parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/444991 | 5 | I am currently reading about the non-Abelian Hodge correspondence. Let $(X,\omega)$ be a compact Kahler manifold. Given a Higgs bundle $(E, D\_0)$ on $X$, we want to construct the corresponding flat bundle, which after fixing a Hermitian metric $K$ on $E$ requires looking at when the curvature $\mathbf{F}\_K$ of the as... | https://mathoverflow.net/users/132110 | Formula in non-Abelian Hodge theory - Hodge-Riemann bilinear relations | I have figured out the issue. I mistakenly claimed that $F\_h^\dagger = - F\_h$. In fact, what we really have is
$$(F\_h^{1,1})^\dagger = -F\_h^{1,1},\quad (F\_h^{2,0})^{\dagger} = F\_h^{0,2}.$$
Thus defining the Hermitian inner product
$$H(\alpha,\beta) = i^k\int\_X\mathrm{tr}(\alpha\wedge\beta^\dagger)\wedge\omega^{n... | 0 | https://mathoverflow.net/users/132110 | 445062 | 179,417 |
https://mathoverflow.net/questions/445004 | 2 | In Knapp's book *Representation Theory of Semisimple Groups: An Overview Based on Examples*, he proves the following theorem of Weyl: If $G$ is a compact connected semisimple Lie group, then $\pi\_1(G)$ is finite. I am confused about a detail in the proof, which I now explain.
First, Knapp shows (by exploiting compac... | https://mathoverflow.net/users/165625 | Knapp's proof that the fundamental group of a compact semisimple Lie group is finite | It seems to me that you are right and that the proof does not need the detour to linear Lie groups. I think the reason why Knapp writes the proof of his Theorem 4.26 in this way is that throughout paragraph IV.5 he assumes that G is a compact linear Lie group. Strictly speaking, Knapp shows the necessary results about ... | 1 | https://mathoverflow.net/users/503099 | 445072 | 179,420 |
https://mathoverflow.net/questions/444681 | 3 | [Faulhaber polynomial](http://en.wikipedia.org/wiki/Faulhaber%27s_formula) of order $p \in \Bbb{N}$ is defined as the unique polynomial of degree $p+1$ satisfying
$$ S\_{p}(n) = \sum\_{k=1}^{n} k^p $$
for $n = 1, 2, 3, \cdots$. For example,
\begin{align\*}
S\_0(x) &= x, \\
S\_1(x) &= \frac{x(x+1)}{2}, \\
S\_2(x) ... | https://mathoverflow.net/users/38620 | maybe Faulhaber polynomial $S_{k}(x)=0$ have only rational roots $0,-\frac{1}{2},-1$ | This is a probably known consequence of the von Staudt-Clausen theorem.
From the exponential generating function
$$G(x,t)~=~ \sum S\_p(x) \frac{t^p}{p!} ~=~ e^t\,\frac{1-e^{x\,t}}{1-e^t}$$
of the polynomials $S\_p(x)$ it follows that a rational zero $x\_0=m/n$ with $n\geq 1$ and $\gcd (n,m)=1$ leads to a zero coefficie... | 3 | https://mathoverflow.net/users/20804 | 445076 | 179,422 |
https://mathoverflow.net/questions/445067 | 8 | Let $G$ be a non-elementary group generated by a finite set $S$. Here, a group is called non-elementary if it is not virtually abelian. Denote $S^{\le n}:=\{g\in G: |g|\_S\le n\}$ for any $n\in \mathbb N$ and $C(H,K):=\{[h,k]: h\in H, k\in K\}$ for any two subsets $H,K\subset G$. It is easy to see that $|S^{\le n}|\ge ... | https://mathoverflow.net/users/397904 | The growth rate of a commutator set in a non-elementary group | You can take $\kappa(n) = n/2$ if $G$ is not virtually nilpotent of class $\le 2$.
Let $B\_n = S^{\le n}$ and $C\_n = \{[b\_1, b\_2] : b\_1, b\_2 \in B\_n\}$. Suppose $|C\_n| < n/2$. By pigeonhole there is some $m < n$ such that $C\_m = C\_{m+2}$. For $[b\_1, b\_2] \in C\_m$ and $s \in S$ we have $[b\_1,b\_2]^s = [b\... | 6 | https://mathoverflow.net/users/20598 | 445080 | 179,423 |
https://mathoverflow.net/questions/445077 | 0 | Let $f:X \to S$ be a proper $C^\infty$-morphism between real manifolds. Assume that each fiber of $f$ is connected of real dimension 2 (the fiber may not be smooth, but it is the union of smooth manifolds). Suppose that $f$ admits 2 sections $s\_1, s\_2: S \to X$.
**Question:** is there a homotopy $H: S \times [0,1] ... | https://mathoverflow.net/users/29730 | Homotopy between sections | Not in general.
Suppose $f: S^1\times T \to S^1$ is the projection, where
$f$ is the first factor projection and $T = S^1 \times S^1$ is the torus.
Then a section amounts to a map $S^1 \to T$ and the homotopy class of a section amounts to an element of $[S^1,T] \cong \Bbb Z \times\Bbb Z$.
So you have an easy counte... | 3 | https://mathoverflow.net/users/8032 | 445081 | 179,424 |
https://mathoverflow.net/questions/445078 | 10 | Let $S\_\omega$ be the group of permutations (bijections) $\varphi:\omega\to\omega$, together with composition as binary operation.
[Zorn's Lemma](https://en.wikipedia.org/wiki/Zorn%27s_lemma) implies that every commutative subgroup of $S\_\omega$ is contained in a maximal commutative subgroup of $S\_\omega$.
**Que... | https://mathoverflow.net/users/8628 | Maximal Abelian subgroups of $S_\omega$ | You can achieve both $2^\omega$ and $\omega$ itself.
Since $\omega$ is in bijection with $\mathbb Z$, you can consider the subgroup $H\_1$ generated by $\phi:n\to n+1$. A couple of lines shows that anything commuting with $\phi$ is a power of $\phi$.
On the other extreme, working in $\mathbb N$, let $S\_i$ denote t... | 16 | https://mathoverflow.net/users/11054 | 445085 | 179,425 |
https://mathoverflow.net/questions/445089 | 7 | I'd like to know when the equality holds in the following inequality
$$
| x - y |^a \le | x - z |^a + | y - z |^a.
$$
More precisely, for which points $x = (x\_1, x\_2)$, $y = (y\_1, y\_2)$ and $z = (z\_1, z\_2)$ in the plane does hold
$$
| x - y |^a = | x - z |^a + | y - z |^a
$$
where $a \in (0,1)$?
Notice that w... | https://mathoverflow.net/users/502917 | When does equality hold in a specific triangle inequality? | For $a\in(0,1)$ and any real $u,v\ge0$,
$$(u+v)^a\le u^a+v^a, \tag{1}\label{1}$$
with the equality if and only if $u=0$ or $v=0$.
Now,
$$|x-y|^a=|(x-z)+(z-y)|^a\le(|x-z|+|z-y|)^a \\
\le|x-z|^a+|z-y|^a
=|x-z|^a+|y-z|^a.$$
So, if $|x-y|^a=|x-z|^a+|y-z|^a$, then $(|x-z|+|z-y|)^a
=|x-z|^a+|z-y|^a$, so that we have the ... | 13 | https://mathoverflow.net/users/36721 | 445104 | 179,430 |
https://mathoverflow.net/questions/444751 | 0 | I would like to find the following limit which is somewhat similar to the usual Vandermonde's convolution for binomial coefficients. Define $L$ as follows.
$$ L \triangleq \lim\_{N\to\infty} \frac{1}{2^{2N-2}\log N} \sum\_{k=1}^{N} \log k \binom{2k-2}{k-1}\binom{2N-2k}{N-k}\, . $$
Note that if we simply use $\log N... | https://mathoverflow.net/users/306951 | Asymptotic approximation of a convolution of binomial coefficients | As $k\to\infty$, $\binom{2k-2}{k-1}\sim 2^{2k-2}/\sqrt{\pi k}$. As $N-k\to\infty$, $\binom{2N-2k}{N-1}\sim 2^{2N-2k}/\sqrt{\pi (N-k)}$.
Now approximate the sum by an integral:
$$\int\_0^N \frac{\ln k}{\pi\sqrt{k(N-k)}\ln N}\,dk
=\frac{\pi \ln N -2\pi -C}{\pi\ln N} \to 1,$$
where $C$ is a particular hypergeometric con... | 0 | https://mathoverflow.net/users/9025 | 445108 | 179,432 |
https://mathoverflow.net/questions/445088 | 3 | Suppose that $A \leq\_a 0^\omega$ (i.e. $A$ is arithmetic in $0^\omega$) does there exist $\widehat{A} \equiv\_a A$ with $\widehat{A} \leq\_T 0^\omega$ [1]?
More generally, say that a set $X$ is aT-complete (better name?) if every arithmetic degree $\mathbf{a} \leq\_a X$ has a representative $A \leq\_T X$. Does every... | https://mathoverflow.net/users/23648 | Does every arithmetic degree below $0^\omega$ have a representative computable in $0^\omega$? | Ok, I'm pretty sure the answer is no to the basic question. We build a set $A$ to satisfy the following conditions.
$P\_e: A \neq \phi\_i(0^{n})$
$R\_{i,j}: X = \phi\_i(0^\omega) \land (\forall i)\left(X(i) = A \models \psi\_j(i)\right) \implies (\exists n)(X \leq\_T 0^{n})$
Where $\phi\_i$ is a Turing reduction an... | 2 | https://mathoverflow.net/users/23648 | 445111 | 179,433 |
https://mathoverflow.net/questions/444984 | 5 | Consider the heat equation $\partial\_t u= \Delta u+\lambda\_1 u$ on a non-compact complete manifold $M$ (with nonpositive curvature) where $\lambda\_1$ is the first eigenvalue and we start with some smooth initial data $u\_0$ at $t=0.$ Then does the heat flow converge to the first eigenfunction on $M$?
| https://mathoverflow.net/users/68232 | Convergence of heat flow on non-compact manifolds? | The following answer only applies if $u\_0\in L^2$, and it mostly relies on abstract operator theoretic arguments. I am sure more could be said by exploiting the geometric structure.
The Laplace-Beltrami operator on a complete manifold is essentially self-adjoint on $C\_c^\infty$. Since $\lambda\_1$ is the bottom of ... | 4 | https://mathoverflow.net/users/95776 | 445147 | 179,443 |
https://mathoverflow.net/questions/445151 | 3 | Do we already know the classification of the finite-dimensional irreducible unitary representations of the modular group $PSL(2,\mathbb{Z})=\mathbb{Z}/2\*\mathbb{Z}/3$?
I'm particularly interested in the following question.
**Q:** Is there a maximal dimension for the finite-dimensional irreducible unitary representat... | https://mathoverflow.net/users/472749 | Irreducible unitary representation of PSL(2,Z) | The answer is **No**.
There is a mod-$p$ map $f:PSL(2,\mathbb Z) \rightarrow PSL(2,\mathbb F\_p)$.
The permutation representation of $PSL(2,\mathbb F\_p)$ on the projective line on $\mathbb F\_p$ with $p+1$ points splits as a $1$-dimensional and a $p$-dimensional irreducible representation, which can be checked by ... | 7 | https://mathoverflow.net/users/125498 | 445153 | 179,444 |
https://mathoverflow.net/questions/445149 | 6 | Let $T\_n$ be the full transformation monoid of an $n$-set $N\_n$ with elements 1,...,n consisting of all functions $f: N\_n \rightarrow N\_n$.
We can associate to each function $f$ a matrix $M\_f$ in the same way as we associate a permutation matrix to a permutation.
>
> Question 1: Let $A\_n$ be the algebra over ... | https://mathoverflow.net/users/61949 | Algebra generated by transformation matrices | Now that I have worked this out, I will rewrite the answer cleanly. I have deleted the old answer to unclutter things.
