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https://mathoverflow.net/questions/446484 | 0 | Hermite polynomials $H\_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula
$$
H\_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) .
$$
Each $H\_k(x)$ is a polynomial of exact degree $k$. The Hermite polynomials are also given by the generating function
$$
e^{2 x w-w^2}=\sum\_{k=0}^{\infty} \... | https://mathoverflow.net/users/172078 | Closed formula for Hermite polynomials | Up to some normalization, the harmonic oscillator $H$ is self-adjoint such that
$$
\langle Hu, u\rangle=\sum\_{k\ge 0}(\frac12+k) \vert u\_k\vert^2,
$$
and thus defining a self-adjoint $A$ by the equality
$$
\langle Au, u\rangle=\sum\_{k\ge 0}(a+k) \vert u\_k\vert^2,
\quad\text{implying}\ A=H+a-\frac12.
$$
As a result ... | 3 | https://mathoverflow.net/users/21907 | 446509 | 179,893 |
https://mathoverflow.net/questions/446437 | 10 | **I. Four quintics?**
The general quintic can be transformed in radicals to at least ***three*** one-parameter forms. For simplicity, assume this free parameter to be some generic "*alpha*". Hence,
$$x^5-10\alpha x^3+45\alpha^2x-\alpha^2=0\tag1$$
$$x^5-5\alpha x -\alpha = 0\tag2$$
$$x^5+5\sqrt{\alpha}\, x^2 -\sqrt{... | https://mathoverflow.net/users/12905 | On Ramanujan's pi formula $\frac 1\pi=\sum_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {Ak+B}{396^{4k}}$ and the solvable quintic $z^5-5z-396 = 0$? | (*This addresses Question 2*.)
**I.** In general, it asks if we can reduce the general quintic to a one-parameter form with a *specified* discriminant $D\_i$ that is different from the other well-known forms. These $D\_i$ involve the integers $\color{blue}{1728, 256, 108, 64}$, numbers which appear per level in *Rama... | 0 | https://mathoverflow.net/users/12905 | 446511 | 179,894 |
https://mathoverflow.net/questions/446486 | 3 | Let $X$ be a Riemann surface with analytic boundary. Assume that $X$ has negative Euler characteristic. Then there exists a conformal hyperbolic metric $X$ such that $\partial X$ consists of geodesics (conformal in the sense of being compatible with the given complex structure on $X$).
Does anyone has a reference (or... | https://mathoverflow.net/users/90076 | Finding a hyperbolic metric with geodesic boundary on a given Riemann surface | A good reference is
W. Abikoff, The real analytic theory of Teichmuller space, Springer, 1980. (Chap. II section 1).
The idea is that you construct the double: it is the result of gluing of your surface with its mirror image. This is a compact surface, it has a hyperbolic metric, and the restriction of this metric ... | 4 | https://mathoverflow.net/users/25510 | 446520 | 179,900 |
https://mathoverflow.net/questions/446154 | 7 | I am currently reading "On Subsets with Cardinalities of Intersections Divisible by a Fixed Integer" by P. Frankl And A. M. Odlyzko. They used the following result without citation:
For each number $l$, we have a decomposition $l=l\_1+\cdots+l\_q$ with $l\_i\geq\epsilon l$ for a fixed constant $\epsilon$, such that t... | https://mathoverflow.net/users/148253 | About a result on Hadamard matrix | This can be proved using the strategy of Fedor Petrov and a theorem from the following paper:
*Haselgrove, C. B.*, [**Some theorems in the analytic theory of numbers**](https://doi.org/10.1112/jlms/s1-26.4.273), J. Lond. Math. Soc. 26, 273-277 (1951). [ZBL0043.04704](https://zbmath.org/?q=an:0043.04704).
Let $63/64... | 5 | https://mathoverflow.net/users/297 | 446536 | 179,904 |
https://mathoverflow.net/questions/446530 | 1 | Let $\mathcal{X}$ be the input or feature space, let $\mathcal{B}$ be Borel $\sigma$-algebra on $\mathcal{X}$ and $P(\mathcal{X})$ denotes the set of all probability measures on $(\mathcal{X},\mathcal{B})$. Let $P\_1, P\_2 \in P(\mathcal{X})$, The total variation is defined by
$$\delta(P\_1, P\_2) = \sup\_{A\in \mathca... | https://mathoverflow.net/users/504474 | Total variation distance | $\newcommand{\X}{\mathcal X}\newcommand{\B}{\mathcal B}\newcommand{\F}{\mathcal F}\newcommand{\De}{\Delta}\newcommand{\de}{\delta}$Let $\mu:=P\_1-P\_2$, so that $\mu$ is a finite signed measure. By the [Hahn--Jordan decomposition](https://en.wikipedia.org/wiki/Hahn_decomposition_theorem), there exist sets $\X^\pm\in\B$... | 3 | https://mathoverflow.net/users/36721 | 446537 | 179,905 |
https://mathoverflow.net/questions/446539 | 1 | Say we have 2 functions $f$ and $g$ such that:
$f(a)<f(b) \Leftrightarrow g(a)<g(b)\;\; \forall a,b \in \mathbb{R}^n$
Is there an accepted name for a couple of functions like these?
Is there a body of research or some known theorems on this kind of functions?
| https://mathoverflow.net/users/504503 | Pair of functions that vary in the same direction | $\newcommand\R{\mathbb R}$Such pairs of functions may be called comonotone -- cf. [comonotone approximation](https://mathworld.wolfram.com/ComonotoneApproximation.html), which, for $n=1$, is an approximation of a piecewise monotonic function by a polynomial with the same monotonicity.
If functions $f$ and $g$ are com... | 1 | https://mathoverflow.net/users/36721 | 446543 | 179,907 |
https://mathoverflow.net/questions/446441 | 1 | Lions and Paul claim in their 1993 paper "Sur les mesures de Wigner" that the Hamiltonian flow
$$\dot{x} = \xi, \quad \dot{\xi} = - \nabla V(x) $$
of the Vlasov equation
$$\partial\_t f + \xi \cdot \nabla\_x f + \nabla\_x V \cdot \nabla\_{\xi} f = 0$$
is well defined if $V\in C^{1,1}(\mathbb{R}^d)$ and $\exists... | https://mathoverflow.net/users/146998 | Why is this Hamiltonian flow of the Vlasov equation well defined? | I have found an answer.
The condition $V\in C^{1,1}$ ensures that $\nabla V$ is Lipschitz which implies the existence of a global solution in the neighborhood of every point for the Cauchy problem of the differential system $\dot{x} = \xi, \dot{\xi} = -\nabla V(x)$.
We have the following inequality for the energy $... | 2 | https://mathoverflow.net/users/146998 | 446544 | 179,908 |
https://mathoverflow.net/questions/445491 | 3 | When underlying $4$ manifold is compact and hyperkähler, the philosophy of infinite dimensional moment map tells us that its instantons moduli space is also hyperkähler. I'm curious about the following two questions.
1. When our hyperkähler $4$ fold is no longer compact, is its instantons moduli space(maybe with some... | https://mathoverflow.net/users/494608 | Hyperkähler structure of framed instantons over $\smash{\overline{\mathbb{C}P}}^2$ |
>
> When our hyperkähler 4-fold is no longer compact, is its
> instantons moduli space (maybe with some constraint about
> decaying condition) still hyperkähler?
>
>
>
The "decay constraint" condition is called "gravitational instanton".
A gravitational instanton is a complete hyperkähler metric on
a 4-manifold ... | 5 | https://mathoverflow.net/users/3377 | 446548 | 179,909 |
https://mathoverflow.net/questions/446534 | 3 | A subgroup $H$ of a group $G$ is *malnormal* if $gHg^{-1}\cap H=\{e\}$ for all $g\in G$ with $g\notin H$. It is *almost malnormal* if we merely require $gHg^{-1}\cap H$ to be finite.
I am wondering whether $\mathrm{SL}(n,\mathbb{Z})$, or $\mathrm{PSL}(n,\mathbb{Z})$, for $n\geq 2$, have (almost) malnormal non-abelian... | https://mathoverflow.net/users/16107 | Does $\mathrm{SL}(n,\mathbb{Z})$ have an (almost) malnormal free subgroup? | First of all, you have to work with $G=PSL(2, {\mathbb Z})$ and not $SL(2, {\mathbb Z})$, for otherwise the claim is clearly false. Then $G$ is a nonelementary hyperbolic group with trivial maximal finite normal subgroup. According to Lemma 8 in
*Minasyan, Ashot; Olshanskii, Alexander Yu.; Sonkin, Dmitriy*, [**Period... | 5 | https://mathoverflow.net/users/39654 | 446556 | 179,910 |
https://mathoverflow.net/questions/446516 | 9 | In type theory, the dependent sum $\sum\_{x : A} T(x)$ and the dependent product $\prod\_{x:A} T(x)$ are defined by their introduction/elimination rules.
In category theory, we use a base-change functor. Given a morphism $f \colon b \to a$, we define the functor $f^\* \colon C/a \to C/b$ between slice categories usin... | https://mathoverflow.net/users/34546 | Dependent sum/product and the base-change functor adjunctions | The morphism $f$ is invisible in type theory because it corresponds to weakening, which in type theory appears as context extension, rather than an explicitly applied substitution.
#### Type-theoretic explanation
More precisely, given a context $\Gamma$ and a type family $\Gamma \vdash A \; \mathsf{type}$
there is ... | 11 | https://mathoverflow.net/users/1176 | 446559 | 179,911 |
https://mathoverflow.net/questions/446540 | 1 | Let $N \geq 1$ be an integer and let $p$ be a prime not dividing $N$. For $r \geq 1$, let $M\_2(\Gamma\_0(Np^r))$ denote the space of weight $2$ modular forms of level $\Gamma\_0(Np^r)$. Let $$U\_p: M\_2(\Gamma\_0(Np^r)) \to M\_2(\Gamma\_0(Np^r))$$ denote the $p$-th Hecke operator, which acts on $q$-expansions by sendi... | https://mathoverflow.net/users/394740 | What is the image of the Hecke operator $U_p$? | The statement, as claimed, is false.
Let $p = 2, N = 11$, and let $f\_0$ be the unique normalised eigenform in $S\_2(\Gamma\_0(11))$; and set $f(\tau) = f\_0(8\tau)$. Then $f \in M\_2(\Gamma\_0(Np^3))$, but $U\_p(f) = f\_0(4\tau)$ is not in $M\_2(\Gamma\_0(Np))$.
However, $U\_p(f)$ is in $M\_2(\Gamma\_0(Np^2))$. Th... | 4 | https://mathoverflow.net/users/2481 | 446563 | 179,913 |
https://mathoverflow.net/questions/446523 | 4 | I would like to know if there is a "moral" reason why in the definition of [triangulated categories](https://en.wikipedia.org/wiki/Triangulated_category#TR_2) the "rotation axiom" TR2 requires that we have to add a negative sign to an arrow when we rotate the triangles? What was (presumably Verdier's) initial motivatio... | https://mathoverflow.net/users/501436 | Moral reason for negative sign in rotation axiom for triangulated categories | One deeper reason is that this is what you get (no convention) when you have a triangulated category that comes from a stable $\infty$-category.
