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https://mathoverflow.net/questions/365420 | 2 | $F$ is a mapping among $\{\theta\_{n\_1n\_2}\}$, with $\eta\_{1/2}$ being arbitrary constants involved.
$F: \theta\_{n\_1n\_2} \rightarrow \theta\_{n\_1+1n\_2}+\theta\_{n\_1n\_2+1}+\eta\_{1}n\_1\theta\_{n\_{1}-1n\_{2}}
+\eta\_{2}n\_2\theta\_{n\_{1}n\_{2}-1}$
so what is $F^k:\theta\_{n\_1n\_2} \rightarrow ?$ ($F$... | https://mathoverflow.net/users/160981 | Search for a general formula from known iterative relation | Consider the generating function:
$$H(x,y) := \sum\_{i,j} \theta\_{i,j} \frac{x^i}{i!} \frac{y^j}{j!}.$$
Extending $F$ by linearity, define
$$F(H)(x,y):= \sum\_{i,j} \left(\theta\_{i+1,j}+\theta\_{i,j+1}+\eta\_{1}i\theta\_{i-1,j}+\eta\_{2}j\theta\_{i,j-1}\right) \frac{x^i}{i!} \frac{y^j}{j!}$$
so that the question amou... | 2 | https://mathoverflow.net/users/7076 | 365486 | 153,535 |
https://mathoverflow.net/questions/365484 | 5 | Let $\mathcal{L}$ be a recursive first-order theory, with a deductive system $\Xi$ (for instance, Hilbert-Ackerman proof system). Let $\phi$ be a formula and let $l=(\psi\_1, \ldots, \psi\_n=\phi)$ be a sequence of formulas.
* Question 1: Suppose we what want to discuss the (asymptotical) computational complexity cos... | https://mathoverflow.net/users/160378 | Computational complexity of proof verification | Your setup doesn't provide any complexity restrictions on determining whether a formula is an axiom or not, beyond demanding that this is computable. Thus, you won't be able to limit the complexity of proof verification either.
Determining the axioms is going to be the only issue, though. Any reasonable proof system ... | 10 | https://mathoverflow.net/users/15002 | 365490 | 153,538 |
https://mathoverflow.net/questions/365482 | 1 | Let $(a,b) \subset (0,1)$. Consider the following transport equation
$$z\_t+z\_x=0, \ (t,x)\in(0,T)\times(0,1), \\z(t,0)=0, \ z(0,x)=z\_0(x).$$
It is clear that the solution to the above equation is given by $z(t,x)=z\_0(x-t),\ \text{if} \ x-t\in (0,1)$ and $0$ otherwise.
I want to prove the following observability ine... | https://mathoverflow.net/users/106804 | Observability inequality for the 1D transport equation | We have $0\le a\le b\le1$ and $T\in(0,\infty)$. We want to know when there is a positive constant $C$ such that
$$\int\_0^T dt\, \int\_a^b dx\, u^2(x-t)\geq C\int\_0^1 dx\,u^2(x) \tag{1}$$
for all measurable functions $u\colon\mathbb R\to\mathbb R$ such that $u(x)=0$ for $x\notin(a,b)$.
The answer is: never. Indeed, ... | 2 | https://mathoverflow.net/users/36721 | 365492 | 153,539 |
https://mathoverflow.net/questions/365504 | 1 | Let the function $f: \mathbb{R} \rightarrow \mathbb{R}$ be convex, differentiable with derivative $f\_x$ and Lipschitz continuous with constant $L$. Then, for $a,b,c,d \in \mathbb{R}$ such that $a \ge b\ge d $ and $ a \ge c\ge d$,
\begin{equation\*}
\begin{split}
& f(\max\{ b,c\}) - f(a) + f(\min\{ b,c\}) - f(d)\\
& ... | https://mathoverflow.net/users/156574 | Baffling proof using function convexity | As [suggested](https://mathoverflow.net/questions/365504/baffling-proof-using-function-convexity#comment922614_365504), assume that $a \ge b \ge c \ge d$ and re-write the desired inequality as
$$
f(b) - f(a) + f(c) - f(d) \le f'(c)(b - d + c - a).
$$
We have $f(a) - f(b) \ge f'(b)(a - b)$, $f(c) - f(d) \le f'(c)(c - d)... | 2 | https://mathoverflow.net/users/2383 | 365506 | 153,544 |
https://mathoverflow.net/questions/365433 | 3 | Let $X$ be a smooth scheme over $\mathbb{C}$.
A $O\_X$-algebra $A$ is called Azumaya algebra on $X$
if locally it's ismorphic to matrix algebra: ie for
every $p \in X$ there exist open $U \subset X$ with
$p \in U$ and $A \vert \_U \cong Mat\_{r}(O\_U) $
for some rank $ r >0 $.
Two Azumaya algebra
$A$ & $B$ are *equiv... | https://mathoverflow.net/users/108274 | Cohomological Brauer group vs classical | Briefly, one uses the exact sequence
$$
H^{1}(X,GL\_{n})\rightarrow H^{1}(X,PGL\_{n})\rightarrow H^{2}(X,\mathbb{G}\_{m})
$$
(etale cohomology). The set $H^{1}(X,PGL\_{n})$ classifies the isomorphism classes of Azumaya
algebras of degree $n^{2}$ over $X$, the set $H^{1}(X,GL\_{n})$ classifies the
isomorphism classes of... | 4 | https://mathoverflow.net/users/149169 | 365508 | 153,545 |
https://mathoverflow.net/questions/365372 | 3 | Is there an r.e. set $A$ such that 0’ is cuppable relative to $A$? What about cappable?
This is equivalent to asking if there is an r.e. $A$ such that 0’ is one half of a pair of $A$ r.e. non-$A$ computable sets whose meet is $A$ and similarly if there is an $A$ such that 0’ can be (non-trivially) joined to $A'$ via ... | https://mathoverflow.net/users/23648 | Cupping and capping for 0’ relative to a recursively enumerable set | You could look at the Jockusch and Shore papers on pseudo jump operators. They showed that for every $e$ there is an r.e. $A$ such that $A+W^A\_e$ is Turing equivalent to $0’$. So $0’$ can have the behaviors that you mentioned relative to r.e. sets.
| 5 | https://mathoverflow.net/users/31026 | 365509 | 153,546 |
https://mathoverflow.net/questions/365483 | 9 | Let $f\colon X\to \mathbb{A}^n\_{\mathbb{C}}$ be a morphism of $\mathbb{C}$-schemes. Suppose $f$ is (a) separated, (b) flat, (c) locally of finite type, (d) all fibers are quasi-compact, is $X$ necessarily quasi-compact?
| https://mathoverflow.net/users/nan | fiberwise-quasi-compact implies quasi-compact? | Here is a counterexample:
**Example.** We will define $X$ as a union of affine varieties
$$U\_0 \subseteq U\_1 \subseteq \ldots$$
as follows: start with $U\_0 = \mathbf A^1 \times (\mathbf A^1 \setminus 0) \subseteq \mathbf A^2 = V\_0$ with its natural projection to $\mathbf A^1$, and let $Z\_0 = \mathbf A^1 \times 0... | 10 | https://mathoverflow.net/users/82179 | 365512 | 153,548 |
https://mathoverflow.net/questions/365519 | 1 | Schur's decomposition says any matrix $A$ is similar to a upper triangular matrix $U$ i.e., there exists unitary $Q$ such that $A = Q^{-1}UQ$. If we split $U$ as $D+N$ where $D$ is the diagonal part and $N$ is the off-diagonal part, then we know $N$ is nilpotent. Any Nilpotent matrix can be brought to Jordan form using... | https://mathoverflow.net/users/160574 | Is it possible to prove the Jordan decomposition starting from Schur's decomposition? | I am not sure I get what you mean by "brought to Jordan form", but if you don't consider the structure of $D$ while changing basis for $N$ then it won't work. Example:
$$
U =
\begin{bmatrix}
1 & 1& 0 & 0\\
0 & 1& 0 & 0\\
0 & 0 & 2 & 1\\
0 & 0 & 0 & 2
\end{bmatrix}
$$
has
$$
N =
\begin{bmatrix}
0 & 1& 0 & 0\\
0 & 0& 0... | 2 | https://mathoverflow.net/users/1898 | 365522 | 153,551 |
https://mathoverflow.net/questions/365516 | 1 | Let $\mu$ be a Borel probability measure on $R^d$. If $\mu$ satisfies $\mu(B(x,r))\le Cr^\alpha$ for any $x\in R^d$ and $r>0$, then Strichartz (Fourier asymptotics of fractal measures, J. Funct. Anal. 1990) proved that
$$\limsup\limits\_{R\to \infty}\frac{1}{R^{d-\alpha}}\int\_{|x|\le R}|\widehat{\mu}(x)|^2dx\le C\_2.$... | https://mathoverflow.net/users/129565 | Does Ahlfors–David regularity of a measure imply its Fourier asymptotic behavior? | (There are some details missing in this answer. Time permitting, I will try to expand it.)
It suffices to find a *non-negative* continuous function $\phi$ with $\phi(0) > 0$ and Fourier transform bounded and supported in $B(0, 1)$. In dimension one, $\phi(x) = (1 - \cos x)/(\pi x^2)$, $\hat\phi(z) = (1 - |z|)\_+$ is ... | 1 | https://mathoverflow.net/users/108637 | 365526 | 153,553 |
https://mathoverflow.net/questions/365529 | 2 | If one takes in general $(\star)\, \,x^2-dy^2=C$ where $d$, $C$ in $\mathbb{N}$.
Taking $d=w^2p^2+p$ with $w\in \mathbb{Q}\ge 1$ and $p\in \mathbb{Z}$ which is verified (explained later), for the matrix $$A=\begin{pmatrix}2w^2p+1&2w(w^2p^2+p)\\2w&2w^2p+1\end{pmatrix}$$ if $X\_0$ is a solution to $(\star)$ then $AX\_0... | https://mathoverflow.net/users/121643 | A Pell like equation | If I understand correctly, your question is the following: suppose that for a given positive integer $d$ the equation
$$\displaystyle x^2 - dy^2 = c \text{ } (\ast)$$
has a solution in integers $x,y$ for some integer $c$. Then does there exist an infinite family of solutions generated by $A^k (x,y)^T$ for some $A \... | 6 | https://mathoverflow.net/users/10898 | 365534 | 153,556 |
https://mathoverflow.net/questions/365092 | 3 | I am interested in vector fields whose Jacobian has orthogonal columns; i.e. if $\mathbf{f}(\cdot):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a function where $\mathbf{f}(\mathbf{x})=[f\_1(\mathbf{x}), f\_2(\mathbf{x}),~\dots, f\_n(\mathbf{x})]^{\rm T}$, I am looking for all such functions that:
$\forall~\mathbf{x... | https://mathoverflow.net/users/160768 | Functions with a Jacobian whose columns are orthogonal | You are asking about the subject of *orthogonal (coordinate) systems*. There is an extensive literature on this subject, in particular by Darboux when $n=3$, and if you search on "triply orthogonal systems", you will see references to Darboux, Eisenhart, etc. plus many more recent references. There are many classical e... | 1 | https://mathoverflow.net/users/13972 | 365535 | 153,557 |
https://mathoverflow.net/questions/364555 | 3 | Let $f:X \to Y$ be a flat, projective morphism with $Y$ integral and every fiber of $f$ normal and integral. Let $F$ be a torsion-free, coherent sheaf on $X$ (not necessarily flat over $Y$). Then, is the function $y \mapsto \chi(F|\_{X\_y})$, upper semi-continuous, where $X\_y:=f^{-1}(y)$?