The algebra $A\_n$ is Morita equivalent to $2\times 2$ upper triangular matrices and the algebra $B\_n$ is Morita equivalent to the square-zero algebra with quiver the Dynkin quiver consisting of a s... | 7 | https://mathoverflow.net/users/15934 | 445158 | 179,445 |
https://mathoverflow.net/questions/445152 | 1 | My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it seems to be true but I don't know much about combinatorics and combinatorial tricks, so I hope the community is able to... | https://mathoverflow.net/users/153916 | Existence of some lattice path connecting all given lattice paths | Let us draw a checkered table with $N$ columns and $n$ rows, such that the cell $(i,j)$ (that is, the one in the $i$th row and the $j$th column) contains the number $q\_i^j$. Paint all cells with zeroes in black, all others are white. The conditions on paths claim that any two cells having a common edge contain numbers... | 3 | https://mathoverflow.net/users/17581 | 445164 | 179,449 |
https://mathoverflow.net/questions/422493 | 24 | This is a [crosspost](https://math.stackexchange.com/questions/4445539/is-there-an-open-subset-a-of-0-12-with-measure-frac1100-that-sati) from MSE.
Can we find for any given $\varepsilon>0$ an open subset $A\subseteq[0,1]^2$ with measure $>\frac{1}{100}$ such that, for any smooth curve $\gamma:[0,1]\to\mathbb{R}^2$ o... | https://mathoverflow.net/users/172802 | Is there an open subset $A$ of $[0,1]^2$ with measure $>\frac{1}{100}$ that satisfies this property? | The answer is no: if $\varepsilon$ is small enough, then for every open $A \subset [0,1]^2$ of measure at least $1/100$, there exists a smooth curve $\gamma$ of length $\leq 1$ such that $\gamma+A$ contains a $\varepsilon$-ball. The idea is to randomly shift by translates first at extremely small scales to cover a lot ... | 21 | https://mathoverflow.net/users/766 | 445176 | 179,453 |
https://mathoverflow.net/questions/445198 | 7 | I've noticed a curious relationship between the coefficient $a\_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum\_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. Then, we have $a\_n = \frac{(-1)^n}{n!}$. The relationship says that
$$\int\_{-\infty}^\infty e^{-x^2} dx = \pi i a\_{\f... | https://mathoverflow.net/users/146528 | If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$ | Let $\phi(z)$ be analytic function defined on the half-plane
$$H(\delta)=\{z\in \mathbb{C}: \operatorname{Re}z\ge -\delta\}$$
for $0<\delta<1$. Suppose that, for some $A<\pi$, $\phi$ satisfies the growth condition
$$|\phi(x+iy)|\le Ce^{Px+ A|y|}$$
for all $z\in H(\delta)$. Then, for any $0<\operatorname{Re} s< \delta$,... | 12 | https://mathoverflow.net/users/152473 | 445204 | 179,462 |
https://mathoverflow.net/questions/443799 | 2 | Given a compact symplectic manifold $(X, \omega)$, are there any invariants (topological or easily computable geometric/analytic ones) which give an estimate of the maximal number of independent constants of motions in $(X, \omega)$?
More precisely, we would like to find a reasonable bound on a number $k \in \mathbb{... | https://mathoverflow.net/users/51112 | Obstructions to maximal number of independent constants of motion in a given symplectic manifold | Any symplectic 2n-dimensional
manifold admits a systems of Poisson-commuting functions $f\_1,..,f\_n$ whose differentials are linearly independent on an open subset of full measure.
The result is proven in the book ''Symplectic geometry" of A.T. Fomenko. The initial russian version and its engish translation have dif... | 2 | https://mathoverflow.net/users/14515 | 445212 | 179,466 |
https://mathoverflow.net/questions/445215 | 2 | **Motivation.** With my younger son I played the following game on a big (dysfunctional) clock which can be modelled as $\mathbb{Z}/12\mathbb{Z}$ : Put the clock hands at number $12 ( = 0)$. Toss a coin, and move the clock hands to the right (i.e. from $n$ to $n+1 \text{ mod } 12$) if the coin toss was tails, and move ... | https://mathoverflow.net/users/8628 | Tossing a coin around $\mathbb{Z}/n\mathbb{Z}$ | Suppose you are at position $a$, and have already visited $\{a-i,\dotsc,a,\dotsc,a+j\}$, where $i,j\geq 0$ and $i+j<n$. The expected time $t\_{ij}$ until you have covered the whole clock must satisfy $t\_{i,n-1-i}=0$ and
$$ t\_{ij} = \frac{1}{2}t\_{(i-1)\_+,\;j+1}+\frac{1}{2}t\_{i+1,\;(j-1)\_+}+1 $$
(where $k\_+$ means... | 10 | https://mathoverflow.net/users/10366 | 445219 | 179,468 |
https://mathoverflow.net/questions/445214 | 0 | I am looking for two undirected graphs $G$ and $H$ of the same order (i.e., they have the same number of vertices) with adjacency matrices $A\_G$ and $A\_H$, respectively, such that
* $G$ and $H$ are *fractional isomorphic*, that is, there exists a doubly stochastic matrix $S$ such that $A\_G\cdot S=S\cdot A\_H$ hold... | https://mathoverflow.net/users/9839 | Two fractionally isomorphic graphs but only one having eigenvalue zero | As mentioned in the comments, it is sufficient to find two regular graphs $G$ and $H$ of the same order and same degree, $G$ having zero as eigenvalue whereas $H$ does not.
* Let $G$ consist of the disjoint union of a $3$-cycle and a $4$-cycle, with eigenvalues $$\{-2.0000,-1.0000,-1.0000,-0.0000,0.0000,2.0000,2.0000... | 3 | https://mathoverflow.net/users/9839 | 445226 | 179,469 |
https://mathoverflow.net/questions/445225 | 0 | Suppose that $X\_1,X\_2,\ldots$ are independent random variables. Denote $F\_t$ the $\sigma$-algebra generated by $X\_1,\ldots,X\_t$. Let $T, M \geq 1$, and let $f\_1,\ldots,f\_M$ be $\mathbb{R} \to \mathbb{R}$ functions such that for every $m =1,\ldots, M$, $t =1,\ldots, T$, $E[f\_m(X\_t) | F\_{t-1}]= 0$.
Denote for e... | https://mathoverflow.net/users/100069 | Extension of Kolmogorov's martingale inequality | Of course not. E.g., suppose that the $X\_i$'s are independent random variables each uniformly distributed on the interval $(0,1)$. Let $f\_m(X\_i):=B\_{i,m}-1/2$, where $B\_{i,m}$ is the $m$th digit in the binary expansion of $X\_i$. Then all your conditions hold, with $\sigma^2=1/4$. Also, the $f\_m(X\_i)$'s are i.i.... | 1 | https://mathoverflow.net/users/36721 | 445233 | 179,472 |
https://mathoverflow.net/questions/445208 | 0 | Let $X$ be a random variable from a normal distribution $N(\mu, \sigma)$. How do we calculate the expectation $E[e^{k\cdot e^{-X}}]$, where $k<0$?
I think we can use the moment generating function
$
E[e^{k\cdot e^{-X}}] = \sum\_{t=0}^{\infty}\frac{k^{t}E[e^{-tX}]}{t!}
$, where $E[e^{-tX}]=e^{-t\mu+\frac{1}{2}\sigma... | https://mathoverflow.net/users/503234 | The expected value of a double exponential function of normal random variable | For simplicity I consider first the case $\mu=0,\sigma=1$. (The more general formula follows at the end.)
A [saddlepoint approximation](https://en.wikipedia.org/wiki/Saddlepoint_approximation_method) of the integral
$$I(k)=(2\pi)^{-1/2}\int\_{-\infty}^\infty e^{-x^2/2}e^{ke^{-x}}\,dx$$
gives for large $-k$ the expres... | 2 | https://mathoverflow.net/users/11260 | 445238 | 179,474 |
https://mathoverflow.net/questions/445235 | 4 | Let $F$ be a global function field, for example $F={\mathbb F}\_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}\_q\,$.
>
> **Question.** What would be an example of a global function field $F$ and a finite Galois extension $E/F$ with non-cyclic Galois group $G=\{1,a,b,ab\}$ o... | https://mathoverflow.net/users/4149 | Biquadratic extension of global function fields with cyclic decomposition groups | $E= \mathbb F\_q ( \sqrt{t}, \sqrt{t^2-1} ) $ over $F =\mathbb F\_q(t)$ does the trick if $q$ is congruent to $1$ mod $4$. It suffices to check that at each place where one of the extensions ramifies, the other is split, as this clearly gives a cyclic decomposition group, and unramified places are always cyclic.
The ... | 8 | https://mathoverflow.net/users/18060 | 445239 | 179,475 |
https://mathoverflow.net/questions/445228 | 5 | Let $F\_n=2^{{2^n}}+1$, $n\geq 1$ ( Fermat numbers) and $p>2$ a prime number sucht that $p|F\_n$
I want to show if true that :
$p$ is Wieferich prime number $\Longleftrightarrow $ $p^2|F\_n$
the sense $\Longleftarrow$
If $p>2$ is a prime number and divides $F\_n$, then $p=1+k2^{n+1}$ for some $k\in \mathbb N$. ... | https://mathoverflow.net/users/126827 | A prime divisor $p$ of Fermat number $F_n$ is a Wieferich prime if and only if $p^2$ divides $F_n$ | For the $\Longrightarrow$ sense, note we have, for some positive integer $m$, that
$$p \mid 2^{2^n} + 1 \;\;\to\;\; p \mid 2^{2^{n+1}} - 1 \;\;\to\;\; mp + 1 = 2^{2^{n+1}} \tag{1}\label{eq1A}$$
From your definition of
$$p = 1 + k2^{n+1} \;\;\to\;\; p - 1 = k2^{n+1} \tag{2}\label{eq2A}$$
we then get
$$\begin{e... | 6 | https://mathoverflow.net/users/129887 | 445253 | 179,478 |
https://mathoverflow.net/questions/445260 | 1 | The hop $H\_e$ is defined by $H\_e(X) = X \oplus W\_e^{X}$. A 2-REA operator (or double hop) $J\_{\langle e,i\rangle}$ is defined by $J\_{\langle e,i\rangle}(X) = H\_e(H\_i(X))$
By a famous result from [Pseudo Jump Operators. I: The R. E. Case](https://doi.org/10.2307/1999041) by Jockusch and Shore it's known that fo... | https://mathoverflow.net/users/23648 | Double Hop Inversion Theorem | Ohh, I think I was being dumb. There is a 2-REA operator $J$ such that $J(X) <\_T X'$ isn't of degree r.e. in $X$. Since $0''$ is of r.e. degree in every $X < 0''$ with $X' \geq\_T 0''$.
| 2 | https://mathoverflow.net/users/23648 | 445261 | 179,480 |
https://mathoverflow.net/questions/445259 | 3 | For a smooth complete variety $X$ over a field $F$, we say that $X$ is incompressible if every rational map from $X$ to $X$ is dominant. If $X$ is a smooth complete variety of dimension $d$ over $F$, let’s say that $X$ satisfies property (P) if for every cycle $\alpha\in \text{CH}^{d}(X\times X)$, then $(p\_1)\_{\*}(\a... | https://mathoverflow.net/users/489806 | Cycles on an incompressible variety | The answer is no, even for $d=1$. Note that for curves over a field $F$ (edit: geometrically connected ones, so that global sections of the curve are just $X$), incompressibility is equivalent to there being no $F$-rational points. On the other hand, there exist curves with no $F$-rational points (hence incompressible)... | 2 | https://mathoverflow.net/users/30186 | 445263 | 179,481 |
https://mathoverflow.net/questions/445252 | 1 | Let me restrict to the case of Hilbert spaces, which seem simplest.
