ in $\mathbb{R}^n$ combinatorially equivalent to $P$, satisfying the following condition: The vertices of $P\_0$ are lattice points and for every facet $F$ of $P\_0$ its vertices $v\_1,\dots,v\_n$ span $\mathbb{Z}^... | https://mathoverflow.net/users/38983 | Simplicial polytope with regular cones | The conditions you pose on $P\_0$ imply that it is a *reflexive* polytope. (That is, a lattice polytope with the origin in its interior and such that its polar dual is also a lattice polytope).
There are finitely many reflexive polytopes in each dimension (modulo $GL(\mathbb Z,n)$), which implies that the answer to y... | 2 | https://mathoverflow.net/users/22608 | 409163 | 167,511 |
https://mathoverflow.net/questions/409107 | 10 | Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p\_0\in P$.
Let $f:X\times P\to Y$ be a *continuous* map such that
for any parameter $p\in P$, $f\_p:= f|\_{X\times \{p\}}:X\to Y$ is *smooth*.
Suppose also that $f\_p\to f\_{p\_0}$ in $C^3\_{loc}(X,Y)$ for $p\to p\_0$ (i.e. once we fix an arbitrary com... | https://mathoverflow.net/users/99042 | Implicit function theorem with continuous dependence on parameter | Assuming $P$ first countable, the standard contraction principle and elementary bounds are sufficient to conclude. You do not need higher regularity:
*Let $X$, $Y$ be Banach spaces, $P$ a topological space, $f:X\times P\to Y$. Assume that*
*i. $f:X\times P\to Y$ is continuous;*
*ii. $f(\cdot,p):X\to Y$ is differe... | 5 | https://mathoverflow.net/users/6101 | 409166 | 167,512 |
https://mathoverflow.net/questions/392731 | 8 | Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S\_n$. As we know from the branching rule, we have an isomorphism of $S\_{n-1}$ modules
$$S^\lambda\cong\bigoplus\_{\mu}S^{\mu}$$
where $\mu$ are the partitions obtained by deleting an outer corner from $\lambda$.
... | https://mathoverflow.net/users/172429 | Branching rule for Specht modules over Kazhdan-Lusztig basis | It turns out that this is indeed true. The relevant proofs are given in Chapter 3 of [this extended abstract](https://arxiv.org/abs/2111.09510).
| 4 | https://mathoverflow.net/users/172429 | 409168 | 167,513 |
https://mathoverflow.net/questions/409165 | -4 | Is the underlying set of every renormalization group countable and finite？
Suppose A is a renormalization group, and the elements of it compose of the set B. Is B the set countable and finite?
| https://mathoverflow.net/users/14024 | Is the underlying set of every renormalization group countable and finite？ | No, the renormalization group of a continuum field theory contains continuously parameterized scale-changing transformations—hence an uncountable number of them.
| 4 | https://mathoverflow.net/users/170778 | 409169 | 167,514 |
https://mathoverflow.net/questions/408575 | 9 | My problem seems elementary. However a post in [SE](https://math.stackexchange.com/questions/4304278/least-common-multiple-of-three-integers) has not got an answer.
Let $a\_1,a\_2,a\_3\geq1$ be integers and let $A=\mathrm{lcm}(a\_1,a\_2,a\_3)$ be their least common multiple. I want to show the following.
>
> If $... | https://mathoverflow.net/users/105537 | Certain property of the least common multiple of three integers | 1. Without loss of generality $d:=\gcd(a\_1,a\_2,a\_2)=1$, else divide $a\_1,a\_2,a\_3$ and $A$ by $d$.
2. Denote $d\_{ij}=\gcd(a\_i,a\_j)$. Now $d\_{12}, d\_{13},d\_{23}$ are mutually coprime, we may write $a\_1=d\_{12}d\_{13}b\_1$ etc, $A=d\_{12}d\_{13}d\_{23}b\_1b\_2b\_3$, $m\_1$ must be divisible by $d\_{23}$ etc, ... | 3 | https://mathoverflow.net/users/4312 | 409179 | 167,516 |
https://mathoverflow.net/questions/409135 | 5 | Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$.
Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of copies of $\mathbb{P}^1\_k$.
Let $T\_C = \mathcal{H}om(\Omega^1\_C,\mathcal{O}\_C)$ be the tangent sheaf of $C$.