Also, given any discrete va... | https://mathoverflow.net/users/45397 | Variation of Euler characteristic when the sheaf is not flat | Kollar in [his article](https://arxiv.org/pdf/1503.08694.pdf) mentions that in general Euler characteristic is lower semi-continuous without the assumption of flatness. However, there is no proof of this statement in the article.
| 2 | https://mathoverflow.net/users/32151 | 365537 | 153,558 |
https://mathoverflow.net/questions/365401 | 10 | Let $M$ be a connected open topological $d$-manifold (without boundary).
Whitehead showed that if $M$ has a PL structure, there exists a subcomplex of dimension $\leq d-1$ onto which $M$ deformation retracts.
Can we still find a homotopy equivalent CW complex of dimension $\leq d-1$ when $M$ is not PL?
| https://mathoverflow.net/users/21848 | Is every open topological $d$-manifold homotopy equivalent to a CW-complex of dimension $\leq d-1$? | $\DeclareMathOperator{\co}{H}
\DeclareMathOperator{\ch}{C}
\newcommand{\zz}{\mathbb{Z}}
\newcommand{\nn}{\mathbb{N}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\DeclareMathOperator{\lf}{lf}$Let me put together an answer following the pointers in the comments. By Whitehead's result stated in the question ... | 10 | https://mathoverflow.net/users/21848 | 365550 | 153,561 |
https://mathoverflow.net/questions/365544 | 4 | I am looking for a proof of the following statement which is known to be true as far as I heard.
Let $g\colon [a,b]\to \mathbb{R}$ be a smooth function. Assume that
$$b-a< \pi.$$
Assume also $$g(a)\geq 0,g(b)\geq 0,$$
$$g''+g\leq 0 \mbox{ on } [a,b].$$
Then $g\geq 0$ on $[a,b]$.
| https://mathoverflow.net/users/16183 | Elementary inequality generalizing convexity of a function on a segment | Write $g=g^+-g^-$ in $[0,\ell]$, multiply $g''+g \le 0$ by $g^-$ (which vanishes at the the endpoints) and integrate. Then we get with $v=g^-$
$$
\int\_0^l v'^2- \int\_0^l v^2 \le 0.$$
Since the first eigenvalue of the Dirichlet laplacian in $[0,\ell]$ is $\pi^2/\ell^2 $ we have also
$$
\int\_0^l v^2 \le \frac{\ell^2}{... | 7 | https://mathoverflow.net/users/150653 | 365553 | 153,563 |
https://mathoverflow.net/questions/365425 | 12 | As you all know, Ronald Graham just passed away. He is famous for many fabulous contributions to finite combinatorics, and much much more, but perhaps none of them is as popular as the infamous Graham number ([see here](https://mathworld.wolfram.com/GrahamsNumber.html)): $g\_{64}$.
This number is truly [huge](https:/... | https://mathoverflow.net/users/15293 | The inconsistency of Graham Arithmetics plus $ \forall n, n < g_{64}$ | At the request of the OP, I’m writing a lengthy nonanswer showing that there are short proofs of inconsistency of similar theories where the “big number” is given by a term in the usual language of arithmetic $L\_{PA}=\{0,S,+,\cdot\}$, possibly expanded by the exponential function. The argument does not work for langua... | 11 | https://mathoverflow.net/users/12705 | 365555 | 153,564 |
https://mathoverflow.net/questions/365525 | 5 | I am looking for a reference for Fourier analysis on compact (Lie) groups. The kind of theorems I would like the book to cover/do are the Peter-Weyl theorem, define Fourier transforms and use the Peter-Weyl theorem to derive the Plancherel theorem.The Peter-Weyl theorem(s?) can be found in multiple references but most ... | https://mathoverflow.net/users/nan | Reference on Fourier analysis on compact groups | Chapter 5 of Folland's *A Course in Abstract Harmonic Analysis* should have what you need -- it is quite a short treatment, but it seems to be complete, provided that one is happy to fill in (routine) details in a narrative rather than go for the style of "Lemma 2.1.2, Lemma 2.1.3, Definition 2.1.4, Proposition 2.1.5, ... | 7 | https://mathoverflow.net/users/763 | 365562 | 153,568 |
https://mathoverflow.net/questions/365573 | 2 | Suppose we have $a^2 \equiv b^2 \bmod 4R$ where $R$ is an integral domain. Under what conditions on $R$ can we conclude that $a \equiv b \bmod 2R$?
This would hold if $2 \in R$ is a prime or the product of distinct comaximal primes. This can fail when $R$ is not integrally closed: For $R = \mathbb{Z}[\sqrt{-3}]$ we h... | https://mathoverflow.net/users/15428 | For an integral domain $R$ when does $a^2 \equiv b^2 \bmod 4R$ imply $a \equiv b \bmod 2R$? | Denote $a-b=x$, then $a^2-b^2=x(x+2b)=4z$ for $z\in R$. Assuming that $R$ is integrally closed, we see that $(x/2)^2+b(x/2)-z=0$, so $x/2$ is an algebraic integer, thus $x/2\in R$.
| 5 | https://mathoverflow.net/users/4312 | 365575 | 153,571 |
https://mathoverflow.net/questions/365188 | 7 | Consider the round metric on $S^n$. The geodesics are (multiples of) great circles, and one can verify that this metric is of Morse-Bott type. The Morse indices of the n-covered great circles are (if I recall correctly) $(n-1), 3(n-1), 5(n-1), \dots, (2k-1)(n-1), \dots$.
Observe in particular that the Morse index of ... | https://mathoverflow.net/users/59235 | Index and length of closed geodesics | This follows from [Bonnet-Myers](https://en.wikipedia.org/wiki/Myers%27s_theorem). For a metric near the round metric (in the $C^\infty$ topology), the sectional curvature will be pinched below by $k > 0$, where $k\thickapprox 1$. Hence a segment of length $>\pi/\sqrt{k}$ of any geodesic will be unstable by Myers' theo... | 6 | https://mathoverflow.net/users/1345 | 365577 | 153,572 |
https://mathoverflow.net/questions/365569 | 40 | I have just graduated from the University of Chicago and no longer have access to online journal resources, but I cannot afford to pay for them directly. Normally, I would be able to access library resources for a small fee in the campus library. However, due to closures due to Covid-19, this is no longer an option.
... | https://mathoverflow.net/users/140709 | Access to journals during pandemic | Let me try to summarize this long discussion in the comments. There are many free resources.
1. [arXiv](https://arxiv.org/). It is true that not all mathematicians post their papers on the arXiv, for various reasons. But some of those who don't, post them on their personal sites. There are also other depositories, fo... | 32 | https://mathoverflow.net/users/25510 | 365585 | 153,574 |
https://mathoverflow.net/questions/365524 | 1 | I want to solve the usual $A x = b$ system. In block form:
$$ \begin{bmatrix} B & c \\ c^{T} & 0 \end{bmatrix} \begin{bmatrix} x' \\ x\_{n+1} \end{bmatrix} = \begin{bmatrix} b' \\ b\_{n+1} \end{bmatrix}$$
where
* $B \in \mathbb{R}^{n \times n} $ is a positive definite matrix
* $c \in \mathbb{R}^{n}$
* $x,b \in \m... | https://mathoverflow.net/users/161031 | Solve linear system with bordered positive definite matrix | By combining the useful comments of [Rodrigo](https://mathoverflow.net/users/91764/rodrigo-de-azevedo) and [Todd](https://mathoverflow.net/users/2926/todd-trimble), the methodology to solve this system is shown here below. One caveat is that the method is probably not very efficient, since you need to use the decomposi... | 2 | https://mathoverflow.net/users/161031 | 365598 | 153,579 |
https://mathoverflow.net/questions/365593 | 1 | **This is a question about coupling times of subordinate Brownian motions.**
We fix $y \in \mathbb{R}^d$ with $y \neq x$ and define a map $R\_{x,y} \colon \mathbb{R}^d \to \mathbb{R}^d$ by
\begin{align\*}
R\_{x,y}(z)=z-2 (z-(x+y)/2,x-y)\frac{x-y}{|x-y|^2},\quad z \in \mathbb{R}^d.
\end{align\*}
We note that $R\_{x,y}... | https://mathoverflow.net/users/68463 | Coupling times of subordinate Brownian motions | Here is how I would approach the problem.
The coupling time $U\_{x,y}$ is not greater than the first exit time from $H\_{x,y}^+$, the half-space bounded by $H\_{x,y}$ and containing $x$, by the process $X\_t^x$. Thus,
$$ I\_{x,y} \le \mathbb{P}^x(\tau\_{H\_{x,y}^+} \ge \tau\_{B(x, r)}) , $$
where $r = |x-y|^\epsilon$... | 1 | https://mathoverflow.net/users/108637 | 365602 | 153,582 |
https://mathoverflow.net/questions/362109 | 1 | In $\mathbb{C}^n,\ n\geq 2$, there is no bijection between unit disk $B^n(0,1)$ and unit polydisk $P^n(0,1)$. But if we wish to find injective holomorphic mapping from unit disk to polydisk(whose image contains origin), inclusion is the obvious mapping or suitable automorphisms of unit disk(which is sort of inclusion, ... | https://mathoverflow.net/users/159066 | injective holomorphic mapping between unit disk and unit polydisk | Take $f(z\_1,z\_2):=(z\_1,sz\_2+(1-s)z\_1)$, $0<s<1$. For example, $s=1/2$. Then $f$ is an injective holomorphic mapping from $B(0,1)$ to $P(0,1)$, ($n=2$), sending $0$ to $0$. Moreover, $\|f(s,0)\|>1$ for $s<1$, near $1$, i.e. $f(s,0)\notin B(0,1)$..
| 2 | https://mathoverflow.net/users/126775 | 365606 | 153,583 |
https://mathoverflow.net/questions/275289 | 14 | Let $X$ be a smooth projective surface. Then, using the compactified moduli space of anti self-dual connections or torsion free sheaves we can construct Donaldson invariants of $X$. Similarly, one can take a CY3-fold and by slanting elements of the universal sheaf with elements of the Chow group of the 3fold, construct... | https://mathoverflow.net/users/80109 | Donaldson and DT invariants | In some sense this is the topic of Vafa-Witten theory for complex surfaces; see the many recent papers of Göttsche-Kool on the subject.
In DT theory the virtual dimension is 0, so you don't usually use insertions (or the slant product) -- you just get one number. It is (a virtual version of) the Euler characteristic ... | 5 | https://mathoverflow.net/users/7653 | 365615 | 153,587 |
https://mathoverflow.net/questions/365608 | 6 | Let $\lambda$ be a partition, represented by a usual Young diagram in which $1\le i\le \ell(\lambda)$ labels the rows and, for each $i$, $1\le j\le \lambda\_i$ labels the columns. For each box $\square$ in the diagram, $c(\square)=j-i$ is its content. The polynomial
$$ P\_\lambda(x)=\prod\_{\square\in\lambda}(x+c(\squa... | https://mathoverflow.net/users/78061 | Binomial theorem for content polynomials of partitions | If you lift this to the level of symmetric functions then the structure constants are uniquely determined. Suppose $x$ denotes a set of $m$ variables and $y$ denotes a set of $n$ variables. Then you can start with the identity of schur polynomials
$$s\_{\lambda}(x,y)=\sum\_{\mu,\nu}c\_{\mu,\nu}^{\lambda}s\_{\mu}(x)s\_{... | 10 | https://mathoverflow.net/users/2384 | 365618 | 153,588 |
https://mathoverflow.net/questions/365611 | 3 | It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the coarsest topology on $\mathcal{P}(S)$, which makes the mappings
$$\mathcal{P}(S) \ni \mu \mapsto \int f d\mu \in \mathbb{R}$$... | https://mathoverflow.net/users/157982 | About the metrizability of the space of Probability measures $\mathcal{P}(S)$ | I'm not really sure what Villani wrote in his monograph, but it is true that one needs to prove that the weak topology is induced by a distance, as a priori it could be another topology with the same converging sequences.