Let $\{H\_n\}$ be a sequence of (possibly infinite dimensional) Hilbert spaces and $\{ \mu\_n \}$ be a sequence of Borel probability measures on them. Then, by the Kolmogorov extension theorem, it is straightforward to see that there exists a Borel p... | https://mathoverflow.net/users/56524 | When is the probability measure on the "direct product" via the Kolmogorov extension theorem supported on the "direct sum"? | It is convenient to restate the main question as follows:
>
> For each natural $n$, let $X\_n$ be a random vector in $H\_n$. Suppose that the $X\_n$'s are independent. Under what further conditions do we have
> $$\sum\_{n=1}^\infty\|X\_n\|^2<\infty\tag{1}\label{1}$$
> almost surely (a.s.)?
>
>
>
The answer to ... | 3 | https://mathoverflow.net/users/36721 | 445265 | 179,482 |
https://mathoverflow.net/questions/445249 | 9 | Every subset of $\mathbb N \times \mathbb N$ can be viewed as a relation on $\mathbb N$. The set $\mathcal P(\mathbb N \times \mathbb N)$ of all relations on $\mathbb N$ has a natural topology with which it is homeomorphic to the Cantor space. A pretty well-known fact from descriptive set theory is:
>
> The subset ... | https://mathoverflow.net/users/70618 | Can you fit a $G_\delta$ set between these two sets? | No, not for $\alpha\geq\omega$.
For let $A$ be $G\_\delta$ and suppose that WO$\_\alpha\subseteq G\_\delta$. Let's show that $A\not\subseteq$ WO. Fix a sequence $\left<A\_n\right>\_{n<\omega}$ of open sets such that $A=\bigcap\_{n<\omega}A\_n$. Instead of directly discussing elements of $\mathcal{P}(\mathbb{N}\times\ma... | 8 | https://mathoverflow.net/users/160347 | 445266 | 179,483 |
https://mathoverflow.net/questions/445247 | 5 | An ordinal $\alpha$ is "meta-definable" by some formula $\varphi$ without free variables if:
$$
\begin{cases}
L\_\alpha \models\varphi \\
\forall\beta < \alpha \, L\_\beta \not\models \varphi
\end{cases}
$$
Is there an usual terminology for such "meta-definable" ordinals?
| https://mathoverflow.net/users/138089 | Terminology for ordinals whose constructible level is the least one satisfying some formula | Let me say first that your concept is similar in spirit to the notion of *sententially categorical* cardinal appearing in my joint paper
* J. D. Hamkins and R. Solberg, [Categorical large cardinals and the tension between categoricity and set-theoretic reflection](https://arxiv.org/abs/2009.07164), arxiv:[2009.07164]... | 6 | https://mathoverflow.net/users/1946 | 445268 | 179,484 |
https://mathoverflow.net/questions/445262 | 8 | Question: Do there exist integers $(x,y,z)\neq (0,0,0)$ such that
$$
13x^4+11y^4=8z^4 ?
$$
Some motivation: This is currently the smallest (in a sense defined here [On the smallest open Diophantine equations: beyond Hilbert's 10 problem](https://mathoverflow.net/questions/411958)) three-monomial homogeneous equation ... | https://mathoverflow.net/users/89064 | Existence of rational points on a generalized Fermat quartic | This question is amenable to the use of the Mordell-Weil sieve. (For a good introduction to this technique, see the paper [here](https://arxiv.org/abs/0906.1934).) In this situation, there is a simple version of it (using a single prime) suffices.
Let $C : 13x^{4} + 11y^{4} = 8z^{4}$, and let $E : y^{2} = x^{3} + 314... | 17 | https://mathoverflow.net/users/48142 | 445272 | 179,485 |
https://mathoverflow.net/questions/445236 | 1 | If I'm given a prime number $p$: is there an upper bound to the number of prime factors of $p−1$? Alternatively, is there a way to calculate the number of prime factors of $p−1$ without actually calculating the factors?
| https://mathoverflow.net/users/503282 | Upper bound of number of prime factors | To reiterate Stanley Yao Xiao's comment, one should not expect that $p-1$ has any more or less prime factors than a typical integer of size $p$. For an arbitrary integer $n$, the bound
$$
\omega(n) \ll \frac{\log n}{\log\log n}
$$
is standard (see Hardy and Wright's "An Introduction to the Theory of Numbers" for a proo... | 4 | https://mathoverflow.net/users/307675 | 445274 | 179,486 |
https://mathoverflow.net/questions/445287 | 0 | Now we define an elliptic curve as "a smooth projective curve of genus one with a specified base point". (A little question by the way: Is the requirement "with a specified base point" necessary? If we do not say that, what's going to happen?)
We know that the Jacobian of an elliptic curve is itself, moreover, it's ... | https://mathoverflow.net/users/502709 | Why an elliptic curve can be defined as an abelian variety of dimension 1? | Abelian varieties are smooth projective, and if they are of dimension $1$ then they are curves.
Check that the tangent bundle of an abelian variety is trivial by differentiating the multiplication map. For curves, this implies genus $1$.
Then the identity gives a base point.
| 4 | https://mathoverflow.net/users/18060 | 445290 | 179,490 |
https://mathoverflow.net/questions/445245 | 9 | Let $E$ be a Euclidean vector bundle over the unit ball centered at the origin $B^n(0)$. Let $\nabla$ and $\nabla'$ be two metric connections such that the curvatures coincide globally, i.e. $F\_\nabla\equiv F\_{\nabla'}$. Is it true that there exists a Gauge transformation mapping one connection into the other? (maybe... | https://mathoverflow.net/users/149381 | Does the curvature locally determine the connection? | The answer is 'not always'. Here is a simple case where you cannot recover the connection up to gauge transformation from the curvature: Let $n=2$, let the rank of $E$ be $m$, and, since $E$ is trivial over $B^2(0)$, we might as well take it to be $E = B^2(0)\times\mathbb{R}^m$, i.e., trivialized. Then $F$ is just a $2... | 18 | https://mathoverflow.net/users/13972 | 445291 | 179,491 |
https://mathoverflow.net/questions/445283 | 6 | Let $G$ be a finite group.
>
> Question 1: What are the fastest available programs to test whether $G$ has a normal $p$-complement (see <https://en.wikipedia.org/wiki/Normal_p-complement> for a definition)? Is there a quick way using for example MAGMA?
>
>
>
Im especially interested in groups of order 256\*3=7... | https://mathoverflow.net/users/61949 | Checking for a normal p-complement with a computer | For Question 2, you are asking whether a Sylow $3$-subgroup is normal, and I would expect the fastest and easiest way to do that would be
$\mathtt{G:=SmallGroup(768,k);\ IsNormal(G,SylowSubgroup(G,3))};$
I tried this on $100000$ groups of order $768$ and it took about $2$ minutes CPU time, so the whole calculation ... | 10 | https://mathoverflow.net/users/35840 | 445292 | 179,492 |
https://mathoverflow.net/questions/445294 | 5 | I am going through V. Hinich's ["Lectures on Infinity Categories"](https://arxiv.org/pdf/1709.06271.pdf) and I have a (possibly trivial) question on Segal spaces.
We define Segal space to be a bisimplicial set $X$ which is fibrant in Reedy model structure and such that the "spine map" $X\_n\to X\_1\times\_{X\_0}X\_1\... | https://mathoverflow.net/users/123432 | Reedy fibrancy and composition in Segal spaces | Given a simplicial object $X$ in a locally small category $\mathcal{M}$ and a simplicial set $S$, define the weighted limit $\{ S, X \}$ to be an object in $\mathcal{M}$ equipped with an isomorphism
$$\textrm{Hom}\_\mathcal{M} (-, \{ S, X \}) \cong \textrm{Hom}\_{\textbf{sSet}} (S, \textrm{Hom}\_\mathcal{M} (-, X))$$
o... | 8 | https://mathoverflow.net/users/11640 | 445296 | 179,493 |
https://mathoverflow.net/questions/445282 | 2 | Suppose we have a **k-hypercube** $(Q\_k)$ where $k$ is an odd integer.
Define $F(A)$ for $A \subseteq V$ as the set of all vertices such that has odd number of edges to the set $A$.
Is it true that $F(F(A)) = A$?
| https://mathoverflow.net/users/501463 | Is $F(F(A)) = A$ for every k-hypercube where k is odd? | * **Lemma 1:** $\space F(U \Delta V) = F(U) \Delta F(V)$.
**Prove:** By using $S\_1 = S\_2 \iff S\_1 \subseteq S\_2 \land S\_2 \subseteq S\_1$ we can prove this equality.
1. Suppose $c \in F(U \Delta V)$, we know $c$ has odd number of neighbors in $U \Delta V$ by the definition of $F$. By parity
and the definitio... | 4 | https://mathoverflow.net/users/501463 | 445302 | 179,496 |
https://mathoverflow.net/questions/445288 | 3 | A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q\_p$, $A$ be a finite $G\_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect pairing $H^i(G\_K,A)\times H^{2-i}(G\_K,A')\to \mathbb Q/\mathbb Z$, where $A'=Hom(A,\overline K^\times)$. In particular, t... | https://mathoverflow.net/users/498590 | Local Tate duality for F-vector space | EDIT: I treat the general case.
Write ${\Bbb F}={\Bbb F}\_q$ where $q=p^l$ for some natural $l$.
Then ${{\Bbb F}}\_q\supseteq {{\Bbb F}}\_p={\Bbb Z}/p{\Bbb Z}$.
Using [a comment of @DavidLoeffler](https://mathoverflow.net/questions/445288/local-tate-duality-for-f-vector-space?noredirect=1#comment1152371_445288),
we o... | 2 | https://mathoverflow.net/users/4149 | 445305 | 179,497 |
https://mathoverflow.net/questions/445312 | 6 | I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
>
> **Theorem.** Let $n$ be positive integer number. A metric space $X$ is isometric to a subspace of the Euclidean space $\mathbb R^n$ if and only if every subspace $A\subseteq X$ of card... | https://mathoverflow.net/users/61536 | A characterization of metric spaces, isometric to subspaces of Euclidean spaces | It is in
*Menger, K.*, [**Untersuchungen über allgemeine Metrik.**](https://doi.org/10.1007/BF01448840), Math. Ann. 100, 75-163 (1928). [EuDML:159284](https://eudml.org/doc/159284). [JFM 54.0622.02](https://zbmath.org/54.0622.02).
A modern exposition can be found in
*Bowers, John C.; Bowers, Philip L.*, [**A Meng... | 11 | https://mathoverflow.net/users/41291 | 445313 | 179,498 |
https://mathoverflow.net/questions/445315 | 0 | Let there be a Lie-group $G$ and its Lie-algebra $g$. Then the Cartan Maurer form is an 1-form $\omega: T\_gG \rightarrow T\_eG$ for which holds:
$$ (L^\ast\_g)\omega = \omega$$
In Shlomo Sternberg's book "Curvature in Mathematics and Physics" it is said on p. 155:
$$d\omega (X,Y) = X\omega(Y) - Y\omega(X)- \omeg... | https://mathoverflow.net/users/170888 | Behaviour of the Cartan Maurer form | One should distinguish between a vector $X\in T\_eG$ and the associated left-invariant vector field $\underline{X}$ on $G$. With this notation, the Maurer-Cartan form is the $T\_eG$-valued differential form on $G$ defined by $\omega(\underline X)=X$ for any $X\in T\_eG$. So $\omega(\underline X)$ is not a vector field,... | 2 | https://mathoverflow.net/users/485324 | 445321 | 179,500 |
https://mathoverflow.net/questions/445276 | 3 | Let $\mathcal A$ be an Abelian category. We can assume it has enough injectives. Let $\operatorname{SES}\_{\mathcal A}$ denote the category, of which objects are short exact sequences in $\mathcal A$, and morphisms are triples $(f,f',f'')$ fitting in the diagram
$$\require{AMScd}\begin{CD}0@>>>E'@>>>E@>>>E''@>>>0\\{}@V... | https://mathoverflow.net/users/105537 | Can we define $\operatorname{Ext}$ groups in the category of short exact sequences? | Unfortunately, short exact sequences don't form an abelian category; the snake lemma shows you where the problem is. However, they do form an exact category, in which the exact sequences are the diagrams with nine terms and exact rows and columns. Ext can be defined in any exact category, via the Yoneda definition. The... | 4 | https://mathoverflow.net/users/460592 | 445344 | 179,506 |
https://mathoverflow.net/questions/445360 | 0 | Cross post [Optimal hypothesis testing uses sufficient statistics?](https://math.stackexchange.com/questions/4647472/optimal-hypothesis-testing-uses-sufficient-statistics).