**Ques... | https://mathoverflow.net/users/110362 | First cohomology of tangent sheaf of rational curve | Let $C$ be the union of 5 lines in general position in $\mathbb{P}^2$ (hence with 10 pairwise intersection points $P\_{ij}$, $1 \le i < j \le 5$) and let $F$ be the equation of $C$. We have the standard exact sequence
$$
0 \to \mathcal{O}\_C(-5) \stackrel{dF}\to \Omega\_{\mathbb{P}^2}\vert\_C \to \Omega\_C \to 0.
$$
Ta... | 10 | https://mathoverflow.net/users/4428 | 409185 | 167,518 |
https://mathoverflow.net/questions/409013 | 7 | In classical homotopy theory, there are a number of spaces which are important because they represent an interesting functor on $\operatorname{Ho(Top)}$; for example, $K(G,n)$ represents singular cohomology and $BG$ represents principal $G$-bundles. However, modern homotopy theorists know that the homotopy category is ... | https://mathoverflow.net/users/158123 | The contravariant mapping space represented by a homotopical classifying space (e.g. BG) | Let $G$ be a topological group and $X$ be a paracompact Hausdorff topological space. For simplicity let us assume that $G$ has the homotopy type of a CW complex, although a lot of this answer does not need it. Then we can define the simplicial category of principal $G$-bundles over $X$ in the following way:
* Its obj... | 5 | https://mathoverflow.net/users/43054 | 409187 | 167,519 |
https://mathoverflow.net/questions/409202 | 2 | Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function of $u\pm v$, i.e, $K=K(u\pm v)$ ?
| https://mathoverflow.net/users/171439 | Can we always find coordinates on a surface such that $K=K(u-v)$? | Curvature is a smooth function on the surface, and locally, any smooth negative function can serve as a curvature of some surface (M. S. Berger, Riemannian structure of prescribed Gaussian curvature for compact 2-manifolds, J. Differential Geom. 5 (1971), 325-332.)
So the question is whether for an arbitrary smooth f... | 2 | https://mathoverflow.net/users/25510 | 409203 | 167,520 |
https://mathoverflow.net/questions/409200 | 8 | This question asks whether there exists an analogue of the Jordan decomposition for an arbitrary ring $R$. This analogue is not necessarily the Jordan-Chevalley decomposition, which is unnecessarily strong. This follows from [this question](https://mathoverflow.net/questions/408349/matrix-decompositions-as-monoid-isomo... | https://mathoverflow.net/users/75761 | For every ring R, is there a block-diagonal canonical form for a square matrix over R? | Your question is equivalent to whether the category $\mathcal{E}$ of pairs $(V,f)$ consisting of a finitely generated free (right) $R$-module and an endomorphism $f$ of $V$ is a *Krull-Schmidt category*, i.e., an additive category where every object decomposes as a direct sum of finitely many indecomposable objects and... | 12 | https://mathoverflow.net/users/86006 | 409211 | 167,522 |
https://mathoverflow.net/questions/409193 | 1 | I am working with the heat kernel on the hyperbolic space explicitly (as you may guess by my previous questions) and I got the desired results when the curvature is $-\kappa=-1$. Now I am trying to do the same for a fixed but arbitrary curvature $-\kappa<0$, so I need to generalize the explicit formulas for the heat ke... | https://mathoverflow.net/users/411616 | Heat kernel on hyperbolic space of variable curvature | The sectional curvature scales like the inverse of the metric.
So fixing a coordinate system on $\mathbb{H}^n(1)$, with metric $g$, the scaled metric $\kappa^{-1} g$ has sectional curvature $-\kappa$.
If $u(t,x)$ solves the heat equation you have
$$ u\_t = \Delta\_g u \iff \kappa u\_t = \kappa \Delta\_g u \iff \kappa... | 2 | https://mathoverflow.net/users/3948 | 409214 | 167,524 |
https://mathoverflow.net/questions/368883 | 9 | I'm searching for a copy of an old paper made by Edmund Landau:
>
> *Zur relativen Wertbemessung der Turnierresultate*, Deutsches Wochenschach, 11. Jahrgang (1895), 366–369.
>
>
>
However, I can't find it anywhere. I looked at some old books written by Landau and did not succeed in my quest.
The Princeton Un... | https://mathoverflow.net/users/109828 | Math history research: a copy of "Zur relativen Wertbemessung der Turnierresultate" , eigenvector centrality by Edmund Landau | The paper which actually was Landau's first scientific paper written at the tender age of 18, was published in his Collected Works, vol. 1. In it, he proposes to rank chess players having played a round robin tournament according to an eigenvector of the results matrix . A much more comprehensive analysis of this metho... | 6 | https://mathoverflow.net/users/96921 | 409217 | 167,525 |
https://mathoverflow.net/questions/409212 | 6 | This question may be related to [this one](https://math.stackexchange.com/q/4310801/498394).
Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem.
