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https://mathoverflow.net/questions/438541 | 1 | We say that a bijection $\varphi:\omega\to\omega$ *parity-separates* $a\neq b\in \omega$ if $\varphi(a)$ is even and $\varphi(b)$ is odd, or vice versa.
Is there a finite set $\Phi$ of bijections such that for all $a\neq b\in\omega$ there is $\varphi\in\Phi$ such that $\varphi$ parity-separates $a,b$? If yes, how sma... | https://mathoverflow.net/users/8628 | Can $\omega$ be parity-separated with finitely many bijections? | No, because if you have $n$ functions, then the number of possible parity patterns to be exhibited by a number with respect to them is $2^n$. So by the pigeon-hole principle there must be infinitely many numbers with the same pattern. So it doesn't separate those numbers.
| 2 | https://mathoverflow.net/users/1946 | 438542 | 177,135 |
https://mathoverflow.net/questions/438330 | 3 | Given symmetric monoidal $\infty$-categories $A, B, C$ and lax symmetric monoidal maps $F:A\to C$, $G:B\to C$, I am curious if the pullback (when I say pullback here I will really mean homotopy pullback) of $F$ along $G$ is necessarily lax symmetric monoidal in a natural way? The pullback is necessarily an $\infty$-ope... | https://mathoverflow.net/users/489806 | Can there be a cospan of symmetric monoidal $\infty$-categories whose maps are lax symmetric monoidal but the pullback is not symmetric monoidal? | Oops, this is actually not hard, just using 1-categories. Explicitly, take two maps from the terminal category to Ab, one landing in $\mathbb{Z}$, one landing in 0, both are lax symmetric monoidal. The pullback (after replacing the first map by an equivalent isofibration but we can mostly ignore this technicality) is e... | 2 | https://mathoverflow.net/users/489806 | 438548 | 177,136 |
https://mathoverflow.net/questions/438522 | 0 | Recall that an operator $T:X\rightarrow Y$ is called absolutely summing if there exists a constant $C>0$ such that
$$\sum\_{i=1}^{n}\|Tx\_{i}\|\leq C \sup\_{x^{\*}\in B\_{X^{\*}}}\sum\_{i=1}^{n}|\langle x^{\*},x\_{i}\rangle|,$$ for all finite families $(x\_{i})\_{i=1}^{n}$ in $X$. The least $C$ for which the above ineq... | https://mathoverflow.net/users/41619 | Absolutely summing operators from $l_{p}$ to $l_{q}$ | As for Question 3, note that an absolutely summing operator is completely continuous, i.e., maps weakly null sequences to null sequences, a property not shared by $i\_{p,q}$ if $p>1$. Therefore this question has a negative answer.
| 3 | https://mathoverflow.net/users/127871 | 438550 | 177,137 |
https://mathoverflow.net/questions/438549 | 6 | The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and functional equation, the $L$-function also inherits those nice properties.
Most expositions of the Langlands-Shahidi m... | https://mathoverflow.net/users/394740 | Langlands-Shahidi method in classical language | Well, as part of a constant term of an Eisenstein series, it would appear in the constant term of an Eisenstein series attached to the 2,1 parabolic (or 1,2...) in GL3, with the cuspform on the GL2 factor of the Levi component. My relatively recent book(s) with Cambridge U Press... also available on-line... treat some ... | 7 | https://mathoverflow.net/users/15629 | 438557 | 177,139 |
https://mathoverflow.net/questions/438246 | 7 | In quantum information, much can be done with the averaging formula
$$
\int\_{\mathbb{C}P^{n-1}} (zz^\*)^{\otimes t} dz = {n + t -1 \choose t}^{-1} \operatorname{\Pi}\_{\mathrm{Sym}^t}$$
Here the integral gives an average over all unit vectors $z \in \mathbb{C}^n$ and $\operatorname{\Pi}\_{\mathrm{Sym}^t}$ denotes ... | https://mathoverflow.net/users/315386 | Average of polynomials over the real sphere | Here is a solution for the case $t=2$.
Let $$T= \int\_{\mathbb{R}P^{n-1}}
(xx^\top)\otimes (xx^\top) dx $$
(I'm gonna assume that the integral is normalized so that the measure of $\mathbb{R}P^{n-1}$ is $1$).
As in the complex case, it's easy to see that $T$ has image in the symmetric subspace $\text{Sym}^2 \mathb... | 4 | https://mathoverflow.net/users/160416 | 438563 | 177,141 |
https://mathoverflow.net/questions/438333 | 19 | Freyd's theorem in classical category theory says that any small category $\mathcal{C}$ admitting products indexed by the set $\mathcal{C}\_1$ of all its arrows is a preorder. I'm interested in whether this is also true for for precategories/univalent categories à lá HoTT Book (§II.9), provided we assume LEM.
I'll re... | https://mathoverflow.net/users/475802 | Small complete categories in HoTT+LEM | Here is an alternative proof based on Russell's paradox rather than cardinality that doesn't require sets cover, although I do need to assume that hom sets are 0 truncated. The rough outline is to modify Gylterud's definition of the cumulative hierarchy by limiting it to sets that can be constructed from $\mathcal{C}$ ... | 6 | https://mathoverflow.net/users/30790 | 438568 | 177,143 |
https://mathoverflow.net/questions/438555 | 3 | Let $p(u,x)=\frac{1}{(4\pi u)^{q/2}}e^{-|x|^2/(4u)},u>0,x \in \mathbb{R}^q.$
1. Prove that for $r\geq 0,c>1$ there exists $C>0$ (depending on $r,c$) such that $$\forall x \in \mathbb{R}^q,u>0,\frac{|x|^{2r}}{u^r}p(u,x) \leq Cp(cu,x).$$
2. Deduce that for $n \in \mathbb{N}^q,k \in \mathbb{N},$ there exists $C'>0$ such... | https://mathoverflow.net/users/138491 | Inequality: multivariate normal distribition | $\newcommand\p\partial$By induction,
$$\p\_u^k p(u,x)=\frac1{u^k}\,P\_k\Big(\frac{|x|^2}u\Big)p(u,x)
=\frac K{u^{k+q/2}}\,P\_k\Big(\frac{|x|^2}u\Big)e^{-|x|^2/(4u)}, \tag{1}\label{1}$$
where $K$ is a real number not depending on $u$ or $x$, and $P\_k(z)$ is a polynomial (with coefficients not depending on $u$ or $x$).
... | 2 | https://mathoverflow.net/users/36721 | 438569 | 177,144 |
https://mathoverflow.net/questions/438574 | -2 | A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around.
I define the function
$$
L\_4^\*(s) = \prod\_{p\in P} \Bigg(1 - \frac{\chi(p)}{p^{s}}\Bigg)^{-1}
$$
with $P=\{p\_1,p\_2,\dots\}$ the set of prime numbers... | https://mathoverflow.net/users/140356 | Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character | The usual Euler product converges absolutely in $\{s:\Re s>1\}$, hence no matter how you permute its factors, you get the same function in $\{s:\Re s>1\}$. This function has a unique analytic continuation to $\mathbb{C}$, which is then $L(s,\chi)$. In short, the new function you are trying to define will either not be ... | 7 | https://mathoverflow.net/users/11919 | 438578 | 177,146 |
https://mathoverflow.net/questions/438579 | 5 | Recall that
$\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the *sfip* which has no infinite pseudo-intersection $\}$.
The cardinal $\mathfrak{q}\_0$ defined as the smallest cardinality of a subset of $\mathbf{R}$ which is not a $Q$-space.
**Q1.** Is it true that $\mathit{MA}(\omega\_1)$ if... | https://mathoverflow.net/users/112417 | Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$? | Q1: No, see [Between Martin's Axiom and Souslin's Hypothesis](https://eudml.org/doc/211011) by Kunen and Tall. Note: Bell proved in [The combinatorial principle $P(\mathfrak{c})$](https://eudml.org/doc/211293) that $\mathfrak{p}>\aleph\_1$ is equivalent to $\mathsf{MA}(\aleph\_1)$ for $\sigma$-centered partial orders.
... | 9 | https://mathoverflow.net/users/5903 | 438583 | 177,148 |
https://mathoverflow.net/questions/438577 | 4 | Let $U,V\in\mathbb{R}^{4\times n}$ such that $UU^T=VV^T=I$, and $A\in\mathbb{R}^{n\times n}$ be an Hermitian matrix.
Is it true that
$$\sqrt{\lambda\_{\text{max}}\left(\left(UAU^T-VAV^T\right)^2\right)}\leq \lambda\_{\text{max}}\left((U-V)A(U-V)^T\right)$$
?
| https://mathoverflow.net/users/103133 | largest eigenvalue of the difference between two quadratic forms | This is false for the following reason. Say that $n=2$. Choose
$$A=\begin{pmatrix} a & b \\ b & 0 \end{pmatrix},\quad U=I\_2,\quad V=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$
Then $(U-V)A(U-V)^T=0\_2$, while
$$UAU^T-VAV^T=\begin{pmatrix} 0 & b \\ b & 0 \end{pmatrix},$$
whose square is non-zero if $b\ne0$.
| 4 | https://mathoverflow.net/users/8799 | 438585 | 177,150 |
https://mathoverflow.net/questions/438536 | 2 | Let $A$ be a PSD random matrix, which has $0$ as one of its eigenvalues. The second smallest eigenvalue of the expectation of $A$ writes as $\lambda\_2(\mathbb{E}(A))>0$.
Why the following statement holds?
If $\|A-\mathbb{E}(A)\|<\lambda\_2(\mathbb{E}(A))$, then $\lambda\_2(A)>0.$
Note that $\|\cdot\|$ denotes th... | https://mathoverflow.net/users/494410 | Deviation of random matrix from its expectation informs the positiveness of its second smallest eigenvalue | The random-matrix connection is a bit of a red herring: Since the weight of $A$ in the statistical average can be arbitrarily small, we might as well replace $\mathbb{E}(A)$ by an arbitrary PSD matrix $B$ with $\lambda\_2(B)>0$.
The statement in the OP
>
> If $\|A-B\|<\lambda\_2(B)$, then $\lambda\_2(A)>0,$
>
>... | 1 | https://mathoverflow.net/users/11260 | 438593 | 177,152 |
https://mathoverflow.net/questions/438573 | 0 | It is one of the concepts used in "ON THE REPRESENTATION OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AS SUPERPOSITIONS OF CONTINUOUS FUNCTIONS OF A SMALLER NUMBER OF VARIABLES", in the second paragraph, highlighted with black background,
[as shown in this image](https://i.stack.imgur.com/7vbus.png).
| https://mathoverflow.net/users/497862 | What does "a universal tree" mean? | In the beginning of Section 6 on p. 318 of Menger's Kurventheorie, referred to in [Kolmogorov's paper](https://cs.uwaterloo.ca/%7Ey328yu/classics/Kolmogorov56.pdf) on the page whose image you linked in your post, we find this:
>
> Wir bezeichen eine Baumkurve $B$ als einen Universalbaum bzw. als Universalbaum $n$-t... | 3 | https://mathoverflow.net/users/36721 | 438597 | 177,153 |
https://mathoverflow.net/questions/438616 | 5 | $\newcommand{\g}{\mathfrak{g}}$Let $G$ be a reductive algebraic group over field $k = \overline{k}$ and consider the characteristic polynomial $\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$ induced by the adjoint action. Assume $|W|$ is invertible in $k$, so Chevalley-Todd-Shepard implies the quotient is affine spac... | https://mathoverflow.net/users/124185 | Geometric properties of the adjoint action of a reductive group | Yes, all these things are true, and 1. and 2. hold for any action of a reductive group $G$ on an affine variety $X$. The key point is that if I have two closed orbits $O\_1,O\_2$, then the functions on their disjoint union is a quotient of functions on $X$ (because $X$ is affine), so there's a function on $X$ that is 1... | 8 | https://mathoverflow.net/users/66 | 438617 | 177,157 |
https://mathoverflow.net/questions/438621 | 2 | I have a sum of $n$ i.i.d random variables $X\_i$ such that $E[X\_i] = 0$,$\mathrm{E}[|X\_i|^{1 + \delta}]$ exists for some $0 < \delta < 1$ but $\mathrm{E}[|X\_i|^{1 + \delta+ \epsilon}]$ does not exists for any $\epsilon > 0$.