In statistical estimation with any convex risk for a model with a sufficient statistic, in seeking optimal estimators it suffices to consider fun... | https://mathoverflow.net/users/138576 | Optimal hypothesis testing uses sufficient statistics? | $\newcommand\th\theta\newcommand\Th\Theta$It is natural, standard, and convenient to allow randomized tests, with values in $[0,1]$, and then your objection will be removed: $E(\phi(X)|T)$ is a randomized test.
Recall that a Neyman--Pearson test for a simple null hypothesis $H\_0\colon\th=\th\_0$ vs. a simple alterna... | 1 | https://mathoverflow.net/users/36721 | 445364 | 179,514 |
https://mathoverflow.net/questions/445361 | 14 | To what extent does the usual Stone-Weierstrass Theorem depend on some form of the Axiom of Choice? There seems to be a lot of literature on constructive versions in toposes, but I have been unable to find a clear statement about the dependence of the usual theorem on the AC, or a counterexample without AC. And what ab... | https://mathoverflow.net/users/19444 | Stone-Weierstrass Theorem without AC | Caveat - I am not well versed in working with mathematics without the axiom of choice, so this answer should be taken with a grain of salt.
This question seems to be sensitive to the precise definition of compactness, continuity, and separation of points. Here I use the following definitions:
* A space $X$ is compa... | 15 | https://mathoverflow.net/users/766 | 445373 | 179,518 |
https://mathoverflow.net/questions/445374 | 0 | **Question:** Let $a\_n$ be a sequence of real numbers. Is it true that for every $\varepsilon > 0$, if
$$\left \lvert \frac{1}{N} \left ( \sum\_{n=0}^{N-1} a\_n \right )\right \rvert < \frac{1}{N^{1+\varepsilon}}$$
for all $N \in \mathbb Z\_+$, then $a\_n$ converges to $0$?
| https://mathoverflow.net/users/173490 | Does rapid convergence of the Cesaro sums imply convergence of the original sequence? | Multiply by $N$ to get that
$$\left \lvert \sum\_{n=0}^{N-1} a\_n \right \rvert < \frac{1}{N^{\varepsilon}}$$
Therefore $\sum\_{n=0}^\infty a\_n=0$ and $\lim\_{n\to\infty} a\_n=0$.
| 3 | https://mathoverflow.net/users/479223 | 445377 | 179,519 |
https://mathoverflow.net/questions/445300 | 6 | Maybe there is no good answer to this, but I'm trying to get a feel for what the $K$-theory of a (permutative or symmetric monoidal $\infty$-)category computes.
In algebraic $K$-theory, we have explicit descriptions for the low $K$-groups, where $K\_0(R)$ is the Grothendieck group of finite projective modules and $K\... | https://mathoverflow.net/users/152554 | $K_1$ of Categories for intuition | For an exact category ${\cal N}$, Dan Grayson (who occasionally shows up on MathOverflow) gave explicit generators and relations for $K\_n$ [here](https://www.ams.org/journals/jams/2012-25-04/S0894-0347-2012-00743-7/S0894-0347-2012-00743-7.pdf).
What follows is essentially quoted directly from that paper except that ... | 3 | https://mathoverflow.net/users/10503 | 445389 | 179,521 |
https://mathoverflow.net/questions/445367 | 4 | Given a complex projective variety $X$, let's define singular cohomology of $K(X)$ (its function field) as the direct limit of cohomology groups over all of its Zariski open subsets. Similarly let's define the singular cohomology of algebraic closure $\overline{K(X)}$ by taking the direct limit of singular cohomology o... | https://mathoverflow.net/users/127776 | Which cohomology classes come from smooth projective varieties? | Donu Arapura's comment contains the right idea: this fails for Hodge-theoretic reasons. (I will also give a more elementary argument below.)
Indeed, each $H^i(U,\mathbf Q)$ carries a mixed Hodge structure [PS08, Thm. 4.2] that is well-defined and functorial in $U$ [PS08, Prop. 4.18]. Thus, $H^i\big(\overline{K(X)},\m... | 3 | https://mathoverflow.net/users/82179 | 445391 | 179,522 |
https://mathoverflow.net/questions/445380 | 5 | Let $G = (V, E)$ be a uniform random graph on $2n$ labeled vertices and let $S \subseteq {V}$ be the set of vertices with degree $\ge n$. Then what happens to $\mathbf{P}(|S|=n)$ as $n \to \infty$?
From the Harris-Kleitman inequality one can obtain $\mathbf{P}(|S|=n) \le c\_n$, where $c\_n \to \frac{1}{2}$. I found a... | https://mathoverflow.net/users/503381 | Probability of the random graph on $2n$ vertices having exactly $n$ vertices with degree $\ge n$ | It certainly tends to $0$. The way to see it almost without any computation is to mark any $n$ disjoint edges. Then, when deciding the fate of the remaining edges, each vertex has $p=2^{2-2n}{2n-2\choose n-1}\asymp n^{-1/2}$ chance to get the degree of exactly $n-1$ and these events for two distinct vertices are almost... | 8 | https://mathoverflow.net/users/1131 | 445392 | 179,523 |
https://mathoverflow.net/questions/445390 | 3 | As explained in the comments of [this answer](https://mathoverflow.net/q/371880/458355), given a smooth Fano 3-fold of index 1 and genus $g \geq 6$, we have two semiorthogonal decompositions $$\langle \text{Ku}(X), \mathcal{E}, \mathcal{O}\_X\rangle$$
and
$$
\langle \text{Ku}’(X), \mathcal{O}\_X, \mathcal{E}^\vee \rang... | https://mathoverflow.net/users/458355 | Should we expect Kuznetsov component to be independent of exceptional collection | In general the orthogonal complement to a maximal exceptional collection does depend on the collection. A simple example of this sort is given here <https://arxiv.org/abs/1304.0903>.
| 5 | https://mathoverflow.net/users/4428 | 445405 | 179,524 |
https://mathoverflow.net/questions/445413 | 2 | I'm interested in the Grothedieck group of the triangulated category $K^b(\text{proj}A)$ when $A$ is a finite dimensional algebra over a field of infinite global dimension.
For this purpose, It would be helpful to know if $K^b(\text{proj}A)$ is Krull-Schmidt. I couldn't find any proof myself nor in the literature.
It... | https://mathoverflow.net/users/476588 | Literature request: $K^b(\text{proj} A)$ Krull-Schmidt for $\text{gl dim}A = \infty$ and general results about its Grothendieck group | I'm assuming your notation denotes the homotopy category of perfect complexes over $A$. In order for this to be Krull-Schmidt, at least the category of finitely generated projective modules has to be Krull-Schmidt, which is often not the case. The bounded derived category is also usually not Krull-Schmidt.
On the oth... | 5 | https://mathoverflow.net/users/460592 | 445416 | 179,527 |
https://mathoverflow.net/questions/445398 | 7 | [On page 16](https://arxiv.org/abs/1412.7559) of this lecture notes and [in this lecture](https://www.youtube.com/watch?v=fTlRYcyISfA), at some point (somewhere at 1:37:45), Rod Gover defined the $\text{conformal metric}$ $\mathbb{g}$ on a conformal manifold $(M, [g])$. He mentioned that this metric is canonical. Does ... | https://mathoverflow.net/users/298774 | Definition of the conformal metric | Let $(M,[g])$ be a conformal manifold;
i.e. $(M,g)$ is a Riemannian manifold and $[g] = \{ u^2g \mathrel{}:\mathrel{} u \in C^\infty(M), u>0 \}$ is the set of Riemannian metrics conformal to $g$.
It is clear that $[g] = [u^2g]$ for all positive $u \in C^\infty(M)$.
Let me now change notation and write $c:=[g]$, to emph... | 10 | https://mathoverflow.net/users/121820 | 445422 | 179,530 |
https://mathoverflow.net/questions/445395 | 10 | Write $f(n)$ for the quotient of $n$ by its largest squarefree divisor. In other words, $f$ is a multiplicative function with $f(p^k) = p^{k-1}$ for all $k \geq 1$.
What, if anything, is known about the asymptotics of
$$(1/X) \sum\_{n=1}^X f(n)$$
or about the behavior of the Dirichlet series
$$\sum\_{n=1}^\infty f(n)... | https://mathoverflow.net/users/431 | Asymptotic behavior of a "strange" arithmetic function | Let $k(n)=\prod\_{p \mid n}p$ be the kernel of the integer, so that $f(n)=n/k(n)$. As indicated in p. 7 of Finch's article,
$$\sum\_{n\le x} \frac{1}{k(n)} = \exp\left( \left( \frac{8\log x}{\log \log x}\right)^{1/2}(1+o(1))\right)$$
was shown by de Bruijn in "On the number of integers $\le x$ whose prime factors divid... | 12 | https://mathoverflow.net/users/31469 | 445424 | 179,531 |
https://mathoverflow.net/questions/445418 | 0 | Let $N\subset M$ be a be factors acting on a Hilbert space $H$.
Denote by $\mathrm{Cyc}(A)\subset H$ the set of cyclic vectors for $A = M,N$.
I am interested in the equality case of the inclusion $\mathrm{Cyc}(N)\subseteq \mathrm{Cyc}(M)$?
If, e.g., $M= M\_2(\mathbb C) \otimes N$, then it is easy to find vectors ve... | https://mathoverflow.net/users/485160 | Cyclic vectors and subfactor inlcusion | No. Let $N$ be any $\rm II\_1$ factor, and let $\alpha: G\to \operatorname{Aut}(N)$ be an outer action. Then $M:= N\rtimes\_\alpha G$ is again a $\rm II\_1$ factor, and $N\subset M$ is irreducible, i.e., $N'\cap M = \mathbb{C}$. Now let $\Omega$ be the image of $1$ in $L^2M$, the GNS space of the trace. Then $\Omega$ i... | 1 | https://mathoverflow.net/users/351 | 445427 | 179,532 |
https://mathoverflow.net/questions/445327 | 1 | I asked this question in stack exchange but have not received an answer, so I am posting it here.
Given a group action on a manifold (e.g. configuration space of coordinates), cotangent-lift it to the phase space (i.e. coordinates and momentum), which is the appropriate cotangent bundle of the base manifold.
For a ... | https://mathoverflow.net/users/90621 | isotropy of the cotangent lift of a group action | Since the projection $\pi: T^\* Q \to Q$ is equivariant, the stabilizer $G\_p$ of a point $p \in T^\*\_q Q$ is indeed a subgroup of the stabilizer of the base point $G\_q$. In fact, $G\_p$ is also the stabilizer of $p$ under the action of $G\_q$ on the fiber $T^\*\_q Q$. But without knowing what momentum you are lookin... | 1 | https://mathoverflow.net/users/17047 | 445441 | 179,534 |
https://mathoverflow.net/questions/445440 | 1 | Let $X$ be a complex variety with a good stratification $S$ and consider the category $Perv\_S(X)$ of sheaves perverse with respect to the given stratification (with middle perversity) lying in $D^b\_S(X,k)$, where $k$ is a field. By good I mean that the derived pushforward from any stratum of a complex with constructi... | https://mathoverflow.net/users/131868 | $\text{Ext}$-groups of perverse sheaves with a fixed stratification | No, this is not a purely $\ell$-adic phenomenon.