This is quite standard, though. The key point is to realize that it is sufficient to check the c... | 7 | https://mathoverflow.net/users/58975 | 365622 | 153,589 |
https://mathoverflow.net/questions/365624 | 5 | Considering the binomial coefficient $\binom{x}{m}$ as a polynomial in $x$, the span of $\binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{d}$ is exactly the polynomials of degree $\le d$. A closely related characterization is that this subspace is the kernel of $\Delta^{d+1}$, where $\Delta : \mathbb{C}[x] \rightarrow \ma... | https://mathoverflow.net/users/7709 | A $q$-analogue of a characterization of polynomials by binomial coefficients | We have
$$\binom{n}d\_q=\frac{(q^n-1)\ldots(q^n-q^{d-1})}{(q^d-1)\ldots(q^d-q^{d-1})}=f\_d(q^n)=g\_d([n]\_q)
$$
where $f\_d$ and $g\_d$ are polynomials of degree $d$ (depending on $q$ of course). Therefore $$\bigl( P([0]\_q), P([1]\_q), \ldots, P([N-1]\_q) \bigr) \in \bigl\langle u^{(0)}\_q, u^{(1)}\_q, \ldots, u^{(d)}... | 3 | https://mathoverflow.net/users/4312 | 365627 | 153,591 |
https://mathoverflow.net/questions/365637 | 5 | How can I compute this integral?
$$
\int\_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx
$$
| https://mathoverflow.net/users/161101 | Compute $ \int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx $ | Let
$$f(a):=\int\_0^\infty x^{a-1}e^{-2x}\,dx.$$
Then
$$f''(1)=\int\_0^\infty \ln^2x\,e^{-2x}\,dx,$$
which is the integral in question.
On the other hand, $f(a)=2^{-a}\,\Gamma(a)$, and hence
the integral in question is
$$f''(1)=\frac{\ln^2 2}2 - \Gamma'(1)\ln2
+ \Gamma''(1)/2
=\frac{\pi^2}{12}+ \frac{(\gamma +\ln2)^2... | 17 | https://mathoverflow.net/users/36721 | 365639 | 153,596 |
https://mathoverflow.net/questions/365646 | 3 | For an abelian torsion group of finite exponent, i.e. there is an integer $n$ such that $g^n=1$ for all $g\in G$, its theory appears to be $\aleph\_0$-categorical by the theorem of Engeler, Ryll-Nardzewski and Svenonius. I want to confirm this fact.
| https://mathoverflow.net/users/120374 | Is an abelian group of bounded exponent $\aleph_0$-categorical | This is a theorem of Rosenstein from the paper *[$\aleph\_0$-categoricity of groups](https://pdf.sciencedirectassets.com/272332/1-s2.0-S0021869300X03328/1-s2.0-0021869373900926/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEO7%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIHZ04EYzAqky510u1oBygW1At1j7x%2B5r3OV5Tdo... | 8 | https://mathoverflow.net/users/38253 | 365653 | 153,603 |
https://mathoverflow.net/questions/365620 | 2 | Assume that $$\left<f,g\right>\_R=\Re \int\_U f(z) \overline{g(z)} \, dx \, dy, f,g \in L^2(U),$$ where $U$ is the unit disk and assume that $A: L^2(U)\to L^2(U)$ is a real-linear operator. Assume also that $A^\*$ is its adoint, with respect to $\left<f,g\right>\_R$, that is $\left<Af,g\right>\_R= \left<f,A^\*g\right>\... | https://mathoverflow.net/users/124426 | Adjoint operator | Yes, this is true, in the sense that if one side is finite then so is the other and they are equal. All you really need is to notice that $\|g\|\_q = \sup \{{\rm Re}\int fg: \|f\|\_p = 1\}$ (since you can multiply any $f$ by a scalar of modulus 1).
I'll show that $\|A^\*\|\_{L^q \to L^q} \leq \|A\|\_{L^p\to L^p}$; th... | 3 | https://mathoverflow.net/users/23141 | 365655 | 153,604 |
https://mathoverflow.net/questions/365652 | 1 | Let $X$ be an absolutely continuous (i.e. its law is absolutely continuous with respect to the Lebesgue measure) random variable with probability density $p$. Its differential entropy is given by
$$h(X) = - \int\_{\mathbb{R}} p(x) \log p(x) \mathrm{d} x$$
with the convention $0 \log 0 = 0$, as soon as the integral is a... | https://mathoverflow.net/users/39261 | Existence of the differential entropy for infinitely divisible laws | For real $t>0$, let
\begin{equation}
p\_t:=e^{-t}e^{\*tf}\*g\_t:=e^{-t}\sum\_{n=0}^\infty\frac{t^n f^{\*n}}{n!}\*g\_t, \tag{0}
\end{equation}
where $f$ is the (bounded by $c:=1/e$) pdf given by
\begin{equation}
f(x)=\frac{1\{x\ge e\}}{x\ln^2 x}, \tag{0.5}
\end{equation}
$f^{\*n}:=f\*\cdots\*f$ ($n$ times, with $f^... | 2 | https://mathoverflow.net/users/36721 | 365657 | 153,605 |
https://mathoverflow.net/questions/364667 | 2 | Suppose $\mathfrak{g}$ is an (untwisted) affine Lie algebra with the normalized invariant form $(\cdot | \cdot)$. Let $\lambda \in \mathfrak{h}^\ast$ be a dominant integral weight such that $\lambda(d)=0$ for $d$ the derivation in the loop algebra construction; that is, $\lambda = c\_0\Lambda\_0 +c\_1\Lambda\_1+\cdots+... | https://mathoverflow.net/users/138296 | An inequality for weights of affine Lie algebras, level, and dual Coxeter number | The answer to this is found as theorem 13.11 in Kac, "Infinite dimensional Lie Algebras". To be specific, we have $2k(\Lambda|\rho) \geq h^{\vee} (\Lambda| \Lambda)$ for all $\Lambda \in P^k\_+$, with equality if and only if $\Lambda = k \Lambda\_j$ mod $\mathbb{C} \delta$. Here $j \in J$, where $J$ is a set depending ... | 2 | https://mathoverflow.net/users/119460 | 365659 | 153,606 |
https://mathoverflow.net/questions/365656 | 5 | *Background:*
* Let $X: \textbf{CRing} \to \textbf{Set}$ be a presheaf on the category of affine schemes and $Z \subseteq X$ a subfunctor. One defines $Z$ to be **closed** if for every ring $A$ and every morphism $f: \text{Hom}(A , -) \to X$ the inverse image $f^{-1}(Z)$ is of the form $R \mapsto \{ \varphi : A \to R... | https://mathoverflow.net/users/134491 | Closure of the product of subfunctors | This is not true even for affine schemes. Let $k = \mathbb{Z}$, let $X = \operatorname{Spec} \mathbb{Z}$, let $Y = \operatorname{Spec} \mathbb{F}\_p$, and let $Z \cong \operatorname{Spec} \mathbb{Z} [ p^{-1}]$. The closure of $Z$ in $X$ is $X$ itself, but $Z \times Y \cong \operatorname{Spec} \{ 0 \}$, which is already... | 4 | https://mathoverflow.net/users/11640 | 365661 | 153,607 |
https://mathoverflow.net/questions/221614 | 25 | Earlier this year it was asked on MO, "[Are there only countably many compact topological manifolds?](https://mathoverflow.net/questions/198098/are-there-only-countably-many-compact-topological-manifolds)" Thanks to Cheeger and Kister, the answer is yes. On the other hand, Manolescu recently debunked the triangulation ... | https://mathoverflow.net/users/68910 | Are compact topological $n$-manifolds recursively enumerable? | In a [note of Freedman and Zuddas](https://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0026/0001/a005/), they show that this is true for dimensions $\geq 4$.
In the "Background" section of the paper, they describe the solution in the higher dimensional case using surgery theory, but without any r... | 10 | https://mathoverflow.net/users/1345 | 365667 | 153,609 |
https://mathoverflow.net/questions/365647 | 2 | I'm exploring the possibility to apply information theory on an uncountably infinite-dimensional scenario. I found the concept of [generalized entropy](https://ieeexplore.ieee.org/abstract/document/59987) for continuous random variables defined on finite-dimensional Euclidean space. I wonder whether there are similar c... | https://mathoverflow.net/users/161104 | Information theory for uncountably infinite-dimensional continuous random variable | You can do this exactly in the same way, except that the right notion is that of *relative* entropy and that you need a reference measure. Let me explain: on an abstract measurable space $(\Omega,\Sigma)$ choose any reference probability measure $R$. The relative entropy of an arbitrary probability measure $P\in\mathca... | 3 | https://mathoverflow.net/users/33741 | 365669 | 153,610 |
https://mathoverflow.net/questions/365559 | 9 | It was proved by W.B. Johnson and H.P. Rosenthal [[Studia Math. 43 (1972), 77–92]](http://matwbn.icm.edu.pl/ksiazki/sm/sm43/sm4317.pdf)
that every Banach space $X$ with $X^{\*\*}$ separable is *hereditarily reflexive*:
every infinite dimensional closed subspace of $X$ contains an infinite dimensional
reflexive subspace... | https://mathoverflow.net/users/39421 | On hereditarily reflexive Banach spaces | The question has a negative answer:
Following the idea in Bill Johnson's comment, I looked at the work of Argyros. In this [**paper**](https://arxiv.org/abs/0807.2392) (see the reference below), there are several examples of hereditarily indecomposable Banach spaces containing no infinite dimensional reflexive subspa... | 6 | https://mathoverflow.net/users/39421 | 365686 | 153,614 |
https://mathoverflow.net/questions/365674 | 12 | If $X$ is a finite set, what is the smallest (in cardinality) family of open subsets $\mathcal U\subseteq 2^X$ such that $\mathcal U$ generates the discrete topology, i.e. if $\mathcal U\subseteq \tau\subseteq 2^X$ and $\tau$ is a topology, then $\tau=2^X$?
| https://mathoverflow.net/users/1626 | Smallest family of subsets that generates the discrete topology | Let $\mathcal{U}=\{A\_1,\ldots,A\_k\}$. Then for any element $x\in X$ there should exist a set $I(x)\subset \{1,\ldots,k\}$ such that $\cap\_{i\in I(x)} A\_i=\{x\}$. Note that $I(x)$ is not contained in $I(y)$ for $x\ne y$. Therefore $|X|\leqslant \binom{k}{\lfloor k/2\rfloor}$ by [Sperner's theorem](https://en.wikiped... | 17 | https://mathoverflow.net/users/4312 | 365694 | 153,618 |
https://mathoverflow.net/questions/365684 | 4 | I was told that if $A$ is the subring of $\mathbb{C}[x\_1,\ldots, x\_n]$ generated by the polynomials $p\_1(x\_1,\ldots, x\_n),\ldots, p\_1(x\_1,\ldots, x\_n)$, then the preimage $p^{-1}(c)$ via the map $p = (p\_1,\ldots, p\_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ is finite for all $c\in \mathbb{C}^n$ if the ring $\ma... | https://mathoverflow.net/users/125534 | Condition such that the fibres of a polynomial map $p :\mathbb{C}^n\rightarrow \mathbb{C}^n$ are finite | $\def\CC{\mathbb{C}}$User "anon" points out to me that this is Proposition 8.28 in [Milne's notes](https://www.jmilne.org/math/CourseNotes/AG.pdf); see also Example 8.36 for a quasi-finite map $\CC^2 \to \CC^2$ which is not finite. The rest of my answer is probably not as useful now that there is a good reference, but ... | 8 | https://mathoverflow.net/users/297 | 365704 | 153,620 |
https://mathoverflow.net/questions/365709 | 0 | Is there a solution for $AX+XA^T+XBX=C$ where $X$, $B$ and $C$ are symmetric?
| https://mathoverflow.net/users/161144 | Solution for $AX+XA^T+XBX=C $ where $X$, $B$ and $C$ are symmetric | That equation is called a (continuous-time) algebraic Riccati equation, and there is ample literature on when they are solvable; just look for this search term. For instance, the book *Algebraic Riccati equations* by Lancaster and Rodman, or *Numerical solution of AREs* by Bini, Iannazzo, Meini.