I would like to ask if it is possible to provide a lower bound for $P(|\sum\_{i=1}^n X\_i... | https://mathoverflow.net/users/477986 | Lower bound on sum of independent heavy-tailed random variables | Certainly. All you need is $EX^2=+\infty$. Then the characteristic function $f\_X(t)$ satisfies $\lim\_{t\to 0}\frac{1-|f(t)|}{t^2}=+\infty$, so for every finite interval $I\subset \mathbb R$, we have $\lim\_{n\to\infty}\int\_I|f(t/\sqrt n)|^n\,dt=0$ and the usual Fourier trickery finishes the story. Of course, the eff... | 5 | https://mathoverflow.net/users/1131 | 438625 | 177,161 |
https://mathoverflow.net/questions/438623 | 7 | Let $\mathcal{M}\mathrm{fld}\_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category whose objects are topological manifolds, and whose hom-spaces are given by $\operatorname{Sing}\operatorname{Emb}(M,N)$, ... | https://mathoverflow.net/users/144250 | Why does the tangent classifier classify the tangent (micro)bundle? | For an $\infty$-category $X$ let me explicitly write $\mathrm{Fun}(X,\mathcal{S})$ for $\mathcal{P}(X)$. The construction of the tangent classifier firsts associates to $M$ the presheaf $\mathrm{Emb}(-,M) \in \mathrm{Fun}(\mathrm{Top}(n),\mathcal{S})$. The result is literally the topological frame bundle $\mathrm{Emb}(... | 7 | https://mathoverflow.net/users/134512 | 438628 | 177,162 |
https://mathoverflow.net/questions/438547 | 1 | A Verra fourfold is a Fano fourfold which is defined as double cover of $\mathbb{P}^2\times\mathbb{P}^2$ with branch divisor to be $(2,2)$-hypersurface of $\mathbb{P}^2\times\mathbb{P}^2$, which is an index two Fano fourfold of Picard rank two. The $(2,2)$-divisor is a Verra threefold.
But it is known from <https://w... | https://mathoverflow.net/users/41650 | There are only one type of Verra fourfold? | I am not aware of Verra fourfolds of second type, and I think the situation with Verra fourfolds is more similar to the situation with GM sixfolds, where also only one type is available.
| 2 | https://mathoverflow.net/users/4428 | 438630 | 177,163 |
https://mathoverflow.net/questions/438126 | 0 |
>
> **Question.** Is it possible to construct a right triangle with a given [cathetus](https://en.wikipedia.org/wiki/Cathetus) $a$ and a given opposite angle $\alpha$ using only compass and ruler in the absolute geometry (so without the axiom of parallels)?
>
>
>
Observe that such a triangle always exists (by th... | https://mathoverflow.net/users/61536 | Is a right triangle with the given cathetus and the opposite angle constructible in the absolute plane? | A right triangle with a given side and opposite angle can be constructed in the hyperbolic plane with the help of a starightedge and a compass. This is done in Construction 7.2.4 of [this Bachelor Thesis](https://studenttheses.uu.nl/bitstream/handle/20.500.12932/40052/thesis.pdf?sequence=1) of Ruben de Vries.
| 0 | https://mathoverflow.net/users/61536 | 438634 | 177,165 |
https://mathoverflow.net/questions/438632 | 1 | Consider a one dimensional random walk, in which the probability of moving left along a line is $q=1/2$ and the probability of moving right is $p=1/2$. The square root error $\langle d\_N \rangle$, which is the expectation of the absolute distance traveled after $N$ steps is known to be $\sqrt{2N/\pi}.$ I am interested... | https://mathoverflow.net/users/130113 | A question about the square root error of one dimensional random walks | Because of the central-limit-theorem, for large $N$ the absolute distance $d\_N$ converges in distribution as
$$P\_N(d\_N/\sqrt N)\to p(|X|),$$
where $X$ is a Gaussian random variable with mean zero and variance $2p=1-r$. Since $\mathbb{E}(|X|)=2\sqrt{p/\pi}$ we conclude that
$$\mathbb{E}(d\_N)\to \sqrt\frac{2(1-r)N}{\... | 1 | https://mathoverflow.net/users/11260 | 438640 | 177,167 |
https://mathoverflow.net/questions/438466 | 1 | I know several specific variations of the domino tiling problem has been determined to be decidable or undecidable, such as the seed domino problem. I have a variation which I have not been able to find, but as I am not familiar with this subject, I thought someone here might know better.
Given a Wang tile $T=(T\_N,T... | https://mathoverflow.net/users/143153 | A variation of domino tiling problem with fusions | Each tiling with tiles from $\mathcal{T}$ gives rise to exactly two tilings with tiles from $\mathcal{T}\_1$ and exactly two tilings with tiles from $\mathcal{T}\_2$, given by taking the two choices of fusing neighbouring pairs (either $2n\longleftrightarrow 2n+1$ or $2n-1 \longleftrightarrow 2n$). This process also pr... | 1 | https://mathoverflow.net/users/21271 | 438647 | 177,169 |
https://mathoverflow.net/questions/438638 | 3 | Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet series
$$L\_m(\chi)=L\_m(1,\chi)=\sum\_{n=1}^m\frac{\chi(n)}{n}.$$
It is well-known that $L\_\infty(\chi)$ is positive, ... | https://mathoverflow.net/users/207039 | Positivity of partial Dirichlet series for a quadratic character? | Theorem 6.18 in the same chapter of Apostol shows, by partial summation, that
$$L\_m(\chi) = L(1,\chi) + O\_{\chi}(m^{-1})$$
where the implied constant is effective and depends on $\chi$. This can be explicated by using the Pólya--Vinogradov inequality, which gives
$$L\_m(\chi) = L(1,\chi)+O( m^{-1} N^{1/2} \log N)$$
w... | 5 | https://mathoverflow.net/users/31469 | 438649 | 177,170 |
https://mathoverflow.net/questions/438594 | 7 | I've always wondered if the *DeMoivre method* to generate an algebraic number $x\_p$,
$$x\_p = u\_1^{1/p}+u\_2^{1/p}$$
of degree $p$ using only quadratic roots $u\_i$ could be generalized using ***cubic*** roots $v\_i$,
$$x\_p = v\_1^{1/p}+v\_2^{1/p}+v\_3^{1/p}$$
I serendipitously found a method which works for... | https://mathoverflow.net/users/12905 | A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics | The following can explain the shape of the expression and also why you always get pairs (why the minimal polynomials for the two elements in a pair differ by few coefficients, I don't know).
Let $p$ be a prime congruent to $1$ mod $6$, and consider the rational function $f(X):=g(X^p)$, where $g(X):=\frac{X^3-3X+1}{X-... | 5 | https://mathoverflow.net/users/127660 | 438651 | 177,171 |
https://mathoverflow.net/questions/438653 | 3 | Theorem 7.5.2 states:
Let $v\_1, \dots, v\_n$ be vectors with $\|v\_i\| \leq 1.$ Let $\epsilon\_1, \dots, \epsilon\_n \in \{-1, 1\}$ be independent with uniform probability and let $X=\|\epsilon\_1 v\_1 + \dots + \epsilon\_n v\_n\|.$ Then $$\Pr[X - \mathbb{E}[X]] > \lambda \sqrt{n}] < e^{-\lambda^2/2},$$ $$\Pr[X - \m... | https://mathoverflow.net/users/497926 | Probabilistic method Alon and Spencer Azuma's inequality | $\newcommand\ep\epsilon$The martingale $(X\_i)$ is given by the formula
$$X\_i:=E(X|\ep\_1,\dots\ep\_i).$$
By the independence of the $\ep\_i$'s,
$$X\_i=g\_i(\ep\_1,\dots,\ep\_i),$$
where
$$g\_i(t\_1,\dots,t\_i):=E\Big\|\sum\_{j=1}^i t\_jv\_j+\sum\_{j=i+1}^i \ep\_jv\_j\Big\|$$
for real $t\_1,\dots,t\_i$.
By the triangl... | 3 | https://mathoverflow.net/users/36721 | 438655 | 177,174 |
https://mathoverflow.net/questions/438659 | 0 | Let $A$ be a real symmetric matrix in $M\_{2n}(\mathbb{R})$with $A^2=I\_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$J=I\_n\oplus (-I\_n)$$
Q. By the eigen values/eigenvectors of $A$, can we find/make some eigenvalues/eigenvectors of t... | https://mathoverflow.net/users/84390 | The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$ | **Original question:** *What information can be extracted concerning the eigenvalues/eigenvectors of the product $JA$?*
The matrix $JA$ is orthogonal,
$$JA(JA)^\top=JA^2J=J^2=I,$$
so its eigenvalues $\lambda\_p$ are complex conjugate pairs $e^{\pm i\phi\_p}$ on the unit circle. The eigenvectors are an orthonormal set... | 1 | https://mathoverflow.net/users/11260 | 438665 | 177,180 |
https://mathoverflow.net/questions/438656 | 7 | The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi\_{2n} = \Psi\_n \left( \Psi\_{n+2} \Psi\_{n-1}^2 - \Psi\_{n-2} \Psi\_{n+1}^2 \right) / \ 2y$$
$$\Psi\_{2n+1} = \Psi\_{n+2} \Psi\_n^3 - \Psi\_{n+1}^3 \Psi\_{n-1}$$
Who originally discovered these relations?
~~For some reason,... | https://mathoverflow.net/users/110923 | Reference request for recurrence relation of division polynomials | These recurrences are stated explicitly in Weber's *Lehrbuch der Algebra* (published in 1908). See [volume 3, section 58, page 200](https://archive.org/details/lehrbuchderalgeb03webeuoft/page/200/mode/2up). Weber doesn't give a citation to them, so it's hard to know if they were worked out by him or not.
I found the ... | 11 | https://mathoverflow.net/users/48142 | 438675 | 177,182 |
https://mathoverflow.net/questions/438622 | 1 | Let $K$ be a Markov kernel from $\mathcal{X}$ to $\mathcal{Y}$, i.e., $K(\cdot|x)$ is a probability measure on $\mathcal{Y}$ for all $x\in \mathcal{X}$.
