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https://mathoverflow.net/questions/414240 | 0 | Can someone please provide a reference (starting point) for analysing recurrence/transience of random walks on graphs with general edge weights? Looking into random walks that are known to be NOT reversible.
| https://mathoverflow.net/users/473751 | Recurrence criterion for non-reversible random walks on general infinite (locally finite) graph with unequal edge weights | The most powerful and flexible method to prove recurrence or transience for nonreversible walks is the method of Lyapunov functions.
One of the origins of the method is Foster's criteria for recurrence and positive recurrence. [1], [2]. See Theorem 1 page 2 in [3] (or page 40 of [9]) for a concise statement of the recu... | 0 | https://mathoverflow.net/users/7691 | 414956 | 169,211 |
https://mathoverflow.net/questions/414082 | 1 | Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ such that $f\_\# \mu(A)=\nu(A)$ for all $A\in \sigma$?
Keep in mind that if $(X,B)$ and $(Y,C)$ are measurable spaces... | https://mathoverflow.net/users/473399 | A question about pushforward measures and Peano spaces | In general the answer to this problem is negative: if the measure $\mu$ has connected support and the measure $\nu$ has disconnected support, then for any continuous map $f:P\to P$ the measure $f\_\# \mu$ will be supported by a connected subset of $P$ and hence will be not equal to the measure $\nu$.
Let us recall th... | 1 | https://mathoverflow.net/users/61536 | 414959 | 169,212 |
https://mathoverflow.net/questions/414770 | 21 | Consider the following partial order. The objects are unordered tuples $\{V\_1,\ldots,V\_m\}$, where each $V\_i \subseteq \mathbf{R}^n$ is a nontrivial linear subspace and $V\_1 \oplus \cdots \oplus V\_m = \mathbf{R}^n$. We say that $\{V\_1,\ldots,V\_m\} \geq \{W\_1,\ldots,W\_i\}$ if for each $j=1,\ldots,m$ there exist... | https://mathoverflow.net/users/1378 | What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$? | Let me write $V$ for a finite-dimensional vector space over some field (the field will not play a role), and $\mathsf{P}(V)$ for the poset described in the question, which I consider as a category. Let me rephrase the order relation: $\{V\_1, \ldots, V\_n\} \geq \{W\_1, \ldots, W\_m\}$ if and only if each $W\_j$ is a d... | 11 | https://mathoverflow.net/users/318 | 414964 | 169,214 |
https://mathoverflow.net/questions/414962 | 3 | Let $\mathbb{N}$ denote the set of non-negative integers. We say that a sequence $f:\mathbb{N}\to \{0,1\}$ is *normal* if every finite $\{0,1\}$-sequence appears in $f$.
Let the *swapping operation* $\sigma:\mathbb{N}\to \mathbb{N}$ be defined by swapping each even integer with its successor - that is, $\sigma(2n) = ... | https://mathoverflow.net/users/8628 | Is normalcy preserved under the swapping operation? | If every string appears in $f$ consecutively, then every string appears consecutively in $f \sigma$.
(So *yes* in answer to the title question, *no* in answer to the question as phrased in the question body :)
(Thanks to Alessandro Della Corte for corrections in the comments below!)
To see this, suppose that ever... | 9 | https://mathoverflow.net/users/2362 | 414966 | 169,215 |
https://mathoverflow.net/questions/414707 | 1 | Let $W$ be a standard Brownian motion and $\mathcal F\_t$ its natural filtration.
Suppose $\theta, A$ are positive $L^1$ random variables independent of $\mathcal F\_t$.
Let $Y\_t$ be the process
$$Y\_t := A \sin \, (\theta t) + W\_t $$
and denote by $\mathcal Y\_t$ its natural filtration.
**Question:** Is it... | https://mathoverflow.net/users/173490 | Filtering the period and amplitude of a sine wave corrupted by noise | By the Levy zero-one law, the question is equivalent to deciding whether $A$ and $\theta$ are measurable with respect to $\mathcal F\_\infty$, the common refinement of all the $\mathcal F\_t$. The answer is positive.
For $\alpha>0,\beta\ge 0$, consider
$$Z\_n=Z\_n(\alpha,\beta):=Y\_{\beta+(2n+1)\alpha }-Y\_{\beta+(2n... | 2 | https://mathoverflow.net/users/7691 | 414967 | 169,216 |
https://mathoverflow.net/questions/414867 | 4 | I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups.
Let me first recall the classical set-up. According to Steinberg's lecture notes, every simple untwisted Chevalley group $G$ can be obtained as follows. Consdider a finite-dimensiona... | https://mathoverflow.net/users/203598 | Twisted root subgroups in twisted Chevalley groups (reference request) | Tom's answer is the complete story with two kinds of exceptions: (a) $G^\sigma$ is a Suzuki-Ree-type group -- ${^2}B\_2(2^a)$, ${^2}F\_4(2^a)$, or ${^2}G\_2(3^a)$; or (b) there exists a root $a\_0$ such that $a\_0+\gamma(a\_0)$ is again a root.
In case (a), the symmetry of order $2$ of the Dynkin diagram leads to an ... | 2 | https://mathoverflow.net/users/99221 | 414968 | 169,217 |
https://mathoverflow.net/questions/414965 | 0 | Let $u,v,x \in \mathbb R^d$ be three unit vectors. I found a very complicated proof that $(v^Tx)^2-(u^Tx)^2 \leq 1-(u^Tv)^2$.
That is $\lVert uu^T-vv^T\rVert^2\_2 = 1-(u^Tv)^2$, or that $f(v,x)\leq f(v,u)+f(u,x)$ where $f(v,u)=1-(v^Tu)^2$ is the squared sine of the angle between $u$ and $v$.