Let $X = \mathbb P^1$, $S$ the stratification with one stratum so sheaves constructible with respect to this stratification are lisse and complexes constructible with respect to this stratification complexes with lisse cohomology.
Let $F = G = \mathbb Q[1]$, clearly ... | 4 | https://mathoverflow.net/users/18060 | 445444 | 179,535 |
https://mathoverflow.net/questions/445402 | 9 | Let $V$ be a finite dimensional vector space over $\mathbb{R}$. Let $S$ be the vector space of multilinear maps from $V\times V$ to $V$. Let $L:\mathbb{I}\rightarrow S$ be a continuous map such that for every $t$ we have that $L(t)$ is a Lie bracket on $V$. For every $t$, let $G\_t$ be the simply connected Lie group wh... | https://mathoverflow.net/users/32135 | Must a continuous variation through compact simply connected Lie groups preserve topology | Yes, in fact in this situation you have more, all the Lie algebras $(V, L(t))$ (which I denote by $\mathfrak{g}\_t$ to abbreviate) are isomorphic:
Let $G$ be the simply connected Lie group integrating a finite dimensional real Lie algebra $\mathfrak{g}$. Let $W$ be a finite dimensional representation of $G$, and writ... | 9 | https://mathoverflow.net/users/104042 | 445449 | 179,537 |
https://mathoverflow.net/questions/445445 | 2 | When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a\_{-1}$ in the Laurent expansion $f(x) = \sum\_{n=-1}^\infty a\_n x^n$ by evaluating the limit $\lim\_{x \to 0} f(x)x$.
I'm interested in an odd sort of edge case, where we force $a\_{-1} = \infty$. For instance, cons... | https://mathoverflow.net/users/146528 | Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$ | There is an asymptotic for sums of the form $$g(t) = \sum\_{n=1}^{\infty} f(nt)$$ given by Zagier for $C^{\infty}$ functions $f$ of sufficiently rapid decay (Proposition 3 of his appendix on the Mellin transform, <https://people.mpim-bonn.mpg.de/zagier/files/tex/MellinTransform/fulltext.pdf>)
If $$f(t) \sim \sum\_{n=... | 3 | https://mathoverflow.net/users/503444 | 445451 | 179,539 |
https://mathoverflow.net/questions/444939 | 0 | On the space $X=C[0,1]$, define a norm $||| f |||^2=\Vert f \Vert\_{\infty}^2 + \Vert f \Vert\_2^2$, where $\Vert \cdot \Vert\_\infty$ is the sup norm on $C[0,1]$ space and $\Vert \cdot \Vert\_2$ is the $L\_2$ norm. This norm is equivalent to that of $\Vert \cdot \Vert\_\infty$ on $C[0,1]$ space and $(X, ||| \cdot |||)... | https://mathoverflow.net/users/494605 | Finding weak LUR property of $C[0,1]$ with an equivalent norm | Your norm, call it $N$ for \TeX-nical simplicity, is not WLUR. You want to know whether $N(f)=1$, $N(f\_n)\to1$, $N(f+f\_n)\to 2$ imply that $f\_n\to f$ weakly. For a counterexample let $f$ be the constant function $f=1/\sqrt2$ and $f\_n(x)=1/\sqrt2$ for $x\ge 1/n$, $f\_n(0)=0$ and $f\_n$ linear in between. (Another ch... | 1 | https://mathoverflow.net/users/127871 | 445452 | 179,540 |
https://mathoverflow.net/questions/445448 | 7 | The question below relates to the paper ["Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier](https://uva.theopenscholar.com/files/ken-ono/files/1902572116.full__8.pdf). I'm asking it here because I am sure the answer is well-known to a specialist, and was unsuccessf... | https://mathoverflow.net/users/6269 | Positivity of the coefficients of Taylor series associated to the Riemann hypothesis | I am no expert but it seems to be known that $\gamma\_n>0$.
The function that defines $\gamma(n)$ is
$(-1+4z^2) \Lambda(1/2+z)=8\xi(1/2+z)$,
where $\xi(s)=(1/2)s(s-1)\Lambda(s)$ is the Riemann $\xi$ function so that
we have
$\gamma(n)=8\frac{n! \xi^{2n}(1/2) }{(2n)!}$
and the fact that $\xi^{2n}(1/2)>0$ fol... | 5 | https://mathoverflow.net/users/112259 | 445470 | 179,543 |
https://mathoverflow.net/questions/445458 | 5 | Let $A, B\in M\_n(\mathbb{C})$ be matrices that commute. We suppose that there exists a vector $v\in\mathbb{C}^{n}$ such that $(\mathbb{C}[A,B]).v$ generates $\mathbb{C}^{n}$. We call such a pair a cyclic pair.
Is there only a finite number of flags that are stabilized both by $A$ and $B$?
Note, that if we already ... | https://mathoverflow.net/users/27398 | Commuting matrices and cyclic modules | Let $A=\left(\begin{smallmatrix}1&0&0\\1&1&0\\0&0&1\end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix}1&0&0\\0&1&0\\1&0&1\end{smallmatrix}\right)$. Then your hypotheses are satisfied, with $v=\left(\begin{smallmatrix}1\\0\\0\end{smallmatrix}\right)$, and there are infinitely many flags stabilised by $A$ and $B$ ... | 5 | https://mathoverflow.net/users/460592 | 445473 | 179,544 |
https://mathoverflow.net/questions/445474 | 2 | Let $\alpha=(\alpha\_1,\ldots,\alpha\_m)\subset\mathbb R^m\_+$ and $\beta=(\beta\_1,\ldots,\beta\_n)\subset\mathbb R^n\_+$ be given and satisfy
$$\sum\_{i=1}^m \alpha\_i =1 = \sum\_{j=1}^n\beta\_j.$$
Define $\Pi(\alpha,\beta)\subset \mathbb R^{mn}\_+$ be the subset consisting $c=(c\_{ij})\_{1\le i\le m, 1\le j\le n... | https://mathoverflow.net/users/493556 | A variant of (discrete) optimal transport problem | The minimum here can be found exactly, in a finite number of steps.
Indeed, the target function is concave. So, its minimum on the (compact) (transportation) polytope $P:=\Pi(\alpha,\beta)$ is attained at one of the extreme points of $P$. For a construction and a characterization of all extreme points of $P$, see [th... | 2 | https://mathoverflow.net/users/36721 | 445488 | 179,548 |
https://mathoverflow.net/questions/445462 | 2 | Consider the fractional Sobolev space $H^{1/2}\_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a\_0+\sum\_{k=1}^{\infty}(a\_k \cos kt+ b\_k\sin kt)$$ satisfy $$\sum\_{k=1}^{\infty}k(a\_k^2+b\_k^2)<\infty.$$
If we also consider $v\in H^{... | https://mathoverflow.net/users/124648 | The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$ | An excellent reference for this, with exercises, full details etc, is
*Demengel, Françoise; Demengel, Gilbert*, [**Functional spaces for the theory of elliptic partial differential equations. Transl. from the French by Reinie Erné**](https://doi.org/10.1007/978-1-4471-2807-6), Universitext. Berlin: Springer (ISBN 978... | 4 | https://mathoverflow.net/users/40120 | 445498 | 179,550 |
https://mathoverflow.net/questions/445493 | 0 | Let us take the Hilbert space $l\_2$ with an equivalent norm
$\Vert x \Vert = \max \{2 \Vert x \Vert\_1, \Vert x \Vert\_2 \}$, where $\Vert x \Vert\_1 =( \sum\_{n=2}^\infty x\_n^2 )^{\frac{1}{2}}$ and $\Vert x \Vert\_2 = ( \sum\_{n=1}^\infty x\_n^2 )^{\frac{1}{2}}$, for $(x\_n)\_{n \geq 1} \in l\_2$.
The unit ball ... | https://mathoverflow.net/users/494605 | Smoothness of a Hilbert space under an equivalent norm | Let $z:=\frac12\,(\sqrt3,1,0,0,\dots)$. For $x=(x\_1,x\_2,\dots)$, let
$$f(x):=2x\_2,\quad g(x):=\tfrac12\,(\sqrt3\,x\_1+x\_2).$$
Then $f\ne g$, $\|z\|=1$, $f(z)=1=g(z)$, and $\|f\|=1=\|g\|$.
So, your space is not smooth.
---
*Details:* For all $x=(x\_1,x\_2,\dots)\in B\_{\|\cdot\|}$ we have
$$f(x)=2x\_2\le1,\q... | 1 | https://mathoverflow.net/users/36721 | 445500 | 179,551 |
https://mathoverflow.net/questions/445479 | 1 | Using result from calculus:(1) $ \log(x\log x)>\log\vartheta \cdot \log(2x\log 2x)$ for $2\leq \vartheta<e$ ($e$-Euler number) and $ x>x(\vartheta )>2$, I am able to prove $ p\_{2n}<\frac{2n}{\log(\vartheta )}\cdot \log(p\_{n})$ for $n>n(\vartheta)$, where $p\_{n}$ is the $n$-th prime number. The proof does not work fo... | https://mathoverflow.net/users/169583 | Estimate for the $2n$-th consecutive prime number | Let's use some standard bounds on the $n$th prime, as found in [this paper](https://www.ams.org/journals/mcom/1999-68-225/S0025-5718-99-01037-6/S0025-5718-99-01037-6.pdf) by Pierre Dusart.
We have
$$
p\_{2n} \leq 2n[\ln(2n)+\ln(\ln(2n))-0.9484] = 2n\left[\ln(n)+\ln\left(\frac{2}{e^{0.9484}}\ln(2n)\right)\right]
$$
us... | 3 | https://mathoverflow.net/users/3199 | 445502 | 179,552 |
https://mathoverflow.net/questions/445497 | 3 | Let $O\_M(\mathbb{R}^n):= \mathcal{S}'(\mathbb{R}^n) \cap C^\infty(\mathbb{R}^n)$ be the space of slowly increasing smooth functions on $\mathbb{R}^n$.
Following p.294 proposition 9.10 of the "Real Analysis" (1999) by Gerald Folland, define the convlution map $\mathcal{S}'(\mathbb{R}^n) \times \mathcal{S}(\mathbb{R}^... | https://mathoverflow.net/users/56524 | What exactly is the topology on $O_M$ that makes the convolution map $S \times S' \to O_M$ hypocontinuous? | The definition is wrong. For example in 1d, the function $\sin(e^x)$ is in the intersection of $S'$ and $C^{\infty}$ but is not in $O\_M$ which is a strict subset (all 3 spaces being seen as subsets of $D'$ otherwise the intersection does not even make sense).
The definition (for the real-valued case) is the space of... | 2 | https://mathoverflow.net/users/7410 | 445505 | 179,553 |
https://mathoverflow.net/questions/445508 | 3 | Let $G$ be the Cayley graph on the alternating group $A\_n\,n\ge4$ with generating set $$S=\begin{cases}\{(1,2,3),(1,3,2),\\(1,2,\ldots,n),(1,n,n-1,\ldots,2)\}, &n\ \text{odd}\\ \{(1,2,3),(1,3,2),\\(2,3,\ldots,n),(2,n,n-1,\ldots,3)\}, &\text{otherwise}. \end{cases}$$ Then, will $G$ be class I, that is, have chromatic i... | https://mathoverflow.net/users/100231 | Edge coloring of a graph on alternating groups | This graph is 4-colourable.
Here is some SageMath code that constructs the graph, forms its line graph and then presents a 4-colouring of the graph that you can check.
I found the colouring using the gCol package.