Let $\mu$ and $\nu$ be two probability measures on $\mathcal{X}$. Moreover, let $K\mu$ and $K\nu$ be the probability measures on $\mathcal Y$ induced by $K$ with input... | https://mathoverflow.net/users/41666 | An inequality relating $\ell_1$ distance of input and output of a Markov krnel | Let me assume that $X$ and $Y$ are topological spaces (satisfying the usual conditions), and denote by $M(X), M(Y)$ the respective Banach spaces of finite (signed) Borel measures endowed with the total variation norm. I will also need the closed codimension 1 subspaces $M\_0(X),M\_0(Y)$ that consist of the measures of ... | 1 | https://mathoverflow.net/users/8588 | 438679 | 177,184 |
https://mathoverflow.net/questions/438682 | 5 | Let G be a discrete or say for sake of simplicity a finite group. In Hatcher's book Algebraic Topology on p 89 the construction of universal bundle $EG$ carries structure of a $\Delta$-complex whose $n$-simplices are the ordered $(n + 1)$ tuples
$[g\_0, ... ,g\_n]$ of elements of $G$. As
quotient space the model of the... | https://mathoverflow.net/users/108274 | Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$ | I think that this works.
EDIT: No, it doesn't. See John Rognes's comment.
Notation: For a point $x\in EG$ we may symbolically write $x=\sum\_{j=0}^nt\_jg\_j$, where $g\_0,\dots ,g\_n$ is an ordered tuple of distinct elements of $G$ and $t\_j\ge 0$ and $\sum\_jt\_j=1$. Here the face relations are accounted for by ag... | 5 | https://mathoverflow.net/users/6666 | 438688 | 177,186 |
https://mathoverflow.net/questions/394142 | 7 | $\newcommand\R{\mathbb R}\newcommand\la\lambda$It is well known and easy to see that the rotationally invariant
product of two probability measures on $\R$ has to be a Gaussian (or Dirac) measure; see e.g. [this answer](https://math.stackexchange.com/a/2599580/96609).
This appears to make the conjecture below somewha... | https://mathoverflow.net/users/36721 | Normal distribution by successive approximation? | In the rewritten form it doesn't require any ingenuity at all. For brevity, introduce the notation
$$
(Th)(s)=\frac 1\pi\int\_0^s\frac{h(s-u)h(u)}{\sqrt{u(s-u)}}\,du=\frac 1\pi\int\_0^1\frac{h(s(1-v))h(sv)}{\sqrt{v(1-v)}}\,dv
$$
Suppose that $h\_0$ is any real valued Lipschitz function on $[0,+\infty)$ (the finiteness ... | 1 | https://mathoverflow.net/users/1131 | 438691 | 177,187 |
https://mathoverflow.net/questions/438390 | 4 | Consider bivariate copulas $C\_1$ and $C\_2$ with $\max\{C\_1(u,v), C\_2(u,v)\}< M\_2(u,v)$ for all $u,v \in(0,1)$, where $M\_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound.
Is there a copula $D$ with $\max\{C\_1(u,v), C\_2(u,v)\}\leq D(u,v) < M\_2(u,v)$ for all $u,v \in(0,1)$?
The problem here is that $... | https://mathoverflow.net/users/497745 | Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound | You can prove more. Let $F(u,v)$ be any $1$-Lipschitz function on $[0,1]^2$ such that $F(u,v)<\min(u,v)$ inside the square. Then there exists a copula $D(u,v)$ such that
$$
F(u,v)\le D(u,v)<\min(u,v)
$$
everywhere inside the square.
The second inequality will be immediate if we just construct the corresponding joint ... | 3 | https://mathoverflow.net/users/1131 | 438715 | 177,192 |
https://mathoverflow.net/questions/438538 | 4 | In his paper "Hecke Algebras of type $A\_n$ (Inv. Math. 1988, [EUDML link](https://eudml.org/doc/143571)) and subfactors", in section 2 "Orthogonal representations...", Wenzl takes the usual third relation of a standard generator of Hecke algebras, namely $g\_i^2=(q-1)g\_i+q$, where $q$ is a complex number that is neit... | https://mathoverflow.net/users/160919 | Spectral projection of an eigenvalue associated to a generator of Hecke algebras | Since $e\_i$ kills any eigenvectors with eigenvalue $q$ (clear) and squares to itself (due to the quadratic relation for $g\_i$ and normalisation), it is a projector onto the $-1$-eigenspace of $g\_i$. For the usual (reducible) representation of $H\_n(q)$ on $V^{\otimes n}$, where $g\_i$ becomes the simple transpositio... | 3 | https://mathoverflow.net/users/45956 | 438717 | 177,193 |
https://mathoverflow.net/questions/438710 | 4 | I was trying to get a short, intuitive proof of Sierpinski’s theorem (gch implies axiom of choice) and I could but only by using the following gch2 for the generalized continuum hypothesis gch.
gch: Let S be an infinite set and S’ be the power set of S. If T is a subset of S’ such that |T| > |S|, i.e., T is not injec... | https://mathoverflow.net/users/480831 | About the relationship between the generalized continuum hypothesis and the axiom of choice | Possibility (1) holds; i.e. ZF + "for all limit ordinals $\lambda$, GCH2($\lambda$) holds" implies choice. For it implies that for cofinally many ordinals $\lambda$, $V\_\lambda$ can be wellordered. For fix an ordinal $\alpha$; then we can easily find a limit ordinal $\lambda>\alpha$ such that $\lambda$ does not inject... | 6 | https://mathoverflow.net/users/160347 | 438728 | 177,196 |
https://mathoverflow.net/questions/438727 | 17 | This is a question about the true number of constraints imposed by the Jacobi identity on the structure constants of a Lie algebra.
For an $n$-dimensional Lie algebra, there are $\frac{n^2(n-1)}{2}$ structure constants $f\_{ab}^c$, where I've accounted for antisymmetry but not the Jacobi identity. Accounting for obvi... | https://mathoverflow.net/users/30496 | Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontrivial Lie algebras of dim >5?) | Linear independence does not really say much.
This algebraic variety is discussed in some detail in an old paper of Kirillov and Neretin: [The variety $A\_n$ of $n$-dimensional Lie algebra structures](https://www.mat.univie.ac.at/%7Eneretin/kiri87a.pdf).
The case of $n=4$ which already shows many interesting phenom... | 12 | https://mathoverflow.net/users/1306 | 438733 | 177,198 |
https://mathoverflow.net/questions/420921 | 6 | Consider the following proposition.
Proposition: let $\lambda$ be a limit ordinal and $V$ be the cumulative hierarchy starting with the null set, and $S$ be a set with
$\vert S\vert<\vert V\_\lambda\vert$. Then there exists an $x\in V\_\lambda$ with $\vert x\vert=\vert S\vert$.
The proposition is trivially true for... | https://mathoverflow.net/users/480831 | When does the cardinality of a set equal the cardinality of an element of $V_\lambda$ for $\lambda$ being a limit ordinal? | The proposition holds for the limit ordinal $\lambda$ iff $V\_\lambda$ is wellorderable.
For if $V\_\lambda$ is wellorderable where $\lambda$ is a limit, then it proposition easily follows at $\lambda$. Conversely, suppose the proposition holds at limit ordinal $\lambda$. (Without using AC) we can specify a certain o... | 4 | https://mathoverflow.net/users/160347 | 438734 | 177,199 |
https://mathoverflow.net/questions/438737 | 2 | Let $G$ be a split reductive group and let $T$ be a split maximal torus whose rank is $l$. Is it possible to find a base $\gamma\_1,..., \gamma\_l$ of the weight lattice $X(T)$ such that the cone $C$ in the coweight lattice $Y(T)$ defined by
$$C=\{\psi\in Y(T)\vert (\psi, \gamma\_i)\geq 0\; i=1,...,l\}$$
forms a comple... | https://mathoverflow.net/users/173314 | Find an analogue of Weyl chamber structure | Saying that the translates of C form a complete fan seems to imply that $C$ is a fundamental domain for the Weyl group: The translates cover space, and they form a fan, meaning the intersection of any two translates has lower dimension.
The Weyl group is generated by reflections. The interior of a fundamental domain ... | 3 | https://mathoverflow.net/users/18060 | 438740 | 177,201 |
https://mathoverflow.net/questions/438729 | 3 | Suppose $M$ and $N$ are complete metric spaces and $f, g: M \to N$ are uniformly continuous maps between them with common modulus of continuity $m$. Further suppose $f$ and $g$ are homotopy equivalent.
Must there be a a homotopy equivalence $\alpha\colon M \times [0, 1] \to N$ between $f$ and $g$ such that $\alpha$ i... | https://mathoverflow.net/users/8106 | Uniformly continuous homotopy equivalence | I think not.
If $M$ is compact, then $\alpha$ must be uniformly continuous; but even then the modulus of continuity can be impossible to preserve; there is some interpretation on what is meant here, depending on the metric you use on the product $M\times [0,1]$, but we can have that there must exist some $t$ such tha... | 3 | https://mathoverflow.net/users/4961 | 438742 | 177,202 |
https://mathoverflow.net/questions/438746 | 1 | [This answer](https://math.stackexchange.com/a/3095770/897367) seems to imply that: for an ordinal $\alpha$, to be recursively inaccessible (i.e. $\alpha$ is admissible and limit of admissible) implies to be not locally countable (i.e. $L\_\alpha \models \exists \beta \ ``\beta \text{ is uncountable"}$). Here is the re... | https://mathoverflow.net/users/138089 | Recursively inaccessible ordinals and non locally countable ordinals | Suppose $L\_\alpha\models$"There is a largest cardinal". Then ($L\_\alpha$ is $\Sigma\_n$-admissible for all $n<\omega$) $\Leftrightarrow$ ($L\_\alpha\models$ ZF$^-$) $\Leftrightarrow$ ($L\_\alpha$ does not project ${<\alpha}$) $\Leftrightarrow$ ($\alpha$ is a cardinal in $L\_{\alpha+1}$). So the least $\beta$ such tha... | 3 | https://mathoverflow.net/users/160347 | 438752 | 177,205 |
https://mathoverflow.net/questions/438719 | 9 | The question is as in the title:
Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup\_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural number $q >1$?
Here, "nonpolynomial" excludes constant functions as well, of course.