In the generic case t... | 4 | https://mathoverflow.net/users/1898 | 365713 | 153,621 |
https://mathoverflow.net/questions/365685 | 6 | A $2n$-dimensional manifold $M$ is said to be *almost symplectic* if it possesses a non-degenerate two-form $\omega \in \Omega^2(M)$. Equivalently, an almost symplectic structure is a $G$-subbundle $P \subset F(M)$ of the frame bundle where $G < GL(2n,\mathbb{R})$ is isomorphic to the symplectic group $Sp(2n,\mathbb{R}... | https://mathoverflow.net/users/394 | Name for a class of almost symplectic manifolds | I'm a bit confused by your question, because I believe that, if one defines an $\omega$-Hamiltonian vector field to be a vector field of the form $X\_f = \omega^\#(\mathrm{d}f)$ where $f$ is a (smooth) function on $M$, then $\omega^{n}$ is *always* invariant under the flow of $X\_f$.
To see this, recall that, when $n... | 4 | https://mathoverflow.net/users/13972 | 365714 | 153,622 |
https://mathoverflow.net/questions/365441 | 7 | Fix $\epsilon>0$. For a finite set of arbitrary-degree polynomials with integer constant term, $p\_1(x), ..., p\_m(x)\in \mathbb{R}[x]$ is it possible to find an $n\in \mathbb{N}$ such that $$\max\_{i=1,...,m}||p\_i(n)||<\epsilon$$ where $||\cdot||$ denotes the distance to the nearest integer?
In the description of t... | https://mathoverflow.net/users/140709 | Simultaneous small fractional parts of polynomials | The claim is true as stated, and can be arrived at using Weyl's criterion which was pointed out in the comments. As I post this answer, there is no consensus on the rate of convergence of the $N$.
We proceed by induction on $k$, the number of polynomials. For $k=1$, either $p\_1$ has only rational coefficients or it ... | 2 | https://mathoverflow.net/users/140709 | 365719 | 153,623 |
https://mathoverflow.net/questions/365648 | 5 | Recall that a (separable) metric space is called *punctiform*, if all its compact subspaces are zero-dimensional. While "natural" spaces would seem to be punctiform if they already themselves zero-dimensional, there are even infinite dimensional punctiform spaces. The constructions I have seen however are still yieldin... | https://mathoverflow.net/users/15002 | Is the Hilbert cube the countable union of punctiform spaces? | The Hilbert cube can be written as the union of two punctiform spaces. Just take any [Bernstein set](https://en.wikipedia.org/wiki/Bernstein_set) $X\subset[0,1]^\omega$ and observe that compact subsets in $X$ and $Y=[0,1]^\omega\setminus X$ are at most countable. So, $X$ and $Y$ are punctiform spaces and $X\cup Y=[0,1]... | 5 | https://mathoverflow.net/users/61536 | 365728 | 153,630 |
https://mathoverflow.net/questions/365725 | 10 | Macpherson in *a survey of homogeneous structures*, states that there are many $\aleph\_0$-categorical structures which are not homogeneous. I'd like to know more of such examples.
**Edit:** The homogeneity mentioned here is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of... | https://mathoverflow.net/users/120374 | Examples of $\aleph_0$-categorical nonhomogeneous structures | Here is my favorite example.
>
> **Theorem.** Fix $n\geq 1$. Then there is a unique (up to isometry) countable metric space $(M\_n,d)$ satisfying the following properties:
>
>
> 1. $d(x,y)\in\{0,1,2,\ldots,n\}$ for all $x,y\in M\_n$
> 2. Any finite metric space with distances in $\{0,1,2,\ldots,n\}$ embeds as a s... | 12 | https://mathoverflow.net/users/38253 | 365729 | 153,631 |
https://mathoverflow.net/questions/365733 | 0 | Is there any elementary way of proving that for all natural numbers $n>1$ there exists a prime $p$ such that $n<p<n^2$. And I mean elementary, not using the Prime Number Theorem or Bertrand's Postulate.
| https://mathoverflow.net/users/160978 | Elementary proof for $n^2>p>n$ for all $n>1$ | Erdos has a nice proof in his paper on the Sylvester Schur theorem from 1934. I will sketch part of it here.
Look at the prime factors of the binomial coefficient $B = \binom{m+n}{n}$. An elementary argument says that the prime $p$ divides $B$ to a power that is less than $m+n$. Therefore $B$ has all its prime factor... | 1 | https://mathoverflow.net/users/3402 | 365736 | 153,633 |
https://mathoverflow.net/questions/365692 | -2 | let $\epsilon >0$, I tried to evaluate $\int\_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ function seems is not easy to get such closed form by this variable change , For one iteration by means $\int\_{0}^{1}\left(\sqr... | https://mathoverflow.net/users/51189 | Is it possible to express $\int_{0+\epsilon}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ in elementary functions? | I think this is
$$
-\int\_{0+\epsilon}^{1-\epsilon}
{\frac {2\;{\rm W} \left(-\frac12\,\ln \left( 1-{x
}^{2} \right) \right)}{\ln \left( 1-{x}^{2} \right) }}\,{\rm d}x
$$
not elementary.
| 2 | https://mathoverflow.net/users/454 | 365737 | 153,634 |
https://mathoverflow.net/questions/364415 | 7 | We say an integer $k$ is [Pell](https://en.wikipedia.org/wiki/Pell%27s_equation#The_negative_Pell_equation) if there exist some integers $p,q$ such that
$$
p^2k-q^2=1
$$
In studying a [physics system](https://arxiv.org/abs/1904.12884) we ended up with two weaker notions of Pell:
1. We say an integer $k$ is pre-Pell... | https://mathoverflow.net/users/131264 | Pell equation and quadratic residues | The answer is yes. As was observed, the condition of being pre-Pell is simply the stipulation that $k$ is a sum of two squares: that is, if $p | k$ and $p \equiv 3 \pmod{4}$ then $p$ must divide $k$ with even multiplicity. If we assume $k$ is square-free, then it is divisible only by $2$ or primes of congruent to $1 \p... | 5 | https://mathoverflow.net/users/10898 | 365746 | 153,637 |
https://mathoverflow.net/questions/365745 | 4 | In algebraic topology, for any space with finite homology type, the universal coefficient theorem states that for any abelian group $G$, we have
$$H^n(X,G)\cong \left( H^n(X,\mathbb{Z})\otimes G\right)\oplus \text{Tor}\_1(H^{n+1}(X,\mathbb{Z}),G).$$
My question is whether the analogous statement is true for the pro-éta... | https://mathoverflow.net/users/152554 | "Universal coefficent theorem" for pro-étale cohomology | The only condition you need on $X$ for such a formula to hold is that it is coherent (=quasi-compact and quasi-separated).
Let $R$ be a discrete ${\mathbf{Z}\_\ell}$-module. We consider the sheaf $\underline{\mathbf{Z}}\_\ell$ on the pro-étale site defined as the limit of the constant sheaves $\mathbf{Z}/\ell^i\mathb... | 6 | https://mathoverflow.net/users/1017 | 365755 | 153,640 |
https://mathoverflow.net/questions/365749 | 9 | Let $Y$ be a complex manifold, $X\subset Y$ a compact submanifold, and $E\to X$ a holomorphic vector bundle. Can $E$ be extended
to a bundle over an open neighborhood of $X$ in $Y$? (Four years ago I have asked this question on MO [Extending the tangent bundle of a submanifold](https://mathoverflow.net/questions/249050... | https://mathoverflow.net/users/9833 | Extending a holomorphic vector bundle: a reference request | It seems to me that your result follows by Proposition 1.1 of the paper
P. A. Griffiths: [**The extension problem in complex analysis. II: Embeddings with positive normal bundle**](http://dx.doi.org/10.2307/2373200), Am. J. Math. **88**, 366-446 (1966). [ZBL0147.07502](https://zbmath.org/?q=an:0147.07502),
that can... | 11 | https://mathoverflow.net/users/7460 | 365758 | 153,641 |
https://mathoverflow.net/questions/365764 | 2 | Let $p\_1, p\_2,\dots, p\_n$ and $q\_1,q\_2,\dots,q\_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying
$$a\_{ij} = \begin{cases} p\_i(x) & \text{ if } i = j, \\ q\_i(x) & \text{ otherwise} \end{cases} .$$
is there any connection between the roots of the polynomials $p\_i$'s and ... | https://mathoverflow.net/users/33047 | Roots of determinant of matrix with polynomial entries | Let $r\_i := p\_i - q\_i$.
$${\bf A} (x) := \begin{bmatrix} p\_1 (x) & q\_1 (x) & \ldots & q\_1 (x)\\ q\_2 (x) & p\_2 (x) & \ldots & q\_2 (x)\\ \vdots & \vdots & \ddots & \vdots\\ q\_n (x) & q\_n (x) & \ldots & p\_n (x)\end{bmatrix} = \mbox{diag} \left( {\bf r} (x) \right) + {\bf q} (x) {\Bbb 1}\_n^\top$$
Using the... | 2 | https://mathoverflow.net/users/91764 | 365772 | 153,643 |
https://mathoverflow.net/questions/365775 | 2 | I have a question about a definition used in nLab article on $n$-groupoids: <https://ncatlab.org/nlab/show/n-groupoid>
What does it mean that "every parallel pair of $j$-morphisms is equivalent for $j>n$? Surely, that's not a research question, but I nowhere found an answer. Does that mean that the corresponending eq... | https://mathoverflow.net/users/108274 | Equivalent parallel pair of $j$-morphisms | It means that between every two $j$-cells with the same domain and codomain ($j > n$), there is a $(j + 1)$-cell between them (which is necessarily an equivalence by the first condition of the definition on that page) exhibiting them as equivalent.
(Meta note: [math.stackexchange](https://math.stackexchange.com/) is ... | 5 | https://mathoverflow.net/users/152679 | 365776 | 153,644 |
https://mathoverflow.net/questions/365757 | 3 | Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map $x\mapsto x^2$ is surjective?
This is certainly true for fields. For DVR's, you can take the strict henselization, and then take the colimit over all extensions taking square roots of the uniformizer.