Ultimately, this boils down to the fact that when you take a loop in some space and reverse its direction, you get minus that loop in $\pi\_1$. The connection to this become... | 3 | https://mathoverflow.net/users/102343 | 446577 | 179,916 |
https://mathoverflow.net/questions/446451 | 11 | In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds:
$$\sqrt{x} = \sum\_{n=0}^{\infty}\frac{\left(\prod\_{k=1}^{n}\left(\frac{3}{2}-k\right)\right)\left(\frac{1}{4}+\frac{x}{2}+\frac{x^2}{4}\right)^{\frac{1}{2}-n}\left(-\frac{1}{4}+\frac{x}{2... | https://mathoverflow.net/users/504411 | New method to compute square roots | Let me "unclutter" the basic formula $S(x,a)=\sqrt{x}$, starting from the definition in the OP,
$$S(x,a) =\sum\_{n=0}^{\infty}\left(\frac{\left(n+1\right)\binom{2n+2}{n+1}}{\left(4n^2-1\right)2^{2n+1}}\sum\_{k=0}^{n}\left(\binom{n}{k}\left(-1\right)^{k+1}a^{1-2k}x^{k}\right)\right).$$
The finite sum over $k$ is the [bi... | 7 | https://mathoverflow.net/users/11260 | 446587 | 179,918 |
https://mathoverflow.net/questions/446512 | 3 | This is a refined version of a [question I have recently posted](https://mathoverflow.net/q/446461/9924).
For a prime $p$, let $\varphi\_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$.
>
> Given an integer $n\ge 3$, what is the smallest $\v... | https://mathoverflow.net/users/9924 | Bins-and-primes (prime divisors of $\prod(a_i-a_j)$, II) | The answer is $\varepsilon(n)=1-\frac{2}{n}$. Clearly, $\lvert\varphi\_p(A)\rvert\ge2$ for all primes $p$. However,
for every $n\ge2$ there is a set $A$ of size $n$ such that $\lvert\varphi\_p(A)\rvert=2$ whenever $\varphi\_p$ is not injective on $A$:
Let $P=\{2,3,\ldots,p\_{n-1}\}$ be the set of the first $n-1$ prim... | 5 | https://mathoverflow.net/users/18739 | 446617 | 179,924 |
https://mathoverflow.net/questions/446611 | 2 | Let $A$ be a finite dimensional algebra over an algebraically closed field. I'm trying to better understand the difference between $A$ being representation infinite and $A$ being $\tau$-tilting infinite when $A$ is a string algebra. So, if $A$ is representation finite, then $A$ is also $\tau$-tilting finite.
I'm wond... | https://mathoverflow.net/users/338456 | Rep infinite, but $\tau$-tilting finite | I think the group algebra of a dihedral $2$-group in characteristic two, mod its socle, is an example. The smallest of these is $k\langle x,y\rangle/(x^2,y^2,xy,yx)$. See Plamondon, "$\tau$-Tilting finite gentle algebras are representation finite."
| 3 | https://mathoverflow.net/users/460592 | 446618 | 179,925 |
https://mathoverflow.net/questions/446565 | 4 | While reading a paper [Hengang Li and Weiping Yan - Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations](https://doi.org/10.1016/j.cnsns.2019.104928), I experienced that my calculation results kept differing from the author's calculation results.
The au... | https://mathoverflow.net/users/113502 | Question about calculation in Schwartz space | The key computation is the commutator $$ \Lambda^s (xf) - x \Lambda^s f. $$ You can check this "classically" in the case $s = 4$ to find $$ (1 - \Delta)^2 (xf) = x (1-\Delta)^2 f - 4 (1-\Delta) f'$$
which differs from was found in the paper (their computation dropped the final commutator term).
In fact, your computat... | 5 | https://mathoverflow.net/users/3948 | 446619 | 179,926 |
https://mathoverflow.net/questions/446613 | 0 | Let $p\_n$ denote the $n$-th consecutive prime number and $g\_n=p\_{n+1}-p\_n$ a prime gap. There are many results about the upper bound for $g\_n$. Some of them still has astatus of conjecture, such as Firoozbakth conjecture (in a prime gap version):
$g\_n<p\_n\left( \sqrt[n]{p\_n}-1\right) $ , $\forall n\in N$, and i... | https://mathoverflow.net/users/169583 | The lower bound for prime gaps | **1.** The Proposition in the post is almost equivalent to the Conjecture, namely it implies that
$$\frac{g\_n}{\log g\_n}\leq (2+o(1))\log n.$$
In particular, the Proposition (hence also the Conjecture) implies that
$$g\_n\ll\log n\,\log\log n.\tag{$\ast$}$$
This would contradict the common expectation that $g\_n\gg (... | 7 | https://mathoverflow.net/users/11919 | 446629 | 179,929 |
https://mathoverflow.net/questions/446631 | 4 | Let $X$ be a random variable and $Y=f(X)$ where $f$ is a deterministic function. Moreover, assume that there exists a deterministic function $g(.)$ such that the following probability is small.
\begin{align}
\mathbb{P}[g(Y)\neq X]\leq\delta.
\end{align}
Assume that $X'$ is another random variable which is close to $X$ ... | https://mathoverflow.net/users/68835 | Effect of small change in probability distribution on error probability | $\newcommand\de\delta\newcommand\ep\epsilon$Let $h:=g\circ f$, so that $g(Y)=h(X)$ and $g(f(X'))=h(X')$. Let $A:=\{x\colon h(x)\ne x)$. Then the condition $P(g(Y)\ne X)\le\de$ can be written as
$$\int\_A p\_X\le\de.$$
So,
$$P(g(f(X'))\ne X')=P(h(X')\ne X')=\int\_A p\_{X'} \\
=\int\_A p\_X+\int\_A (p\_{X'}-p\_X)
\le\i... | 4 | https://mathoverflow.net/users/36721 | 446635 | 179,932 |
https://mathoverflow.net/questions/446388 | 2 | I've been struggling a bit with a double sum that arose as the trace of an operator:
$$\sum\_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$
where $n$ is an even natural number. **Is there a closed form for this sum?**
Even in the case of $n=2$, I'm a bit miffed. If there were no numerator, I would ... | https://mathoverflow.net/users/504359 | 2D lattice sum with numerator | The $n=4$ sum is accessible as follows. Expand $(j+k)^4$; by symmetry
the odd terms $4j^3k$ and $4jk^3$ cancel out so we need only sum
$(j^4 + 6j^2k^2 + k^4) / (j^2+k^2)^4$. You say you already know
the sum of $1 / (j^2+k^2)^2$, which is $(j^4 + 2j^2k^2 + k^4) / (j^2+k^2)^4$ [see also the "P.S." paragraph below];
so we... | 4 | https://mathoverflow.net/users/14830 | 446640 | 179,934 |
https://mathoverflow.net/questions/446644 | 3 | Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
>
> Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ contained in $[-2/N, 2/N]$ and $\int\_{\mathbb R} h(x) dx \leq N$? if so how?
>
>
>
My attempts: Take $\widehat{f}= \c... | https://mathoverflow.net/users/173418 | How to choose some $h$ so its Fourier transform supported in some set? | No. It is already impossible for $h$ to be "band-limited" (i.e. with $\widehat h$ of compact support) and constant on an interval. Indeed by the Fourier inversion formula a band-limited function is analytic, so if $h=1$ on an interval then $h=1$ on all of ${\mathbf R}$, whence $h$ does not have a Fourier transform at a... | 8 | https://mathoverflow.net/users/14830 | 446646 | 179,936 |
https://mathoverflow.net/questions/445814 | 3 | Let $a(n,k)$ be the sequence of $k$-Dowling numbers (for more information see [A007405](https://oeis.org/A007405) and its *CROSSREFS* section) with e.g.f.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
Let
$$\ell(n)=\left\lfloor\log\_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$
$$T(n,k)=\left\... | https://mathoverflow.net/users/231922 | Sequences that sum up to Dowling numbers | Cleaning up the notation a bit,
$$b\_{m,k}(n) = m\, b\_{m,k}(n-2^{\ell(n)}) + k \sum\_{j=0}^{\ell(n)-1} [n \,\&\, 2^j = 0] \,b\_{m,k}(n - 2^{\ell(n)} + 2^j)$$ where $\&$ is bitwise AND.
$$s\_{m,k}(n) = \sum\_{j=0}^{2^n-1} b\_{m,k}(j)$$
---
Let $\operatorname{wt}(n)$ be the Hamming weight of $n$, and for an arbi... | 1 | https://mathoverflow.net/users/46140 | 446656 | 179,937 |
https://mathoverflow.net/questions/446325 | 0 | For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M\_{\phi}:=Y\cup\_{\phi} (X \times \{ 1\})$. Denote $\pi\_n (M\_{\phi},X \times \{ 1\} )$ by $\pi\_n (\phi)$. The map $\phi$ is called $n$-connected if $X$ and $Y$ are connected and $\pi\_i (\phi)=0$ for $1\leq i\leq n$.... | https://mathoverflow.net/users/114476 | Explaining some detail in Wall's paper of CW-complexes | As to (1):
If we choose a basepoint in $K$, then $\phi$ can be viewed as a map of based spaces. Let $F$ be the homotopy fiber of $\phi$. Then there is
a well-defined action $\Omega K \times F \to F$ which induces a $\Bbb Z[\pi\_1(X)]$-module structure on $H\_\*(F)$.
As $F$ is $1$-connected (by the assumptions), the... | 3 | https://mathoverflow.net/users/8032 | 446669 | 179,941 |
https://mathoverflow.net/questions/446654 | 5 | *(For brevity, the level-6 functions have been migrated to [another post](https://mathoverflow.net/q/448777/12905).)*
**I. Level-10 functions**
Given the Dedekind eta function $\eta(\tau)$. To recall, for level-6,
$$j\_{6A} = \left(\sqrt{j\_{6B}} + \frac{\color{blue}{-1}}{\sqrt{j\_{6B}}}\right)^2 =\left(\sqrt{j\_... | https://mathoverflow.net/users/12905 | On level $10$ of the McKay-Thompson series of the Monster | For $s\_{10C}$, Maple finds this $7$-term recurrence:
`{(15625*n^3 + 46875*n^2 + 46875*n + 15625)*u(n) + (11250*n^3 + 61875*n^2 + 115625*n + 73125)*u(n + 1) + (4575*n^3 + 36600*n^2 + 99150*n + 90950)*u(n + 2) + (1116*n^3 + 11718*n^2 + 41434*n + 49322)*u(n + 3) + (183*n^3 + 2379*n^2 + 10371*n + 15157)*u(n + 4) + (18*n... | 3 | https://mathoverflow.net/users/454 | 446673 | 179,942 |
https://mathoverflow.net/questions/446663 | 3 | Let $u: D\_1 \to \mathbf{R}$ be a smooth function defined on the unit disk $D\_1 \subset \mathbf{R}^2$ which describes the minimal graph $G$. Suppose that at the origin $G$ is tangent to the horizontal plane $\{ x^{3} = 0 \}$, that is: $u(0) = 0$ and $Du(0) = 0$. Suppose that in a small disk $D\_r$ around the origin, t... | https://mathoverflow.net/users/103792 | 'Degenerate' tangent point of a minimal graph | Yes, this is possible. Consider the intersection of the helicoid $z = \tan^{-1}(y/x)$ with its tangent plane $z = y$ at $(1,0,0)$. The projection of the intersection curves to $\{z = 0\}$ consists of the line $y = 0$ and the curve $x = y/\tan(y) \sim 1 - y^2/3$. After a rigid motion so that the tangent plane is horizon... | 3 | https://mathoverflow.net/users/16659 | 446681 | 179,945 |
https://mathoverflow.net/questions/446691 | 2 | I was trying to solve some integrals that appear in quantum electrodynamics but I was not able to do it on my own.
$$1/6\int\_0^1 \int\_0^1 { u^3 z^2(1-z^2/3) \over [u^2(1-z^2)+4(1-u)]}dudz $$
I know the answer should be $$(\pi^2 / 18) - (115 / 216)$$
Can anyone help me solve this or at least point me to a book t... | https://mathoverflow.net/users/504630 | Integral in the Lamb shift calculation – fourth order | The double integral in question is
$$I:=\int\_0^1 du\, J(u),$$
where
$$J(u):=\int\_0^1 dz\,{ u^3 z^2(1-z^2/3) \over {u^2(1-z^2)+4(1-u)}} \\
\text{[which is a standard integral, with a denominator of the integrand quadratic in $z$]} \\
=\frac{3 \left(u^3-6 u+4\right) \ln(1-u)+u \left(-5 u^2-12
u+12\right)}{9 u^2} \\ ... | 2 | https://mathoverflow.net/users/36721 | 446697 | 179,948 |
https://mathoverflow.net/questions/446378 | 16 | (cross-posted from [this math.SE question](https://math.stackexchange.com/questions/4679118/does-a-completely-metrizable-space-admit-a-compatible-metric-where-all-intersect))
It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with re... | https://mathoverflow.net/users/121875 | Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty? | Let us say that a topological space $X$ is *spherically completely metrizable* if the topology of $X$ is generated by a spherically complete metric.