I think nonpolynomial functions "... | https://mathoverflow.net/users/56524 | Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$? | The answer is no. From [Taylor's theorem with remainder](https://en.wikipedia.org/wiki/Taylor%27s_theorem#Explicit_formulas_for_the_remainder), we see that for any integer $q>2$, we have
$$ f^{(2)}(x) = \sum\_{j=0}^{q-3} \frac{f^{(2+j)}(0)}{j!} x^j + O( \frac{(\ln q)^{-q} |x|^{q-2}}{(q-2)!} )$$
and thus for $|x| \leq \... | 17 | https://mathoverflow.net/users/766 | 438755 | 177,208 |
https://mathoverflow.net/questions/438766 | 0 | $\mathcal{S}^{1/2}\_{1/2}(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such that
\begin{equation}
\sup\_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq CA^kB^q k^{k/2} q^{q/2}
\end{equation}
where $k$ and $q$ are nonnegative integers and $A,B,C$ are positive constants de... | https://mathoverflow.net/users/56524 | Existence of smooth functions $f$ satisfying $\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq C B^q k^{1/8} q^{q/2}$ | Start from the estimate (with $q = K$ and $k = K+1$) (valid for $x > 0$)
$$ |f^{(K)}(x)| \leq C B^{K} (K+1)^{1/8} K^{K/2} x^{-K-1}$$
Integrate back from infinity $K$ times, you get
$$ |f(x)| \leq C \frac{B^K (K+1)^{1/8} K^{K/2}}{K!} \frac{1}{x} $$
Using Stirling's approximation
$$ |f(x)| \lesssim \frac{e^K B^K (K+1)^{1... | 4 | https://mathoverflow.net/users/3948 | 438768 | 177,212 |
https://mathoverflow.net/questions/438744 | 3 | I need the result that for all $t$,
$$\sum\_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i+1} \binom{t-i}{i} C\_{t-i-1} = 0,$$
where $C\_{t-i-1}$ is the $(t-i-1)$-th Catalan number. I've checked for $t$ up to 1000 using Python and the result holds, but I don't really have an intuition for why it would be true. The terms of th... | https://mathoverflow.net/users/497997 | Why is this alternating sum involving Catalan numbers $\sum_{i=0}^{\lfloor t/2 \rfloor} (-1)^{i+1} \binom{t-i}{i} C_{t-i-1} = 0$ for all $t$? | The formula (which holds for $t>1$ but not for $t=1$), is equivalent to
$$\sum\_{t=1}^\infty C\_{t-1}\bigl(x(1-x)\bigr)^t = x,$$
which follows directly from the generating function
$$\sum\_{t=1}^\infty C\_{t-1}x^t = \frac{1-\sqrt{1-4x}}{2}.$$
| 4 | https://mathoverflow.net/users/10744 | 438771 | 177,213 |
https://mathoverflow.net/questions/437803 | 2 | Let $A$ be a Cartan matrix, i.e. a $n\times n$ matrix with integer entries such that $A\_{ii}=2$ and $A\_{ij}\leq0$ for $i\neq j$. Then the corresponding Kac-Moody Lie algebra has a Cartan subalgebra $\mathfrak{h}$, and a root system $\Delta\subseteq\mathfrak{h}^\*$, along with a Weyl group $W$ acting on $\mathfrak{h}^... | https://mathoverflow.net/users/97652 | Real roots along root strings | The characterisation you are looking for is given by the following theorem of Jun Morita (*Root strings with three or four real roots in Kac-Moody root systems*, Tohoku Math. J. 40 (1988), p. 645-650):
**Theorem** (Morita): Let $A = (a\_{ij})$ be an $n\times n$ generalized Cartan matrix, $\Delta$ the associated root ... | 1 | https://mathoverflow.net/users/106751 | 438791 | 177,220 |
https://mathoverflow.net/questions/438666 | 5 | I have recently stumbled across the problem of describing the characteristic subgroups of a finite abelian group. With some discussions with some mathematicians in my lab, I managed to obtain a "visual" description of such, and wrote it down.
Nonetheless, I know that some of this work has already been clear by Kaplan... | https://mathoverflow.net/users/242651 | Characteristic subgroups of a finite abelian $2$-group | It seems to me that there is the requisite description here Kerby, B. L., Rode, E. Characteristic Subgroups of Finite Abelian Groups. Communications in Algebra, 2011, 39:4, 1315-1343 (Section 2).
Theorem 2.9 describes characteristic subgroups of abelian 2-groups.
| 5 | https://mathoverflow.net/users/173068 | 438796 | 177,222 |
https://mathoverflow.net/questions/438793 | 4 | Let $n$ be a positive integer and $1\leq j\leq n$. Consider the following polynomial:
$$p\_{n,j}(x)=\frac{\prod\limits\_{i=1}^{n+1}\frac{x^{i}+1}{x+1}}{\prod\limits\_{i=1}^{j}\frac{x^{i}+1}{x+1}\prod\limits\_{i=1}^{n-j+1}\frac{x^{i}+1}{x+1}}\in\mathbb{F}[x]$$
This polynomial can be computed for each $j,n$ given. Wh... | https://mathoverflow.net/users/482329 | Vanishing of a product of cyclotomic polynomials in characteristic 2 | *Claim:* $p\_{n,j}(1)$ vanishes modulo $2$ if and only there is a carry when adding $j$ and $n+1-j$ in base-$2$.
*Proof:* Let us write $x^n+1 \equiv x^n-1= \prod\_{d \mid n} \Phi\_d(x)$ where $\Phi\_d$ is the $d$th cyclotomic polynomial. Then, since there are $\lfloor m/d\rfloor$ multiplies of $d$ in $\{1,2,\ldots,m\... | 5 | https://mathoverflow.net/users/31469 | 438803 | 177,224 |
https://mathoverflow.net/questions/438785 | 1 | A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a *[Brjuno number](https://en.wikipedia.org/wiki/Brjuno_number)* if $$\sum\_{i=1}^\infty\frac{\log q\_{i+1}}{q\_i} < \infty$$ where $q\_i>0$ is the denominator of the $i$th [convergent](https://en.wikipedia.org/wiki/Continued_fractio... | https://mathoverflow.net/users/8628 | Measurability of Brjuno numbers | I replace $\alpha$ by $x$ below.
Restricting ourselves to positive numbers is useless, since the property considered does not depend on $a\_0(x) = \lfloor x \rfloor$. Therefore, $B$ is the union of all images of $B \cap [0,1[$ by integer translations.
Now, call $G$ the Gauss-Kuzmin map from $[0,1[ \setminus \mathbb... | 1 | https://mathoverflow.net/users/169474 | 438804 | 177,225 |
https://mathoverflow.net/questions/438761 | 9 | Infinitesimal analysis (by which I mean that originating from topos theory---not the nonstandard analysis of Robinson) seeks to recover the pre-limit notions of calculus (which are sufficiently useful to persist in applications and intuition) in a formal way by throwing out the law of the excluded middle, which led to ... | https://mathoverflow.net/users/160917 | Are the models of infinitesimal analysis (philosophically) circular? | It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of hyperbolic geometry. Rather, these interpretations are important because they establish the basic coherence of the other ... | 21 | https://mathoverflow.net/users/1946 | 438808 | 177,227 |
https://mathoverflow.net/questions/438757 | 3 | Suppose $X\_n$ $n = 1, 2, \ldots$ are i.i.d random variables with $\mu := \mathbb{E}[X\_n]$ > 0. (although they are not necessarily non-negative). Then if $S\_n = \sum\_{k=1}^n X\_k$ and $\tau\_a$ = $\inf \{n \geq 1 : S\_n \geq a\}$ - so that $\tau$ is the first time that the random sum exceeds the value a. Does there ... | https://mathoverflow.net/users/497999 | First time random sum exceeds value | $\newcommand\al\alpha$Your desired bound is easy to get if we assume that $\alpha\_p:=E|X\_1-\mu|^p<\infty$ for some real $p\in(2,3)$.
Indeed,
$$E\tau\_a=E\sum\_{n=0}^{\tau\_a-1} 1=E\sum\_{n=0}^\infty 1(\tau\_a>n)=\sum\_{n=0}^\infty P(\tau\_a>n). \tag{1}\label{1}$$
Next, if $n\ge2a/\mu$, then
$$P(\tau\_a>n)\le P(S\_n... | 3 | https://mathoverflow.net/users/36721 | 438813 | 177,229 |
https://mathoverflow.net/questions/438811 | 1 | Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are
$$
\mathbf{V} = \begin{bmatrix} \mathbf{v}\_1 & \mathbf{v}\_2 & \cdots & \mathbf{v}\_p \end{bmatrix}.
$$
Denote its submatrix as
$$
\mathbf{V}\_r = \begin{bmatrix} \mathb... | https://mathoverflow.net/users/159685 | Convergent condition of the high-dimensional submatrix of some orthogonal matrix | For a given matrix $V$ this number $Z=p^{-1} \mathbf{1}^\top \mathbf{V}\_r \mathbf{V}\_r^\top \mathbf{1}$ will converge to $1$ if you take the limit $p\rightarrow\infty$ at fixed $r$. If you average over $V$ (with the Haar measure), the convergence to $(p-r+1)/p$ will apply also for finite ratio $p/r$.
At the left yo... | 1 | https://mathoverflow.net/users/11260 | 438817 | 177,230 |
https://mathoverflow.net/questions/438815 | 0 | Let $X$ and $Y$ be continuous random variables with finite first and second moments. Then, is it true that $Var[X]\geq Var[E(X|Y)]$?
| https://mathoverflow.net/users/490493 | Is the unconditional variance of a RV an upper bound for the variance of any conditional expectation of the RV? | It is enough to show
$$
Var[X] - Var[E[X|Y]] \ge 0.
$$
By the fact $Var[X] = E[X^2] - E[X]^2$,
$$
Var[X] - Var[E[X|Y]] = E[X^2] - E[X]^2 - E\big[ E[X|Y]^2 \big] + E \big[E[X|Y]\big]^2 \\
= E\big[ X^2 - E[X | Y]^2 \big] \\
= E \big[ E\big[X^2 - E[X|Y]^2 \big| Y\big] \big] = E[Var[X|Y]\big] \ge 0,
$$
where the second and... | 2 | https://mathoverflow.net/users/159685 | 438819 | 177,232 |
https://mathoverflow.net/questions/438807 | 4 | Is $d^2+d+1$ prime for infinitely many $d\in \mathbb{Z}\_{>0}$?
This is expected by the [Bunyakovsky conjecture](https://en.wikipedia.org/wiki/Bunyakovsky_conjecture) which says that, under some conditions, given a polynomial $p(x) \in \mathbb{Z}[x]$ we have $p(d)$ prime for infinitely many $d\in \mathbb{Z}\_{>0}$. I... | https://mathoverflow.net/users/100511 | Primes of the form $d^2+d+1$ | To answer the question and summarizing the comments: the answer is no, there is no known proof of this conjecture.
| 7 | https://mathoverflow.net/users/10898 | 438820 | 177,233 |
https://mathoverflow.net/questions/438798 | 2 | I am currently reading volume 2 of "Generalized Functions" by Gelfand and $\mathcal{S}^{2}\_0(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such that
\begin{equation}
\sup\_{x \in \mathbb{R}}\lvert x^k f^{(q)}(x) \rvert \leq CA^kB^q q^{2q}
\end{equation}
where $k$ and $q$ are ... | https://mathoverflow.net/users/56524 | Description of $\mathcal{S}^{2}_0(\mathbb{R})$ and certain class of functions inside it | $\newcommand\S{\mathcal S}\newcommand\R{\mathbb R}$For $x\in(0,1)$, let $g(x):=\frac1x$, and then let
$$f(x):=e^{-g(x)-g(1-x)}$$
for $x\in(0,1)$, with $f(x):=0$ for real $x\notin(0,1)$.
Then $f\in\S\_0^3(\R)$ -- see the details on this at the end of the answer.
As noted in [Willie Wong's comment](https://mathoverfl... | 2 | https://mathoverflow.net/users/36721 | 438823 | 177,235 |
https://mathoverflow.net/questions/438702 | 2 | In an arXiv preprint [[2108.05125v1]](https://export.arxiv.org/abs/2108.05125v1), the authors use the following vertical Fourier decomposition (page 7 therein).