Is there a one-liner proo... | https://mathoverflow.net/users/476110 | Prove that $(v^Tx)^2-(u^Tx)^2 < 1-(u^Tv)^2$ for any unit vectors $u$, $v$, $x$ | The inequality as stated is false, but it is true that
$$\langle v,x\rangle^2+\langle v,u\rangle^2\leq 1+|\langle x,u\rangle|.$$
Moreover, the right-hand side is optimal in the sense that it is the maximum of the left-hand side over the unit vectors $v$. More generally, if $u\_1,\dots,u\_R$ are any vectors in a Hilbert... | 3 | https://mathoverflow.net/users/11919 | 414969 | 169,218 |
https://mathoverflow.net/questions/414961 | 1 | The generator an OU process is given by
$$A = \operatorname{tr}(QD^2)+\langle Bx,D\rangle.$$
This one possesses an invariant measure given by
$$d\mu(x) = b(x) \ dx \text{ with } b(x) = \frac{1}{(4\pi)^{N/2} \vert Q\_{\infty} \vert^{1/2}} e^{-\langle Q\_{\infty}^{-1}x,x \rangle /4},$$
where $Q\_{\infty}= \int\_0^{\i... | https://mathoverflow.net/users/150549 | Adjoint operator of OU generator | The argument is a bit tricky. Assume for simplicity that $Q=I$, then $BQ\_\infty+Q\_\infty B^\*=-I$. Next decompose $A$ as a self-adjoint part plus a remainder, namely introduce the form $a(u,v)=\int\_{\mathbb R^n} \nabla u \nabla v \, d\mu$ which corresponds to $A\_1=\Delta-\frac 12 Q\_\infty ^{-1} x \cdot \nabla$ and... | 1 | https://mathoverflow.net/users/150653 | 414972 | 169,219 |
https://mathoverflow.net/questions/414973 | 17 | [Van der Waerden's theorem](https://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem) states that any colouring of the integers in a finite number of colours has monochromatic arithmetic progressions of arbitrary length. [Szemerédi's Theorem](https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem) is a dramatic stre... | https://mathoverflow.net/users/7113 | A proof of Van der Waerden's theorem using a weakened form of Szemeredi's theorem | As (implicitly) observed already in Szemerédi's celebrated paper
*Szemerédi, Endre*, [**On sets of integers containing no (k) elements in arithmetic progression**](http://dx.doi.org/10.4064/aa-27-1-199-245), Acta Arith. 27, 199-245 (1975). [ZBL0303.10056](https://zbmath.org/?q=an:0303.10056).
(and perhaps previousl... | 36 | https://mathoverflow.net/users/766 | 414978 | 169,220 |
https://mathoverflow.net/questions/414981 | 1 | Let $X$ be a one dimensional Ito diffusion given by
$$X\_t = b \,W\_t$$
where $b$ is a constant, and $W$ is a standard Brownian motion.
Let $B$ be another Brownian motion independent of $W$, and define the *observation process* $Y$ by
$$Y\_t = X\_t + B\_t.$$
Fix $T > 0$. A choice of *sampling times* is simply... | https://mathoverflow.net/users/173490 | What are the optimal times to sample a process? | Write $$Z\_t = W\_t - b B\_t,$$ so that $Y\_t$ and $Z\_t$ are independent Brownian motions, $$X\_t = b W\_t = b \cdot \frac{b Y\_t + Z\_t}{1 + b^2} \, ,$$ and the question asks for the distance between $X\_T$ and
$$ \mathbb E[X\_T | \sigma(Y\_{t\_1},\ldots,Y\_{t\_n})] = b \cdot \frac{b Y\_{t\_n} + 0}{1 + b^2} \, . $$
T... | 3 | https://mathoverflow.net/users/108637 | 414985 | 169,222 |
https://mathoverflow.net/questions/410224 | 2 | **Short version.** If $X$ is a diffusion with generator $L$ and the Lebesgue measure is invariant for $X$, then $L^\*$ has no term of order zero and it corresponds to another diffusion $X^\*$. Denoting by $p$ and $p^\*$ their kernels,
>
> do we have $p^\*\_t(x,y)=p\_t(y,x)$?
>
>
>
I am interested in full answe... | https://mathoverflow.net/users/129074 | Is $p_t(y,x)$ the kernel of the time reversal of the diffusion $X$, for $p_t(x,y)$ the kernel of $X$? | This is an immediate corollary of Theorem 3.50 in
>
> D. W. Stroock, [An introduction to the analysis of paths on a Riemannian manifold](https://bookstore.ams.org/surv-74-s) (2000).
>
>
> Mathematical Surveys and Monographs. 74. Providence, RI: American Mathematical Society (AMS). xvii, 269 p.
>
>
>
See also... | 1 | https://mathoverflow.net/users/129074 | 415001 | 169,225 |
https://mathoverflow.net/questions/414940 | 1 | I've posted this question [Tangent space of $G \times\_H M$](https://math.stackexchange.com/questions/4365925/tangent-space-of-g-times-h-m) in MSE, but didn't get any answer. My question is the following:
Let $G$ be a Lie group and let $H$ be a Lie subgroup of $H$. Let $M$ be a smooth manifold on which $H$ acts from ... | https://mathoverflow.net/users/172459 | The tangent space of $G \times _H M$ | When moving to the tangent space level, the general rule is to take the derivative (of everything you can). Just from this I'd expect something like the equivalence on $T\_gG\oplus T\_mM$ should be such that $gX - Xm = 0$ for every $X\in \mathfrak h$.
(As a side note, it may be better to write $(g,m).h$ because it's ... | 3 | https://mathoverflow.net/users/145436 | 415020 | 169,231 |
https://mathoverflow.net/questions/415021 | 4 | Let $a,b \in \mathbb R$ and consider the functional $J$ on $X$:
$$J[u] = \int\_0^1 \left( (u'(x))^2 -a)^2 + b \ln (1+ u^2(x))\right) dx$$
Providing reasons specify if the $\inf J$ over $X$ is attained of not
1. $a>0, b \geq 0, X =\{ u \in C^1([0,1]); u(0)=0, u(1)=0 \}$
2. $a>0, b > 0, X =\{ u \in AC([0,1]); u(0)=... | https://mathoverflow.net/users/471464 | Which set of functions admits the existence of the minimizer? | The answer to (3) is yes. Indeed, then all the conditions of what you call "special version of Tonelli’s theorem" (proved in [this answer](https://mathoverflow.net/a/413645/36721)) are satisfied.