... | https://mathoverflow.net/users/88840 | Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map is surjective? | The following works over any ring $R$: Take a family $\underline{X}:=(X\_a)\_{a\in R}$ of indeterminates indexed by $R$, and put $R\_1:=R[\underline{X}]/I$ where $I$ is generated by $(X\_a^2-a)\_{a\in R}$. Then $R\_1$ is free as an $R$-module (you can view it as $\bigotimes\_{a\in R}R[X\_a]/(X\_a^2-a)$) and every eleme... | 8 | https://mathoverflow.net/users/7666 | 365780 | 153,646 |
https://mathoverflow.net/questions/365766 | 5 | I am looking for examples of amenable Banach algebras which have non-amenable subalgebra
I know
>
> 1: Each amenable Banach algebra has a bounded approximate identity
>
>
>
>
> 2: If $I$ be a closed ideal in an amenable Banach algebra, then
>
>
>
>
> $I$ amenable if and only if $I$ has a bounded app... | https://mathoverflow.net/users/52860 | Examples of amenable Banach algebras which have non-amenable subalgebra | Mateusz's answer mentions lots of good mathematics but I feel obliged to point out that **the** fundamental example which answers your original question in the negative is $M\_2({\bf C})$. (Banach algebras behave very differently from ${\rm C}^\*$-algebras and $L^1$-group algebras.)
The point is that the algebra
$$
{... | 12 | https://mathoverflow.net/users/763 | 365783 | 153,647 |
https://mathoverflow.net/questions/364974 | 1 | (this is an attempt to refine [a previous question](https://mathoverflow.net/questions/363930/gaussian-bounds-with-exponential-decay-for-discrete-graph-dirichlet-heat-kerne); I was told that it would be better to create a new question than edit the previous one, I hope this is the correct ettiquete.)
Let $\Omega$ be ... | https://mathoverflow.net/users/24122 | Gaussian bounds for discrete (graph) Dirichlet heat kernel | You can get a reasonable estimate by splitting time in three pieces, say [0,t/3], [t/3,2t/3], [2t/3,t] and estimating from above by the product of
--- the probability starting at x of staying in the domain up to time t/3
--- the probability starting at y of the the reversed walk staying in the domain up to time t/3... | 2 | https://mathoverflow.net/users/82209 | 365800 | 153,652 |
https://mathoverflow.net/questions/365797 | 2 | Trying to study isomorphism classes of certain commutative Artinian $\mathbb{C}$-algebras I was lead to the following problem about matrices.
Suppose you have a (non-zero) nilpotent matrix $A\in M\_n(\mathbb{C})$. Think of matrix algebras over the Artin algebra $\mathbb{C}[A]$. To be more specific, if another (linear... | https://mathoverflow.net/users/50468 | Commuting nilpotent matrices and conjugation isomorphisms | Decided to turn my comment into an answer since it seems to be independent of the ambiguity noticed by @LSpice.
Take
$$
A=
\begin{pmatrix}
0&1&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0
\end{pmatrix},
$$
$$
B\_1=
\begin{pmatrix}
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&0&0\... | 3 | https://mathoverflow.net/users/41291 | 365806 | 153,654 |
https://mathoverflow.net/questions/365808 | 11 | Sorry if something like this has already been asked, I searched but I couldn't find anything similar to my question.
I'm a senior undergraduate and currently doing my senior thesis. My senior thesis is not original work, however it's quite demanding and I'm learning a lot of high level topics. I have been lurking aro... | https://mathoverflow.net/users/155670 | Publishing undergraduate research | There are many undergraduate journals that would not be likely to reject your work as "not profound enough." Basically, if it's written while the author was an undergraduate, and contains anything novel at all (at the level one would expect of an undergraduate), then a journal can be found for it. This includes well-wr... | 23 | https://mathoverflow.net/users/11540 | 365812 | 153,656 |
https://mathoverflow.net/questions/365802 | 6 | [Wolfram MathWorld](https://mathworld.wolfram.com/ExactSolution.html) says
>
> As used in physics, the term “exact” generally refers to a solution that captures the entire physics and mathematics of a problem as opposed to one that is approximate, perturbative, etc. Exact solutions therefore need not be closed-form... | https://mathoverflow.net/users/38783 | What is an "exact solution" to a PDE? | It depends on context. In the physics literature, there is a term "exactly solvable" meaning that a closed form for the solution can be written; it is never used to indicate that the solution exists in an abstract sense. E. g., see Baxter's classical book "Exactly solvable models in Statistical mechanics". So, in this ... | 6 | https://mathoverflow.net/users/56624 | 365813 | 153,657 |
https://mathoverflow.net/questions/365814 | 7 | [It is well known](https://en.wikipedia.org/wiki/Cayley%27s_formula) that the number of labelled trees on $n$ vertices is equal to $n^{n-2}$.
1. We do not expect any such exact formula for [the number of isomorphism types](https://oeis.org/A000055) of trees on $n$ vertices. But what are the sharpest asymptotics, or b... | https://mathoverflow.net/users/4558 | How does the number of trees on $n$ vertices *up to isomorphism* grow as $n \to \infty$? | For Q1 the answer is known to be $\sim C\_1C\_2^n n^{-5/2} $ for $C\_1\approx 0.5349496061...$ and $C\_2\approx 2.9955765856...$. This can be found in Flajolet and Sedgewick's "Analytic Combinatorics" (see p.481) with the main ingredients being singularity analysis and the relation
$$I(z)=H(z)-\frac{1}{2}\left(H(z)^2-H... | 17 | https://mathoverflow.net/users/2384 | 365818 | 153,660 |
https://mathoverflow.net/questions/365761 | 6 | Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $\omega$ along $c$ as
$$I(\omega, c) = \int \_0 ^1 \omega\_{c(t)} (\dot c (t)) \ \mathrm d t \ .$$
This is clear, bu... | https://mathoverflow.net/users/54780 | An abstract characterization of line integrals | I'll suggest here another possible characterisation, expanding on a suggestion of the OP in [one of the comments](https://mathoverflow.net/questions/365761/an-abstract-characterization-of-line-integrals#comment923647_365779). Again, this is an assertion that certain known properties of line integration characterise it ... | 2 | https://mathoverflow.net/users/126183 | 365849 | 153,668 |
https://mathoverflow.net/questions/365867 | 11 | I am wondering if anyone can provide information on the [Indiana University Mathematics Journal](http://www.iumj.indiana.edu/), as I have been able to find very little (ie aims and scope) on their website. I have the following list of questions.
1. What kinds of papers do they "like" to publish? Should submitted pape... | https://mathoverflow.net/users/129192 | What kinds of papers does the Indiana University Mathematics Journal publish? | The key portion of the FAQ reads:
>
> The initial review is handled by the Managing Editor and by members of the Editorial Board and/or other departmental reviewers, depending on the area of expertise. This initial review is usually completed within weeks and, for most manuscripts, this is when it is determined whe... | 21 | https://mathoverflow.net/users/22 | 365869 | 153,673 |
https://mathoverflow.net/questions/365716 | 0 | I am trying to solve the following recurrence relation
$4 \leq n \;\; \; \; \; \;$ and $\; \; \; \; 2\leq i \leq \lfloor{\frac{n}{2}}\rfloor$
$F(2i,n)=$
$\begin{cases}
\frac{1}{2(2i)-5}F(2i-2,2i-1),& \text{if } n=2i\text{, } i\geq3\\
\frac{n}{2n-5}F(4,n-1)=\frac{n!\*3}{4!(2n-5)!!},& \text{if } i=2 \text{, } n\geq... | https://mathoverflow.net/users/160832 | How to solve this conditional recurrence relation?(two variable and conditions) | Under the assumption that $F(2i,n) = 0$ when $i<4$ or $2i>n$, the first and second cases in the recurrence become partial cases of the third one. Hence, the recurrence reduces to
$$F(2i,n) =
\begin{cases}
1, & \text{if } i=2,\ n=4;\\
0, & \text{if } i<2\text{ or } 2i>n;\\
\frac{2i+n-4}{2n-5}F(2i,n-1)+\frac{n-2i+1}{2n-... | 2 | https://mathoverflow.net/users/7076 | 365883 | 153,677 |
https://mathoverflow.net/questions/365882 | -1 | So I've been trying for a while to write an article on 3 applications of some theorem for a small journal. But after finding my second example I realized that I didn't need the theorem to prove it. Now, the two proofs are linked by another theorem which I used in both for some constructions, but I use it just so I coul... | https://mathoverflow.net/users/160978 | Can it be an application of a theorem if I only use it to generalize? | It's really hard to answer this question when you phrase things so vaguely. I take your question to mean:
* There is a theorem, let's call it Theorem A.
* You thought you had three new results that are applications of Theorem A.
* You realized you could prove those results via a generalization of Theorem B instead.
... | 0 | https://mathoverflow.net/users/11540 | 365884 | 153,678 |
https://mathoverflow.net/questions/365896 | 5 | What is the group cohomology $H^{d}(\mathbb{Q}/\mathbb{Z}, \mathbb{Z})$ with trivial action?
Can it be computed succinctly using the short exact sequence $0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$?
| https://mathoverflow.net/users/161264 | Group cohomology of Q/Z | Short answer:
$$
H^i(\mathbb{Q}/\mathbb{Z}) =
\begin{cases}
\mathbb{Z}, & i = 0,\\
0, & i \equiv 1 \mod 2,\\
A, & i\equiv 0 \mod 2, i>1,
\end{cases}$$
where $A$ fits into a short exact sequence $\mathbb{Z}\to A\to H^2(\mathbb{Q})$.
Explanation:
We can use the short exact sequence to compute the cohomology of $\mat... | 3 | https://mathoverflow.net/users/161272 | 365907 | 153,684 |
https://mathoverflow.net/questions/365917 | 5 |
>
> Let $E$ be a Banach space and $T: E\rightarrow E$ be a mapping. $T$ is said to be *subcontinuous* if for any sequences $(u\_n)\_{n\in\mathbb{N}}$ in $E$ that converge strongly to $u$ the sequence $(T(u\_n))\_{n\in\mathbb{N}}$ converges weakly to $T(u)$.
>
>
>
I am looking for a subcontinuous function which i... | https://mathoverflow.net/users/102228 | A subcontinuous function, which is not continuous | Let $(e\_n)$ be the standard orthonormal basis of $\ell^2$: recall that, as a sequence, $(e\_n)$ converges weakly to $0$. Now define a map $f\colon\mathbb{R} \to \ell^2$ by $f(\frac{1}{n})=e\_n$ and $f(t)=0$ if $t\leq 0$, and interpolating linearly between $\frac{1}{n}$ and $\frac{1}{n+1}$: this is continuous at every ... | 10 | https://mathoverflow.net/users/17064 | 365923 | 153,690 |
https://mathoverflow.net/questions/365857 | 10 | Let $w(n,l)$ denote the number of closed walks of length $2l$ from a given vertex of the $n$-cube. Then, it is well-known that
$$\cosh^n(x)=\sum\_{l=0}^{\infty}\frac{w(n,l)}{(2l)!}x^{2l}.$$
Differentiating both sides, we get
$$n \cdot \cosh^{n-1}(x)\cdot \sinh(x) = \displaystyle\sum\_{l=1}^{\infty}\frac{w(n,l)}{(2l... | https://mathoverflow.net/users/149093 | Closed walks on an $n$-cube and alternating permutations | This is a kind of inclusion-exclusion related to the identity
$$
\sum\_{k=1}^m (-1)^{k+1} \binom{2m-1}{2k-1}A(2k-1)=1 \quad\quad(1)
$$
for all $m=1,2,\ldots$.