Let $(M,g)$ be a Riemannian surface and $SM$ be its unit tangent bundle. Denote by $V$ the vertical vector field on $SM$, i.e. $V = \partial/\partial \theta$ g... | https://mathoverflow.net/users/117619 | Vertical Fourier decomposition for skew-Hermitian 1-forms | If $SM$ is trivial (such that you can view it as $M\times \mathbb S^1$ and use the angle $\theta$ to describe the second variable), the $k$th Fourier mode of $\mathbb A\in C^\infty(SM,\mathbb C^{n\times n})$ is given by $$
\mathbb A\_k(x)=\int\_{0}^{2\pi}\mathbb A(x,\theta)e^{-ik\theta}d\theta.
$$
Take the complex conj... | 2 | https://mathoverflow.net/users/126651 | 438827 | 177,236 |
https://mathoverflow.net/questions/438814 | 0 | Does there exist a function which is holomorphic in $|z|<1,$ continuous in $|z|\leq1$ and such that the series $\sum |a\_n|$ is divergent, where $a\_n$'s coefficients in the Taylor series expansion of $f?$
| https://mathoverflow.net/users/143655 | A holomorphic function in the open unit disk satisfying certain properties | Yes there are lots of classical examples by Hardy and Littlewood like say $f(z)=\sum \_{n \ge 1}e^{in \log n}\frac{z^n}{n^{3/4}}$.
Using the second derivative test for exponential sums with $f(u)=(u\log u+u\theta)/(2\pi)$, so $f''(u) \sim 1/M$ when $u \sim M$, the exponential sum $\sum\_{n=M}^{2M}e^{in\log n}e^{in\th... | 3 | https://mathoverflow.net/users/133811 | 438829 | 177,237 |
https://mathoverflow.net/questions/438842 | 3 | Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N\_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ is a line, then $N\_{H,X}$ splits as a sum of line bundles. Does there exists an example with $\dim(H) \geq 2$ in whi... | https://mathoverflow.net/users/14514 | Normal bundle of a linear subspace | For instance, if $X$ is a smooth 4-dimensional quadric in $\mathbb{P}^5$ and $H = \mathbb{P}^2$, the normal bundle fits into the exact sequence
$$
0 \to N\_{H/X} \to \mathcal{O}(1)^{\oplus 3} \to \mathcal{O}(2) \to 0,
$$
which implies that
$$
N\_{H/X} \cong \Omega\_H(2).
$$
In particular, it does not split.
| 11 | https://mathoverflow.net/users/4428 | 438848 | 177,243 |
https://mathoverflow.net/questions/438288 | 2 | Let $\Delta:=\partial\_z\,\partial\_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\in\mathbb{C} $.
| https://mathoverflow.net/users/497676 | Linear elliptic equation | The equation is happily linear, so depending on the domain you may find separable analytical solutions to the Dirichlet problem thanks to Sturm–Liouville.
For instance, let's take the domain to be a unit circle and set $u(r=1,\theta)=f(\theta)$ as boundary value. Construct the solution as
\begin{align}
u(r,\theta)=... | 1 | https://mathoverflow.net/users/171439 | 438849 | 177,244 |
https://mathoverflow.net/questions/438741 | 17 | For real variable $x$, the function
\begin{equation}
f(x):=\sum\_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n
\end{equation}
clearly has infinite radius of convergence and defines a $C^\infty$ function on $\mathbb{R}$.
However, I wonder if this $f(x)$ is a bounded function on $\mathbb{R}$ as well. Also, is it possible to es... | https://mathoverflow.net/users/56524 | Evaluating the sum $f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n$ and estimating bounds | As already stated by @Noam and @Alexandre in the comments above,
$$
f(x)= - x \int\_0^1 \mathrm d\tau \sqrt{-8\log \tau} \, J\_1\big(x \tau \sqrt{-8\log \tau}\big),\tag{1}
$$
with Bessel function $J\_1$. Here, I did one further substitution in order to simplify the expression. The function $f(x)$ oscillates at large $x... | 10 | https://mathoverflow.net/users/90413 | 438852 | 177,245 |
https://mathoverflow.net/questions/434233 | 6 | In 1-category theory a representation of a 1-functor $F:C\to Set$ is a 0-cell $X$ in $C$ together with a universal element $u\in FX$ such that the transformation $C(X,-)\to F$ is an isomorphism (=a 1-equality) in the 1-category of functors $Fun(C,Set)$. It is well known how unique a representation $(X,u)$ is: unique up... | https://mathoverflow.net/users/219922 | Do the representations of a 2-functor naturally form a contractible 2-category? | All my 2-categories, 2-functors, 2-(co)limits etc. are per default weak (or pseudo- or bi- if you so like).
One can unfold a 2-functor $M: K\to Cat$ into the 2-category $el(M)$ of its elements, and any representation of $M$ will be a 2-initial object in that 2-category. Unfortunately, and in contrast to the 1-categor... | 2 | https://mathoverflow.net/users/219922 | 438854 | 177,246 |
https://mathoverflow.net/questions/438837 | 9 | $\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an abelian group treated as a trivial $G$- and $H$-module.
Further on, there is a double complex $C^{p,q}:= C^p(H,C^q(G,M))$... | https://mathoverflow.net/users/123432 | Comparing cohomology of a total complex with the cohomology of semidirect product | Let me change the notation a little: let $\phi \colon H \to \operatorname{Aut}(N)$ be a group homomorphism, and consider $G = N \rtimes\_\phi H$ (we will drop the subscript $\phi$ in the rest of this post). As usual, write $^hn$ for $\phi(h)(n)$ if $h \in H$ and $n\in N$.
The point is that the double complex $C^{\bul... | 3 | https://mathoverflow.net/users/82179 | 438859 | 177,248 |
https://mathoverflow.net/questions/438732 | 7 | *Throughout, work in $\mathsf{ZFC}$ + large cardinals (let's say a proper class of Woodin limits of Woodins but I'm happy to go higher if that would help).*
Let $\mathcal{R}=(\mathbb{R};+)$ be the additive group of real numbers. We have $\mathit{Aut}(\mathcal{R})^{L(\mathbb{R})}\not\cong\mathit{Aut}(\mathcal{R})^V$ d... | https://mathoverflow.net/users/8133 | Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals? | Assuming also CH, the answer to the more general question is yes,
there is a structure in $L(\mathbb{R})$ (in fact, just the set $\mathbb{R}$, with no additional structure), whose automorphism group in $V$ does not have an isomorphic copy in $L(\mathbb{R})$. (I haven't really thought about the original question, i.e. $... | 4 | https://mathoverflow.net/users/160347 | 438860 | 177,249 |
https://mathoverflow.net/questions/437766 | 1 | Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
Recall that $u$ is called plurisubharmonic (psh) if its restriction to any complex line is subharmonic.
Any psh function $u$ satisfies the following property: for any point $x\in \Omeg... | https://mathoverflow.net/users/16183 | A characterization of plurisubharmonic functions | You can consult [Harvey and Lawson](http://www.math.stonybrook.edu/%7Eblaine/MA8.pdf), sections 5 and 6, on that matter. Especially Lemma 5.5 and point (6) on p. 19 (note that for smooth $\phi$ condition you gave is equivalent to having complex hessian non negative).
| 2 | https://mathoverflow.net/users/85450 | 438866 | 177,252 |
https://mathoverflow.net/questions/438806 | 1 |
>
> Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves?
>
>
>
The following is a formal capture of that idea:
To the language of $\sf ZF$ (i.e., mono-sorted $\sf FOL(=,\in)$) add primitive partial unary functions $W$ and $j$.
To the axi... | https://mathoverflow.net/users/95347 | Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy? | $\newcommand\Ord{\mathit{Ord}}$The edit to the question has changed it enough so I think it deserves its own answer.
I assume that in reflection we have $\forall \vec{x} \in W\_\alpha \, (\phi \to \phi^{W\_\alpha})$ holds for all ordinals $\alpha$.
The theory with this strong reflection is inconsistent in a strong ... | 3 | https://mathoverflow.net/users/113405 | 438882 | 177,255 |
https://mathoverflow.net/questions/438881 | 6 | The ladder operator in quantum mechanics are the operators
$$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$
and
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\right).$$
They are differential operators on $\mathbb R.$ If one writes them in the Hermite basis, then
$$a^\dagger = \... | https://mathoverflow.net/users/496243 | Spectrum of operator involving ladder operators | **Q:** *Does anybody know how to numerically overcome this pseudospectral effect?*
The key idea is "normal ordering". Rewrite the problem in such a way that annihilation operators $a$ appear to the right of creation operators $a^\ast$. In this particular case, first notice that $H$ has chiral symmetry, if $\lambda$ i... | 6 | https://mathoverflow.net/users/11260 | 438883 | 177,256 |
https://mathoverflow.net/questions/438895 | 0 | I'm looking for a citable reference for the following identity involving the Stirling numbers of the second kind $S(n, k)$ stated in [Equation (27)](https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html): For $n \geq 2$,
$$
\sum\_{m=1}^n S(n, m) (-1)^m (m-1)! = 0.
$$
Thank you.
| https://mathoverflow.net/users/479711 | Identity involving Stirling number of the second kind | The first formula in Section 24.1.4.I.B in Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover edition, 1965 (Library of Congress Catalog Card Number: 65.12253) by Abramowitz and Stegun is
$$
x^n=\sum\_{m=0}^n S(n,m) x(x-1)\cdots(x-m+1). \tag{1}\label{1}
$$
Assuming here $n>1$ (so t... | 4 | https://mathoverflow.net/users/36721 | 438902 | 177,260 |
https://mathoverflow.net/questions/438782 | 3 | Let $\ p\_n\ $ be the consecutive primes starting with
$\ p\_0:=2.\ $ Let $\ M(n)\ $ be the multiplicative monomial
generated by $\ \{p\_k:\ k=0\ldots n\}\ $ (of course $\ 1\in M(n)$).
Could you prove or disprove:
$$ \forall\_{n\in\mathbb Z\_0}\, \exists\_{K\ L\in M(n)}
\ \left( \prod\_{k=0}^n p\_k|K\cdot L\ \text{... | https://mathoverflow.net/users/110389 | Generating prime $\ p_{n+1}\ $ (the complete version) | Assuming that the triple $(a,b,c)=(2,3^{10}\cdot 109,23^5)$ found by Eric Reyssat is the one with the highest quality $q=\log(c)/\log(\text{rad}(abc))=1.6299\ldots$ for the ABC conjecture, one quickly computes that there is indeed no solution for $p\_{n+1}=31$. Setting $Q=\text{rad}(KLp\_{n+1})=\prod\_{k=0}^{n+1}p\_k$,... | 8 | https://mathoverflow.net/users/18739 | 438905 | 177,261 |
https://mathoverflow.net/questions/438694 | 0 | Let $R$ be a regular local ring and let $I \subset R$ be an almost complete intersection ideal, that is, $\mu(I)=\text{ht}(I)+1$ where $\mu(I)$ is the number of minimal generators of $I$ and $ht(I)=\text{dim}(R)-\text{dim}(R/I)$. Is it true that $\text{depth}(R/I)=\text{dim}(R/I)-1$? If not, is there any other relation... | https://mathoverflow.net/users/113200 | Depth of almost complete intersection rings | In general, depth can be zero. Let me give an example.
Let $X\subset\mathbb{P}^n$ be a smooth variety of dimension $d$ which is subcanonical (this means $K\_X=O\_X(\*)$ for some integer $\*$) and NOT arithmetically normal (this means the map $H^0(O\_{\mathbb{P}^n}(l))\to H^0(O\_X(l))$ is not onto for some $l$).