Let $f$ be some one-sided probability distribution defined on $[0,\infty)$, and $\hat{f}$ be its Laplace transf... | https://mathoverflow.net/users/159685 | Convergence speed of the tail of distribution using Tauberian remainder theorem | Let $f$ be a pdf on $[0,\infty)$. Let $\hat f$ be the Laplace transform of $f$, so that
\begin{equation}
\hat f(s)=\int\_0^\infty f(x)e^{-sx}\,dx
\end{equation}
for real $s\ge0$.
Suppose that
\begin{equation}
|\hat f(s)-\hat f(0)|\le Cs^a
\end{equation}
for some real $a,C>0$ and all real $s\ge0$.
Then for all rea... | 3 | https://mathoverflow.net/users/36721 | 409219 | 167,527 |
https://mathoverflow.net/questions/409233 | 11 | To quote [Kerodon](https://kerodon.net/tag/0004):
>
> In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces.
>
>
>
A similar quote can be found in the mathscinet review for Kan's *On c. s. s. complexes*:
... | https://mathoverflow.net/users/469290 | Algebraic topology and homotopy theory with simplicial sets instead of topological spaces | It depends on what you mean by "all results". Of course results regarding manifolds or vector bundles do not admit statements completely internal to the world of simplicial sets (although most of them are just an application of $\operatorname{Sing}$ away from the world of simplicial sets).
But if one concentrates one... | 16 | https://mathoverflow.net/users/43054 | 409236 | 167,533 |
https://mathoverflow.net/questions/409140 | 2 | $\DeclareMathOperator\Hom{Hom}$Assume $F$ and $M$ are respectively right and left modules over a ring $R$ and let $I^\bullet$ be a left-bounded exact complex of $R$-$R$-bimodules. We know there is a natural map of complexes $\varphi: F\otimes\_R \Hom(I^\bullet, M)\longrightarrow \Hom\_R(\Hom\_R(F, I^\bullet), M)$ which... | https://mathoverflow.net/users/466540 | Is a certain map a quasi-isomorphism? | This isn’t true when $R=\mathbb{Z}$, $F=\mathbb{Q}$, $M=\mathbb{Z}$ and $I^\bullet$ is the complex
$$\cdots\to0\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}\to0\to\cdots$$
with $\mathbb{Z}$ in degree zero.
Then $\varphi$ is a map from $\mathbb{Q}$, as a complex concentrated in degree zero, to the zero complex.
... | 2 | https://mathoverflow.net/users/22989 | 409241 | 167,536 |
https://mathoverflow.net/questions/409237 | 5 | Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so that it is a non trivial torsor for it's Picard group).
Can I still embed $C$ into the projective plane? I guess not ... | https://mathoverflow.net/users/58001 | Embedding torsors of elliptic curves into projective space | Suppose that $C \subset X$ is a smooth projective curve of genus $1$ embedded in a Brauer-Severi surface over a field $k$. We have $C^2 = 9$ since this holds after passing to the algebraic closure, where it is embedded as a curve of degree $3$ in the projective plane. So we deduce that $C$ admits a divisor of degree $9... | 11 | https://mathoverflow.net/users/5101 | 409248 | 167,539 |
https://mathoverflow.net/questions/409252 | 3 | In the plane, two figures are called congruent exactly if one can be transformed into the other by translation, rotation, and reflection. What if reflection is excluded, that is, preservation of orientation is required? Is there a term for the resulting equivalence relation?
| https://mathoverflow.net/users/25527 | What is the term for two figures being congruent and of same orientation? | While perhaps not widespread, the term “direct congruence” is used for this equivalence relation.
| 2 | https://mathoverflow.net/users/25527 | 409254 | 167,540 |
https://mathoverflow.net/questions/409208 | 4 | A tower $\Delta\_g:=\{(x,n)\in X \times \{0,1,2,\cdots\}: n < R(x)\}$
where $R:X \to \{1,2,3,\cdots\}$ is a $L^1$ function on a probability space $(X,\mu)$, $g: X \to X$ is mixing and $\gcd \{R\}=1$,
A map $f: \Delta \to \Delta$ is defined as: $f(x,n)=(x,n+1)$ if $n < R(x)-1$ and $f(x,n)=(g(x),0)$ if $n=R(x)-1$.
... | https://mathoverflow.net/users/124254 | a mixing property on a tower | There are counterexamples. The easiest way to “cheat” is to let the height function be cohomologous to a constant.