>
> **Theorem.** Every closed subspace $X$ of the countable product of locally compact metrizable spaces is spherically completely metrizable.
>
>
>
*Proof.* We l... | 10 | https://mathoverflow.net/users/61536 | 446702 | 179,949 |
https://mathoverflow.net/questions/446671 | 10 | I have discovered a pertinent solution to my problem in the article *On the Kinetic Theory of Rarefied Gases* by Harold Grad and the book *Thermodynamik und Statistik* by Arnold Sommerfeld, both of which present the same proof for the issue I was addressing. The proof assumes the continuity of the function $f$.
---... | https://mathoverflow.net/users/478784 | Proving the simple form of a function from statistical mechanics | We can indeed prove this for reasonable functions, $\log f\_0\in C^2$, say.
Let me write $F=\log f\_0$. By replacing $F$ by $F(v)-C-d\cdot v$, we can also assume that $F(0),\nabla F(0)=0$.
If $a,v$ are orthogonal, then, by assumption,
$$
F(v)+F(a)=F(v+a)+F(0)=F(v+a) ,
$$
and for small $a$, we have $F(v+a)\simeq F(v... | 8 | https://mathoverflow.net/users/48839 | 446732 | 179,955 |
https://mathoverflow.net/questions/446737 | 6 | Let $f\colon M\to N$ a smooth surjective map of compact oriented manifolds of the same dimension. Then there is a map $f\_!\colon H\_i(N)\to H\_i(M)$ obtained from the induced map on cohomology combined with Poincaré duality. This map has several names. I have seen it called the transfer, umkehr or wrong-way map. And i... | https://mathoverflow.net/users/36563 | Geometric interpretation of transfer map on homology | Let $K\subset N$ be a compact oriented smooth submanifold of codimension $k$ in an oriented smooth manifold $N$, $T\subset N$ a tubular neighbourhood of $K$ and $\tau$ a $k$-form on $N$ supported on $T$ such that the restriction of $\tau$ to the fibre of $T$ over any point of $K$ (which is of course diffeomorphic to $\... | 8 | https://mathoverflow.net/users/485324 | 446744 | 179,958 |
https://mathoverflow.net/questions/446742 | 8 | In the Author Commentary to the reprint of the paper paper *[Diagonal Arguments and Cartesian Closed Categories](http://www.tac.mta.ca/tac/reprints/articles/15/tr15abs.html)* in *Theory and Applications of Categories* Bill Lawvere wrote:
>
> Although the cartesian-closed view of function spaces and functionals was ... | https://mathoverflow.net/users/73577 | Mention of Bernoulli principle by Bill Lawvere | Since the topic of Lawvere paper is differential geometry, the Bernoulli is likely [Jacob Bernoulli,](https://en.wikipedia.org/wiki/Jacob_Bernoulli) or his brother [Johann,](https://en.wikipedia.org/wiki/Johann_Bernoulli) and refers to their calculus of variations and the principle of virtual work. Further evidence is ... | 5 | https://mathoverflow.net/users/11260 | 446749 | 179,960 |
https://mathoverflow.net/questions/446658 | 2 | As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, model categories (as an independent concept, not a tool) are of no interest and behave rather strangely (starting with the zigzag... | https://mathoverflow.net/users/148161 | Why do we need enriched model categories? | To me, the interest in model categories stems from Quillen's observation that the tools of topology (e.g., CW approximation) can be applied in so many different settings, especially in algebra. But not all of those settings are simplicial model categories. Many of them are dg-model categories, i.e., enriched in the cat... | 9 | https://mathoverflow.net/users/11540 | 446755 | 179,964 |
https://mathoverflow.net/questions/446739 | 4 | Consider an $n$-dimensional convex set $K \subset \mathbb{R}^n$ and let $\mu$ denote the Gaussian measure with density
$$
\gamma(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}} e^{-\lVert \mathbf{x} \rVert^2/2}.
$$
I am trying to figure out if, as I slide the convex body $K$ along a straight line, its Gaussian measure, viewed as ... | https://mathoverflow.net/users/100355 | Sliding a convex body over a Gaussian measure | $\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$We have
$$\mu(\u+t\v+K)=f(t):=\int\_{\R^n}d\x\,F(\x,t),$$
where
$$F(\x,t):=\ga(\x)\,1(\x\in\u+t\v+K).$$
The function $F\colon\R^n\times\R\to\R\_+$ is log concave, as the product of two log-concave funct... | 7 | https://mathoverflow.net/users/36721 | 446756 | 179,965 |
https://mathoverflow.net/questions/446711 | 3 | I am looking for a sequence of topological spaces $X\_n$, $n\in\mathbb N$, with the following property. Let $\tilde{K}^0(X\_n)$ be the complex reduced $K$-theory group of $X\_n$ (with respect to some choice of base point). I would like, for each $n$, for there to be a class $\xi\_n\in\tilde{K}^0(X\_n)$ such that $\xi\_... | https://mathoverflow.net/users/78729 | "High-dimensional" classes in topological $K$-theory | Let $X\_n = S^{2n+2}$.
Since $\operatorname{ch} : K(S^{2n+2})\otimes\_{\mathbb{Z}}\mathbb{Q} \to H^{\text{even}}(S^{2n+2}; \mathbb{Q})$ is an isomorphism, there is a complex vector bundle $E \to S^{2n+2}$ of rank $n + 1$ with $\operatorname{ch}\_{n+1}(E) \neq 0$, i.e. $c\_{n+1}(E) \neq 0$. Let
$$\xi\_n = E - \varep... | 3 | https://mathoverflow.net/users/21564 | 446758 | 179,966 |
https://mathoverflow.net/questions/446775 | 4 | Let $A$ be a complete, Noetherian, local ring with finite residue field of characteristic $p$. If $F$ is a non-Archimedean local field, then we will denote the ring of integers of $F$ by $\mathcal{O}\_F$. Let $I$ denote the intersection of kernels of all (local) morphisms $A\to \mathcal{O}\_F$ where $F$ runs over all n... | https://mathoverflow.net/users/149460 | Is a complete local ring determined by its values in local fields? | The paper you cite itself cites Corollary 10.5.8 of EGA 4, part III. Corollary 10.5.9 says the points of dimension $1$ in $\operatorname{Spec} A$ are dense in $\operatorname{Spec} A - \mathfrak m$. Each point of dimension $1$ corresponds to a homomorphism to a complete local ring of dimension $1$, whose field of fracti... | 6 | https://mathoverflow.net/users/18060 | 446782 | 179,974 |
https://mathoverflow.net/questions/446788 | 5 | Let $L\_n(p)$ be the $2n+1$ dimensional Lens space
$$
S^{2n+1}/\mathbb{Z}\_p
$$
where the action is given as $z\_i\rightarrow e^{\frac{2\pi}{p}}z\_i$, $i=1,...,n+1$, with $z\_i$ the coordinates of $\mathbb{C}^{n+1}$ such that $S^{2n+1}$ is $|z\_1|^2+...+|z\_{n+1}|^2=1$. For $k\neq 0,2n+1$ the homology groups with coeff... | https://mathoverflow.net/users/495347 | Computation of the linking invariant on Lens spaces | In my thesis, I gave the calculation of the linking form on homology, which is equivalent to the question you asked. I credited the calculation to de Rham ([Sur L'analysis situs des varietés a n dimensions](http://www.numdam.org/item/THESE_1931__129__1_0.pdf), J. Math. Pures et Appl., 10 (1931), 115-200.) See Propositi... | 5 | https://mathoverflow.net/users/3460 | 446802 | 179,978 |
https://mathoverflow.net/questions/446790 | 3 | Let $(X, g)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms on $X$. Then the Hodge-star operator decomposes $\Lambda^2$ into the space of self-dual and anti-self-dual 2-forms
$$\Lambda^2 = \Lambda^2\_+ \oplus \Lambda^2\_- .$$
Both $\Lambda^2\_+$ and $\Lambda^2\_-$... | https://mathoverflow.net/users/24965 | Do we have $d(P_{+}(\omega\wedge \theta))=d_{+}\omega\wedge \theta-\omega\wedge d_{+}\theta$ on a self-dual manifold? | No. If this formula were true, then we would have
$$
\mathrm{d}\bigl(P\_+(\mathrm{d}f\wedge\mathrm{d}g)\bigr) = 0
$$
for all smooth functions $f$ and $g$, since $\mathrm{d}\_+(\mathrm{d}f) =P\_+\bigl(\mathrm{d}(\mathrm{d}f)\bigr) = 0$.
Now, consider $\mathbb{R}^4$ with its standard flat metric $g = (\mathrm{d}x\_1)^2... | 5 | https://mathoverflow.net/users/13972 | 446803 | 179,979 |
https://mathoverflow.net/questions/446701 | 2 | Let $\mathbb{S}^2\_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$, and the associated eigenfunctions are the coordinate functions $x,y,z$ restricted to the sphere. I wonder if the fun... | https://mathoverflow.net/users/85934 | Are these the only first eigenfunctions on a hemisphere? | As Christian Remling already indicated in the comments: one can use reflection techniques and thus show: eigenfunctions of the Laplace-Beltrami operator on the $n$-dimensional hemisphere with Neumann boundary conditions are in 1-to-1 correspondence to eigenfunctions on the $n$-dimensional sphere that are invariant unde... | 2 | https://mathoverflow.net/users/110127 | 446815 | 179,985 |
https://mathoverflow.net/questions/446830 | 2 | I am struggling to find a reference for the following statement, which I still believe to be true.
"Let $(\Omega\_1, \mathcal{A}\_1, \mu\_1), (\Omega\_2, \mathcal{A}\_2, \mu\_2)$ be finite measure spaces. Furthermore, let $(\Omega\_1\times\Omega\_2, \mathcal{A}\_1\otimes\mathcal{A}\_2, \mu\_1\otimes\mu\_2)$ the usual... | https://mathoverflow.net/users/504794 | Product sigma-algebra: approximating elements arbitrary good using the generating sets | $\newcommand\Om\Omega\newcommand\A{\mathcal A}\newcommand\I{\mathbb I}$Your original statement is trivial: Take $B\_1^i=\Omega\_1$ and $B\_2^i=\Omega\_2$ for all $i$.