Let... | 2 | https://mathoverflow.net/users/9502 | 438910 | 177,263 |
https://mathoverflow.net/questions/438873 | 8 | *I have asked [the same question on math.SE](https://math.stackexchange.com/questions/4620954/is-the-n-2-th-heat-kernel-coefficient-topological), without much success so I'm trying my luck here too.*
Let $M$ be an $n$-dimensional manifold, with $n$ *even* and consider the heat kernel of the Laplacian on $M$:
$$K\_M(t... | https://mathoverflow.net/users/498081 | Is the $n/2$-th heat kernel coefficient topological? | For $n=2$, the answer is given by $\frac{E}{6}$, where $E$ is the Euler characteristic of $M$, see McKean Jr, H. P., & Singer, I. M. (1967). Curvature and the eigenvalues of the Laplacian. Journal of Differential Geometry, 1(1-2), 43-69.
For $n>2$, if it is topological, then it is none of what you conjectured. Consid... | 6 | https://mathoverflow.net/users/56624 | 438923 | 177,268 |
https://mathoverflow.net/questions/438889 | 5 | This question came about when I was naively considering to what extent generating sets for finitely-generated groups are unique.
Let $G$ be a group and let $\phi\_1, \phi\_2 : F\_k \to G$ be two surjective homomorphisms from a free group on $k$ letters $F\_k$. Given a natural number $K \geq k$ let the surjective homo... | https://mathoverflow.net/users/419791 | Stable equivalence of generating sets of a finitely-generated group? | The answer is $K=2k$, which is an exercise. Here is a sketch: Let $f\colon \langle x\_i\rangle\to G$ be a surjective homomorphism, and $h\colon \langle x\_i\rangle\*\langle y\rangle\to G$ the map defined by $h(x\_i)=f(x\_i)$, $h(y)=e$. Let $g\in G$ be arbitrary, and choose $w\in\langle x\_i\rangle$ such that $f(w)=g$. ... | 9 | https://mathoverflow.net/users/2225 | 438951 | 177,279 |
https://mathoverflow.net/questions/438878 | 5 | For a link $L\subset S^3$ and two *Seifert surfaces* (edit: a better name would be slice surfaces as the comments below [1](https://mathoverflow.net/questions/438878/are-two-slice-surfaces-with-minimal-genus-isotopic#comment1131877_438878) [2](https://mathoverflow.net/questions/438878/are-two-slice-surfaces-with-minima... | https://mathoverflow.net/users/117639 | Are two slice surfaces with minimal genus isotopic? | The question is very loaded and the question would almost require a survey...
Anyway, the answer to your questions is mostly no. Let $S \subset S^4$ be a 2-knot (i.e. an embedded 2-sphere), $p \in S$ a point, and remove a small ball around $p$. The complement of $S$ in the complement of the ball is a slice disc $D$ f... | 9 | https://mathoverflow.net/users/13119 | 438960 | 177,282 |
https://mathoverflow.net/questions/438701 | 3 | **I. Comparison**
It doesn't seem to be well-known that the generic cubic (prominent in this [MO post](https://mathoverflow.net/questions/438594/)) for $C\_3 = A\_3$,
$$x^3-nx^2+(n-3)x+1 = 0$$
has the nice property that its roots $a,b,c$, if in correct order, obey,
$$(a^2b)^{1/3}+(b^2c)^{1/3}+(c^2a)^{1/3} = 0$$... | https://mathoverflow.net/users/12905 | A similar relationship between the generic cubic and the Lehmer quintic? | The map $s(r) = \frac{n+2 + nr - r^2}{1 + (n+2)r}$ cyclically permutes the roots. This map is given in [2], and I found it through the reference in [1]. It turns out to give the correct order.
Explicitly, if we pick one root $x\_1$ we have
\begin{eqnarray\*}
D &=& n^3 + 5n^2 + 10n + 7 \\
Dx\_2 &=& (n^2 + 4n + 4)x\_... | 3 | https://mathoverflow.net/users/46140 | 438962 | 177,284 |
https://mathoverflow.net/questions/438374 | 2 | Let $\mathfrak {g}$ be a non-degenerate triangular Lie bialgebra with the non-degenerate triangular structure $r \in \bigwedge^2 \mathfrak {g}.$ Then how does it induce $r^{-1} \in \bigwedge^2 \mathfrak {g}^{\ast}\ ?$
Since $r$ is a non-degenerate triangular structure on $\mathfrak {g}$ it induces an isomorphism $r^{... | https://mathoverflow.net/users/491136 | How to define inverse of a non-degenerate triangular structure? | The answer to your last question is yes.
Note that the property of being triangular for $r$ reads as $[r,r]=0$, where $[-,-]$ is the Lie bracket of $\mathfrak{g}$ that one extends to $\wedge^\bullet\mathfrak{g}$ by the graded Leibniz rule.
This condition translates on $\omega:=r^{-1}$ in the following way: $d\_{CE}... | 2 | https://mathoverflow.net/users/7031 | 439006 | 177,305 |
https://mathoverflow.net/questions/439005 | 8 | One can consider a variant of the Dedekind-Peano axioms
in which one replaces the assumption that every number
has exactly one successor by the assumption that every
number has **at most** one successor, leaving the other
axioms the same (and in particular retaining the
second-order induction axiom). The models of this... | https://mathoverflow.net/users/3621 | Dedekind-Peano axioms, but numbers have at most one successor | Victoria Gitman and I recently looked at the first-order version of this theory, which we called fPA, to contrast it with the theory FPA, which Woodin has looked into, which is a possibly stronger version.
[**Update** As explained by Emil Jeřábek in the comments below and in Ali Enayat's answer, this theory has been ... | 14 | https://mathoverflow.net/users/1946 | 439012 | 177,311 |
https://mathoverflow.net/questions/439018 | 4 | I am wondering if there are some know results for the non-trivial roots at ${\rm Re}(s) = \frac{1}{2}$, even maybe a table of the first few roots with $t>0$. This sister function
$$
L\_4^\* (s,\chi\_4) = \prod\_{p} \Big(1 + \frac{\chi\_4(p)}{p^{s}}\Big)^{-1} = \left(1-\frac{1}{4^s}\right)
\frac{\zeta(2s)}{L\_4(s,\chi... | https://mathoverflow.net/users/140356 | Zeros of Dirichlet function $L(s,\chi_4)$ | There is no need to use the subscript $4$ on the $L$-function: just write
$L(s,\chi\_4)$ and $L^\*(s,\chi\_4)$. The first nontrivial zero in the upper half of the critical strip is $1/2 + it$ where
$t \approx 6.020948i$.
Data for $L(s,\chi\_4)$ is available on the LMFDB: see <https://www.lmfdb.org/L/1/2e2/4.3/r1/0/0>... | 11 | https://mathoverflow.net/users/3272 | 439019 | 177,313 |
https://mathoverflow.net/questions/438993 | 1 | Let $\mathcal{X}$ be some nice space, let $\Phi$ be some ordered space, and let $K :\mathcal{X} \times \mathcal{X} \times \Phi \to \mathbf{R}$ be a positive-semidefinite kernel indexed by a hyperparameter $\phi$, i.e. for any $\phi \in \Phi$, the mapping $K(\cdot, \cdot, \phi) :\mathcal{X} \times \mathcal{X} \to \mathb... | https://mathoverflow.net/users/121692 | Monotonicity of kernel matrices with respect to hyperparameters | This example may be a little bit ridiculous, but suppose we take $\mathcal{X}=\mathbb{R}$ and let $\Phi$ be any parametric subset of the set of PSD kernels itself. We define
$$ \mathbf{K}(\phi)\_{i,j} = \sum\_{i,j}\phi(x\_i, x\_j) $$
We will say that $\phi\_1 \preceq \phi\_2$ iff for any $N$, $x\_1,...,x\_N$, and $c\... | 1 | https://mathoverflow.net/users/8938 | 439021 | 177,314 |
https://mathoverflow.net/questions/439028 | 8 | Let $M$ be the Mandelbrot set.
**Question:** Is the interior of $M$ dense in $M$?
My intuition is that this is true, and moreover that hyperbolic components of the interior are dense in $M$ as well, and moreover that this is known (as it is not very close to the Hyperbolicity Conjecture and thus not too hard). Is t... | https://mathoverflow.net/users/22930 | Does the Mandelbrot set have dense interior? | The answer is positive and this is not difficult (a normal families argument). The boundary
of the Mandelbrot set is the set of $J$-instability. Every point $c\_0$ of this set is a limit of $c\_n$ such that $z\mapsto z^2+c\_n$ has a superattracting cycle. So actually hyperbolic components accumulate
to every boundary p... | 10 | https://mathoverflow.net/users/25510 | 439039 | 177,321 |
https://mathoverflow.net/questions/439030 | 7 | In some constructive systems, [every function from $\mathbb{R}\to\mathbb{R}$ is continuous](https://math.stackexchange.com/a/176285/24563) (roughly speaking from the classical fact that computable functions are continuous). More weakly, in Bishop's constructive approach [one cannot prove](https://mathoverflow.net/q/164... | https://mathoverflow.net/users/36972 | Continuous nowhere differentiability and constructive mathematics | The usual proofs are either constructive or can be made constructive fairly easily, sometimes by a slight weakening of the theorem. For example, let us read through [this note](https://math.berkeley.edu/%7Ebrent/files/104_weierstrass.pdf) by Brent Nelson. (Please read the four page proof before reading the rest of the ... | 8 | https://mathoverflow.net/users/1176 | 439047 | 177,324 |
https://mathoverflow.net/questions/439051 | 10 | [Baker's theorem](https://en.wikipedia.org/wiki/Baker%27s_theorem) in transcendental number theory states that
$$
\left|\beta\_0 + \sum\_{i=1}^n \beta\_i \log \alpha\_i\right| > H^{-C}
$$
where
* $\beta\_0, \ldots, \beta\_n$ are algebraic numbers, not all zero,
* $\alpha\_1, \ldots, \alpha\_n$ are multiplicatively in... | https://mathoverflow.net/users/4758 | Baker's theorem for integer combinations of logarithms of integers? | The main difficulty in proving Baker's theorem is in estimating $C$. If you don't care about those estimates, then the proof is not difficult. For example, Chapter 7 of Waldschmidt's *Diophantine approximation on linear groups* gives a simple proof for the homogeneous case $\beta\_0=0$. Some technicalities (like using ... | 14 | https://mathoverflow.net/users/1811 | 439054 | 177,328 |
https://mathoverflow.net/questions/439046 | 4 | The “second Kleene algebra” $\mathcal{K}\_2$ is defined, e.g. [here on nLab](https://ncatlab.org/nlab/show/Kleene%27s+second+algebra), or in section 1.4.3 of van Oosten's book *Realizability: an Introduction to its Categorical Side* (2008), or as example 3.4 of [the notes “Realizability” by Thomas Streicher (2017–2018)... | https://mathoverflow.net/users/17064 | Intuition behind Kleene's “second algebra” $\mathcal{K}_2$ | There's a bit of explanation in my [lecture notes on intuitionistic logic](https://awswan.github.io/teaching/intuitionisticlogic/index.html), in particular the [section on function realizability](https://awswan.github.io/teaching/intuitionisticlogic/lecturenotes/section14.pdf).
The idea is that you want to encode con... | 4 | https://mathoverflow.net/users/30790 | 439062 | 177,330 |
https://mathoverflow.net/questions/439063 | 4 |
>
> Can we have a sequence of transitive sets $\langle\mathcal V\_0, \mathcal V\_1, \mathcal V\_2,...\rangle$, all modeling $\sf ZF$, such that $\mathcal P(V\_n) \subset \mathcal V\_{n+1}$, and the cardinality of each is inaccessible from the cardinalities of the sets preceding it, and where choice fails inside each ... | https://mathoverflow.net/users/95347 | Can we have this sequence where choice fails and returns? | Sure.