For a route on the $n$-cube with first step being vertical we label other $2k-1$ vertical steps, take a weight $(-1)^{k+1}A(2k-1)$ for such a configuration and... | 3 | https://mathoverflow.net/users/4312 | 365931 | 153,695 |
https://mathoverflow.net/questions/365807 | 6 | Let $\kappa$ be an infinite cardinal. We call a cardinal $\lambda \leq 2^\kappa$ *intersecting* if there is ${\cal C}\subseteq {\cal P}(\kappa)$ such that
1. for every $A\in {\cal C}$ we have $|A|=\kappa$,
2. $|A\_0\cap A\_1|<\lambda$ whenever $A\_0\neq A\_1\in {\cal C}$, and
3. $|{\cal C}| > \kappa$.
We denote the... | https://mathoverflow.net/users/8628 | "Intersection number" of a cardinal | Since this question is still unanswered I thought I might write down some of what you can get out of [Baumgartner's paper](https://pdf.sciencedirectassets.com/272681/1-s2.0-S0003484300X0023X/1-s2.0-0003484376900188/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEEYaCXVzLWVhc3QtMSJIMEYCIQCS2%2FrnWTtDAo53EJ7yg3Rbm3hXn2D%2... | 7 | https://mathoverflow.net/users/11233 | 365942 | 153,700 |
https://mathoverflow.net/questions/365887 | 3 | For some work in equivariant stable homotopy, I am trying to understand the family of finite $p$-groups $P$ with derived subgroup $P'$ of order $p$. There is a 1999 J. Algebra paper by Simon Blackburn ([Groups of prime power order with derived subgroup of prime order](https://doi.org/10.1006/jabr.1998.7909)) that gives... | https://mathoverflow.net/users/102519 | Construction of finite $p$-groups with derived subgroup of order $p$? | As I said in my comment, I am not completely sure whether I understand your construction in (c), but the following example is an interesting test case.
Start with an extraspecial group $\langle a,b,c \rangle$ of order $p^3$ and exponent $p$ (with $p$ odd), with $[a,b]=c$ and $c$ central of order $p$.
Now let $A = C... | 2 | https://mathoverflow.net/users/35840 | 365945 | 153,701 |
https://mathoverflow.net/questions/365947 | 61 | I think a related question might be this ([Set-Theoretic Issues/Categories](https://mathoverflow.net/questions/94794/set-theoretic-issues-categories/94823#94823)).
There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance [Shulman - Set theory for category ... | https://mathoverflow.net/users/160378 | When size matters in category theory for the working mathematician | Very often one has the feeling that set-theoretic issues are somewhat cheatable, and people feel like they have eluded foundations when they manage to cheat them. Even worse, some claim that foundations are irrelevant because each time they dare to be relevant, they can be cheated. What these people haven't understood ... | 86 | https://mathoverflow.net/users/104432 | 365951 | 153,703 |
https://mathoverflow.net/questions/365683 | 2 | By the converse of the strong law of large numbers, we know that, given a sequence of i.i.d random variables $X\_1,X\_2,\dots$ such that $\mathbb{P}(X\_1 \ge 0)=1$ and $\mathbb{E}X\_1= \infty$,
then I have
$$
S\_N:=\frac{1}{N}\sum\_{i=1}^N X\_i \longrightarrow \infty \quad \mathbb{P}\textit{-a.s}.
$$
I suppose that,... | https://mathoverflow.net/users/52960 | On the speed of divergence of the converse of the Strong law of large numbers | Such a sequence $a\_n$ does not exist even for a well studied example like returns to the origin of simple random walk in one dimension. If $X\_i$ denotes the number of steps from the $i-1$ time the walk returned to the origin to the $i$'th time, then $X\_i$ are i.i.d. and their sum $S\_n$ is the number of steps until ... | 2 | https://mathoverflow.net/users/7691 | 365954 | 153,705 |
https://mathoverflow.net/questions/365892 | 2 | Given some $d$ dimensional torus, (i.e. just a $d$-dimensional hypercube with periodic boundary conditions) I'll call $\Omega$, and a group of transformations $G$ of $\Omega$, I want to find the smallest sub-region in the hyper-torus such that the rest of the torus can be generated by the action of the group elements o... | https://mathoverflow.net/users/161263 | Finding an irreducible region of a space given a group of transformations | I assume that $G$ acts isometrically on the flat torus $T^n$. Then the standard construction of $\tilde\Omega$ proceeds as follows. Pick a point $x\in T^n$ not fixed by any $g\in G$ and consider its $G$-orbit $Gx= \{gx: g\in G\}$. Take the Voronoi tiling of $T^n$ corresponding to this subset, let $D\_x$ be the tile "ce... | 2 | https://mathoverflow.net/users/39654 | 365959 | 153,707 |
https://mathoverflow.net/questions/365946 | 6 | I have the following function
$$
\int\_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it}
{ \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt
$$
where $1<\beta<2$, $v>0$. Need to show it is positive.
The inverse Mellin transform of
$$
\left\{2^{1/\beta-1/2} \right\}^{it}
{ \Gamma\{(it+1)/\beta\}... | https://mathoverflow.net/users/161301 | Positivity of $ \int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it} { \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt$ | $\newcommand\Ga\Gamma
\newcommand{\R}{\mathbb{R}}
\newcommand{\de}{\delta}
\newcommand{\ga}{\gamma}
\newcommand{\Si}{\Sigma}$
We have to show that for $a:=-\ln(2^{1/b-1/2}/v)\in\R$ and $b:=\beta\in(1,2)$,
\begin{equation\*}
I(a):=\int\_{-\infty}^{\infty} e^{-iat}R(t)\,dt>0, \tag{1}
\end{equation\*}
where
\begin{equatio... | 11 | https://mathoverflow.net/users/36721 | 365977 | 153,714 |
https://mathoverflow.net/questions/365955 | 4 | In the paper "[The display of a formal $p$-divisible group](https://smf.emath.fr/node/42448)" Zink defines some objects and calls them $3n$-display. A $3n$-display over $R$ is a quadruple $P$, $Q$, $F$, $F^1$ such that $P$ is a projective $W(R)$ module, $Q$ is a submodule and $F:P\to P$, $F^1 :Q\to P$ are functions sat... | https://mathoverflow.net/users/65846 | What is the reason behind the name 3n-display? | I have not found a source where Thomas Zink explains the name, however [arXiv:1906.00899](https://arxiv.org/abs/1906.00899) explains it as an abbreviation of *“not-necessarily-nilpotent”* (or $3n$-) displays. See also [Travaux de Zink](http://www.numdam.org/article/SB_2005-2006__48__341_0.pdf), page 343.
| 6 | https://mathoverflow.net/users/11260 | 365992 | 153,719 |
https://mathoverflow.net/questions/365994 | 9 | The following ratio:
$$\frac{\Gamma(2/5)^3}{\pi\Gamma(1/5)}$$
has kept appearing in my research, and the only thing I know about its value is that it is $\cong 0.7567213$, whence the following two questions:
Is the value of this ratio an algebraic number?
What is the exact value of this ratio?
| https://mathoverflow.net/users/105094 | Algebraicity of a ratio of values of the Gamma function | This number is expected to be transcendental. [This answer](https://mathoverflow.net/a/344203/6506) gives a conceptual framework for studying the algebraicity of such $\Gamma$ ratios, and in fact a completely explicit criterion (which is only conjectural, when it comes to establishing transcendence).
Your number is e... | 8 | https://mathoverflow.net/users/6506 | 365995 | 153,720 |
https://mathoverflow.net/questions/365981 | 0 | Let $b = [b\_1,b\_2,b\_3,...b\_n]^T$
$A = [a\_{ij}]\_{n \times n}$ such that $a\_{i,j} = 1\forall 1\le i,j \le n$
$C = [c\_{ij}(\lambda)]\_{n \times n}$ such that $c\_{ij}(\lambda) = O(\frac{1}{\lambda})$
$A+C$ is known to be a symmetric positive semi definite matrix
$I\_n$ is an $n\times n$ Identity matrix
$e ... | https://mathoverflow.net/users/14414 | Asymptotic expansion involving a matrix equation | Re [your answer](https://mathoverflow.net/a/365990/1898):
Statement 1 follows from [Weyl's inequalities](https://en.wikipedia.org/wiki/Weyl%27s_inequality#Weyl%27s_inequality_about_perturbation) and $\rho\_1(C) = O(\lambda^{-1})$.
$2 \implies 3$ seems problematic; what if $C = \frac{1}{\lambda^3}I$, which should be... | 2 | https://mathoverflow.net/users/1898 | 365999 | 153,723 |
https://mathoverflow.net/questions/365984 | 2 | Let $u=\int e^{\dot{\imath}K(x,y)} f(y) dy$. My advisor told me that we can disprove an integrability estimate
$$\|u\|\_{L^p}\lesssim \|f\|\_{L^{1}}\label{1}\tag{1}$$
by disproving it when $f=\delta$, the Dirac Delta distribution.
When I asked him for the reasoning for this, he told me $\delta$ is the limit of a sequ... | https://mathoverflow.net/users/116555 | If an estimate is false on $L^{1}$, then it is false for the $\delta$ distribution? | First of all, observe that your "subquestion" is ill-posed because $\int\_{\mathbb{R}^{d}}|f\_{n}-\delta|$ does not make any sense: the Dirac delta distributions is of course not an $L^1$ function, it is merely a distribution $\mathcal D'$.
The right answer goes as follows (well, more or less, you should actually giv... | 2 | https://mathoverflow.net/users/33741 | 366000 | 153,724 |
https://mathoverflow.net/questions/365982 | 1 | Macpherson in *a survey of homogeneous structures*, states that there are many $\aleph\_0$-categorical structures which are not homogeneous. Here homogeneity is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of a structure $M$ can be extended to an automorphism of $M$. $\omeg... | https://mathoverflow.net/users/120374 | Question on $\aleph_0$-categorical nonhomogeneous structures | You are confusing several notions of homogeneity. Saturated structures, and therefore also $\aleph\_0$-categorical structures, are *homogeneous*, but not necessarily *ultrahomogeneous*. This means that every finite partial *elementary mapping* extends to an automorphism.
$\omega$-homogeneity is in fact an even weaker... | 8 | https://mathoverflow.net/users/12705 | 366002 | 153,725 |
https://mathoverflow.net/questions/365919 | 5 | It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$
We then define the finite-difference Laplacian
$$(\Delta\_hu)(x):=\frac{\left(\sum\_{i=1}^3 f(x+he\_i)+f(x-he\_i)\right)-6 f(x)}{h^2}... | https://mathoverflow.net/users/150564 | Approximate Sobolev embedding | Yes, this is true, and there is a proof which closely tracks your intuition. As you know, this estimate can be proved in the continuum by applying the Sobolev embedding twice, first to get $\nabla u \in L^p$ for $p<\frac{2d}{d-2}=6$, and then once more to get $u\in L^\infty$. So for simplicity let me discuss how to get... | 5 | https://mathoverflow.net/users/5678 | 366003 | 153,726 |
https://mathoverflow.net/questions/366008 | 10 | A finitely generated abelian group $A$ is isomorphic to a direct sum of cyclic groups. I am interested in an extension of this result on couples of abelian groups $(A,B),$ where $B$ is a subgroup of $A.$ Consider the category of such couples $(A,B),$ where morphism $f:(A,B)\to (A',B')$ is a homomorphism $f:A\to A'$ suc... | https://mathoverflow.net/users/23310 | Classification of subgroups of finitely generated abelian groups | The answer to Question 1 is no.
Let $A=\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$
and let $B$ be the subgroup generated by $(2,1)$.