The answer to (2) is no. Indeed, for natural $n$ and $x\in[0,1]$, let
$$u\_n(x):=\sqrt a\,d(x,E\_n),$$
where $E\_n$ denote... | 3 | https://mathoverflow.net/users/36721 | 415026 | 169,232 |
https://mathoverflow.net/questions/415012 | 5 | In the previous post [What is the smallest set of real continuous functions generating all rational numbers by iteration?](https://mathoverflow.net/q/412923/7113) I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous functio... | https://mathoverflow.net/users/7113 | Is $\mathbb{Q}$ the orbit of a continuous function that is computable when restricted to $\mathbb{Q}$? | Yes. This answer is based on the answers to your previous question.
Start with a computable ergodic map $T$ (D. Thomine constructs an example [here](https://mathoverflow.net/a/404564)). For every basic open neighborhood $B\_i$, the set $D\_i = \bigcup\_n T^{-n}(B\_i)$ is a dense effectively open set, uniformly in $i$... | 6 | https://mathoverflow.net/users/32178 | 415027 | 169,233 |
https://mathoverflow.net/questions/415007 | 23 | Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property:
its integral $\int\_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int\_{-\infty}^\infty F(x)\,dx=1$? It definitely can be seen as a limit of smooth functions, and its Fourier transform will be a finction, equal to $1$ at... | https://mathoverflow.net/users/10059 | Anti-delta function? | There's quite a bit on this in Harold Jeffreys's unusual book *Theory of Probability*, in which such a function is one of the more prominent examples of “improper priors.” In [Robert, Chopin, and Rousseau – Harold Jeffreys's Theory of Probability Revisited](https://arxiv.org/abs/0804.3173) we find this:
>
> For a 2... | 22 | https://mathoverflow.net/users/6316 | 415029 | 169,234 |
https://mathoverflow.net/questions/415022 | 17 | In Cohen's article, [The Discovery of Forcing](https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-32/issue-4/The-Discovery-of-Forcing/10.1216/rmjm/1181070010.full), he says that "one cannot prove the
existence of any uncountable standard model in which AC holds, and
CH is false,"
and offers... | https://mathoverflow.net/users/3106 | Must uncountable standard models of ZFC satisfy CH? |
>
> Remarks (2) and (3) are added in this edit.
>
>
>
What Cohen's quoted proof outline is leaving implicit is the following statement in which $\mathrm{Con}(T)$ means "$T$ is consistent".
$(\*)$ Assuming $\mathrm{Con(ZF + SM)}$, $\mathrm{V} \neq \mathrm {L}$ is not provable from $\mathrm{ZF + SM}$, where $\ma... | 23 | https://mathoverflow.net/users/9269 | 415033 | 169,235 |
https://mathoverflow.net/questions/415050 | 1 | Do there exist continuous functions $c\_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim\_{x\to\infty}\gamma\left(x\right)=0$ and the following equation is true for all $\alpha,x>0$
$$
e^{-\alpha x}=\sum\_{n=1}^{\infty}c\_n\left(\alpha\right)\gamma\left(x\right)^n
$$
If yes, how... | https://mathoverflow.net/users/104719 | Can we write $e^{-\alpha x}$ as $\sum_{n=0}^\infty c_n\left(\alpha\right)\gamma\left(x\right)^n$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ | The answer is **no**.
Suppose that $c\_n(\alpha)$ and $\gamma(x)$ with the desired properties exist. Necessarily $\gamma$ is one-to-one, and hence $$e^{-\alpha \gamma^{-1}(x)} = \sum\_{n = 0}^\infty c\_n(\alpha) x^n.$$ Since the right-hand side converges for some $x > 0$, it defines a function $$f\_\alpha(x) = \sum\_... | 3 | https://mathoverflow.net/users/108637 | 415053 | 169,239 |
https://mathoverflow.net/questions/414943 | 2 | Let $k$ be a field and $A$ be a central simple algebra over $k$. It's known that $A$ has a splitting field (i.e. a field $K/k$ such that $A\_K\cong M\_n(K)$ for some $n$) which is finite and Galois.
This allows us to define a *reduced norm* $N:A\to k$ which is given by the determinant $M\_n(K)\to K$ and then descende... | https://mathoverflow.net/users/131975 | Existence of reduced norms for CSAs using fpqc descent | You can define the reduced norm using fpqc descent. Have a look at chapter III, section 1.2 in Knus' "[Quadratic and Hermitian Forms over Rings](https://doi.org/10.1007/978-3-642-75401-2)", for instance.
| 2 | https://mathoverflow.net/users/86006 | 415065 | 169,242 |
https://mathoverflow.net/questions/415071 | 2 | Let $T$ be a given (finite) tree.
**Question 1:** Is it always possible to add edges to $T$ to obtain a $2$-connected outerplanar supergraph $G$?
**Question 2:** If the answer to Question #1 is negative, can the trees for which it is possible be characterized?
**Question 3( Defect form of Question 1):** Let $T$ b... | https://mathoverflow.net/users/22051 | Completing a tree to a 2-connected outerplanar graph | Yes. Pick an arbitrary vertex to be the root. Consider the sequence of vertices $v\_1, v\_2, \ldots$ produced by a pre-order traversal of the rooted tree, adding edges $v\_i - v\_{i+1}$ where they don't already exist. Finally, add an edge back from the last vertex to the root, if it doesn't already exist. This cycle de... | 2 | https://mathoverflow.net/users/46140 | 415075 | 169,246 |
https://mathoverflow.net/questions/414315 | 6 | A follow-up question to [Alternating subgroups of $\mathrm{SU}\_n $](https://mathoverflow.net/questions/414265/alternating-subgroups-of-mathrmsu-n).