If you additionally require that $B\_1^i\times B\_2^i\subseteq A$ for all $i$, then the statement will become false in general. For instance, for $j=1,... | 0 | https://mathoverflow.net/users/36721 | 446837 | 179,994 |
https://mathoverflow.net/questions/446705 | 7 | Consider the quantum group $U\_q(\mathfrak{sl}\_2)$, with generators $E,F,K$ such that $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$. Write $[n]=\frac{q^n-q^{-n}}{q-q^{-1}}$, and $[n]!=[n][n-1]\dotsm[1]$.
In [Quantum deformations of certain simple modules over enveloping algebras](https://doi.org/10.1016/0001-8708(88)90056-4), L... | https://mathoverflow.net/users/138150 | What is the ring structure on Lusztig's integral form of quantum $\mathfrak{sl}(2)$? | I figured it out, but I would really appreciate a reference!
The formula is quite nice. For $c,d\ge0$ such that $c\le t $ and $d\le s$, the following holds:
$$ \newcommand\qbinom{\genfrac[]0{}}\qbinom{K;\ c}t\qbinom{K;\ d}s=\sum\_{i\ge 0}\qbinom{t-c+d}{i-c}\qbinom{s-d+c}{i-d}\qbinom{K;\ i}{t+s}.$$
(Here, the binomi... | 3 | https://mathoverflow.net/users/138150 | 446852 | 179,999 |
https://mathoverflow.net/questions/446738 | 2 | Recently I was thinking if there is a way to do the following: assuming I have some sampled points of distribution $\mathcal{X}$ and distribution $\mathcal Z$ (whose MGF I do not have in closed form) and I know that $\text{law}(\mathcal X)\cdot\text{law}(\mathcal Y) = \text{law}(\mathcal Z)$ can I obtain $\mathcal{Y}$?... | https://mathoverflow.net/users/504682 | Approximation to ratio distribution | $\newcommand\R{\mathbb R}\newcommand\Z{\mathbb Z}$After [James Martin's clarifying comment](https://mathoverflow.net/questions/446738/approximation-to-ratio-distribution?noredirect=1#comment1154351_446738), the question becomes as follows:
>
> Suppose that $Z=XY$, where $X$ and $Y$ are independent positive random v... | 2 | https://mathoverflow.net/users/36721 | 446854 | 180,000 |
https://mathoverflow.net/questions/446848 | 7 |
>
> Is there a cubic polynomial $c(x,y)$ with exactly 3 saddle point critical points?
>
>
>
In other words, can a cubic polynomial in two variables have three critical points, all of which are saddle points? Or could it have the maximum 4 critical points (as per Bézout's theorem), with only one of them being a l... | https://mathoverflow.net/users/497175 | Can a cubic polynomial in two real variables have three saddle points? | The cubic $x^3 - xy^2 - 2x^2 + x$ has critical points in $(1,0)$, $(0,-1)$, $(0,1)$ and $(1/3, 0)$. The determinant of the Hessian matrix is $-4(3x^2 + y^2 - 2x)$. It assumes the values $-4$, $-4$, $-4$ and $4/3$ in these four critical points. Thus the first three of them are saddle points.
**Added later:** A simpler... | 14 | https://mathoverflow.net/users/18739 | 446858 | 180,001 |
https://mathoverflow.net/questions/442076 | 0 | I have a 600-cell, whose coordinates are given by
$$\begin{array}{ccc}
\text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\
\text{16 vertices} & \frac{1}{2}\left(\pm1,\pm1,\pm1,\pm1\right),\\
\text{96 vertices} & \frac{1}{2}\left(\pm\tau,\pm1,\pm\tau^{-1},0\right) & \text{even permutations}.
\end{... | https://mathoverflow.net/users/83165 | How can I find the hyperplane passing through a 600-cell | You already could have considered your provided vertices within layers according to their last coord values. Within decreasing order you get:
$(0, 0, 0; 1)$: the single point `o3o5o` at the north pole
$\frac12(0,\pm\tau^{-1},\pm1; \tau)$: a full icosahedron `v3o5o` (with $\tau^{-1}$-sized edges)
$\frac12(\pm1, \p... | 1 | https://mathoverflow.net/users/118679 | 446863 | 180,003 |
https://mathoverflow.net/questions/446851 | 3 | For integral homology groups there is the notion of linking form (<http://www.map.mpim-bonn.mpg.de/Linking_form>)
$$
Tor(H\_{l}(X,\mathbb{Z}))\times Tor(H\_{n-l-1}(X,\mathbb{Z}))\rightarrow \mathbb{Q}/\mathbb{Z}
$$
for the torsion part of the homology groups. This can be defined by using the Bockstein map associated wi... | https://mathoverflow.net/users/495347 | Linking form for homology with general coefficients | Expanding on Ryan's comment, one way to get the torsion linking form you mention (at least for compact oriented manifolds) is through the sequence of isomorphisms $$Tor(H\_l(X;\mathbb{Z}))\cong Tor(H^{n-l}(X;\mathbb{Z})) \cong Ext(H\_{n-l-1}(X);\mathbb{Z})\cong Ext(Tor(H\_{n-l-1}(X));\mathbb{Z})\cong Hom(Tor(H\_{n-l-1}... | 3 | https://mathoverflow.net/users/6646 | 446873 | 180,004 |
https://mathoverflow.net/questions/446868 | 1 | Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it is well-known that
\begin{equation}
\text{Var}(f(X)) \leq E[\bigl(f'(X)\bigr)^2]
\end{equation}
and is called the Gaussian Po... | https://mathoverflow.net/users/56524 | Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$? | Yes, Gaussians also satisfy a Poincaré inequality with $p = 1$ (such an inequality is equivalent to what is called a "Cheeger inequality"). More generally, E. Milman has shown that for log-concave measures, all $(p, q)$-Poincaré inequalities are equivalent:
Milman, E. On the role of convexity in isoperimetry, spectra... | 1 | https://mathoverflow.net/users/37014 | 446874 | 180,005 |
https://mathoverflow.net/questions/446872 | 7 | It is well-known that for a finite group $G$ and field $k$ of characteristic 0, the linearization morphism $B(G) \to R\_k(G)$ has in most cases nontrivial kernel, and this can be used to find permutation modules which admit non-isomorphic permutation bases (as $G$-sets). I'm hoping to find a relatively easy-to-state ex... | https://mathoverflow.net/users/152544 | Examples of permutation $\mathbb{Z}G$-modules which admit non-isomorphic permutation bases? | A similar question was asked on [math.stackexchange](https://math.stackexchange.com/q/2730330) a few years ago, and I posted the following answer. I've just looked again at Conlon's paper, and I'm afraid it's a bit short on explicit examples.
*===========================================================*
There are f... | 8 | https://mathoverflow.net/users/22989 | 446883 | 180,009 |
https://mathoverflow.net/questions/446877 | 5 | I asked this question some time ago in MSE but I didn't recieved any feedback.
<https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions>
This problem arised to me when I was trying to find an analog to orthogonal reference frames for singular metric tensors.
Let $U\subs... | https://mathoverflow.net/users/148711 | Diagonalization of symmetric matrices of functions | In general, this cannot be done. For example, in dimension $2$ in coordinates $(x,y)$, let
$$
G(x,y) = \left[\begin{matrix}x&y\\y&-x\end{matrix}\right].
$$
If $G$ could be diagonalized by a differentiable invertible matrix $A(x,y)$, i.e., if
$$
A^T G A = \left[\begin{matrix}\lambda\_1&0\\ 0&\lambda\_2\end{matrix}\right... | 8 | https://mathoverflow.net/users/13972 | 446894 | 180,012 |
https://mathoverflow.net/questions/446843 | 1 | Let $B: C^{\infty}([0,1]^3)$ satisfy
$$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$
Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation:
$$ \int\_0^1 f(t,x)\,dx + \int\_0^t\left(\int\_0^1 f(s,x)\,B(t,s,x)\,dx\right)\,ds =0, \quad \forall\, t\in (0,1).$$
Does it follow that
$$ \int\_0^1 f(t,x)\... | https://mathoverflow.net/users/50438 | On an integral equation | The answer is no. A counterexample is
$$
f(t,x) = x - \frac{1}{2} -\frac{1}{24} t^2
$$
$$
B(t,s,x) = \left( x-\frac{1}{2} \right) (t-s)
$$
(Method: I obtained this by expanding $f$ and $B$ into power series in the arguments $t$ and $s$, considering the integral equation order by order in $t$, fiddling a bit with how fe... | 3 | https://mathoverflow.net/users/134299 | 446912 | 180,020 |
https://mathoverflow.net/questions/446919 | -3 | The starting point of this question is the observation that in ${\sf (ZFC)}$, all ordinals $\alpha < \omega\_1$ can be [order-embedded in](https://math.stackexchange.com/questions/408300/countable-ordinals-are-embeddable-in-the-rationals-bbb-q-proofs-and-their) $\mathbb{R}$.
Let $\omega^\omega$ denote the set of all ... | https://mathoverflow.net/users/8628 | Order-embeddability of ${\frak b}$ and ${\frak d}$ in $\mathbb{R}$ | Assume that $i : \alpha \to \mathbb{R}$ is an order-embedding of some ordinal $\alpha$ into $(\mathbb{R},<)$. We can modify $i$ to yield an order embedding $j : \alpha \to \mathbb{Q}$ by induction over $\beta < \alpha:
The interval $[i(\beta), i(\beta + 1)]$ is a non-degenerate interval in $\mathbb{R}$, which contain... | 1 | https://mathoverflow.net/users/15002 | 446920 | 180,023 |
https://mathoverflow.net/questions/446888 | 2 | Problem :
Show that :
$$\frac{1}{\zeta(3)}<2C-1$$
Where we can see the zeta function and the Catalan's constant .
After a bounty on Maths Stack Exchange there is no satisfying answer .
See <https://math.stackexchange.com/questions/4693000/show-that-frac1-zeta32c-1>
My last attempt was to use Cauchy-Schwarz ... | https://mathoverflow.net/users/147649 | An inequality related to Catalan's constant and $\zeta(3)$ | One has
\begin{equation}
C=\sum\_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}
=\sum\_{k=0}^\infty f\_1(k)=S\_{1,c}(f\_1)+S\_{2,c}(f\_1),
\end{equation}
where
\begin{equation}
S\_{1,c}(f):=\sum\_{k=0}^{c-1}f(k),\quad S\_{2,c}(f):=\sum\_{k=0}^\infty f(c+k),
\end{equation}
\begin{equation}
f\_1(x):=\frac{1}{(4 x+1)^2}-\frac{1}... | 6 | https://mathoverflow.net/users/36721 | 446926 | 180,025 |
https://mathoverflow.net/questions/441631 | 5 | Does every triangulable manifold have a vertex-transitive triangulation?
When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically as a simplicial complex such that the automorphism group of the simplicial complex is transitive on the vertices or 0... | https://mathoverflow.net/users/495429 | Does every triangulable manifold have a vertex-transitive triangulation? | There [exists](https://mathscinet.ams.org/mathscinet-getitem?mr=953960) many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence [trivial mapping class group](https://www.ams.org/journals/jams/1997-10-01/S0894-0347-97-00206-3/). $M$ cannot be homeomorphic to a simplicial complex $\tau$ wh... | 5 | https://mathoverflow.net/users/1345 | 446933 | 180,030 |
https://mathoverflow.net/questions/446911 | 11 | The eigenvalue map in question is
$\sigma: {\mathfrak gl}(\mathbb{C}, n) \to S\_n \backslash \mathbb{C}^n$,
from $n$ by $n$ complex matrices to $\mathbb{C}^n$ vectors modulo permutation of entries by $S\_n$, and is known to be *continuous*.