Since $B$ is cyclic of order $4$, if it were contained in a proper direct summand of $A$ then it would be contained in a cyclic subgroup of $A$ of order $8$, and so $(2,1)$ wo... | 20 | https://mathoverflow.net/users/22989 | 366010 | 153,728 |
https://mathoverflow.net/questions/365962 | 17 | Consider a family of flabby (= flasque) sheaves $(\mathcal F\_i)\_{i\in I}$ of abelian groups on the topological space $X$.
My question : is their direct sum sheaf $\mathcal F=\oplus \_{i\in I} \mathcal F\_i$ also flabby?
Here is the difficulty:
Given an open subset $U\subset X$ a section $s\in \Gamma(U,\mathca... | https://mathoverflow.net/users/450 | Is a direct sum of flabby sheaves flabby? | No, a direct sum of flabby sheaves need not be flabby.
Take $X=\{1,1/2,1/3,1/4,\dots\}\cup\{0\}$ with the subspace topology from $\mathbb R$, and let $\mathcal F$ be the sheaf whose sections over an open $U\subseteq X$ are the functions $U\to\mathbb F\_2$ (not necessarily continuous). This is a flabby sheaf. I claim ... | 21 | https://mathoverflow.net/users/126183 | 366023 | 153,735 |
https://mathoverflow.net/questions/366015 | 1 | I have asked the following question at [MSE](https://math.stackexchange.com/questions/3042104/in-mathbbcx-y-if-langle-u-v-rangle-is-a-maximal-ideal-then-langle) and got one answer. Any further ideas are welcome:
Let $u=u(x,y), v=v(x,y) \in \mathbb{C}[x,y]$, with $\deg(u) \geq 2$ and $\deg(v) \geq 2$.
Let $\lambda, \m... | https://mathoverflow.net/users/72288 | In $\mathbb{C}[x,y]$: If $\langle u,v \rangle$ is a maximal ideal, then $\langle u-\lambda,v-\mu \rangle$ is a maximal ideal? | While it may be, in general it is not. Consider as an example, $u=x+y+yp(x), v=x+yp(x)$ where$ \deg p(x)\geq 2$. Then $(u,v)=(x,y)$ and so maximal. Notice that $u=y+v$. So, $$(u-a, v-b)=(u-v+b-a, v-b)=(y+b-a, v-b)=(y+b-a, x-b+(a-b)p(x))$$
and so most pairs of values of $a,b$, it is not maximal, as long as $\deg (x+(a-b... | 14 | https://mathoverflow.net/users/9502 | 366026 | 153,736 |
https://mathoverflow.net/questions/365912 | 8 | [This question](https://math.stackexchange.com/questions/3758097/are-any-interesting-classes-of-polynomial-sequences-besides-sheffer-sequences-gr) on math.stackexchange.com has 35 views, three up-votes, and not a word from anybody, so I'm posting it here.
Let us understand the term **polynomial sequence** to mean a s... | https://mathoverflow.net/users/6316 | Are any interesting classes of polynomial sequences besides Sheffer sequences groups under umbral composition? | Another equivalent characterization of Sheffer sequences is that they fit into a generating function of the form
$$\sum\_{n=0}^{\infty}\frac{p\_n(x)}{n!}t^n=f(t)e^{xg(t)}.$$
Most of the results on Sheffer sequences apply to a more general setting where we work with a function $\Psi(x)=\sum\_{n\geq 0}x^n/c\_n$ and defin... | 8 | https://mathoverflow.net/users/2384 | 366031 | 153,739 |
https://mathoverflow.net/questions/365326 | 0 | Let $S$ be an infinite set of positive integers, $N\_S(z)$ be the number of elements of $S$ less than or equal to $z$, and let
$$D\_S(z, n, p)= \sum\_{k\in S,k\leq z}\chi(k\equiv p\bmod{n}).$$
Here $\chi$ is the indicator function, and $z, p, n$ are positive integers, with $p<n$ and $n>1$. If
$$\lim\_{z\rightarro... | https://mathoverflow.net/users/140356 | Congruential equidistribution, prime numbers, and Goldbach conjecture |
>
> If $S$ is congruentially equidistributed and contains enough elements .... is it true that $S+S$ contains all the positive integers except a finite number of them?
>
>
>
Let $S=\bigcup\_{n=1}^\infty \{2^{2n},2^{2n}+1,\dots, 2^{2n+1}-1\}.$ It is easy to show that $S$ is congruentially equidistributed and $S+S... | 5 | https://mathoverflow.net/users/43954 | 366041 | 153,743 |
https://mathoverflow.net/questions/366056 | 3 | This is probably a really basic result that I'm forgetting but if $M \models \text{PA}$ and $M \models \phi$ for some $\Sigma^0\_1$ sentence $\phi$ such that $\mathbb{N} \models \lnot \phi$ does it follow that there is some consistent sentence $\psi$ such that $M \models \lnot \text{Con}(\psi)$? I vaguely feel like it ... | https://mathoverflow.net/users/23648 | Model of PA with false $\Sigma^0_1$ sentence but no false Con sentence? | This essentially boils down to the provable $\Sigma^0\_1$-completeness of PA. Suppose $\phi$ is a $\Sigma^0\_1$ sentence false in $\mathbb N$ but true in some other model $M$ of PA. Then, by $\Sigma^0\_1$-completeness, $M$ also satisfies "$\phi$ is provable in PA" and therefore also satisfies $\neg\text{Con}\_{PA}(\neg... | 6 | https://mathoverflow.net/users/6794 | 366061 | 153,748 |
https://mathoverflow.net/questions/366059 | 11 | Two possibly noncommutative rings are called Morita equivalent if their left-module categories are equivalent. In the commutative case, Morita equivalence is nothing more than ring isomorphism. Otherwise, there are many known examples where this does not hold.
That means modules alone are not enough to characterize t... | https://mathoverflow.net/users/124549 | Does Morita theory hint higher modules for noncommutative ring? | Yes. The trick is to use not just categories, but *pointed categories*, which are categories equipped with a choice of object (the "pointing"). Given any ring $R$, the category $\mathrm{Mod}(R)$ is naturally pointed by the rank-1 free module, i.e. $R$-as-an-$R$-module, which I will write as $R\_R$. Then it is almost tr... | 20 | https://mathoverflow.net/users/78 | 366063 | 153,749 |
https://mathoverflow.net/questions/366049 | 15 | Suppose I have a smooth Riemannian manifold $X$ with induced distance function $d$, and a bi-Lipschitz (with respect to $d$) homeomorphism
$$\phi: X \to X.$$
Under what circumstances could $\phi$ be smoothable to a diffeomorphism? By "smoothable" in this case I mean "homotopic to a diffeomorphism through bi-Lipschitz... | https://mathoverflow.net/users/43158 | When is a bi-Lipschitz homeomorphism smoothable? | Any self-homeomorphism of a manifold of dimension $\neq 4$ is topologically isotopic to a bi-Lipschitz homeomorphism, see lemma 2.4 in [Lipschitz and quasiconformal approximation of homeomorphism pairs](https://www.sciencedirect.com/science/article/pii/S0166864199001455)
by Jouni Luukkainen.
There are exotic spheres ... | 13 | https://mathoverflow.net/users/1573 | 366065 | 153,750 |
https://mathoverflow.net/questions/365269 | -1 | I have been trying to solve a research problem for a while now and in doing so, I stumbled upon the following integral:
$$\int\_0^{\infty } r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^r-1\right)^{\frac{d}{2}-1}}{b^d \Gamma (d)} \, dr.$$
However, I've got no idea how to solve that. Therefore, I'd like t... | https://mathoverflow.net/users/103291 | What is the integral of $r \frac{2^{r-1} \log (2) e^{-\frac{\sqrt{2^r-1}}{b}} \left(2^r-1\right)^{\frac{d}{2}-1}}{b^d \Gamma (d)}$? | An solution for this integral can be found at [Mathmatica.SE](https://mathematica.stackexchange.com/a/225672), which is reproduced next.
After applying the change of variable technique with $x=2^r-1$ we get
$$f=\frac{e^{-\frac{\sqrt{r}}{b}} r^{\frac{d}{2}-1} \log \_2(r+1)}{2 \left(b^d \Gamma (d)\right)} $$
$$\tex... | 0 | https://mathoverflow.net/users/103291 | 366067 | 153,751 |
https://mathoverflow.net/questions/366082 | 1 | We know that the number of decomposition as a sum of four squares of $n\in\mathbb{N}$ such that $n=a^2 + b^2 + c^2 + d^2$ is :
$$ r\_4(n) = 8 \sum\_{d\mid n, 4\nmid d}{d} $$
And there is a more general one from [this answer](https://mathoverflow.net/a/154134).
But is there any restriction of this function to $a,b,c,d... | https://mathoverflow.net/users/153306 | Restriction of Jacobi's four-squares theorem | By inclusion-exclusion principle, the number of representations of $n$ as the sum of squares of four nonzero integers equals:
$$\sum\_{k=0}^4 \binom4k (-1)^k r\_{4-k}(n).$$
Formulae for $r\_k(n)$ are given in [this article](https://mathworld.wolfram.com/SumofSquaresFunction.html) at MathWorld.
If one wants to further... | 4 | https://mathoverflow.net/users/7076 | 366089 | 153,757 |
https://mathoverflow.net/questions/366074 | 2 | this is Rajeev Srivastava and his colleague Ravinder Padmanabha, specializing in computational geometry algorithms for application in science and research. We would like to include methods from algebraic graph theory in our research.
May we ask a conceptual question about conjugacy in algebra. In group theory, two el... | https://mathoverflow.net/users/161330 | Interpretation around conjugacy classes in group theory | Regarding your first question, I think the comment of Henrik Rüping gives the best answer.
Regarding your second question, I am not sure about a geometric interpretation, but maybe the following perspective is a helpful starting point for your intuition: look at the *center* of the group.
Let's call your group $G$ ... | 5 | https://mathoverflow.net/users/156936 | 366090 | 153,758 |
https://mathoverflow.net/questions/366068 | 7 | Let $g:\mathbb{T}\to\mathbb{R}$ and is given as $$g(x) = \sum\limits\_{\eta\in\mathbb{Z}}\frac{1}{1+\gamma \eta^2}\cos{2\pi\eta x}$$
Consider the matrix $$G\_{\gamma} = [g(x\_i-x\_j)]\_{1\le i,j\le n}$$
where $x\_1,x\_2,...x\_n \in (0,1)$ and pairwise distinct.
Due to Bochner's theorem, $g(x)$ is a positive semi ... | https://mathoverflow.net/users/14414 | Bounding the smallest eigenvalue of a matrix generated by a positive definite function | Given any $n$ distinct points $\{x\_i/x\_i\in(0,1)\}$ which are pairwise distinct. For any $c\_i,i = 1,2,3,...n$, and not all zeros.
Using the given expression for $g(x)$ we can deduce that
$$\sum\_{i=1}^n\sum\_{j=1}^nc\_ic\_jg(x\_i-x\_j) = \sum\_{\eta\in\mathbb{Z}} \left(\frac{1}{1+\gamma\eta^2} \left|\sum\_{i=1}^... | 0 | https://mathoverflow.net/users/14414 | 366093 | 153,761 |
https://mathoverflow.net/questions/365925 | 20 | I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's *Graphs on surfaces* from 2001 a great reference as well.