$\DeclareMathOperator\PU{PU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PU\_m $ be the projective unitary group, a compact simple adjoint Lie group... | https://mathoverflow.net/users/387190 | Finite simple groups and $ \operatorname{SU}_n $ | One can consult the tables of Bray-Holt-Roney Dougal to work out the subgroups of $\mathrm{PSU}\_m$.
1. For $m=5$, we have copies of $PSL\_2(11)$ and $PSU\_4(4)$.
2. For $m=6$ we have $PSL\_2(11)$, $PSL\_3(4)$ and $PSU\_4(9)$.
3. For $m=7$ we have $PSU\_3(9)$.
4. For $m=8$ we have $PSL\_3(4)$.
5. For $m=9$ we have $P... | 8 | https://mathoverflow.net/users/152674 | 415082 | 169,248 |
https://mathoverflow.net/questions/415062 | 4 | Let finite group $G$ act on a finite set $X$ and hence on colorings $Y^X$, where $Y=\{1,2,\ldots,k\}$ is a set of colors. The Burnside-Pólya-Redfield-etc. counting theorem says that the number of orbits of $G$ acting on $Y^X$ is
$$ \frac{1}{\#G} \sum\_{g \in G} k^{c(g)},$$
where for $g\in G$, we use $c(g)$ to denote th... | https://mathoverflow.net/users/25028 | Integer-valued polynomials from Pólya counting | Okay, I understand Nate's comment now and am posting this elucidation of it as a community wiki answer.
Recall that an *ordered set partition* of $X$ into $j$ blocks is an ordered tuple $(T\_1,T\_2,\ldots,T\_j)$ of non-empty subsets $\varnothing \neq T\_i \subseteq X$ that are pairwise disjoint and whose union is all... | 8 | https://mathoverflow.net/users/25028 | 415089 | 169,250 |
https://mathoverflow.net/questions/415092 | 17 | Suppose that $a\_1,\dots,a\_n,b\_1,\dots,b\_n$ are iid random variables each with a symmetric non-atomic distribution.
Let $p$ denote the probability that there is some real $t$ such that $t a\_i \ge b\_i$ for all $i$.
It [was shown](https://mathoverflow.net/a/415074/36721) that
$$p=\frac{n+1}{2^n}.$$
Can this be pro... | https://mathoverflow.net/users/36721 | Can this probability be obtained by a combinatorial/symmetry argument? | If I understand correctly, $c\_i := b\_i/a\_i$ should also be symmetric and non-atomic.
Then the result holds if there exists $t$ so that for all $i$
* $t \geq c\_i$ if $a\_i > 0$;
* $c\_i \geq t$ if $a\_i < 0$.
Reorder the indices so that $|c\_1| < |c\_2| < \dots < |c\_n|$.
There are $2^n$ ways to assign signs to ... | 25 | https://mathoverflow.net/users/7717 | 415102 | 169,253 |
https://mathoverflow.net/questions/415043 | 1 | Let $\Omega(n)$ be the total number of prime factors of $n$ (e.g $\Omega(12)=\Omega(2\*2\*3)=3$).
Consider these probabilities (probably they have a name):
* $P\_{>}(n)$ the probability that $\Omega(n+1) > \Omega(n)$
* $P\_{<}(n)$ the probability that $\Omega(n+1) < \Omega(n)$
* $P\_{=}(n)$ the probability that $\O... | https://mathoverflow.net/users/35419 | Probability of an equal number of factors of a successor | Loosely speaking, $\Omega(n)$ is approximately normal with mean and variance $\log\log n$. (See <https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem>) We expect $\Omega(n)$ and $\Omega(n+1)$ to behave like independent random variables. So we expect that all your probabilities containing an equal sign have a li... | 3 | https://mathoverflow.net/users/12947 | 415110 | 169,254 |
https://mathoverflow.net/questions/415039 | 5 | Consider a (countable) group $G$, a subgroup $H\leq G$ of finite index, and a unitary representation $\pi:G\to \mathcal{U}(\mathcal{H})$.
>
> If the center of the von Neumann algebra $\pi(H)''$ is finite dimensional, does this imply that the center of $\pi(G)''$ is finite dimensional as well?
> Is it at least true ... | https://mathoverflow.net/users/104498 | The center of a representation von Neumann algebra, and finite index subgroups | The answer is YES and it follows from the general theory of finite-index inclusions of von Neumann subalgebras (a la Jones, Pimsner, Popa...), which says that finite-dimensionality of the center is preserved under taking a finite-index subalgebra/extension. This is perhaps an overkill and so I will try a layman's proof... | 7 | https://mathoverflow.net/users/7591 | 415115 | 169,255 |
https://mathoverflow.net/questions/415120 | 7 | It is well known that the codomain functor $$cod:\mathcal{C}^\to\to\mathcal{C}$$ from the [arrow category](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/arrow+category) of a category $\mathcal{C}$ to itself is a [fibration](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/Grothendieck%20fibration) iff $\... | https://mathoverflow.net/users/92164 | A fibration equivalent to having a terminal object | A category $C$ has a terminal object if and only if the canonical functor $C \star \mathbf{1} \to \mathbf{1} \star \mathbf{1} = \mathbf{2}$ is a fibration.
| 10 | https://mathoverflow.net/users/57405 | 415123 | 169,257 |
https://mathoverflow.net/questions/414631 | 0 | Let $E\to X$ be a holomorphic vector bundle. Denote by $\mathbb{P}(E)\to X$ its projectivisation and $\mathcal{O}\_E(1)\to \mathbb{P}(E)$ the associated tautological line bundle.
I would like to know whether we can characterize the fact that $E$ is Hermitian-Eintein using the bundle $\mathcal{O}\_E(1)$.