```
A6 = AlternatingGroup(6)
G = A6.cayley_graph(generators=[A6((1,2,3)), A6((2,3,4,5,6))])
G = G.t... | 4 | https://mathoverflow.net/users/1492 | 445523 | 179,556 |
https://mathoverflow.net/questions/445519 | 6 | A pythagorean triple is a triple of integers $(a,b,c)$ with $a^2 + b^2 = c^2$. Given a triple, $(a/c, b/c)$ is a point on the unit circle, so we may associate to it the normalized angle
$$\theta\_{a,b} := \frac{\tan^{-1}(b/a)}{2\pi}\in (-1/4, 1/4)$$
Let $A$ denote the $\mathbb{Q}$-vector space spanned by $\theta\_{a,b}... | https://mathoverflow.net/users/88840 | $\mathbb{Q}$-rank of the space of angles of pythagorean triples | Yes, the dimension is infinite.
${\bf Z}$-linear relations on $\frac1{2\pi}\tan^{-1}(b/a)$
correspond to monomials in $(a+ib)/c$ that equal $1$.
But for each prime $p \equiv 1 \bmod 4$ we may choose integers $m,n$
such that $p = m^2 + n^2$ and let $(a+ib)/c = (m+in)/(m-in)$;
these are multiplicatively independent by ... | 12 | https://mathoverflow.net/users/14830 | 445528 | 179,558 |
https://mathoverflow.net/questions/445540 | 1 | The following startles me. Let $\Omega \subseteq \mathbb R^n$ and write $W^{\sigma,1}(\Omega)$ for the fractional Sobolev space with norm
$$|u|\_{W^{\sigma,1}(\Omega)} := \iint \frac{|u(x) - u(y)|}{|x-y|^{n+\sigma}} dx dy$$
Now, suppose that $u \in W^{\sigma,1}(\Omega)$ and that $v \in C^{0,\sigma}(\Omega)$ is boun... | https://mathoverflow.net/users/2082 | Is the product of $u \in W^{\sigma,1}(\Omega)$ and $v \in C^{0,\sigma}(\Omega)$ again in $W^{\sigma,1}(\Omega)$? | **It is not true that $W^{\sigma,1} \times C^{0,\sigma} \hookrightarrow W^{\sigma,1}$ with $\Omega = \mathbb{R}^n$.**
My reference for such questions is
>
> Thomas Runst, and Winfried Sickel. *Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations.* de Gruyter, 1996.... | 4 | https://mathoverflow.net/users/50777 | 445544 | 179,563 |
https://mathoverflow.net/questions/445545 | 1 | Let $u$ be a harmonic function defined on $B\_1(0)\subset\mathbb{R}^2$, $u(0)=0$, and $\{x\in B\_1(0):u(x)>0\}$ is simply connected. Is there a universal constant $c>0$ satisfying that
$$
c\leq \frac{m(B\_1(0)\cap\{u(x)>0\} )}{m(B\_1(0)\cap\{u(x)<0\} )}\leq c^{-1}
$$
| https://mathoverflow.net/users/478411 | Ratio of measure of level region for harmonic functions | The answer is negative. To construct a counterexample, use an entire function $f$ which satisfies $|f(z)|<1$ in ${\mathbf{C}}\backslash D$, where $D$ is a half-strip $\{ x+iy:x>0,|y|<\pi \}$. Then $v={\mathrm{Re}}f-1$ is harmonic, and negative in ${\mathbf{C}}\backslash D$. Let $D\_1=\{ z: u(z)>0\}\subset D$, and
$z\_1... | 2 | https://mathoverflow.net/users/25510 | 445546 | 179,564 |
https://mathoverflow.net/questions/445256 | 3 | I am reading "*Dirac Operator in Riemannian Geometry*" by T. Friedrich, where he writes that (the total space of) the frame bundle $R$ of the tangent space of $\mathbb{C}P^n$ is:
$$ R = SU(n+1) \times\_{\sigma} SO(2n) $$
where $SU(n+1)$ act transitively on $\mathbb{C}P^n \simeq SU(n+1)/S(U(n) \times U(1))$, with $S(U(n... | https://mathoverflow.net/users/74372 | Frame bundle of $\mathbb{C}P^n$ as homogeneous space | There's a more general description of frame bundles on homogeneous spaces here: if you take $G/H$ and give it a $G$-invariant Riemannian metric, then $H$ preserves the identity coset, and so acts on the tangent space there $T\_{eH}G/H=\mathfrak{g}/\mathfrak{h}$, preserving the metric $g$; actually this is induced by th... | 2 | https://mathoverflow.net/users/66 | 445554 | 179,565 |
https://mathoverflow.net/questions/325926 | 1 | I've been trying to be more diligent about reading motivic homotopy theory, and have been reading Levine's 'An Overview of Motivic Homotopy Theory.' I think the subject is fascinating, and I've developed a question during my reading.
Levine defines $\mathcal{X}:\mathbf{Sm}/S^{\text{op}}\rightarrow\mathbf{Spc}$, where... | https://mathoverflow.net/users/123309 | Motivic knot embedding | Apparently, [there is now](https://arxiv.org/abs/2210.11048)! Clémentine Lemarié-Rieusset, a PhD student of Frédéric Déglise and Adrien Dubouloz at Université de Bourgogne is developing the theory.
| 2 | https://mathoverflow.net/users/133676 | 445564 | 179,568 |
https://mathoverflow.net/questions/445550 | 1 | Let $k$ be an algebraically closed field and $X$ be a normal variety over $k$.
I am trying to show that there is a surjective group homomorphism $G\_0(X)\rightarrow \mathbb{Z}\oplus \mathrm{Cl}(X)$, by using Quillen’s spectral sequence for the $G$-theory of a Noetherian scheme of finite Krull dimension.
But I am not su... | https://mathoverflow.net/users/477848 | Relation between $G_0(X)$ and $\mathrm{Cl}(X)$ for a normal variety $X$ | You can construct the homomorphism directly by
$$
F \mapsto (\mathrm{rank}(F), c\_1(F)).
$$
Here $c\_1(F)$ is defined by
$$
c\_1(F) =
c\_1(F\vert\_{X\_0}) \in
\mathrm{Pic}(X\_0) =
\mathrm{Cl}(X),
$$
where $X\_0 \subset X$ is the smooth locus of $X$.
| 2 | https://mathoverflow.net/users/4428 | 445572 | 179,570 |
https://mathoverflow.net/questions/404581 | 4 | Suppose that $M$ is a von Neumann algebra with no minimal projections. Let $p$ be a nonzero projection in $M$ and $\rho$ be a normal state on $M$.
For any $\epsilon>0$, can we find a projection $e$ in $M$ such that $0\leq e\leq p$ and $\rho(e)=\epsilon \rho(p)$?
| https://mathoverflow.net/users/153196 | Projections in a diffuse von Neumann algebra | I think the proof goes something like this. We need to assume that $\epsilon\le 1$ as pointed out.
Let $M$ be a von Neumann algebra with no minimal projections. Let $\rho$ be a normal state on $M$.
Claim-1: Given $\epsilon>0$, there exists a non-zero projection $p\in \text{Proj}(M)$ such that $\rho(p) <\epsilon$.
... | 1 | https://mathoverflow.net/users/40212 | 445577 | 179,573 |
https://mathoverflow.net/questions/444102 | 5 | I'm trying to get a digital copy of the article "C. Pixley and P. Roy, Uncompletable Moore spaces,
Proc. Auburn Univ. Conf. (Auburn, Alabama, 1969), 75-85." but I have not been successful. Could you please share a copy of that article with me?
| https://mathoverflow.net/users/146942 | Pixley and Roy article request | I was visiting Auburn today and obtained a scan.
<https://github.com/StevenClontz/research/blob/master/miscellaneous/SKM_C650i23042612550.pdf>
| 10 | https://mathoverflow.net/users/73785 | 445581 | 179,574 |
https://mathoverflow.net/questions/445582 | 3 | Let $X\_1,X\_2\subset\mathbb{P}^{n+1}$ be two smooth complex cubic hypesurfaces, then I know the following Torelli theorems:
[$n=1$] In this case, $X\_1\cong X\_2$ if and only if there exists a Hodge isometry $H^1(X\_1,\mathbb{Z})\cong H^1(X\_2,\mathbb{Z})$.
[$n=3$] In this case, $X\_1\cong X\_2$ if and only if the... | https://mathoverflow.net/users/nan | Torelli theorem for smooth complex cubic surfaces? | The Hodge structures of cubic surfaces do not allow to distinguish them. But there are some ways around. For instance, given a cubic surface $X \subset \mathbb{P}^3$, you can associate with it a cyclic covering $Y(X) \to \mathbb{P}^3$ of degree 3 ramified over $X$ (thus, $Y(X)$ is a cubic threefold) and use the Hodge s... | 6 | https://mathoverflow.net/users/4428 | 445583 | 179,575 |
https://mathoverflow.net/questions/445561 | 3 | Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times 3n}$ matrix is Schur stable (all eigenvalues in open unit disk):
$$
A=\begin{bmatrix}
I & 0&-\alpha I\\
I&0&0\\
(M+H... | https://mathoverflow.net/users/252894 | Condition for 3×3 block matrix to be stable | $\newcommand{\al}{\alpha}\newcommand\la\lambda\newcommand\R{\mathbb R}$Such a construction of $M$ and $\al$ is always possible.
Indeed, take any complex $\la$. Rearranging columns and rows of the matrix $A-\la I\_{3n}$, we see that $\la$ is an eigenvalue of $A$ iff
\begin{equation\*}
D(\la):=\begin{vmatrix}
-\la I&... | 4 | https://mathoverflow.net/users/36721 | 445585 | 179,576 |
https://mathoverflow.net/questions/445362 | 12 | **Motivation.** This weekend, my children took part in a soccer tournament consisting of $n$ teams, each of which playing once against every other team. As there was only one soccer field, the schedule consisted of $n\choose 2$ games played one after the other. Some teams protested that they were scheduled to play in t... | https://mathoverflow.net/users/8628 | Optimal schedule for a soccer tournament | Here is a streamlined description of the optimal strategies that have been found. In both strategies, there is a team that plays a special role -- call them $x$ -- and the other teams are numbered modulo $n-1$.
---
With $n=2k$ teams: We play $2k-1$ rounds of $k$ games each. In the $i$-th round, the games are
$$(x... | 5 | https://mathoverflow.net/users/297 | 445590 | 179,579 |
https://mathoverflow.net/questions/445613 | 4 | An $R$-module I is called faithfully injective if it is injective and the functor $Hom\_R(-, I)$ has the image of a complex being exact if and only if the original complex is exact.
I wonder if it is known when, for a finite dimensional algebra, there exists a faithfully injective module which is also a projective mo... | https://mathoverflow.net/users/503624 | Faithfully injective projective modules | There are certainly finite dimensional algebras for which there are no projective modules that are also injective. An example is the $D\_4$ quiver algebra with two incoming and two outgoing arrows to the central vertex. $\begin{smallmatrix}\bullet\\\downarrow\\\bullet\to\bullet\to\bullet\\\downarrow\\\bullet\end{smallm... | 12 | https://mathoverflow.net/users/460592 | 445618 | 179,585 |
https://mathoverflow.net/questions/445541 | 6 | We consider the following combinatorial game (with two players
alternatively playing optimally). Posititions are given by heaps containing $b\geq 0$ black and $w\geq 0$ white stones and are
encoded by $(b,w)$ in $\mathbb N^2$ where $\mathbb N=\lbrace 0,1,\ldots\rbrace$.
Players are allowed to remove either a unique b... | https://mathoverflow.net/users/4556 | A combinatorial game with seemingly curious arithmetic properties | Consider the case where $(0,0)$ and $(0,1)$ are winning (i.e. P) positions. There's a biphasic structure.
Let $\operatorname{off}\_w(b) = 3 \lfloor \frac b2 \rfloor + (b \bmod 1)$. Then if $w < \operatorname{off}\_w(b)$ you look at the periodic table $$\begin{matrix} P & P & N & N & N & N \\ N & N & N & P & P & N\end... | 3 | https://mathoverflow.net/users/46140 | 445624 | 179,588 |
https://mathoverflow.net/questions/443156 | 1 | The following is a result from Shub's monograph "Global Stability of Dynamical Systems".