This map can be constructed by first considering the root map $\rho: \mathbb{... | https://mathoverflow.net/users/12170 | Is the eigenvalue map open? | Yes. Write $D(\lambda\_1, \ldots, \lambda\_n)$ as the diagonal matrix with diagonal entries $(\lambda\_1, \ldots, \lambda\_n)$.
Let $A$ be any complex $n \times n$ matrix. We can upper-triangularize $A$ as $A = S (D(\lambda\_1, \ldots, \lambda\_n)+N) S^{-1}$ where $N$ is upper triangular. Let $U$ be an open ball arou... | 13 | https://mathoverflow.net/users/297 | 446934 | 180,031 |
https://mathoverflow.net/questions/446932 | 4 | If $(P,\leq)$ is a partially ordered set and $a,b\in P$ we set $[a,b]:=\{x\in P: a\leq x\leq b\}$. We say that $P$ is *fractal* if whenever $a,b\in P$ and $[a,b]$ contains more than one element, then $[a,b]\cong P$.
So for instance, $[0,1]$ and $[0,1]\cap \mathbb{Q}$ are fractal with their usual linear orderings.
F... | https://mathoverflow.net/users/8628 | Is ${\cal P}(\omega)/\text{(fin)}$ a fractal poset? | Yes, $P(\omega)/\text{fin}$ is fractal. If $A\subseteq^\* B$ but not equivalent, then the interval $[A,B]$ in $P(\omega)/\text{fin}$ consists of the sets that almost contain $A$ and are almost contained in $B$, and this is isomorphic to $P(B-A)/\text{fin}$, which is isomorphic to $P(\omega)/\text{fin}$, since $B-A$ wil... | 7 | https://mathoverflow.net/users/1946 | 446938 | 180,032 |
https://mathoverflow.net/questions/446937 | 1 | Let $0 < a < 1$ be an irrational number. Is it true that
$$\liminf\_{n \in \mathbb N, n \to \infty} n \{na\} = 0?$$
*Note: Here $\{\cdot\}$ denotes the fractional part.*
| https://mathoverflow.net/users/173490 | The liminf of an expression involving an irrational rotation | The answer is no, because for some real $c>0$ and all integers $q>0$ we have
$$q\{q\sqrt2\}=q(q\sqrt2-\lfloor q\sqrt2\rfloor)
=q|q\sqrt2-\lfloor q\sqrt2\rfloor| \\
\ge q\,\inf\_{p\in\mathbb Z}|q\sqrt 2-p|
=q^2\,\inf\_{p\in\mathbb Z}|\sqrt 2-p/q|\ge c.$$
So, $\liminf\limits\_{q\to\infty} q\{q\sqrt2\}\ge c>0$.
More spe... | 6 | https://mathoverflow.net/users/36721 | 446941 | 180,034 |
https://mathoverflow.net/questions/446740 | 1 | I understand the concept of the 1 dimensional Brownian bridge with the form of:
$$dx\_t=\frac{-1}{1-t}x\_t \, dt + dw\_t$$
s.t. $x\_0=0$ and $x\_1=0$
where $dw\_t$ is a Wiener process.
I am thinking about the Brownian bridge in the phase space with arbitrary boundary conditions:
$$dx\_t=v\_t \, dt$$
$$dv\_t... | https://mathoverflow.net/users/504685 | Phase space Brownian bridge | I use capital letters for random variables and small letters for possible values.
Let $W$ be a brownian motion, defined on the canonical space $\mathcal{C}(\mathbb{R}\_+)$ endowed with the Wiener measure $\mathbb{P}$ and with the canonical filtration $\mathcal{F}$. For $t>0$, call $p\_t$ the density of the random var... | 0 | https://mathoverflow.net/users/169474 | 446953 | 180,038 |
https://mathoverflow.net/questions/446813 | 13 | I am looking for an example of a locally compact group $G$ and a continuous $G$ module $M$, which also is locally compact, such that the continuous cochain cohomology differs from group cohomology (ignoring the topologies). In the literature, I find many results, giving criteria when they agree, but I could not find an... | https://mathoverflow.net/users/473423 | Example of continuous cohomology vs cohomology | Consider the circle $G=\mathbb{R}/\mathbb{Z}$ and its trivial representation on $M=\mathbb{R}$.
Since $G$ is abelian, $H^1(G,M)\cong \text{Hom}(G,M)$ and $H^1\_c(G,M)\cong \text{Hom}\_c(G,M)$ (these are the continuous versions).
Clearly, $\text{Hom}\_c(G,M)=0$.
However, as abstract groups $\mathbb{R}\cong \mathbb{Q}\op... | 8 | https://mathoverflow.net/users/89334 | 446956 | 180,039 |
https://mathoverflow.net/questions/446948 | 2 | (*Note*: This third method continues from [this post](https://mathoverflow.net/q/446886/12905).)
There are level-$7$ pi formulas based on the McKay-Thompson series $T\_{7A}$ and Cooper's $s\_7$ sequence in this [paper](https://www.researchgate.net/publication/257642843_Sporadic_sequences_modular_forms_and_new_series_... | https://mathoverflow.net/users/12905 | Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$ | For Question $3$ about the recurrence relations, using my code from
[MMA question 285008](https://mathematica.stackexchange.com/q/285008/)
for $a\_n := T\_{7A}(n)$ I used
`findseqrecur[4, 4, Array[t7A, 33, 1], 1, "a", k, -1]` to get
$$ 0 = 14(n+1)(n+2)(2n+3) a\_n \\
-3(n+2)(19n^2+76n+80) a\_{n+1} \\
+ 5(2n+5)(3n^2+... | 2 | https://mathoverflow.net/users/113409 | 446969 | 180,041 |
https://mathoverflow.net/questions/446962 | 3 | Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H\_1, H\_2, H\_3)$ with $H\_i \in H^0(\mathcal{O}\_{\mathbb{P}^8}(1))$ a hyperplane section such that $V \cap H\_1 \cap H\_2 \cap H\_3$ is a proper inters... | https://mathoverflow.net/users/45397 | Segre embedding and intersections by hyperplanes | This is a standard projective duality argument.
Let $W = H^0(\mathcal{O}\_{\mathbb{P}^8}(1))$. Consider the variety $X$ of tuples
$$
(P,H\_1,H\_2,H\_3) \in V \times W^{\oplus 3}
$$
such that $V \cap H\_1 \cap H\_2 \cap H\_3$ is singular at $P$. Then it is easy to see that the projection
$$
X \to V,
\qquad
(P,H\_1,H\_... | 5 | https://mathoverflow.net/users/4428 | 446974 | 180,043 |
https://mathoverflow.net/questions/446959 | 1 | Let $G$ be a finite abelian group and denote by $G^{\vee}=\mathrm{Hom}(G,U(1))$ its Pontryagin dual. For any positive integer $n$ one can define a homomorphism of abelian groups
$$
f:H^{n}(G,G^{\vee})\rightarrow H^{n+1}(G,U(1))
$$
Its action on the chains is
$$
f(c)(g\_1,...,g\_{n+1})=\langle c(g\_1,....,g\_n),g\_{n+1}... | https://mathoverflow.net/users/495347 | A map in group cohomology from $H^n(G,G^{\vee})$ to $H^{n+1}(G,U(1))$ | Let $I\_G$ be the augmentation ideal of $\mathbb{Z}G$. Given that $G$ is abelian, we have $I\_G/I\_G^2\cong G$ by the Hurewicz isomorphism sending the coset of $g-1$ to $g$. Then your map is induced by the surjection $I\_G\to I\_G/I\_G^2\cong G$ in the following sense. We can regard $H^n(G,G^\vee)$ as $\underline{Hom}(... | 5 | https://mathoverflow.net/users/460592 | 446976 | 180,045 |
https://mathoverflow.net/questions/446979 | 9 |
>
> By Bézout's theorem a quartic polynomial $p(x,y)$ can have at most 9 isolated critical points. Can all of them be saddle points?
>
>
>
In case of a cubic polynomial there is a mechanical way to answer this type of questions: One can find a general form of such polynomials with critical points at three given ... | https://mathoverflow.net/users/497175 | How many saddle points can a quartic polynomial in two real variables have? All 9? | By (3.1) of [Counting Critical Points of Real Polynomials in Two Variables by Alan Durfee, Nathan Kronefeld, Heidi Munson, Jeff Roy, Ina Westby](https://www.jstor.org/stable/2324459?seq=7) a degree $d$ polynomial with only nondegenerate critical points can contain at most $d(d-1)/2$ saddle points. For $d=4$ this gives ... | 14 | https://mathoverflow.net/users/18060 | 446982 | 180,047 |
https://mathoverflow.net/questions/446978 | 36 | In a recent [talk](https://agenda.unige.ch/events/view/36495) at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online.
If I'm reading the abstract right, his conjecture (which, as I understand from it, is p... | https://mathoverflow.net/users/497064 | Clausen's modified Hodge Conjecture | It's a bit of a long story, but I can at least give the idea. Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hodge conjecture says that for all $p\geq 0$, the cycle class map
$$Ch^p(X)\_{\mathbb{Q}} \to Hdg^p(X)\_{\mathbb{Q}}$$
from codimension p algebraic cycles to Hodge classes is surjective, with ... | 59 | https://mathoverflow.net/users/3931 | 446992 | 180,049 |
https://mathoverflow.net/questions/446993 | 3 | Let $(\mathcal T, \otimes)$ be a tensor tringulated (tt-)category. Balmer defined a functor from the category of tt-categories to the category of locally ringed spaces, called the Balmer spectra or tt-spectra. He showed that the functor is not essentially injective if we set the target to be the category of topological... | https://mathoverflow.net/users/177839 | "Essential injectivity" of Balmer spectra | No this is not true in general. In [Tensor Triangulated Categories in
Algebraic Geometry](https://www.math.uni-hamburg.de/home/sosna/diplom-online.pdf), Prop 4.0.9, Sosna shows that if X is a connected noetherian scheme then it is possible to put a tt-structure $\boxtimes$ on the derived category of $X\amalg X$ such th... | 5 | https://mathoverflow.net/users/44499 | 447011 | 180,056 |
https://mathoverflow.net/questions/447014 | 13 | Assuming the negation of CH, let $\omega\_1$ be the first uncountable ordinal, $\mathfrak{c}$ be the cardinality of the continuum. Does there exist a map $f: \omega\_1 \times [0, 1] \rightarrow \mathfrak{c}$ s.t. for all $t \in \mathfrak{c}$, we have $t \in f(\omega\_1 \times \{s\})$ for Lebesgue measure a.e. $s \in [0... | https://mathoverflow.net/users/504602 | Almost everywhere “filling” of the continuum by the first uncountable cardinal without CH | The existence of such an $f$ is consistent with $\mathsf{ZFC+\neg CH}$. Suppose $\mathfrak{c}=\aleph\_2$ and every set of size $\le\aleph\_1$ is null *(this is consistent with $\mathsf{ZFC}$; it follows, for example, from [$\mathsf{2^{\aleph\_0}=\aleph\_2+MA}$](https://en.wikipedia.org/wiki/Martin%27s_axiom))*. Fix a b... | 15 | https://mathoverflow.net/users/8133 | 447016 | 180,058 |
https://mathoverflow.net/questions/447024 | 6 | A poset $(P,\leq)$ is *homogeneous* if $P\cong [a,b]$ for all $a,b\in P$ with $a<b$ (where $[a,b] := \{x\in P: a\leq x\leq b\}$).