* On one ha... | https://mathoverflow.net/users/102114 | Characterize Hermitian-Einstein metric on $E$ using the tautological bundle $\mathcal{O}_E(1)$ | This is exactly the proposition 3.5 of the mentioned paper:
*If $\mathcal{O}\_{P(E^\*)}(1)$ admits a geodesic-Einstein metric, then the induced $L^2$ metric on $E$ is a Hermitian-Einstein metric.*
| 0 | https://mathoverflow.net/users/102114 | 415136 | 169,261 |
https://mathoverflow.net/questions/415139 | 6 | *Motivation: define a concrete Abelian category as a category with a univalent and injective functor in $\mathrm{Ab}^I$ (such that all homological concepts in it coincide with simple set-theoretic concepts $\mathrm{Ab}^I$)*
Is it true that any abelian category is isomorphic to a subcategory $\mathrm{Ab}^I$ for some s... | https://mathoverflow.net/users/148161 | Is any abelian category a subcategory of $\mathrm{Ab}^I$? | Observe that $\bigoplus\_I : \textbf{Ab}^I \to \textbf{Ab}$ is a conservative exact functor: it is right exact by general nonsense, it preserves monomorphisms (because e.g. $\bigoplus\_{i \in I} A\_i$ is naturally a subgroup of $\prod\_{i \in I} A\_i$), and it is conservative because $\bigoplus\_{i \in I} A\_i \cong 0$... | 5 | https://mathoverflow.net/users/11640 | 415150 | 169,265 |
https://mathoverflow.net/questions/414747 | 2 | I was reading the following [paper](https://eudml.org/doc/286980) which claims to generalize Borel spectral sequence for non-compact Stein fibers. However, I don't understand how the following bundle fits into the picture:
$$
(\mathbb{C}^\times)^2\xrightarrow{\mathbb{C}} \mathbb{T}
$$
$\mathbb{T}$ is a compact torus.... | https://mathoverflow.net/users/143549 | Borel spectral sequence with non-compact fibers | The second page of the Borel spectral sequence in your example is
$$
E\_2^s=\bigoplus\_{p,q} \mathrm H^{s-q,q}(\mathbf T,\Omega^{p+q-s}(\mathbf C))=\bigoplus\_{p,q} \mathrm H^{s-q,q}(\mathbf T)\otimes\_{\mathbf C}\Omega^{p+q-s}(\mathbf C)
$$
and comprises Dolbeault cohomology of the compact complex torus $\mathbf T$ wi... | 3 | https://mathoverflow.net/users/85592 | 415152 | 169,266 |
https://mathoverflow.net/questions/415151 | 4 | Let $f=(u,v)\in \mathscr{D}'(U,\mathbb{C})$ be a distribution, where $U\subset\mathbb{C}=\mathbb{R}^2$ is an open set and $u$ and $v$ are the projection of $f$ onto the real and imaginary axis (ie $\langle f,\phi\rangle=\langle u,\phi\rangle+i\langle v,\phi\rangle$). Suppose that
$$
\frac{\partial}{\partial \overline{z... | https://mathoverflow.net/users/351083 | $\frac{\partial f}{\partial \overline{z}}=0$ in distributional sense implies $f$ is holomorphic | I've just realized that, if $f$ is in $L^1\_{loc}$ and not just in $\mathscr{D}'$, my question can be answered using Weyl's lemma for harmonic functions. Indeed, from $\frac{\partial f}{\partial \overline{z}}=0$ it easily follows that $\Delta u=\Delta v=0$, and then Weyl's lemma implies that $u$ and $v$ are smooth. But... | 4 | https://mathoverflow.net/users/351083 | 415153 | 169,267 |
https://mathoverflow.net/questions/415159 | 15 | $\DeclareMathOperator\Hom{Hom}$Fix a commutative ring $R$. For $R$-modules $M$ and $N$, there is an inclusion of abelian groups $\Hom\_R(M,N) \to \Hom\_{\mathbb{Z}}(M,N).$ Are there conditions on $R$ that will ensure this is an inclusion of a direct summand (preferably with a section natural in $M$ and $N$)?
(I have ... | https://mathoverflow.net/users/131360 | R-module hom a direct summand of Z-module hom? | There exists a natural section if and only if $R$ is *separable* over $\mathbb Z$ (more generally over $k$, if you have a $k$-algebra $R$ and are asking about the morphism $\hom\_R(M,N)\to \hom\_k(M,N)$). If you're only asking for a section, no naturality, then I don't know any reasonable condition, but maybe someone c... | 21 | https://mathoverflow.net/users/102343 | 415162 | 169,269 |
https://mathoverflow.net/questions/415165 | 2 | Consider an arbitrary finite set of orthogonal-projection matrices (symmetric, idempotent, etc.) in $\mathbb{R}^{n\times n}$.
We draw two matrices $Q,P$ uniformly and i.i.d. from this set.
**Question:** Is the following expected matrix $\mathbb{E}\_{P,Q} \left[I-2P+QPQ\right]$ necessarily positive semi-definite?
... | https://mathoverflow.net/users/100796 | Expected matrix created from two random orthogonal-projection matrices | Found a counter example.
If we take the set to be $\left\{
\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix},
\frac{1}{2}\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}
\right\}$, then we have
$\mathbb{E}\_{P,Q} \left[I-2P+QPQ\right]=
\frac{1}{16}\begin{bmatrix}1 & -5 \\ -5 & 11\end{bmatrix}$, whose eigenvalues are $-0.06694174,... | 0 | https://mathoverflow.net/users/100796 | 415166 | 169,270 |
https://mathoverflow.net/questions/415145 | 1 | Suppose we have an equivalence of triangulated categories $\Phi : \mathcal{A} \to \mathcal{B}$. Let $G$ be a finite group. Are there any methods/conditions for specifying when one has an induced equivalence $\Phi^G : \mathcal{A}^G \to \mathcal{B}^G$ of the associated $G$-equivariant categories?