Start with countably many inaccessible cardinals, $\kappa\_n$, and now take the full support product adding $\kappa\_n^+$ subsets to each $\kappa\_n$. Then the $n$th model is the symmetric extension given by adding all the generics for the first $n-1$ coordinates, and violating choice in the $n$th one (e.g. a C... | 8 | https://mathoverflow.net/users/7206 | 439064 | 177,331 |
https://mathoverflow.net/questions/438850 | 1 | If $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots\geq\lambda\_k)\vdash n$ is an integer partition of $n$ then $\lambda\_1$ is its largest part and $k$ is its length, $\ell(\lambda)$.
Define the statistic $c\_n(\lambda)=\max\{\lambda\_1,\ell(\lambda)\}$ for the above partition. Also consider the polynomial (in $t$),
$$Q\... | https://mathoverflow.net/users/66131 | Largest part and length of a partition in play | Your $c\_n(\lambda)$ is closely related to what [Billey, Konvalinka, Swanson 2020](https://arxiv.org/abs/1905.00975) call the aft of a partition:
$$ \text{aft}(\lambda) = | \lambda | - \max\{\lambda\_1, \lambda'\_1\}$$
where $\lambda'$ is the conjugate of $\lambda$, so $\lambda'\_1 = \ell(\lambda)$.
Following up on M... | 3 | https://mathoverflow.net/users/14807 | 439084 | 177,340 |
https://mathoverflow.net/questions/438875 | 2 | Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\sum\_{k=i}^{n+i}s\_k^{m+1-j}e^{-s\_k}\left[\prod\_{l=i,l\neq k}^{n+i}(s\_k-s\_l)\right]^{-1}\right]\_{i,j=1\ldots m+1}$... | https://mathoverflow.net/users/153070 | A Vandermonde like determinant with exponentials | After slight renumbering $(m,n)\mapsto (m,n)-1$ and horizontal reflection $j\mapsto m+1-j$, let's call the $m\times m$ matrix in the OP's determinant
$$
A=\left[\sum\_{k=i}^{i+n-1}s\_k^{j-1}e^{-s\_k}\prod\_{l=i,l\neq k}^{i+n-1}(s\_k-s\_l)^{-1}\right]\_{\,i,j=1}^{\,m},\tag{1}
$$
and define $r=m+n-1$. Note that this answ... | 1 | https://mathoverflow.net/users/90413 | 439088 | 177,343 |
https://mathoverflow.net/questions/436657 | 5 | The maximum symmetry metric on real projective space $ \mathbb{RP}^n $ is the round metric.
The maximum symmetry metric on complex projective space $ \mathbb{CP}^n $ is the Fubini–Study metric; see [Maximum symmetry metric on $ \mathbb{C}P^n $](https://mathoverflow.net/questions/433847/maximum-symmetry-metric-on-math... | https://mathoverflow.net/users/387190 | Maximum symmetry metric on Cayley plane $ F_4/{\operatorname{Spin}(9)}$ | $\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$A maximum symmetry metric on $M:=\mathbb{O}P^2$ must be equivalent to the one you described.
Here's one way to see it.
Suppose $G$ is the isometry group of some fixed Riemannian metric on $M$ of maximal symmetry. Then $G$ acts effectively on $M$. From Theo... | 4 | https://mathoverflow.net/users/1708 | 439091 | 177,344 |
https://mathoverflow.net/questions/439095 | 3 | Let $C$ be a smooth genus two hyperelliptic curve and $\mathcal{M}\_C$ be a moduli space of stable rank two vector bundles of fixed degree(or fixed determinant). Then is $\mathrm{Aut}(\mathcal{M}\_C)\subset\mathrm{Aut}(C)?$. The motivation is following. Let $Y$ be an index two prime Fano threefold of degree $4$, one ca... | https://mathoverflow.net/users/41650 | Automorphism of moduli space of stable vector bundles over a curve | The moduli space of rank 2 vector bundles on $C$ with fixed determinant of odd degree is a smooth complete intersection of two quadrics in $\ \mathbb{P}^5\qquad$ (P. Newstead, Topology 7 (1968), 205-215); in an appropriate system of coordinates, it is given by $\sum X\_i^2=\sum \lambda \_iX\_i^2=0$, for $\lambda\_0,\ld... | 11 | https://mathoverflow.net/users/40297 | 439096 | 177,346 |
https://mathoverflow.net/questions/365879 | 3 | I have arrived at needing SPDEs and encountered a strange thing. In the literature, two approaches are mentioned: One where the equation is thought of as an SDE in an infinite dimensional space; an other where the solution is thought of a random field which changes over time (?). Now, I have read that these two approac... | https://mathoverflow.net/users/160649 | Two approaches two SPDEs not equivalent? | Indeed in the reference ["Stochastic integrals for spde’s: A comparison"](https://www.sciencedirect.com/science/article/pii/S0723086910000435),
they show that the two approaches
* Walsh random field approach
* the framework of the stochastic evolution in Hilbert spaces
are equivalent. However, as explained eg. ["In... | 1 | https://mathoverflow.net/users/99863 | 439104 | 177,348 |
https://mathoverflow.net/questions/439105 | 39 | Is the number of "breakthroughs" in mathematics decreasing, as it is claimed to be in other sciences?
Background for the question:
1. Park, M., Leahey, E. & Funk, R.J. Papers and patents are becoming less disruptive over time. Nature 613, 138–144 (2023). <https://doi.org/10.1038/s41586-022-05543-x>
2. [What Happene... | https://mathoverflow.net/users/498238 | Is the number of "breakthroughs" in mathematics decreasing, as it is claimed to be in other sciences? | **Q:** *Is the number of "breakthroughs" in mathematics decreasing?*
To get some quantitative feel for the question I considered the [Timeline of mathematics](https://en.wikipedia.org/wiki/Timeline_of_mathematics) on Wikipedia. Not all entries are "breakthroughs", but most could be considered as such. Here is a plot ... | 19 | https://mathoverflow.net/users/11260 | 439118 | 177,353 |
https://mathoverflow.net/questions/297638 | 15 | I am looking for a reference to the bipartite version of the Schoenberg's criterion of embeddability into a Hilbert space. The Schoenberg criterion is formulated as Proposition 8.5(ii) of the book "[Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss](https://books.google.pl/books/about/Geometric_No... | https://mathoverflow.net/users/61536 | A reference to a characterization of metric spaces admitting an isometric embedding into a Hilbert space | In "[Geometry of cuts and metrics](https://www.researchgate.net/publication/242569526_Geometry_of_Cuts_and_Metrics)" by Deza and Laurent, your inequality is called *pure inequality of negative type*.
In Section 6.1.1, it is stated that pure inequalities of negative type imply all inequalities of negative type.
In T... | 6 | https://mathoverflow.net/users/1441 | 439119 | 177,354 |
https://mathoverflow.net/questions/439110 | 1 | I want to draw a matrix $A\in \mathbb{R}^{n\times k}$ uniformally at random from the Stiefel manifold $\mathbb{V}\_k(\mathbb{R}^n)$, that is from the collection of all $n\times k$ matrices $A$ such that $A^TA=I\_{k\times k}$.
Is this true that generating a matrix $X\in\mathbb{R}^{n\times n}$ with iid standard normal en... | https://mathoverflow.net/users/103133 | Is svd of a Gaussian iid matrix corresponds to Haar measure on the Stiefel manifold? | Yes, this is correct; the probability distribution of $X=U\Lambda V^\top$ is
$$P(X)\propto \exp\left(-\tfrac{1}{2}\,{\rm tr}\,XX^\top\right)=\exp\left(-\tfrac{1}{2}\,{\rm tr}\,\Lambda^2\right),$$
so it is independent of the orthogonal matrices $U$,$V$. These are therefore distributed uniformly in $O(n)$, and identifyin... | 1 | https://mathoverflow.net/users/11260 | 439140 | 177,363 |
https://mathoverflow.net/questions/301577 | 15 | This is a followup question to [Does foundation/regularity have any categorical/structural consequences, in ZF?](https://mathoverflow.net/questions/300046/does-foundation-regularity-have-any-categorical-structural-consequences-in-zf)
As shown in answers to that question, the axiom of foundation (AF, aka regularity) h... | https://mathoverflow.net/users/2273 | Consequences of foundation/regularity in ordinary mathematics (over ZF–AF)? | I think Frucht's theorem, the statement "every group is isomorphic to the automorphism group of a graph", is a great example of what you're looking for; from the statement, it'd be hard to tell that any set theory is relevant in the proof.
Frucht's theorem is true in ZF (and ZFC-AF), but independent from ZF-AF. I'll ... | 12 | https://mathoverflow.net/users/498259 | 439141 | 177,364 |
https://mathoverflow.net/questions/439101 | 5 | Consider 6 people $p\_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p\_i$ again. Suppose no pair of these points $p\_i$ are antipodal. At each point $p\_i$ each person divides the sphere in two 4 identical regions by two orthogonal planes passing through the center of the sph... | https://mathoverflow.net/users/115905 | Six people standing on earth | It is possible for all pairs of these people to see each other in different colors.
Consider six people all on the arctic circle, with longitudes 0°, 61°, 122°, 183°, 244° and 305° east of the prime meridian. Suppose they all color the directions northwest and southeast of them red, and they all color the directions ... | 1 | https://mathoverflow.net/users/nan | 439146 | 177,365 |
https://mathoverflow.net/questions/439103 | 4 | I posted this on Stackexchange but got no responses or comments.
Consider the following integral, for $\epsilon\ne 0:$
$$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int\_{\Omega}yb\,e^{\frac{i}{\epsilon}[-ay+bx-yb]}\,dx\,da\,dy\,db\,,$$
where $\Omega$ is some compact neighborhood of the origin (you can assume the i... | https://mathoverflow.net/users/40323 | How to compute the asymptotics of this oscillatory integral? | As Fedor Petrov suggested, you can diagonalize the quadratic form.
Writing $A = a+y$, $B = (a-y)$, $C = (b+x-y)$ and $D = (b-x +y)$, up to some negligible constants (and restoring the smooth cut-off) your integral is
$$ \int \varphi \cdot (C+D)(B-A) e^{\frac{i}{\epsilon} (- A^2 + B^2 + C^2 - D^2)} $$
This you can... | 4 | https://mathoverflow.net/users/3948 | 439154 | 177,369 |
https://mathoverflow.net/questions/439168 | 2 | Let $D$ be a diagonal matrix in $M\_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0
Suppose that $e\_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is
$$e\_k=(0,\cdots 0,1,0,\cdots,0)^t$$ where $1$ is in the $k^{th}$ position.
Let us consider the following vectors for $k=1,\cdots,n$.
$$v\_k=e\_{2k}~~,... | https://mathoverflow.net/users/84390 | The eigenvalues of the product $WD$ for some particular matrices | I noticed that the $n\times n$ matrix $M=WD$ is "periodic", $M^p=\pm M$ for some even integer $p\leq n$. This identifies the eigenvalues of $M$ as $p$-th roots of $\pm 1$.