Examples of homogeneous posets include $[0,1]$, $[0,1]\cap \mathbb{Q}$, and [the Boolean algebra ${\cal P}(\omega)/(\text{fin})$](https://mathoverflow.net/questions/446932/is-cal-p-omega-t... | https://mathoverflow.net/users/8628 | Is every homogeneous poset a lattice? | **Counterexample.** Let
$$P=\{(x,i)\in\mathbb Q\times\{0,1\}:0\le x\le1,\ x\ne i\}$$
be ordered so that
$$(x,i)\lt(x',i')\iff x\lt x'.$$
| 11 | https://mathoverflow.net/users/43266 | 447025 | 180,059 |
https://mathoverflow.net/questions/447023 | 1 | Edges of the complete graph on $2n$ vertices can be colored with $2n-1$ colors such that only edges of different colors intersect.
*Can this always be done such that for every pair of different colors the set of edges of these two colors form a unique cycle?*
Reformulation: Are there $2n-1$ fixed-point-free involut... | https://mathoverflow.net/users/4556 | Hamiltonian edge colouring of complete graphs with even numbers of vertices | This is called a "perfect 1-factorisation". Existence is incompletely determined.
<https://core.ac.uk/download/pdf/82799957.pdf> is an older source.
<https://math.stackexchange.com/questions/3455298/perfect-1-factorization-of-k-2n> contains some more recent refs. Searching on the name will find many more.
| 3 | https://mathoverflow.net/users/9025 | 447026 | 180,060 |
https://mathoverflow.net/questions/447032 | -2 | I am looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:
1. $f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.
2. $f(m,n) \le f(r,s)$ if and only if $m n \le r s$, with $m,n,r,s$ integers $> 0$.
I would like to know if functions with properties 1... | https://mathoverflow.net/users/116669 | Two-variable continuous function which results in an integer if and only if arguments are integer | Such functions as you require do not exist. Your requirements impose that $f(1,1) < f(1,2) < f(2,2)$ (for example, $f(1,1) \leq f(1,2)$ by (2), but $\neg (f(1,2) \leq f(1,1))$ also by (2), so we have $f(1,1) < f(1,2)$). Now consider the continuous function $x \mapsto f(x,x)$ on $[1,2]$: by the intermediate value theore... | 8 | https://mathoverflow.net/users/17064 | 447033 | 180,062 |
https://mathoverflow.net/questions/446951 | 8 | Consider a rigid braided monoidal category, with braiding $\beta\_{x,y} : x \otimes y \cong y \otimes x$, and every object has a dual such that $\epsilon\_x : 1 \to a \otimes a^\*, \bar\epsilon\_x : a^\* \otimes a \to 1$ satisfying the zig-zag identities. Now we have the Reidemeister moves, e.g.
$$ (c \otimes \beta\_{a... | https://mathoverflow.net/users/136535 | Is there a notion of "knot category"? | To expand on my comment, this connection is indeed well-known and the key concept is that of ribbon category. A standard textbook reference is Turaev, Quantum Invariants of Knots and 3-Manifolds.
Braided and rigid is not enough to get links invariants, because RI will not hold in general (and in fact pretty much neve... | 3 | https://mathoverflow.net/users/13552 | 447039 | 180,064 |
https://mathoverflow.net/questions/447021 | 0 | Let $M$ be a von Neumann algebra and $\psi$ a normal faithful semifinite weight on $M$. Then one should be able to form the object
$$\iota \otimes \psi: (M\overline{\otimes} M)\_+ \to \widehat{M\_+}.$$
This is for example used in the theory of locally compact quantum groups (in the sense of Vaes-Kustermans). I have bee... | https://mathoverflow.net/users/216007 | Tensor product of operator values weights (in the theory of locally compact quantum groups) | The approach I had in mind is the following. For a reference, see Section 4, Chapter IX of Takesaki 2. The extended positive part $\widehat{M\_+}$ is by definition the space of positive-homogeneous, additive, lower semi-continuous maps $M\_\*^+\rightarrow [0,\infty]$. Given $x\in (M\bar\otimes M)\_+$ define $(\iota\oti... | 3 | https://mathoverflow.net/users/406 | 447042 | 180,065 |
https://mathoverflow.net/questions/447030 | 5 | Take $G=\operatorname{GL}\_3$, defined over the algebraic closure of a finite field $\mathbb{F}\_q$ and let $X$ be the set of Borel subgroups of $G$.
The Frobenius morphism $F:G\to G$ induces a map $F:X\to X$, as $G/B\cong X$ for any chosen Borel subset $B$.
The Weyl group $W$ of $G$ is isomorphic to the symmetric grou... | https://mathoverflow.net/users/504919 | An example of a Deligne–Lusztig variety for a general linear group | Let $\mathcal{F}\colon V\_1\subseteq\dotsb\subseteq V\_n$ and $\mathcal{F}'\colon U\_1\subseteq\dotsb\subseteq U\_n$ be two complete flags. Then $\mathcal{F}$ and $\mathcal{F}'$ are said to be in relative position $w$, where $w\in S\_n$ is a permutation, if
$$\dim V\_i\cap U\_j=\#(\{ 1,\dotsc,i \}\cap \{ w(1),\dots,w(j... | 9 | https://mathoverflow.net/users/56217 | 447051 | 180,068 |
https://mathoverflow.net/questions/447040 | 1 | This question is an extension of the one I posted on ME: <https://math.stackexchange.com/questions/4701500/if-alpha-nx-int-lvert-x-y-rvert-leq-1-n-lvert-x-y-rvert2-d-muy>
It might be elementary for here, but I would deeply appreciate any help.
Let for each $n \in \mathbb{N}$, let $r\_n : \mathbb{R}^m \to (0,\infty)... | https://mathoverflow.net/users/56524 | A generalized form of the approximation to identity? | $\newcommand\al\alpha\newcommand\be\beta\newcommand\R{\mathbb R}$This is not true in general. E.g., suppose that $m=1$, $\mu=N(0,1)$, and $r\_n(x)=r:=1/n$ for all $n$ and $x$. Let $f$ be the standard normal pdf.
Then for each real $x$ (and $n\to\infty$) we have
$$\al\_n(x)\sim\int\_{x-r}^{x+r}(y-x)^2\,dy\,f(x)=\frac{... | 3 | https://mathoverflow.net/users/36721 | 447054 | 180,070 |
https://mathoverflow.net/questions/447047 | 5 | The constructible universe $L$ has some nice properties:
1. $L$ has a $\mathit{\Delta}^1\_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
2. For any $\mathit{\Sigma}^1\_2$ formula $\varphi(x)$ and a real $r \in \mathbb{R}\cap L, \ \varphi(r) \iff \varphi(r)^L$. (Shoenfield)
3. For any $\mathit{\Sigma}^1\_2$ s... | https://mathoverflow.net/users/141146 | Inner model with a $\mathit{\Delta}^1_3$-good well-ordering of the reals | The situation is a bit more complicated than you might hope because of the periodicity phenomena in the projective hierarchy. For odd $n$, assuming $\mathbf{\Delta}^1\_{n-1}$-determinacy, the set $Q\_n$ of reals that are $\Delta^1\_n$ in a countable ordinal is $\Pi\_n^1$ definable, so its complement is a $\Sigma\_n^1$ ... | 8 | https://mathoverflow.net/users/102684 | 447056 | 180,071 |
https://mathoverflow.net/questions/447059 | 0 | The motivation for my current question arises from [this MO post by R. Stanley](https://mathoverflow.net/questions/430741/number-of-coefficients-equal-to-k-in-certain-fibonacci-polynomials). *Caveat.* There's a slight alteration.
With the convention $F\_1=F\_2=1$ for the Fibonacci numbers, define the polynomials $f\_n(... | https://mathoverflow.net/users/66131 | Fibonacci and product polynomials | Question 2 follows from Theorem 6.1 of [arXiv:2101.02131](https://arxiv.org/pdf/2101.02131.pdf). (In this reference, I consider $\prod\_{i=1}^n(1+x^{F\_{i+1}})$ rather than $\prod\_{i=1}^n(1+x^{F\_i})$, but the proof still works.) The result holds for any positive integer $p$, not just primes. The proof is constructive... | 4 | https://mathoverflow.net/users/2807 | 447063 | 180,073 |
https://mathoverflow.net/questions/447062 | 3 | I am reading the paper by Jardine and Goerss, Localization theories for simplicial presheaves and having troubles with understand an argument. In this paper, the two authors considered $\mathcal{C}$ to be a small Grothendieck site (so in particular the underlying class of objects is a set and morphisms between two obje... | https://mathoverflow.net/users/482398 | Injective model structure for simplicial presheaves | To answer the question as it is stated: $U$ is an object in a locally presentable category, therefore $U$ is a small object, hence the corepresentable functor of $U$ preserves $α$-filtered colimits for some regular cardinal $α$. More precisely, the image of $U$ in such a colimit has a cardinality bounded by the product... | 1 | https://mathoverflow.net/users/402 | 447075 | 180,076 |
https://mathoverflow.net/questions/447067 | 3 | Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is equipped with the standard basis. This will be our starting frame of reference, so to speak. Let $S \subseteq V$ be an arbitra... | https://mathoverflow.net/users/1849 | Recovering a set from its projections in varying coordinate systems - a projection hull? | $\newcommand{\R}{\mathbb R}\newcommand{\tU}{\tilde U}$Suppose that $U$ is connected. Then all its projections are connected. So, all one-dimensional projections of $U$ will be convex. So, $\tilde U$ will be convex.
Now take any non-convex connected $U$. Then $\tilde U\ne U$ and hence $\tilde{U} \supsetneqq U$.
--... | 3 | https://mathoverflow.net/users/36721 | 447077 | 180,078 |
https://mathoverflow.net/questions/447090 | 2 | A *Halin graph* is a graph constructed by embedding a tree with no vertex of degree
two in the plane and then adding a cycle to join the tree’s leaves.
We found a list of the number of Halin graphs within $14$ vertices on the website <https://oeis.org/A346779>.
```
n a(n)
1 0
2 0
3 0
4 ... | https://mathoverflow.net/users/171032 | Is there an algorithm to generate non-isomorphic Halin graphs? | There is a theorem due to Jordan which is useful for enumerating or generating trees: every tree has a centre or a bicentre. To enumerate/generate suitable bicentral trees, enumerate/generate suitable rooted trees grouping them by height and join two rooted trees of the same height with an edge between their roots. To ... | 2 | https://mathoverflow.net/users/46140 | 447101 | 180,083 |
https://mathoverflow.net/questions/441734 | 1 | I am looking for a reference proving the existence of the minimal Steiner tree in the Euclidean Steiner tree problem:
Given N points in the d-dimensional Euclidean space, the goal is to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either direct... | https://mathoverflow.net/users/47256 | Existence of minimal Steiner tree | The existence theorem for a finite number of points is more-or-less obvious, so classical sources do not give it as a statement. But you can extract them from the very classical paper Gilbert E. N., Pollak H. O. Steiner minimal trees //SIAM Journal on Applied Mathematics. – 1968. – Т. 16. – №. 1. – С. 1-29, or from any... | 3 | https://mathoverflow.net/users/479618 | 447111 | 180,084 |
https://mathoverflow.net/questions/447114 | 7 | Let me begin by mumbling some abstract nonsense, and then attempt to be concrete.
The category of groups inherits the structure of a strict 2-category from the 2-category of small categories.