The particular case I'... | https://mathoverflow.net/users/142073 | Inducing an equivalence of $G$-equivariant categories | You need some assumptions. But in the case of admissible subcategories, if their equivalence is given by a Fourier-Mukai functor $\Phi\_K$ and the object $K \in D^b(X \times X')$ is $G$-equivariant, then $\Phi^G$ can be defined.
See arXiv:1403.7027, Theorem 6.9 for a general statement of this sort.
| 0 | https://mathoverflow.net/users/4428 | 415168 | 169,271 |
https://mathoverflow.net/questions/415182 | 1 | This seemingly simple problem is doing my head in. I have tried doing a proof by induction over the edges of the graph but I just cannot seem to get it to work.
I am trying to prove that in a finite directed graph $G = (V, E)$ that obeys the following rules, any mode number of outgoing edges cannot exceed 2.
The ru... | https://mathoverflow.net/users/82417 | How to prove that the modal number(s) of outgoing edges cannot exceed 2? | The sum of outdegrees is at most $n$. Thus if certain value $d\geqslant 3$ appears, say, $k>0$ times, the value 0 must appear at least $(d-1)k>k$ times, and do $d$ is not modal.
| 4 | https://mathoverflow.net/users/4312 | 415185 | 169,275 |
https://mathoverflow.net/questions/415187 | 2 | Let $A$ and $B$ denote $C^{\ast}$-algebras. Let $\lVert\cdot\rVert\_h$ and $\lVert\cdot\rVert\_{\text {max}}$ denote the Haagerup norm and max $C^\*$-norms on $ A \otimes B$, respectively. I am looking for a reference/proof for the following result:
>
> $\lVert\cdot\rVert\_{\text{max}} \leq \lVert\cdot\rVert\_h$.
>... | https://mathoverflow.net/users/129638 | Need reference for: $\lVert\cdot\rVert_{\text{max}} \leq \lVert\cdot\rVert_h$ | Let $v \in A \otimes B$, where $A\otimes B$ is the algebraic tensor product.
The Haagerup norm of $v$ is the infimum of the expressions $\sqrt{\|\sum\_{i=1}^{r} x\_{i} x\_{i}^{\ast}\|}\cdot \sqrt{\|\sum\_{i=1}^{r} y\_{i}^{\ast} y\_{i} \|}$, taken over all decompositions $v=\sum\_{i=1}^{r} x\_{i} \otimes y\_{i}$ ($r$ ... | 7 | https://mathoverflow.net/users/24953 | 415195 | 169,277 |
https://mathoverflow.net/questions/415096 | 5 | Classification of simple (or simple-restricted) Lie algebras over algebraically closed fields in positive characteristic is studied for a long time. Today, we know all finite-dimensional simple (or simple-restricted) Lie algebras over algebraically closed fields of characteristic $p \ge 5$. But, how about over **finite... | https://mathoverflow.net/users/167704 | Classification of simple Lie algebras over finite fields | The classification of simple finite-dimensional Lie algebras over finite fields is a very hard task and only few results are known. However, I suggest to have a look at the following paper by Bettina Eick and references therein: [Some new simple Lie algebras in characteristic 2](https://doi.org/10.1016/j.jsc.2010.05.00... | 4 | https://mathoverflow.net/users/14653 | 415206 | 169,282 |
https://mathoverflow.net/questions/415203 | 4 | I came across this equation in my research (related to reaction diffusion system):
$$\frac{d^2y}{dr^2}+B\operatorname{sech}^2(r) \frac{dy}{dr} + Cy = 0$$
where $B$ and $C$ are constants. Can it be solved analytically?
| https://mathoverflow.net/users/476359 | Non-constant-coefficient second-order linear ODE | Making the change of the independent variable $x=e^{it}$, we obtain
$$x^2w''+xw'+\frac{4iBx^3}{(x^2+1)^2}w'-Cw=0.$$
This equation has $2$ regular singularities $0,\infty$, and irregular
at $\pm i$. Therefore
for generic $B$ and $C$ its solutions cannot be expressed in terms of functions
of hypergeometric type, or any o... | 8 | https://mathoverflow.net/users/25510 | 415216 | 169,285 |
https://mathoverflow.net/questions/415225 | 5 | Say we're working in a symmetric monoidal $\infty$-category $\mathcal{S}$, and $A$ is an associative algebra in it. For instance,
$$\mathcal{S}\ =\ \text{dg vector spaces},\ \ \ A\ =\ \text{a dg algebra}.$$
Then:
1. $\text{Mod}(A)=\text{Mod}\_\mathcal{S}(A)$ is just a plain category, no extra structure (i.e. an $E\_0... | https://mathoverflow.net/users/119012 | $\text{Mod}(A)$ is an $E_n$ category $\Leftrightarrow$ $A$ is an ??? algebra | $E\_0$ is not quite "no extra structure" : you know who $A$ is inside $Mod(A)$, so it's a *pointed* category (more generally, an $E\_0$ object in $\mathcal S$ is an object with a "unit" $\mathbb 1\to X$).
For a different version of your question, the answer is "$A$ is $E\_{n+1}$ iff $Mod(A)$ is $E\_n$", where "iff" c... | 8 | https://mathoverflow.net/users/102343 | 415233 | 169,290 |
https://mathoverflow.net/questions/415232 | 1 | Let $k$ be a field and $A$ a finite dimensional $k$-algebra. Given a sequence of inclusions $M\_1 \subseteq M\_2 \subseteq \dots$ of $A$-modules consider the direct limit $M:= \bigcup\_{i=1}^\infty M\_i$. For a finite dimensional module $X$ suppose we have $X\subseteq M$. Then $X$ is generated by finitely many elements... | https://mathoverflow.net/users/145920 | Epimorphism going out of an inverse limit into a finite dimensional module | The answer is no. First, let me point out that the $A$ here plays no role: because each $M\_{i+1}\to M\_i$ is an epimorphism, $M\to M\_i$ is one too, and so if there is a $k$-linear factorization $M\to M\_i\to X$, then it is automatically $A$-linear.