I dabble in the proof, and it appears to me that the existence of $W^{\rm cu}\_{\rm loc}$ does not rely on the assumption that $Df(0)$ is invertible, while that of the existence of $W^{\rm cs}\_{\rm loc}$ does.
Question: Can $W^{... | https://mathoverflow.net/users/102458 | Existence of center-stable manifold when the Jacobian is singular? | For both manifolds we do not need the strong invertibility. Moreover, for $W^{cs}$ this is stated in Exercise III.2, p.68 from the mentioned monograph. However, I will give below a more geometrical view on the problem.
In fact, to construct $W^{ss}$ (strongly stable), $W^{su}$ (strongly unstable), $W^{cs}$ (center-st... | 1 | https://mathoverflow.net/users/85336 | 445632 | 179,591 |
https://mathoverflow.net/questions/445629 | 0 | Let $(a\_n)\_{n \geq 1}$ be random variables taking values on a finite subset $B$. Assume that $\nu\_l(b) \le P[a\_n = b\mid a\_1,\ldots,a\_{n-1}] \le \nu\_u(b)$ almost surely for every $n \ge 1$ and $b \in B$. Then, it can be shown that
$$
\frac{1}{n}\sum\_{k=1}^n 1\_{[a\_k = b]}-\frac{1}{n}\sum\_{k=1}^n P[a\_k = b\mi... | https://mathoverflow.net/users/42412 | Bound the expectation of an average | Of course, this is not true. E.g., suppose that the $a\_n$'s are independent random variables each uniformly distributed on the finite set $B$, of cardinality $|B|\ge1$. (With $\nu\_l(b)$ and $\nu\_u(b)$ completely unspecified, the condition $\nu\_l(b) \le P[(a\_n = b|a\_1,\ldots,a\_{n-1}) \le \nu\_u(b)$ is not a restr... | 3 | https://mathoverflow.net/users/36721 | 445641 | 179,592 |
https://mathoverflow.net/questions/445601 | 6 | All the proofs of the high-dimensional Pythagorean theorem that I know are based on induction or the additivity of the dot product. Is there any geometric construction that's similar to the well-known planar proofs, ones that are based on clever divisions of squares or similarities between certain triangles?
| https://mathoverflow.net/users/2158 | Geometric proof of the three-dimensional Pythagorean theorem | In a rectangular parallelepiped with edges $|AB|=a$, $|BC|=b$, $|CD|=c$ and space diagonal $|AD|=d$ let us draw altitudes $BX$ and $CY$ from the vertices to $AD$. Then using similarity arguments it is easy to see that $|AX|=a^2/d$ and $|YD|=c^2/d$. Using, say, a parallel translation and a similarity argument one also s... | 2 | https://mathoverflow.net/users/19864 | 445657 | 179,596 |
https://mathoverflow.net/questions/445654 | 7 | Here $\operatorname{cof}(\mathcal{L})$ refers to the largest cardinal characteristics of the continuum in Cichon's diagram. My question is:
* Is the theory $\mathsf{ZFC}+ \operatorname{cof}(\mathcal{L})=\aleph\_1 + 2^{\aleph\_0}=\aleph\_3$ known to be consistent?
| https://mathoverflow.net/users/141146 | $\operatorname{cof}(\mathcal{L}) = \aleph_1$ and $2^{\aleph_0} = \aleph_3$ | Yes, this is consistent. Probably the simplest way to get a model of this theory is to begin with a model of GCH, and then force with a countable support product of $\aleph\_3$ copies of the Sacks poset. This is sometimes called the "side-by-side" Sacks model, and you can get any value of $\mathfrak{c}$ you like in a m... | 11 | https://mathoverflow.net/users/70618 | 445661 | 179,599 |
https://mathoverflow.net/questions/445663 | 2 | Computations with Maple suggest the following binomial identity
\begin{equation\*}
\forall{p,j}: \sum\_{k=j+1}^{p+1} (-1)^j \dfrac{1}{k}\binom{k-1}{j} =
\sum\_{k=j+1}^{p+1} (-1)^{k-1} \dfrac{1}{k}\binom{p+1}{k}
\end{equation\*}
which is insofar remarkable as on the LHS the summation runs over the upper
index of t... | https://mathoverflow.net/users/161310 | Proof of a binomial identity | $\newcommand{\bi}{\binom} $Denote the left- and right-hand sides of the identity by $l\_{j,p}$ and $r\_{j,p}$, respectively.
Note that
\begin{equation\*}
l\_{j,p+1}-l\_{j,p}=\frac{(-1)^j}{p+2}\,\bi{p+1}j
\end{equation\*}
and
\begin{equation\*}
r\_{j,p+1}-r\_{j,p}=\frac{(-1)^{p+1}}{p+2}+s\_{j,p},
\end{equation\*}
wher... | 3 | https://mathoverflow.net/users/36721 | 445667 | 179,601 |
https://mathoverflow.net/questions/443343 | 1 | Let $\mathbb{F}$ be a field of characteristic $2$, $n$ be a positive integer and $f\_n:\bigoplus\limits\_{i=1}^n\mathbb{F}\sigma\_{i}\mapsto \bigoplus\limits\_{i,j=1,i<j}^n\mathbb{F}\sigma\_{i,j}$ be a linear map, where we identify $\sigma\_{i,j}=\sigma\_{j,i}~\forall~i,j\in\lbrace 1,\dots,n\rbrace$, which is defined a... | https://mathoverflow.net/users/482329 | Dimension of a kernel of a linear map | Let me start with the conclusion. @მამუკა ჯიბლაძე has the correct guess: if $n+1$ has two bits in its binary representation, then the kernel has dimension $2$; otherwise, the kernel is exactly one-dimensional. The only nonzero element in the kernel actually has a nice expression in binary representation. Suppose that $... | 2 | https://mathoverflow.net/users/500054 | 445668 | 179,602 |
https://mathoverflow.net/questions/445630 | 4 | Let $C$ be a category internal to a category $K$. It is well known (for example see **Proposition 2.4** in the paper **Higher Dimensional Algebra VI: Lie 2-Algebra** by *Baez and Crans* <https://digitalcommons.lmu.edu/cgi/viewcontent.cgi?article=1068&context=math_fac>) that there is a strict 2-category $K$Cat consistin... | https://mathoverflow.net/users/86313 | What is a correct notion of an internal pseudofunctor? | Let $\mathcal{C}$ be a category with pullbacks, with ${\sf C}$ an internal category in $\mathcal{C}$. We define a category $${\sf Ext( C})$$ called the *externalization* of ${\sf C}$ as follows:
1. The objects of ${\sf Ext(C)}$ are generalized elements of ${\sf Ob\_C}$, that is arrows $x:X\to{\sf Ob\_C}\in{\bf Hom}\_... | 2 | https://mathoverflow.net/users/92164 | 445669 | 179,603 |
https://mathoverflow.net/questions/445665 | 1 | If we assume that $\int\_0^\infty e^{-sx}\mu\_n(dx)\to \int\_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu\_n\to\mu$ vaguely. Where $\mu\_n$ is a measure. Please check here for [vague convergence](https://en.wikipedia.org/wiki/Vague_topology).
If it is true, how to prove it? Otherwise pleas... | https://mathoverflow.net/users/147009 | Vague convergence VS Laplace transform convergence? | $\newcommand\R{\mathbb R}$We have
\begin{equation\*}
L\_n(s)\to L(s) \tag{1}\label{1}
\end{equation\*}
(as $n\to\infty$) for each $s\in\R\_+:=[0,\infty)$, where
\begin{equation\*}
L\_n(s):=\int\_{\R\_+}e^{-sx}\mu\_n(dx),\quad L(s):=\int\_{\R\_+}e^{-sx}\mu(dx).
\end{equation\*}
From the context, $\mu$ and the $\mu\... | 2 | https://mathoverflow.net/users/36721 | 445676 | 179,604 |
https://mathoverflow.net/questions/445677 | 6 | An [old chestnut](https://mathoverflow.net/questions/25873/geometric-interpretation-of-filtered-rings-and-modules/25888#25888) is that filtered objects are the same as sheaves over $\mathbb A^1 / \mathbb G\_m$.
**Question:** Is there a similar description of chain complexes?
More precisely, if $\mathcal C$ is a cat... | https://mathoverflow.net/users/2362 | Is there a Hopf algebra-style description of chain complexes? | Shouldn't it just be $\operatorname{Spec}(\Lambda)/\mathbb{G}\_m$, where $\Lambda = \mathbb{Z}[d]/d^2$ and the $\mathbb{G}\_m$ action encodes the grading with $d$ in degree $-1$? This is just the observation that chain complexes are the same as graded modules over an exterior algebra in a degree $-1$ generator, similar... | 8 | https://mathoverflow.net/users/39747 | 445683 | 179,606 |
https://mathoverflow.net/questions/445679 | 5 | **I. Level 7**
In Klein's "[*On the Order-Seven Transformations of Elliptic Functions*](http://library.msri.org/books/Book35/files/klein.pdf)", he gave two *elegant* resolvents of degrees 8 and 7 in pages 306 and 313. Translated to more understandable notation, we have,
$$x^8+14x^6+63x^4+70x^2-7 = x\sqrt{j(\tau)-17... | https://mathoverflow.net/users/12905 | Generalizing Klein's order 7 formula $y\left(y^2+7\Big(\tfrac{1-\sqrt{-7}}{2}\Big)y+7\Big(\tfrac{1+\sqrt{-7}}{2}\Big)^3\right)^3 = j$ to order 13? | The smallest $m$ for which $\mathrm{PGL}(2,13)$ acts faithfully and transitively on a set of $13m$ elements is $m=6$, with point stabiliser a dihedral group of order $28$. I think you may be seeing a manifestation of Camille Jordan's theorem that $\mathrm{PSL}(2,p)$ has a subgroup of index $p$ if and only if $p\leqslan... | 8 | https://mathoverflow.net/users/460592 | 445686 | 179,607 |
https://mathoverflow.net/questions/445684 | 3 | Let $A$ be an additive category with kernels and cokernels. A morphism $f$ is called strict if the natural morphism from the coimage to the image is an isomorphism.
In *Schneiders: Quasi-abelian categories and sheaves* one finds a proof that the composition of strict morphisms is strict, so the strict morphisms form a ... | https://mathoverflow.net/users/473423 | Is the subcategory of strict morphisms abelian? | I do not believe that the composition of strict morphisms is strict, proposition 1.1.7 of Schneider's *Quasi-Abelian Categories and Sheaves* claims this only for strict epimorphisms as well as for strict monomorphisms.
I think that rather the opposite of the OP's claim is true:
>
> If all compostions of strict mo... | 9 | https://mathoverflow.net/users/21051 | 445691 | 179,608 |
https://mathoverflow.net/questions/445693 | 8 | Let $({M},g)$ be a connected and non-compact Riemannian manifold without boundary. If $L:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a linear second order elliptic operator on some smooth $\mathbb{R}$-bundle $E$ over ${M}$, is it then true that
$$Lu=0$$
for $u\in\Gamma^{\infty}\_{c}(E)$ implies that $u=0$, or in o... | https://mathoverflow.net/users/199422 | Compactly-supported harmonic tensors | The unique continuation is valid for generalized Laplacians. This follows from Hörmander's result in
>
> *Hörmander, Lars*, [**Uniqueness theorems for second order elliptic differential equations**](https://doi.org/10.1080/03605308308820262), Commun. Partial Differ. Equations 8, 21-64 (1983). [ZBL0546.35023](https:... | 13 | https://mathoverflow.net/users/20302 | 445696 | 179,610 |
https://mathoverflow.net/questions/445652 | 1 | I am aware that the dual of $C^\infty(\mathbb{R}^n)$ is the space of distributions (not necessarily tempered) with compact support.
However, if we fix a compact set $K \subset \mathbb{R}^n$, is the space of distributions supported on $K$ the dual of some test function space?