For some $n$ I find $p=n$ and the eigenvalues are all distinct: $e^{2k\pi i/n}$ if $M^n=M$ or $e^{(2k+1)\pi i/n}$ if $M^n=-M$, with $k=0,1,2,\ldot... | 2 | https://mathoverflow.net/users/11260 | 439175 | 177,373 |
https://mathoverflow.net/questions/439164 | 2 | The universal covering group $G$ of $\mathrm{SL}\_2({\mathbb R})$ has infinite center. Is there an irreducible unitary representation $\pi$ of $G$, whose central character is injective? Or does every $\pi$ factor through a finite cover of $\mathrm{SL}\_2({\mathbb R})$?
| https://mathoverflow.net/users/473423 | Unitary dual of universal cover | Yes.
(a) This is equivalent to ask about the existence of an extremal normalized positive-definite function $\phi$ on $G$ that is "faithful" on its infinite cyclic center $Z$, that is, $\phi^{-1}(\{1\})\cap Z=\{0\}$. This is just because these functions are precisely the functions of the form $x\mapsto\phi(x)=\langle... | 2 | https://mathoverflow.net/users/14094 | 439180 | 177,374 |
https://mathoverflow.net/questions/437097 | 1 | It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus.
Does the general formula for the $n$th derivative of the power-exponential function $x^x$ exist somewhere?
What is the general formula for the $n$th derivative of the power-exponential function $x^... | https://mathoverflow.net/users/147732 | Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$? | The Stirling numbers of the first kind $s(n,k)$ for $n\ge k\ge0$ can be analytically generated by
\begin{equation\*}%\label{1st-stirl-gen-funct}
\frac{[\ln(1+t)]^k}{k!}=\sum\_{n=k}^\infty s(n,k)\frac{t^n}{n!},\quad |t|<1,
\end{equation\*}
which can be rearranged as Maclaurin's power series expansion of the power functi... | 2 | https://mathoverflow.net/users/147732 | 439188 | 177,376 |
https://mathoverflow.net/questions/426487 | 1 | I am the author of the package for tomographic reconstruction <https://github.com/kulvait/KCT_cbct> I have implemented CGLS/CGNR , algorithm which applies conjugate gradients on normal equation
$$
A^\top A x = A^\top b
$$
I use CGLS/CGNR as a default algorithm for many tomographic problems. Let's say $x\_i$ is the so... | https://mathoverflow.net/users/69346 | Does norm of discrepancy decrease monotonously in CGLS/CGNR | Let $(x\_i)$ be sequence of solutions to CGLS/CGNR and let $x^\*$ be a solution of
$$A^\top A x^\* = A^\top b$$
AS CG minimizes $A$ norm of error over growing Krylov subspace, CGLS/CGNR minimizes $A^\top A$ norm of error which is
$$\|x^\* - x\_i\|\_{A^\top A} = \|A x^\* - A x\_i\|$$
thus we have
$$\|A x^\* - A x\_{... | 0 | https://mathoverflow.net/users/69346 | 439193 | 177,377 |
https://mathoverflow.net/questions/439158 | 5 | It is relatively easy to prove that if there exists a positive definite matrix $Q$ such that $Q - A^{H}QA$ is positive definite, where $A^{H}$ means the conjugate transpose of $A$, then the spectral radius of $A$ is less than $1$. Just look at every eigenpair $(v,\lambda)$. But as for the reverse problem, I am wonderin... | https://mathoverflow.net/users/498280 | If the spectral radius of matrix $A$ is less than $1$, how to construct a positive definite $Q$ such that $Q - A^{H}QA$ is also positive definite? | I'll follow the suggestions in the comments and post my comment as an answer:
Take your favourite positive definite matrix $P$ and define $Q := \sum\_{k=0}^\infty (A^{\operatorname{H}})^k P A^k$; note that this series converges since, as the spectral radius of $A$ is $<1$, there exist numbers $\delta \in [0,1)$ and $... | 3 | https://mathoverflow.net/users/102946 | 439200 | 177,382 |
https://mathoverflow.net/questions/439182 | 1 | I'm reading through Jannsen's paper [Motives, numerical equivalence, and semi-simplicity](https://epub.uni-regensburg.de/26642/1/jannsen10.pdf) and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, and $M$ is a **simple** numerical motive **of weight zero** ov... | https://mathoverflow.net/users/497064 | About simple motives | Your second question is easier to answer. The idempotent does not need to be central.
The subset $p A^d(X \times X, F) p $ of $A^d(X\times X, F)$ is closed under addition, additive inverses, and multiplication, and contains the additive identity, but not (usually) the multiplicative identity. However, $p1p=p$ is itse... | 1 | https://mathoverflow.net/users/18060 | 439201 | 177,383 |
https://mathoverflow.net/questions/439190 | 1 | **Statement.** To ensure the rank of $\operatorname{ddiag}(AQQ^T)-\sigma\Delta=n$, it is sufficient to require $\min\_i(\operatorname{diag}(AQQ^T))\_i>\sigma\lVert\Delta\rVert$.
**Note:** $Q\in\mathbb{R}\_{n\times 2}$, $\sigma$ is a scalar constant, $\Delta$ is $n\times n$ random matrix. The operator $\operatorname{d... | https://mathoverflow.net/users/494410 | Control the summation of a diagonal matrix and another matrix to be full rank | You must assume $\sigma > 0$. Then it's just the fact that the if $D$ is a diagonal $n \times n$ matrix with minimum diagonal entry $m > \|B\|$, where $B$ is an $n \times n$ matrix, $\|D^{-1} B\| \le \|D^{-1}\| \|B\| < 1$, so $I - D^{-1} B$ is invertible, and then $D - B = D (I - D^{-1} B)$ is invertible.
| 1 | https://mathoverflow.net/users/13650 | 439209 | 177,385 |
https://mathoverflow.net/questions/439137 | 8 | Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f\_i:Z\_i\rightarrow X\}\_{i\in I}$ on an $S$-scheme $X$ is a collection of **closed immersions** of $S$-schemes such that $\lvert X\rvert=\bigcup\_i f\_i(\lvert Z\_i\rvert)$ (notice that $\lvert Y\rvert$ denotes the... | https://mathoverflow.net/users/35397 | The Grothendieck topology of closed immersions on schemes | The topology you mention is called the "closed topology" $J\_\mathrm{cl}$ in ["Points in algebraic geometry"](https://arxiv.org/abs/1407.5782) by Gabber–Kelly. See also the [comparison diagram](https://pbelmans.ncag.info/topologies-comparison/) by Pieter Belmans.
Note that Gabber and Kelly additionally ask that the cov... | 5 | https://mathoverflow.net/users/37368 | 439211 | 177,386 |
https://mathoverflow.net/questions/439124 | 11 | Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$).
Can we find an open set $O'\subseteq T$ such that $O' \cap K = O$? Can we also find such an $O'$ which is convex?
Edit: The convexity of $K$... | https://mathoverflow.net/users/498220 | Existence of an open convex set | A convex $O'$ need not exist: a counterexample is given by setting $K=[-1,1]\times[0,2]\subseteq\mathbb{R}^2$ and $O=\{(x,y)\in K;y>x^3\}$. Indeed, any open $O'$ with $O'\cap K=O$ would contain some nhood of $(-\frac{1}{2},0)$, so it would contain some point $q=(-\frac{1}{2},-\varepsilon)$ with $\varepsilon>0$. Note th... | 9 | https://mathoverflow.net/users/172802 | 439224 | 177,391 |
https://mathoverflow.net/questions/438331 | 7 | Consider two independent copies of IID random walk on ${\bf Z}$ starting from $0$, and let $N\_1(t)$ (resp. $N\_2(t)$) denote the number of times, up to time $t$, that the first (resp. second) walker has returned to $0$. We know that $N\_1(t),N\_2(t) \rightarrow \infty$ almost surely and that $N\_1(t)/t,N\_2(t)/t \righ... | https://mathoverflow.net/users/3621 | Counting returns in null-recurrent random walk | You can easily derive everything from the first principles by the following back of envelope computation:
If we have a random walk starting *anywhere*, then after $t$ steps the expected number $EN(t)$ of returns is at most $\sum\_{m=1}^t \frac 1{\sqrt m}\approx \sqrt t$ and $E[N(t)^2]\le\sum\_{1\le m< M\le t} \frac 1... | 0 | https://mathoverflow.net/users/1131 | 439225 | 177,392 |
https://mathoverflow.net/questions/439227 | 4 | Let $p$ be a prime, and let $K/\mathbb{Q}\_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (with $\mathbb{Q}\_p\subset L\subset K$) of degree $q$ over $\mathbb{Q}\_p$?
---
Some partial progress, following Hasse... | https://mathoverflow.net/users/127215 | Existence of intermediate field extensions for tamely ramified p-adic extensions | The answer is no. For example the group $S\_3\times C\_3$ occurs (as TransitiveGroup(6,5)) as the Galois group of an (automatically tame) extension of $\mathbb{Q}\_5$ (and more generally over $\mathbb{Q} \_p$ for all odd primes $p\equiv 2$ mod $3$, by taking the compositum of the unramified degree-$3$ extension with th... | 2 | https://mathoverflow.net/users/127660 | 439231 | 177,395 |
https://mathoverflow.net/questions/438690 | 2 | Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi\_1$ and $\pi\_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows?
$$E\_2^{p,q} = \operatorname H^p(G/H,\operatorname{Ext}\_H^q(\pi\_1,\pi\_2)) \implies \operatorname{Ext}\_G^{p+q}(\pi\_1,\... | https://mathoverflow.net/users/125617 | For locally profinite groups $H\lhd G$, is there a spectral sequence $\newcommand\@[2]{{\rm Ext}_#1^{#2}(\pi_1,\pi_2)}H^p(G/H,\@Hq)\implies\@G{p+q}$? | I found a way to prove the existence of such a spectral sequence, at least when $H$ is open in $G$. In order not to leave this question unanswered, let me write things down.
If $G$ is a locally profinite group, let $\mathcal R(G)$ denote the category of smooth representations of $G$. If $V$ is an abstract representat... | 3 | https://mathoverflow.net/users/125617 | 439236 | 177,396 |
https://mathoverflow.net/questions/439223 | 2 | Is it possible to describe, up to isomorphism, all Lie groups $G$ whose underlying manifold is diffeomorphic to $\mathbb{R}^n$ (with its standard smooth structure)?
In fact, I haven't found any such group yet, other than the standard $(\mathbb{R}^n, +)$.
| https://mathoverflow.net/users/148161 | Classification of Lie group structures on $\mathbb{R}^n$ | YCor claims [here](https://math.stackexchange.com/questions/475385/under-what-conditions-is-the-exponential-map-on-a-lie-algebra-injective/1592257#1592257) that the contractible Lie groups (this is equivalent to being diffeomorphic to $\mathbb{R}^n$, since a connected Lie group is diffeomorphic to the product of a Eucl... | 5 | https://mathoverflow.net/users/290 | 439240 | 177,398 |
https://mathoverflow.net/questions/420645 | 2 | Suppose the beth function is defined as follows:
beth[a]=|V[a]|, for all ordinals a. Here V[a] is the ath level of the cumulative hierarchy, and || is the cardinality function defined as in for example the Levy Basic Set Theory book.
The above definition of beth is equivalent to the usual one in the presence of the a... | https://mathoverflow.net/users/480831 | Is the beth function continuous without the axiom of choice? | By the [answer](https://mathoverflow.net/a/438728) of Farmer S to my latest question, the answer to this question is no. Lub property on limit ordinal stages of cumulative hierarchy implies the axiom of choice. Thanks Farmer!
| 1 | https://mathoverflow.net/users/480831 | 439252 | 177,403 |
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