So we can simply focus on $k$-vector spaces and worry about the ana... | 2 | https://mathoverflow.net/users/102343 | 415238 | 169,292 |
https://mathoverflow.net/questions/415237 | 1 | Let $M$ be a compact riemannian manifold equipped with a geodesic distance and let $\mathcal{B}(M)$ be the borel sigma algebra generated by the geodesic distance. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space that arises from a compact metric space $(\Omega,d)$ and $\mathcal{F}$ is the borel sigma algebr... | https://mathoverflow.net/users/297294 | Extension of measurable function from dense subset | $\newcommand\R{\mathbb R}\newcommand\om\omega\newcommand\Om\Omega\newcommand\tom{{\tilde\omega}}\newcommand\tOm{\tilde\Omega}\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}\newcommand\ol\overline\newcommand{\N}{\mathbb N}\newcommand{\Q}{\mathbb Q}$Take any $m\in\text{Im}(f)$ and then let... | 1 | https://mathoverflow.net/users/36721 | 415242 | 169,294 |
https://mathoverflow.net/questions/415244 | 2 | By trying to extend certain limit properties of sequences from compact subsets to the entire set, I cam up with something that can be formed in the following question.
Let $a\_{mn}$ be a double sequence of nonnegative real numbers. I want to be able to switch order of iterated limits in the following form
$$\limsup\_... | https://mathoverflow.net/users/121671 | Switching order of limits in double sequences | $a\_{mn}=1(m\ge n)$ is a counterexample. (Here, $1(A)$ is the indicator of an assertion $A$. That is, $1(A):=1$ if $A$ is true and $1(A):=0$ if $A$ is false.)
If you insist on understanding "increasing" in the strict sense, then $a\_{mn}=1(m\ge n)-1/m-1/n$ is a counterexample.
| 7 | https://mathoverflow.net/users/36721 | 415245 | 169,295 |
https://mathoverflow.net/questions/415246 | 2 | I am looking for a reference to the Epstein zeta function. For the Riemann zeta function, there is Titchmarsh's treatment. However, I do not know of any references regarding the Epstein zeta function and I'll be thankful to anyone who points me to a seminal paper/book that talks about properties of the Epstein zeta fun... | https://mathoverflow.net/users/392272 | Resources and outstanding conjectures about the Epstein zeta function | I would recommend Harold Stark's papers on the Epstein Zeta function, such as "[L-functions and character sums for quadratic forms, I](https://doi.org/10.4064/aa-15-3-307-317)", Acta Arith. vol 14 (1967/68), PP. 35–50, and "[On the zeros of Epstein's zeta function](https://doi.org/10.1112/S0025579300008007)", Mathemati... | 4 | https://mathoverflow.net/users/6756 | 415248 | 169,296 |
https://mathoverflow.net/questions/415250 | 2 | The starting point of my question is the following fact: suppose $G$ is a finite group and let $H,K \leq G$ be arbitrary subgroups, then there exists an isomorphism of $G$-sets as follows
\begin{equation}
G/H \times G/K \cong \coprod\_{H \backslash G/K}G/(H^g\cap K).
\end{equation}
The proof is not difficult: given an ... | https://mathoverflow.net/users/131453 | Double coset decomposition for compact Lie groups | Your question (1) may not be true as stated. One approach that I like is to think of the product of two $G$-CW complexes first as a $(G\times G)$-CW complex, which works because $G/H\times G/K \approx (G\times G)/(H\times K)$. Then you can consider it as a $G$-space by restricting along the diagonal $G\to G\times G$. H... | 4 | https://mathoverflow.net/users/58888 | 415260 | 169,300 |
https://mathoverflow.net/questions/415236 | 3 | Theorem 1 of Milnor's paper "A procedure for killing homotopy groups of differentiable manifolds" states that two manifolds are in the same cobordism class if and only if they can be obtained from each other by a sequence of surgeries (meaning they are $\chi$-equivalent, following Milnor's terminology).
In his proof ... | https://mathoverflow.net/users/475761 | A question on the proof of Theorem 1 in Milnor's "Killing Homotopy Groups" | If you want to avoid the language of attaching handles there is the treatment in Milnor's Lectures on the h-cobordism theorem. He proves that a cobordism with a Morse function with only one critical point (an `elementary cobordism') gives rise to a surgery, which is the statement you're looking for. To a modern reader,... | 3 | https://mathoverflow.net/users/3460 | 415267 | 169,303 |
https://mathoverflow.net/questions/415221 | 2 | Let $G$ be a $d$-regular graph. Let $d= \lambda\_1 \ge \lambda\_2 \ge \dots \ge \lambda\_n \ge -d $ be the eigenvalues of the adjacency matrix of $G$, and set $\lambda = \max (|\lambda\_2| , |\lambda\_n|) $ We say that $G$ is a *Ramanujan graph* if $\lambda \le 2 \sqrt{d-1}$.
[According to Wikipedia](https://en.wikip... | https://mathoverflow.net/users/4558 | Ramanujan graphs from varieties over finite fields | The assumptions mean that the eigenvalues of the Cayley graph are of the form $\sum\_{x\in S} \psi (x)$ for $\psi \colon \mathbb F\_q\to \mathbb C^\times$ an additive character, which is $\sum\_{x\in \mathbb F\_q}\psi(f(x))$ since $S$ is taken with multiplicity.
For sums of this type, we have the bound
$$\left|\sum\_... | 3 | https://mathoverflow.net/users/18060 | 415271 | 169,305 |
https://mathoverflow.net/questions/415266 | 1 | So, I've been reading about the lonely runner conjecture, but I have to admit that my knowledge of diophantine approximation is so limited that I wouldn't be able to properly explain the term.
So let's consider the simplest case, which is $n$ "real" "linear" runners in a circle ($=\mathbb R/\mathbb Z$).
Let me try ... | https://mathoverflow.net/users/58976 | The lonely runner conjecture and equidistribution on tori | A corrected version of this argument is contained in Section 4 of [Six Lonely Runners](https://doi.org/10.37236/1602) by Bohman, Holzman, and Kleitman.