If so, the test function space must be e... | https://mathoverflow.net/users/56524 | Given a compact set $K \subset \mathbb{R}^n$, is the space of distributions supported on $K$ the dual of some test function space? | A few reminders:
[1] The dual space of $\mathscr D(\mathbb R^n)=
C^\infty\_c(\mathbb R^n)$ ($C^\infty$ functions with compact support) is
$\mathscr D'(\mathbb R^n)$ (distributions on $\mathbb R^n$).
[2] The dual space of $\mathscr S(\mathbb R^n)$
($C^\infty$ functions rapidly decreasing) is
$\mathscr S'(\mathbb R^n)$... | 2 | https://mathoverflow.net/users/21907 | 445703 | 179,614 |
https://mathoverflow.net/questions/445617 | 4 | Lurie introduced in subchapter 1.2.12 of his [Higher Topos Theory](https://arxiv.org/abs/math/0608040)
the notion of *final* and *strongly final* objects:
**Definition 1.2.12.1.** let $\mathcal{C}$ be a topological category (e.g. simplicial cats,
simplicial set).
An object $X \in \mathcal{C}$ is *final* if for each $... | https://mathoverflow.net/users/501436 | Final and strongly final objects in Higher Topos Theory | It's good to ask this kind of questions on a critical reading! (They are also great exercises in unwinding the definitions to learn to work with simplicial sets, although in the first case I couldn't quite work it out.)
1. I actually think that this argument is not complete: if $\mathcal C$ is a simplicial set such t... | 2 | https://mathoverflow.net/users/82179 | 445704 | 179,615 |
https://mathoverflow.net/questions/445728 | 8 | So I need to writeup some old results of Harrington's which imply various results about admissible ordinals. I've never really learned admissible recursion theory so what's a good reference?
| https://mathoverflow.net/users/23648 | Good source for admissible set theory? | I **strongly** recommend Sacks' book *[Higher recursion theory](https://projecteuclid.org/ebooks/perspectives-in-logic/Higher-Recursion-Theory/toc/pl/1235422631)* in conjunction with Barwise's *[Admissible sets and structures](https://projecteuclid.org/euclid.pl/1235418470)* (each of which is freely and legally availab... | 10 | https://mathoverflow.net/users/8133 | 445729 | 179,624 |
https://mathoverflow.net/questions/365828 | 1 | Some context: I am going through some literature on empirical risk minimization for bipartite ranking [1] that shows how certain "low-noise" conditions lead to fast rates of convergence of the excess risk to $0$ of the empirical risk minimizer. The low noise condition at hand is:
There exists constants $c>0$ and $\al... | https://mathoverflow.net/users/153334 | Fast rates in ERM: Extreme case of low-noise assumption implies non-differentiability | This can be shown using Jensen's inequality. For all fixed $x$:
$$\mathbb{E}\_X \frac{1}{\left| \eta(x) - \eta(X) \right|}
\\= \mathbb{E}\_X \left [ \frac{1}{ \eta(x) - \eta(X) } \mid \eta(x) > \eta(X) \right]\mathbb P(\eta(x) > \eta(X))
\\+ \mathbb{E}\_X \left [ \frac{1}{ \eta(X) - \eta(x) } \mid \eta(x) \leq \eta(... | 0 | https://mathoverflow.net/users/153334 | 445739 | 179,629 |
https://mathoverflow.net/questions/445711 | 4 | Let $G$ be a finite semigroup (or monoid if that helps) and $K$ a field.
>
> Question: When is the semigroup algebra $KG$ local?
>
>
>
Here local means that there is a unique maximal right (or left) ideal.
When $G$ is a group this is true if and only if $G$ is a $p$-group when $K$ has characteristic $p$ (whe... | https://mathoverflow.net/users/61949 | When is semigroup algebra local? | Question 1 either has a trivial answer or it is answered in my paper (with coauthors). [REPRESENTATION THEORY OF FINITE SEMIGROUPS,
SEMIGROUP RADICALS AND FORMAL LANGUAGE THEORY](https://www.ams.org/journals/tran/2009-361-03/S0002-9947-08-04712-0/S0002-9947-08-04712-0.pdf) depending on what you require for a local ring... | 6 | https://mathoverflow.net/users/15934 | 445743 | 179,633 |
https://mathoverflow.net/questions/445733 | 7 | Let $G$ be a finite group and $k$ a field of characteristic $p$ dividing $|G|$. A perfect complex of $kG$-modules is by definition a finite complex of finitely generated projective ($=$ injective $=$ flat) $kG$-modules. A semi-projective complex of $kG$-modules is a filtered colimit of perfect complexes.
My question ... | https://mathoverflow.net/users/460592 | Semi-projective complexes of modules over a finite group | I think I have a counterexample.
Let $\operatorname{char}(k)=3$ and let $G$ be the symmetric group $S\_{3}$.
Then $kG$ has two simple modules: the trivial module $k$ and another
one-dimensional module $S$.
The projective covers $P$ and $Q$ of these simples
are uniserial with composition series
$P=
\begin{matrix}
... | 7 | https://mathoverflow.net/users/22989 | 445753 | 179,636 |
https://mathoverflow.net/questions/445763 | 2 | Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive roots by
$$
\rho := \frac{1}{2}\sum\_{\alpha \in \Delta\_+} \alpha.
$$
What can one say about the scalar
$$
(\alpha,\rho)?
$... | https://mathoverflow.net/users/491434 | Pairing a root with the half-sum of positive roots | I'm sorry, my original answer was about the expansion of the *highest root* $\theta$ into the fundamental weights $\omega\_1,\ldots,\omega\_n$. For the Weyl vector $\rho$, something much simpler is true: we have $\rho=\sum\_{i=1}^{n} \omega\_i$. It is easy to see this since the simple reflection $s\_{\alpha\_i}$ sends ... | 3 | https://mathoverflow.net/users/25028 | 445765 | 179,638 |
https://mathoverflow.net/questions/445725 | 10 | Let $f(x)$ be a function such that for all $T \leq f(x), \ T / x \to \infty$ we have
$$
\sum\_{x \leq n < 2 x} e^{2 \pi i T / n} = o(x).
$$
How fast can $f(x)$ grow?
I can show that for any $\varepsilon > 0$, $f(x) = e^{(\log x)^{2 - \varepsilon}}$ works, and that any such function $f$ must satisfy $f(x) \leq e^{x^{\... | https://mathoverflow.net/users/88679 | Cancellation in a very rapidly oscillating exponential sum | I doubt that one is able to get as far as $T = \exp(\log^{2-\varepsilon} x)$ with Weyl differencing. Standard Weyl differencing arguments, such as that in Theorem 8.4 of
*Iwaniec, Henryk; Kowalski, Emmanuel*, [**Analytic number theory**](https://mathoverflow.net/posts/comments/1144815), Colloquium Publications. Ameri... | 9 | https://mathoverflow.net/users/766 | 445767 | 179,639 |
https://mathoverflow.net/questions/445731 | 1 | Suppose $X$, $Y$, $X'$ and $Y'$ are random variables whose probability density follows the following relations.
\begin{align}
\|p\_X-p\_{X'}\|\_{\mathrm{TV}}&\leq\epsilon\_1,\\
\|p\_Y-p\_{Y'}\|\_{\mathrm{TV}}&\leq\epsilon\_2,
\end{align}
where $\|\cdot\|\_{\mathrm{TV}}$ is the total variation distance. Moreover, $X$ ... | https://mathoverflow.net/users/68835 | The effect of a small change of the probability distribution on the output of the function | $\newcommand{\TV}[2]{\left\| #1 - #2\right\|}$
Sure. You know that $X,Y$ are independent, as are $X',Y'$. So
\begin{align\*}
\TV{p\_{X,Y}}{p\_{X',Y'}}
&=\TV{p\_Xp\_Y}{p\_{X'}p\_{Y'}}\\
&\leq \TV{p\_X}{p\_{X'}}+\TV{p\_Y}{p\_{Y'}}\\
&\leq \epsilon\_1+\epsilon\_2.
\end{align\*}
But $(Z,X)$ is a function of $(X,Y)$, an... | 2 | https://mathoverflow.net/users/5784 | 445776 | 179,644 |
https://mathoverflow.net/questions/441361 | 4 | Is it true that given a fusion category $\mathcal{C}$ and its Drinfel'd center $Z(\mathcal{C})$, there is a fully faithful functor $F:\mathcal{C}\hookrightarrow Z(\mathcal{C})$? I.e. can $\mathcal{C}$ can be identified with a full monoidal subcategory of $Z(\mathcal{C})$?
Judging by the mention of a 'restriction func... | https://mathoverflow.net/users/135817 | Relationship between fusion category and its Drinfel'd center | The short answer is no.
Suppose you have a fully faithful monoidal functor $(F,J):\mathcal C\to\mathcal B$, where $\mathcal C$ and $\mathcal B$ are fusion and $J\_{X,Y}:F(X)\otimes F(Y)\to F(X\otimes Y)$ is the monoidal structure map (sometimes called a tensorator). If $\{b\_{U,V}:U\otimes V\to V\otimes U\}\_{U,V\in\... | 3 | https://mathoverflow.net/users/125022 | 445787 | 179,649 |
https://mathoverflow.net/questions/445792 | 1 | I am looking for ways to do this integration analytically
\begin{equation}
\int\_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}
\end{equation}
For context:
I encountered an integral of the form
$$
E\_b=\pi+\int \frac{i\left(e^{c|p|}+e^{2|p| \sin (\phi)}\right) e^{-|p|... | https://mathoverflow.net/users/163076 | Analytic expression for $\int_0^\infty \mathrm{d}p\frac{e^{-p \sin (\phi )} \sin (p \cos (\phi ))}{p \left(e^{c p}+1\right)}$ | We seek \begin{align}I&=\int\_0^\infty\frac{e^{-st}\sin t}{t(e^{at}+1)}\,dt\\&=\sum\_{n\ge0}(-1)^n\int\_0^\infty\frac{e^{-(s+a(n+1))t}\sin t}t\,dt=-\sum\_{n\ge1}(-1)^n\arctan\frac1{s+an}\end{align} which has a "closed form" involving complex log-gamma; for instance, [see the case $(s,a)=(0,1)$](https://math.stackexchan... | 5 | https://mathoverflow.net/users/113397 | 445796 | 179,651 |
https://mathoverflow.net/questions/445705 | 2 | Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T\_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K, t \geq 0 \right\}).
$$
My question is about the cases where $K$ is a closed convex **cone**.
The equation right aft... | https://mathoverflow.net/users/488776 | Tangent cone of a closed convex cone | The equality
$$T\_K(u)=\{v-tu\colon v\in K,\, t\ge0\}$$
is false in general, because $T\_K(u)$ is closed by definition, whereas
$$S\_K(u):=K-\mathbb R\_+ u=\{v-tu\colon v\in K,\, t\ge0\}$$
can be not closed.
A counterexample is provided by [this answer](https://math.stackexchange.com/a/894816/96609). Indeed, let
$$K=... | 3 | https://mathoverflow.net/users/36721 | 445797 | 179,652 |
https://mathoverflow.net/questions/445788 | 4 | Suppose $(P, <)$ is a poset of cofinality $\aleph\_2$ and additivity (least cardinality of an unbounded subset) $\aleph\_1$. Can we conclude the existence of a cofinal subset of order-type $\omega\_1 \times \omega\_2$? Are there special conditions under which we get that conclusion? What about higher cofinalities and a... | https://mathoverflow.net/users/495743 | Cofinal rectangles in poset | Let $P$ be the set of all countable subsets of $\omega\_2$ ordered by set inclusion. If $2^{\aleph\_0}\le\aleph\_2$ then $P$ has cofinality $\aleph\_2$ and additivity $\aleph\_1$ and contains no chain of order type $\omega\_1+1$.
| 9 | https://mathoverflow.net/users/43266 | 445803 | 179,653 |
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