Explicitly, a 2-morphism between $\varphi$ and $\varphi'\colon H \to G$ is an element $g \in G$ such that for each $h \in H$, we... | https://mathoverflow.net/users/135175 | Why does the 2-category of groups have (some, strict) coinserters but not (strict) inserters? | One viewpoint goes as follows: the 2-categorical structure on groups can be seen as coming from inner automorphisms, so that a 2-cell is given by an inner automorphism that translates one map to the other. Now, inner automorphisms of an object can be defined in any category (see e.g. [this paper](https://doi.org/10.101... | 9 | https://mathoverflow.net/users/136562 | 447116 | 180,086 |
https://mathoverflow.net/questions/447099 | 9 | **I. Some functions**
As these will be used in the continued fraction evaluations below, recall the *Riemann zeta function* $\zeta(s),$ and *[Dirichlet beta function](https://mathworld.wolfram.com/DirichletBetaFunction.html)* $\beta(s),$
$$\beta(s) = \sum\_{n=1}^\infty\frac{(-1)^{n-1}}{(2n-1)^s}$$
and special cas... | https://mathoverflow.net/users/12905 | On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences" | We have $$C\_2(-17,-6,-72)=-(5/8)L(\chi\_{-3},2)$$ and
$$C\_2(10,3,-9)=(1/2)L(\chi\_{-3},2)$$ so both are proportional to what you
call Gieseking's constant but which is simply the value at 2 of the
L function of the nontrivial character modulo 3, close analogue to Catalan's
constant which is the same with the nontrivi... | 12 | https://mathoverflow.net/users/81776 | 447123 | 180,088 |
https://mathoverflow.net/questions/447100 | 2 | The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \left( n, \frac{1}{\sqrt[4]{n}} \right)$, I am looking for a preferably slick and short formal argument that: $$\text{P... | https://mathoverflow.net/users/475708 | Simple anticoncentration bound for binomially distributed variable | Let $Z\sim N(0,1)$, $p\_n:=n^{-1/4}$, $q\_n:=1-p\_n$. By the [Berry--Esseen inequality](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem#Identically_distributed_summands),
$$P(X\ge EX)\ge P(Z\ge0)-0.5\frac{n(p\_nq\_n^3+q\_np\_n^3)}{(np\_nq\_n)^{3/2}}=\frac12-o(1)$$
as $n\to\infty$. $\quad\Box$
More explicit... | 9 | https://mathoverflow.net/users/36721 | 447124 | 180,089 |
https://mathoverflow.net/questions/446908 | 3 | Let $[1]^n=\{0<1\}^n$ equipped with the product order. I consider the small category $\widehat{\square}$ of the category of partially ordered sets generated by the *coface maps* $\delta^\epsilon\_i:[1]^{n-1}\to [1]^n$ with $\epsilon=0,1$ defined by $$\delta^\epsilon\_i:(x\_1,\dots,x\_{n-1}) \mapsto (x\_1,\ldots,x\_{i-1... | https://mathoverflow.net/users/24563 | Removing the symmetry maps from a small category of cubes | The naive idea has to be slightly modified. The point is not to sort out all terms (it is a wrong intuition), but only where the variables $x\_i$ are "alone". For example, the map $$(x\_1,x\_2,x\_3,x\_4)\mapsto (x\_2,x\_1,\max(x\_3,x\_4),\min(x\_3,x\_4))$$ is not kept in the subcategory because $x\_2$ which is alone is... | 0 | https://mathoverflow.net/users/24563 | 447128 | 180,090 |
https://mathoverflow.net/questions/447107 | 2 | Suppose that $e\_1, \cdots, e\_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.
Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te\_k=e\_{k-1}$ if $k\geq2$ and $Te\_1=e\_n$.
Let $A:\mathbb{R}^n\to \mathbb{R}^n$ be the operator given by $Ae\_3=Ae\_4=e\_1$ and ... | https://mathoverflow.net/users/84390 | The eigenvectors of adding a particular rank one matrix to the circulant matrix | $\newcommand{\la}{\lambda}\newcommand{\R}{\mathbb R}$For any $x=(x\_1,\dots,x\_n)\in\R^n$ we have $Tx=(x\_2,\dots,x\_n,x\_1)$ and $Ax=(x\_3+x\_4,0\dots,0)$, so that for $U:=T+A$ we have $Ux=(x\_2+x\_3+x\_4,x\_3,\dots,x\_n,x\_1)$.
Let now $(x\_1,\dots,x\_n)\in\R^n$ be an eigenvector of $U$ belonging to an eigenvalue $... | 3 | https://mathoverflow.net/users/36721 | 447130 | 180,091 |
https://mathoverflow.net/questions/446780 | 7 | Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\rangle = \int\_X \vert g(z,x)\vert^2 d\mu(x).$
Then one has by a fairly explicit computation
$$ \partial\_z \partia... | https://mathoverflow.net/users/496243 | Log-convexity of determinant | Your claim can be deduced from the case $n=1$.
Let $H$ be the Hilbert space you start with.
Let $H^{\otimes n}=H\otimes\dots\otimes H$ be the $n$-th tensor power. Equip it with the usual inner product and complete to get a Hilbert space $V$ again. Let $g=g\_1\wedge\dots\wedge g\_n$ be the function
$$
g=\sum\_\sigma\mat... | 5 | https://mathoverflow.net/users/473423 | 447132 | 180,092 |
https://mathoverflow.net/questions/446991 | 14 | $\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A\_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ matrices $A$ such that
$$cI\le A\le CI,$$
where $I$ is the $d\times d$ identity matrix and $A\le B$ for $d\times d$ matrices $A$ ... | https://mathoverflow.net/users/36721 | Lipschitz property of the determinant | The best constant is $C^{d-1}\sqrt{d}$.
Write $D(A)=\det A$. We can rephrase the inequality as the claim that $\|D'\|\_F\le L$ for $c\le A\le C$. (It's perhaps best to think of the matrices as long column vectors and the Frobenius norm as the Euclidean norm, and then $D'$ can be viewed as the gradient of $D$.)
By [... | 7 | https://mathoverflow.net/users/48839 | 447138 | 180,095 |
https://mathoverflow.net/questions/447106 | 7 | For elliptic curves over $\mathbb{Q}$ the Mumford-Tate group is either $\mathrm{GL}\_2$ or $\mathrm{Res}\_\mathbb{Q}^F (\mathbb{G}\_m)$ if it has CM with the imaginary quadratic field $F$. In this case the $\mathbb{Q}$-endomorphism algebra completely determines the Mumford-Tate group. On the automorphic side, having CM... | https://mathoverflow.net/users/496065 | Automorphic classification of different types of abelian surfaces | Yes, knowing the endomorphism algebra of $A$ (conjecturally) translates to certain properties of an associated automorphic representation $\pi$.
First, you should look at the *Galois type*, which is labelled (A)-(F) in [FKRS's paper on Sato-Tate groups](https://arxiv.org/abs/1110.6638). (These labels also include non... | 4 | https://mathoverflow.net/users/6518 | 447142 | 180,097 |
https://mathoverflow.net/questions/447118 | 3 | I am looking for further proofs, preferably in the literature, of the following result:
**Proposition:** Let $R$ be a unital commutative $\mathbb{Z}$-torsion free ring. If $f(x) \in R[x]$ with $f'(x) \in U(R[x])$, then there exists $g(x) \in R[x]$ such that $f(g(x)) = g(f(x)) = x$ in $R[x]$.
For example, $R = \math... | https://mathoverflow.net/users/1849 | The $1$-dimensional Jacobian Conjecture over $\mathbb{Z}$-torsion free rings | Here is a straightforward commutative algebra argument. We have to show the following:
**Lemma.** *Let $R$ be a $\mathbf Z$-torsion free ring and $f \colon R[x] \to R[x]$ an étale homomorphism of $R$-algebras. Then $f$ is an isomorphism.*
*Proof.* As noted, this is trivial if $R$ is reduced: then $f'(x) \in R[x]^\t... | 5 | https://mathoverflow.net/users/82179 | 447147 | 180,099 |
https://mathoverflow.net/questions/404849 | 2 | Let $\sigma(x)=\sigma\_1(x)$ be the *classical sum of divisors* of the positive integer $x$.
It is known that
$$\gcd(\sigma(q^k),\sigma(n^2))=\frac{\bigg(\gcd(n,\sigma(n^2))\bigg)^2}{\gcd(n^2,\sigma(n^2))}$$
if $q^k n^2$ is an odd perfect number with special prime $q$.
Hence, **if it is known that $n \mid \sigma(n^... | https://mathoverflow.net/users/10365 | Does $n \mid \sigma(n^2)$, if $q^k n^2$ is an odd perfect number? | Let $p^k m^2$ be an odd perfect number with special prime $p$.
It follows that
$$\frac{\sigma(m^2)}{p^k}\cdot\frac{\sigma(p^k)}{2}=m^2.$$
Let $t\_1 = \sigma(m^2)/p^k$, $t\_2 = \sigma(p^k)/2$. It follows that $m^2 = t\_1 t\_2$.
Now define the GCDs
\begin{align\*}
G&=\gcd(\sigma(p^k),\sigma(m^2))=\gcd(\sigma(p^k)/2... | -1 | https://mathoverflow.net/users/10365 | 447149 | 180,100 |
https://mathoverflow.net/questions/446799 | 1 | A famous theorem of Beilinson gives a finite, locally free resolution of the diagonal for $\mathbf{CP}^{n}$ by exterior tensor products of locally free sheaves on $\mathbf{CP}^{n}$: for $1 \leq k \leq n$, the $k$th component of the resolution is given by $\mathcal{O}(-k) \boxtimes \Omega^{k}(k)$ where $\Omega^{k}(k) :=... | https://mathoverflow.net/users/504744 | Resolution of the diagonal for projective hypersurface | If there is a resolution of the diagonal of a smooth projective variety $Y$ with terms direct sums of $F'\_i \boxtimes F''\_i$, then it is easy to see that the Grothendieck group $K\_0(Y)$ is generated by the classes of $[F'\_i]$ (or of $F''\_i$). To see this just consider the Fourier--Mukai transform given by the stru... | 3 | https://mathoverflow.net/users/4428 | 447152 | 180,102 |
https://mathoverflow.net/questions/447155 | 2 | Let $f: X \to Y$ be a surjective morphism of normal projective varieties with connected fibers (in my case, $X$ is $\mathbb{Q}$- factorial also). Let $E$ be an irreducible $f$-exceptional divisor (i.e. the codim of its image is at least $2$) and $W:= f(E)$. Suppose the irreducible components of $f^{-1}W$ are $E$ and $S... | https://mathoverflow.net/users/150655 | Removing irreducible components from fibers of projective morphisms | No. If this happens then $X = X' \cup S$ where $X'$ and $S$ are closed subsets, because every projective subset of a projective variety is closed.
This means $X$ is not irreducible, meaning $X$ is not a variety (or, depending on your definitions, at least not normal).
| 3 | https://mathoverflow.net/users/18060 | 447159 | 180,105 |
https://mathoverflow.net/questions/444946 | 6 | Let $f:M\to Y$ be a continuous proper bijective map from a metrizable space $M$ onto a $T\_1$-space $Y$. The properness of $f$ means that for every compact subspace $K\subseteq Y$ the preimage $f^{-1}[K]$ is a compact subset of $M$.
Let $g:H\to Y$ be any continuous map from a compact Hausdorff space $H$ into $Y$.
... | https://mathoverflow.net/users/61536 | The continuity of certain maps on compact Hausdorff spaces | The answer to this question is affirmative.