>
> Could you recommend more current references, ideally with focus on ope... | https://mathoverflow.net/users/161287 | Reference for topological graph theory (research / problem-oriented) | Maybe this is another useful reference for you, now I found the link:
Ralucca Gera, Stephen Hedetniemi, Craig Larson, Teresa W. Haynes (editors) (2018): *Graph Theory: Favorite Conjectures and Open Problems*
It is actually two volumes, and obviously more recent than the other reference I mentioned. It covers graph ... | 15 | https://mathoverflow.net/users/156936 | 366099 | 153,764 |
https://mathoverflow.net/questions/366097 | 0 | I'm trying to analytically find the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$
where $a$ and $b$ are constant values, $\mathcal{Q}$ is the Gaussian Q-function, which is defined as $\mathcal{Q}(x) = \frac{1}{2 \pi}\int\_{x}^{\infty} e^{-u^2/2}du$ and $\gamma$ is a... | https://mathoverflow.net/users/103291 | Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v | $\newcommand\Ga\Gamma$
Without loss of generality $a=1$. Let then $Q:=\mathcal Q$, $k:=\kappa>0$, and $t:=\theta\sqrt b>0$, so that $\sqrt b\,\gamma$ has the gamma distribution with parameters $k,t$. Let also $c:=\Ga(k)t^k$. Then, letting $f$ denote the standard normal pdf, we have
$$c\,EaQ(\sqrt b\,\gamma)=\int\_0^... | 1 | https://mathoverflow.net/users/36721 | 366108 | 153,768 |
https://mathoverflow.net/questions/366111 | 8 | Let $d \times d$ matrices $A, B$ be positive definite. Is there a closed form solution for the following quadratic equation in $X$?
$$X A X^{T} = B$$
Thank you.
| https://mathoverflow.net/users/80819 | Closed form solution for $XAX^{T}=B$ | $B^{-1/2}XAX^TB^{-1/2}=I$, so $B^{-1/2}XA^{1/2}=Q$ must be orthogonal. On the other hand, for any orthogonal $Q$, it is simple to verify that $X = B^{1/2}QA^{-1/2}$ solves the equation, so this is a complete parametrization of the solutions.
Here $A^{1/2}$ is the symmetric square root of $A$ (if you prefer you can wo... | 17 | https://mathoverflow.net/users/1898 | 366116 | 153,770 |
https://mathoverflow.net/questions/366070 | 109 | In a lot of computational math, operations research, such as algorithm design for optimization problems and the like, authors like to use $$\langle \cdot, \cdot \rangle$$ as opposed to $$(\cdot)^T (\cdot)$$
Even when the space is clearly Euclidean and the operation is clearly the dot product. What is the benefit or a... | https://mathoverflow.net/users/120345 | What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$? | Mathematical notation in a given mathematical field $X$ is basically a correspondence
$$ \mathrm{Notation}: \{ \hbox{well-formed expressions}\} \to \{ \hbox{abstract objects in } X \}$$
between mathematical expressions (or statements) on the written page (or blackboard, electronic document, etc.) and the mathematical o... | 328 | https://mathoverflow.net/users/766 | 366118 | 153,771 |
https://mathoverflow.net/questions/366083 | 4 | I did quite a few numerical computations and think the following is true, but I cannot prove it:
Let $\varphi(x):=\sum\_{i=1}^n \varphi\_i(x\_i)$ where $x=(x\_1,...,x\_n) \in \mathbb{R}^n$ and $\varphi\_i \in C^{\infty}$ are even scalar convex functions such that $\varphi''$ is strictly increasing on $[0,\infty).$
... | https://mathoverflow.net/users/119875 | Variance of random variable decreasing in parameter | Your probability measure is a product measure, so by
$$\text{Var}\_y(\langle z,X\rangle) = \sum\_{i=1}^nz\_i^2\text{Var}\_{y\_i}(X\_i)$$
everything reduces to the 1d case. Let $q\_y(dx)=e^{xy-\varphi(x)-C(y)}dx$ be one of the marginals, where $y\in\mathbb{R}$ and $C(y)$ is chosen such that $q\_y$ is normalized, and den... | 4 | https://mathoverflow.net/users/69603 | 366119 | 153,772 |
https://mathoverflow.net/questions/366055 | 11 | Quantum Field Theory is a branch of mathematical physics which is begging for a better understanding.
In fact there are no rigorous constructions of interacting QFT in four dimensions. By a rigorous
construction I mean a construction of the quadruple $(\mathcal{H},U,\Omega,\phi)$ where $\mathcal{H}$ is
the Hilbert spac... | https://mathoverflow.net/users/24078 | How do you know that you have succeeded-Constructive Quantum Field Theory and Lagrangian | The axioms don't tell you what theory you constructed. For that you need to go beyond the construction of correlation functions of the elementary field $\phi$ (the basic chapter on renormalization in QFT textbooks) and produce, e.g., by a point-splitting procedure, correlations with insertion of composite fields like $... | 8 | https://mathoverflow.net/users/7410 | 366125 | 153,774 |
https://mathoverflow.net/questions/366127 | 3 | This question is inspired from the post linked below:
[Can an algebraic number on the unit circle have a conjugate with absolute value different from 1?](https://mathoverflow.net/questions/38680/can-an-algebraic-number-on-the-unit-circle-have-a-conjugate-with-absolute-value)
What I am curious about is the following... | https://mathoverflow.net/users/157984 | In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same? | Zero is the only algebraic integer which has *all* its conjugates strictly inside the complex unit circle. (Look at the norm.)
For explicit examples with conjugates on either side of the unit circle, you can start with a real quadratic field with a totally positive unit that isn't already a square in this field, such... | 12 | https://mathoverflow.net/users/49003 | 366128 | 153,775 |
https://mathoverflow.net/questions/366124 | 2 | (Asked in [MSE](https://math.stackexchange.com/questions/3760125) but got no response.)
The generating function $\frac{1}{(1-t)^N}=\sum\_k {N+k-1\choose k}t^k=\sum\_k h\_k(1)t^k$ and the Jacobi–Trudi formula $s\_{\lambda/\mu}=\det(h\_{\lambda\_i-i-\mu\_j+j})$ tell me that the value of the skew Schur function at the i... | https://mathoverflow.net/users/83671 | On the value of a skew Schur function at the identity | You might want to read up on the [principal specialization](https://www.symmetricfunctions.com/standardSymmetricFunctions.htm#standardSpecialization).
It specializes a symmetric function into a formal power series.
For example, the symmetric function $s\_1(x) = x\_1+x\_2+ \dotsb$
has principal specialization $s\_1(1,... | 3 | https://mathoverflow.net/users/1056 | 366138 | 153,778 |
https://mathoverflow.net/questions/366132 | 7 | Let $\Sigma\_g$ be a Riemannian surface of genus $g$. Let $M^4$ be a surface bundle over surface: $\Sigma\_g \to M^4 \to \Sigma\_h$. $\Sigma\_g$ is the fiber and $\Sigma\_h$ is the base space.
My question: *is there a surface bundle over surface $M^4$ such that it has a 2-cocycle $c \in H^2(M^4;Z)$ satisfying
(1) $\i... | https://mathoverflow.net/users/17787 | Intersection form of surface bundle over surface | Yes, such a thing exists, but I don't know an explicit example.
To see that it exists, it is clearest to me to consider the universal situation. For any $k \in \mathbb{Z}$ there is a space $\mathcal{S}\_g(k)$ which classifies oriented surface bundles
$$\Sigma\_g \to E \overset{\pi}\to B$$
equipped with a class $c \in... | 9 | https://mathoverflow.net/users/318 | 366142 | 153,780 |
https://mathoverflow.net/questions/366117 | 1 | Assume that the algebraically independent polynomials $f\_1,\ldots, f\_n\in\mathbb{C}[x\_1,\ldots, x\_n]$ are such that the Jacobian matrix $\text{Jac}\_{x\_1,\ldots, x\_n}^{f\_1,\ldots, f\_n}\in\mathbb{C}\setminus\{0\}$.
Is it true that every derivation $D$ of algebra $\mathbb{C}[f\_1,\ldots, f\_n]$ can be continued... | https://mathoverflow.net/users/100359 | Continuations of derivations of Jacobian subring | Let $R=\mathbb{C}[f\_1,\ldots,f\_n]\subset\mathbb{C}[x\_1,\ldots, x\_n]=A$. A derivation $U$ of $A$ is completely determined by $U(x\_i)$, since, then $U(P(x\_1,\ldots,x\_n))=\sum\frac{\partial P}{\partial x\_i} U(x\_i)$ for any $P\in A$. In particular, one has $U(f\_i)=\sum \frac{\partial f\_i}{\partial x\_j}U(x\_j)$.... | 0 | https://mathoverflow.net/users/9502 | 366146 | 153,782 |
https://mathoverflow.net/questions/366144 | 2 | Let $X \subseteq \mathbb{C}^n$ be a complex affine variety and $\tilde{X} \to X$ a surjective proper morphism where $\tilde{X}$ is smooth. Is it true that every morphism $\mathbb{C} \to X$ can be lifted to $\mathbb{C} \to \tilde{X}$?
I'm particularly interested in the case where $X$ has an isolated singularity and $\... | https://mathoverflow.net/users/123207 | Lifting property for proper morphism | Regarding your general question, the answer is *no*.
Take any hyperbolic projective variety $Y$ (for instance, a ball quotient) of dimension $n$, and project it generically onto $\mathbb{P}^n$. Removing a hyperplane from $\mathbb{P}^n$ and its preimage from $Y$, we get a surjective finite morphism $f \colon Y^{\circ}... | 4 | https://mathoverflow.net/users/7460 | 366148 | 153,783 |
https://mathoverflow.net/questions/366150 | 2 | Let $Y$ be an index $2$, degree $5$, Picard number $1$ Fano threefold, i.e $Y$ is a linear section of Grassmannian $\operatorname{Gr}(2,5)$. Let $\Sigma(Y)$ be the Hilbert scheme of lines on $Y$, it is isomorphic to $\mathbb{P}^2$. Let $\mathcal{B}\in \lvert\mathcal{O}\_Y(2)\rvert$ be a smooth quadric hyersurface, it i... | https://mathoverflow.net/users/41650 | The locus of lines intersecting with another fixed line on a Fano threefold | **Question 1**. Let $I(Y) \subset \Sigma(Y) \times \Sigma(Y) \cong \mathbb{P}^2 \times \mathbb{P}^2$ be the incidence scheme (parameterizing pairs of intersecting lines). Then $I(Y) \cong \mathrm{Fl}(1,2;3) \subset \mathbb{P}^2 \times \mathbb{P}^2$; I think you can find this in Sanna, Giangiacomo. Small charge instanto... | 4 | https://mathoverflow.net/users/4428 | 366163 | 153,788 |
https://mathoverflow.net/questions/366164 | 2 | Let $X$ be a metric space, $\nu,\mu$ be Borel measures on $X$, $f:X\times \mathbb{R}\rightarrow [0,\infty)$ be a measurable function. Under what conditions is the integral functional $F\_f$, defined by:
$$
\begin{aligned}
F\_f: L^1(X) & \rightarrow [0,\infty]
\\
g&\mapsto \int\_{x \in X} f(x,g(x))d\mu(x)
\end{aligned}
... | https://mathoverflow.net/users/36886 | Conditions for continuity of an integral functional | Besides some condition ensuring that the map is well defined a simple and natural condition for the continuity is a Lipschitz condition with respect to the second variable, i.e., $$|f(x,y)-f(x,z)|\le c|y-z|$$ for some constant $c$ independent of $x,y,z$. This is a standard assumption for (a version of) the Picard-Linde... | 2 | https://mathoverflow.net/users/21051 | 366172 | 153,791 |
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