As an example, let $T$ be an ergodic transformation of a space $X$. Let $A$ be a subset of $X$ such that $A$ and $T^{-1}A$ are disjoint. Now define $g(x)=1$ if $x\in A$, $g(x)=3$ if $x\in T^{-1}A$ and $2... | 3 | https://mathoverflow.net/users/11054 | 409259 | 167,542 |
https://mathoverflow.net/questions/409269 | 3 | I am looking into a question of how many points can be put on a plane that pairwise distances between them are as close to a constant as possible. As the first step, it was rather easy to figure out that only 3 points can be put on a plane so that all pairwise distances between them are equal. As the next step, I think... | https://mathoverflow.net/users/101533 | 4 points on a plane with (almost) equal pairwise distances | It is $\sqrt{2}$. For 4 vertices of a square, you get this value. For proving that it is always not less than $\sqrt{2}$, note that one of angles $\angle A\_iA\_jA\_k$ is not less than $\pi/2$ (if $A\_1A\_2A\_3A\_4$ is a convex quadrilateral, the sum of angles equals $2\pi$, thus one of them is at least $\pi/2$; if $A\... | 6 | https://mathoverflow.net/users/4312 | 409271 | 167,546 |
https://mathoverflow.net/questions/409270 | -3 | Knowing the value of $S=\sum\_{k=1}^n s\_k$ with $s\_k\geq 0$,
is it possible to obtain an upper bound on $\sum\_{k=1}^n\sqrt{s\_k}$ better than $n \times \sqrt{\max\_{1\leq k\leq n} s\_k}$ ?
| https://mathoverflow.net/users/148279 | Bounding sum of square roots in function of the sum value | Yes: by the [generalized mean inequality](https://en.wikipedia.org/wiki/Generalized_mean#Generalized_mean_inequality) (or, more specifically, by the [AM--QM inequality](https://en.wikipedia.org/wiki/HM-GM-AM-QM_inequalities)), $\sqrt{nS}$ is an upper bound on $\sum\_{k=1}^n\sqrt{s\_k}$, which is better than
$n\sqrt{\ma... | 2 | https://mathoverflow.net/users/36721 | 409282 | 167,549 |
https://mathoverflow.net/questions/409292 | 3 | Let $R$ be a commutative (noetherian, if needed) ring, let $f\_1，\ldots，f\_r\in R[x\_1，\ldots，x\_n]$ and $A=R[x\_1，…，x\_n]/(f\_1，\ldots，f\_r)$, when is $A$ flat over $R$?
I found a nice answer for the case $n=r=1$ [here](https://math.stackexchange.com/a/3667137/95662), but I don't know how to formulate a characterisa... | https://mathoverflow.net/users/42571 | Flatness of finitely presented algebras | There is a nice criterion that is usually applied in this situation.
A is flat over $R$ if the Krull dimension of the fiber ring $A\_{\mathfrak{P}}/\mathfrak{P}$ for all prime ideals $\mathfrak{P} \subset R$ is $n-r$ (or the fiber is empty). In scheme theoretic language, the fibers either have the expected dimension ... | 3 | https://mathoverflow.net/users/339730 | 409293 | 167,551 |
https://mathoverflow.net/questions/409287 | 11 | Let $f(z)=\sum a\_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies
$$
|f(z)|\le \frac{1}{(1-|z|)^{k}}
$$
for some fixed $k>0$.
**Question:** What can I deduce about the growth of the Taylor coefficients $a\_n$?
**Partial result:** By judiciously selecting the location of the contour in the formula ... | https://mathoverflow.net/users/5690 | Estimating the growth of the Taylor coefficients given the growth of the function at the boundary | The optimal exponent is $k$. Such examples are given by sparse power series. This is actually trivial in the case $k=0$ (which was not included in the OP). Then we can simply take $f(z)=\sum j^{-2} z^{N(j)}$, say. This is obviously bounded, and the coefficients $a\_n$ will not satisfy $|a\_n|\lesssim n^{-\epsilon}$ for... | 12 | https://mathoverflow.net/users/48839 | 409301 | 167,554 |
https://mathoverflow.net/questions/409295 | 3 | I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian:
$ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ when $\mu=g(K)\in\mathbb{R}^{n\times 1}$.
I have three problems with this: 1. The dimensions are not adding up for me. 2... | https://mathoverflow.net/users/301078 | Derivative of an integral of a Gaussian | **Q1:** $dZ/dK$ is a $p\times q$ matrix with elements
$$\bigl[dZ/dK\bigr]\_{ij}=\sum\_{k=1}^n\frac{\partial \mu\_k}{\partial K\_{ij}}\,\mathbb{E}[f(x)\,\Sigma^{-1}\cdot(x-\mu)]\_k.$$
**Q2:** To evaluate this you will have to specify the function $f(x)$. For some $f$ you will be able to evaluate the expectation value sy... | 1 | https://mathoverflow.net/users/11260 | 409307 | 167,556 |
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