>
> **Proposition.** Let $p:X\to Y$ be a proper bijective map from a Hausdorff topological space $X$ onto a $T\_1$-space $Y$. Then for every continuous map $f:K\to Y$ from a compact Hausdorff space $K$, the map $p^{-1}\circ f:K\to X$ is continuous.
>
>
>
*Proof.* By ... | 1 | https://mathoverflow.net/users/61536 | 447162 | 180,106 |
https://mathoverflow.net/questions/447160 | 1 | Let $G$ be a directed graph and let $P\_i$
be its vertices. Let $A$
be the corresponding adjacency matrix of $G$, i.e. $a\_{i,j}=1$
if and only if there is a directed edge from $P\_i$
to $P\_j$, ($a\_{i,j}=0$
otherwise).
Q. Any characterization for directed graphs whose adjacency matrix admits only 0 as the eigenvalu... | https://mathoverflow.net/users/84390 | Directed graph whose adjacency matrix admits only 0 as eigenvalue | $0$ is the only eigenvalue of $A$ if and only if $A$ is nilpotent, which is equivalent to $A^n=0$. But $A^n=0$ expresses that for all $i$ and $j$, there is no path of length $n$ from $P\_i$ to $P\_j$. This in turn is equivalent to the graph containing no circles.
| 5 | https://mathoverflow.net/users/18739 | 447163 | 180,107 |
https://mathoverflow.net/questions/447164 | 3 | It is proved [here](https://mathoverflow.net/questions/447134/can-the-equation-1zz2-zn-have-multiple-complex-roots) that the equation $1+z+z^2=z^n$ have no multiple complex roots.
>
> Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any characterization for the pair ... | https://mathoverflow.net/users/84390 | Can the equation $1+z+z^q=z^n$ have multiple complex roots $z$? | The roots are always simple, this follows without further reasoning from the old paper [On the irreducibility of certain trinomials and quadrinomials](https://doi.org/10.7146/math.scand.a-10593) by Ljunggren. Set $f(z)=z^n-1-z^p-z^q$. Then Ljunggren proves (Theorem 1) that $f(z)=Q(z)g(z)$, where the roots of $Q$ are ro... | 7 | https://mathoverflow.net/users/18739 | 447168 | 180,108 |
https://mathoverflow.net/questions/445930 | 4 | Let $A$ be a Hopf algebra (over a field). Consider a unital subalgebra $B\subseteq A$ with $\Delta(B)\subseteq B\otimes A$. Put
$$B^+:= B\cap \ker(\epsilon).$$
It can be shown that $B^+$ is a two-sided coideal of $A$, so that $AB^+$ is also a two-sided coideal.
**Question**: If $x\in AB^+$, is it true that $S(x) \in ... | https://mathoverflow.net/users/216007 | Hopf algebra and coideal question | This is indeed true, and it forms part of the proof of a result known as ***Koppinen's Lemma***.
The argument is as follows: Let $\{x\_j\,|\, j \in J\}$ be a basis of $B^+$, and take an element $x \in B^+$. It follows from the counit axiom of a Hopf algebra, and linear independence of our basis, that we can write the... | 3 | https://mathoverflow.net/users/3072 | 447174 | 180,111 |
https://mathoverflow.net/questions/447202 | 4 | In the proof of Corollary 5.7 in the following link:
<https://arxiv.org/pdf/1610.05200.pdf>
the author claims that $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ for the standard normal distribution on $\mathbb{R}^n$.
I wonder if this result is still valid for cenetered Gaussian distributions on any infinite... | https://mathoverflow.net/users/56524 | For centered Gaussian measures, is $E[\lVert X\rVert^2] \lesssim E[\lVert X\rVert ]^2$ true in infinite dimensions as well? | $\newcommand{\si}{\sigma}$Yes, we have
\begin{equation\*}
E\|X\|^2\le c(E\|X\|)^2 \tag{1}\label{1}
\end{equation\*}
for
\begin{equation\*}
c:=1+2\pi
\end{equation\*}
and any centered Gaussian random vector $X$ in any separable Banach space $V$.
This follows from the [Borell--Tsirelson--Ibragimov--Sudakov inequality... | 6 | https://mathoverflow.net/users/36721 | 447209 | 180,116 |
https://mathoverflow.net/questions/447213 | -1 | Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are Borel sets. Let $u: X\times A\rightarrow\mathbb{R}$ be a continuous function. Let $\pi\in \Delta(X\times A)$ be a probabi... | https://mathoverflow.net/users/30374 | Conditional expectation: commuting integration and supremum | Note: This answer used to be a counterexample that missed the mark.
The way to get around he definitional issues with conditional expectations is to work with [regular conditional probabilities in product form](https://doi.org/10.1214/aop/1176993081), which guarantee that all conditional expectations fit together wel... | 2 | https://mathoverflow.net/users/35357 | 447222 | 180,121 |
https://mathoverflow.net/questions/445065 | 2 | There seems to be many valid ways of generalizing the notion of the spectral radius $\rho(A)$ of a complex matrix $A$ to spectral radii of multiple operators. I am wondering if there is an abstract theory of what it means to be a multi-spectral radius $\rho(A\_1,\dots,A\_r)$ of complex matrices $A\_1,\dots,A\_r$.
Exa... | https://mathoverflow.net/users/22277 | Is there an abstract theory of multi-spectral radii? | I claim that there is a somewhat abstract notion of a multi-spectral radius and that there is probably an abstract theory behind this abstract notion. I will try to justify this abstract multi-spectral radius by showing that it captures the specific examples of multi-spectral radii that I have mentioned in the question... | 0 | https://mathoverflow.net/users/22277 | 447226 | 180,122 |
https://mathoverflow.net/questions/447220 | 2 | Is there a $\Pi^0\_2$ singleton that forms a minimal pair with $0''$? That is, is there a set $X$ such that $X$ is the unique solution to $\forall x \exists y \phi(X|\_y, x)$, $X$ and $0''$ are incomparable and if $Y \leq\_T 0'' \land Y \leq\_T X$ then $Y \leq\_T 0$.
For some motivation, note that the most common exa... | https://mathoverflow.net/users/23648 | $\Pi^0_2$ singleton forming minimal pair with $0''$ | Maybe I should give a more detailed answer.
Harrington proved (or claimed) the following result in his handwritten draft.
>
> **Theorem** There is a $\Pi^0\_2$-singleton $x$ so that $\forall n<\omega (x^{(n)}\equiv\_T x\oplus \emptyset^{(n)}\wedge \forall m\geq n \forall z (z\leq\_T x^{(n)}\wedge z\leq\_T \emptys... | 4 | https://mathoverflow.net/users/14340 | 447231 | 180,125 |
https://mathoverflow.net/questions/447225 | 2 | Since $\exp(\cdot)$ is locally Lipschitz, the following SDE has a strong solution:
$$
\mathrm{d}X\_s=\exp(X\_s) \, \mathrm{d}B\_s,\quad X\_0=1,
$$
where $B$ is a standard Brownian motion. I wonder if the following expression holds:
$$\mathbb{E}\int\_0^T\exp(2X\_s) \, \mathrm d s<\infty.$$
| https://mathoverflow.net/users/484728 | Existence of solution for a non-linear SDE | Let us show that
\begin{equation\*}
E\int\_0^T e^{2X\_t}\,dt=\infty \quad\text{for real }T\ge T\_\*:=e^{-2}/2. \tag{1}\label{1}
\end{equation\*}
Indeed, letting $Y\_t:=e^{2X\_t}$, we have
$Y\_0=e^2$ and, by [Itô's lemma](https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma#Mathematical_formulation_of_It%C3%B4%27s_lem... | 2 | https://mathoverflow.net/users/36721 | 447238 | 180,126 |
https://mathoverflow.net/questions/447191 | 1 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). The *chromatic number* $\chi(H)$ is the smallest cardinal $\kappa \leq |V|$ such that there is a map $c:V\to \kappa$ such that for all $e\in E$ having more than one element, the restriction $c|\_e: e\to \kappa$ is not constant.
**Question.** I... | https://mathoverflow.net/users/8628 | Is the chromatic number of hypergraphs downward continuous? | Fred Galvin had conjectured that the answer is "yes" for *graphs* in [1] (conjecture 2), in his paper he showed that the variation of the problem to *induced* graphs is consistently false: Assume $2^{\aleph\_0}=2^{\aleph\_1}<2^{\aleph\_2}$ there exists a graph $(V,E)$ with a chromatic number of $\aleph\_2$ but for no s... | 2 | https://mathoverflow.net/users/113405 | 447242 | 180,128 |
https://mathoverflow.net/questions/447241 | 53 | String theory (and related areas of purely theoretical quantum gravity, like loop quantum gravity) has a unique position within the academic physics community. Many academic physicists don't really consider string theory to be physics at all (due to its disconnect with any experimental evidence) and think that it shoul... | https://mathoverflow.net/users/95043 | Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community? | There are some speculative mathematical concepts that come to mind, such as the [field of one element](https://en.wikipedia.org/wiki/Field_with_one_element) or [motives](https://en.wikipedia.org/wiki/Motive_(algebraic_geometry)), though perhaps these are more classifiable as "potential future mathematics" rather than "... | 59 | https://mathoverflow.net/users/766 | 447246 | 180,131 |
https://mathoverflow.net/questions/447019 | 12 | *This question is of course related to [this earlier MO question](https://mathoverflow.net/questions/1058/when-does-cantor-bernstein-hold), but I don't believe is answered by the posts there.*
My favorite proof of the Cantor-Schroeder-Bernstein theorem actually establishes something stronger: if $f:A\rightarrow B$ an... | https://mathoverflow.net/users/8133 | The scope of a "strong Cantor-Bernstein" property | For infinite-dimensional vector spaces over any field, there is no such isomorphism. We will handle the countable-dimensional case, as the others are similar. Let $V$ be generated by the linearly independent vectors $v\_0,v\_1,\ldots$, and similarly let $W$ be generated by the linearly independent vectors $w\_0,w\_1,\l... | 3 | https://mathoverflow.net/users/3199 | 447260 | 180,135 |
https://mathoverflow.net/questions/447050 | 4 | Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any case, it is a real line bundle and is flat, where the locally constant transition functions are given by the sign of the J... | https://mathoverflow.net/users/41686 | Orientation bundle and its flat connection | There is a different construction of orientation bundles. One considers the $\{\pm1\}$-principal bundle $o(TM)$ of fibrewise orientations of $TM$. The associated real line bundle $o(TM)\times\_{\pm 1}\mathbb R$ carries a flat connection which is natural under local diffeomorphisms.
This bundle can be described as hav... | 3 | https://mathoverflow.net/users/70808 | 447272 | 180,138 |
https://mathoverflow.net/questions/447274 | 5 | I asked Chat GPT to suggest a number theoretic conjecture.
It came up with the following interesting conjecture:
**Conjecture (Chat GPT):** For each even natural number $n$, there is a prime $p$ such that $p+n^2$ is also prime.
(I am not sure it stated *even* but this is clearly necessary, e.g., take $n=5$.)
Equi... | https://mathoverflow.net/users/2415 | A number theoretic conjecture by Chat GPT | Ancient Greeks conjectured that there are infinitely many pairs of primes which differ by 2 (twin primes). A natural widely believed generalization is that 2 may be replaced by every even number. Moreover, it is expected that for every integer $T>0$ there are about $C n/\log^2 n$ numbers $p\leqslant n$ for which $p$ an... | 10 | https://mathoverflow.net/users/4312 | 447276 | 180,139 |
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