Their argument shows, using equidistribution, that the case of the lonely runner conjecture with rational speeds implies the general case.
| 5 | https://mathoverflow.net/users/18060 | 415276 | 169,307 |
https://mathoverflow.net/questions/415284 | 3 | As in [Deformations of Hirzebruch surfaces and toric action](https://mathoverflow.net/questions/69339/deformations-of-hirzebruch-surfaces-and-toric-action),
the Hirzebruch surface $F\_n$ can be deformed into $F\_{n-2m}$ ($0<2m\leq n$) under the fibration given by
$$
M=\{([x\_0:x\_1],[y\_0:y\_1:y\_2],t)\in \mathbb{P}^1\... | https://mathoverflow.net/users/476418 | Where's the negative section of a deformation of a Hirzebruch surface? | You can take $y\_0 = 0$, $y\_1 = -tx\_1^m$, $y\_2 = x\_0^m$.
EDIT. Here is a simple computation of the normal bundle of the curve
$$
C\_t = \{([x\_0:x\_1],[0:-tx\_1^m:x\_0^m])\} \subset \mathbb{P}^1 \times \mathbb{P}^2
$$
in the surface
$$
S\_t = \{x\_0^ny\_1−x\_1^ny\_0+tx\_0^{n−m}x\_1^my\_2=0\}
$$
for $t \ne 0$. Fir... | 6 | https://mathoverflow.net/users/4428 | 415287 | 169,309 |
https://mathoverflow.net/questions/415277 | 2 | Consider the category of chain complexes over a ring $R$.
We can show that $\text{Ext}^1(M, N)$ classifies extensions using the triangulated category structure: the homotopy kernel of a map $N \rightarrow M[1]$, $K$, gives rise to the designated triangle $M \rightarrow K \rightarrow N$. This gives a simple picture of... | https://mathoverflow.net/users/30211 | Yoneda Ext theorem and extensions | We have that $\mathrm{Ext}^n(M, N) = \mathrm{Hom}(M, N[n])$, in other words, an element of $\mathrm{Ext}^n(M, N)$ is a map
$$
M \longrightarrow I\_N[n]
$$
where $N \to I\_N$ is an injective resolution. From this we obtain a distinguished triangle,
$$
M \longrightarrow I\_N[n] \longrightarrow C \overset{+}\longrightarro... | 3 | https://mathoverflow.net/users/6348 | 415295 | 169,311 |
https://mathoverflow.net/questions/412526 | 2 | The following paper is about how a K-flow is produced from a K-induced map, but it is written in Russian. Does someone know where to find its English version? Do some textbooks include this topic?
B. Gurevič, [Certain conditions for the existence of K-decompositions for special flows](http://mi.mathnet.ru/eng/mmo192)... | https://mathoverflow.net/users/124254 | K-flows reference | The Gurevich paper you cite was translated as "Some Existence Conditions for K-Decompositions for Special Flows" in Trans. Moscow Math. Soc. 17 (1967), 99-128 (if you don't have access to a university library I can send a copy to you; you can find my university email address by searching my name). There is also the Tot... | 2 | https://mathoverflow.net/users/66883 | 415315 | 169,323 |
https://mathoverflow.net/questions/415312 | 4 | In my earlier (soft) [MO post](https://mathoverflow.net/questions/415230/on-a-variation-of-the-vandermonde-matrix), an elementary [response](https://mathoverflow.net/questions/415230/on-a-variation-of-the-vandermonde-matrix#comment1064945_415230) was given by Ofir Gorodetsky in regard to the determinant of the symbolic... | https://mathoverflow.net/users/66131 | Yet, another numerical variant of the Vandermonde matrix | This is true and this is not about numbers $j^j$ at all. Consider a more genral matrix $(c\_j-i^j)\_{1\leqslant i,j\leqslant n}$. Denote $f\_j(x)=c\_j-x^j$, and find the numbers $\beta$, $\alpha\_1,\ldots,\alpha\_{n-1}$ such that $f\_n(x)+\sum\_{j=1}^{n-1}\alpha\_j f\_j(x)=\beta$ for all $x\in \{1,2,\ldots,n\}$. Then t... | 6 | https://mathoverflow.net/users/4312 | 415317 | 169,325 |
https://mathoverflow.net/questions/415328 | 1 | For the Lie algebra $\frak{sl}\_{n+1}$ we denote its fundamental irreducible representations by $V(\pi\_i)$, with $i=1, \dots, n$. Where can I find a table of the character formula (in other words a list of all the non-zero weight spaces of $V(\pi\_i)$ together with multiplicities)?
P.S. From this answer
[Fundament... | https://mathoverflow.net/users/378228 | Character formula for the fundamental representations of $\frak{sl}_n$ | These are examples of so called *minuscule* weights which are defined through the property that all the weights are given by $w\lambda$ where $\lambda$ is the highest weight and $w$ is an element of the Weyl group. So, in the $\mathfrak{sl}\_{n+1}$, to get all the weights in "the $\epsilon$ basis" you just take all pos... | 4 | https://mathoverflow.net/users/6818 | 415329 | 169,327 |
https://mathoverflow.net/questions/415341 | 4 | A positive integer $n$ is called $k$-rough if all of the prime factors of $n$ strictly exceed $k$.
For $k$ fixed and $n$ large, what's the shortest interval $[n,n+t]$ that contains a $k$-rough number ?
| https://mathoverflow.net/users/476460 | Rough numbers in short interval | Being $k$-rough is the same as being relatively prime to the product of primes up to $k$. Hence if $P(k)$ denotes that product, and $j$ denotes the [Jacobsthal function](https://oeis.org/wiki/Jacobsthal_function), then $t=j(P(k))-1$ works for every $n$, while $t=j(P(k))-2$ fails for infinitely many $n$'s. In particular... | 7 | https://mathoverflow.net/users/11919 | 415342 | 169,332 |
https://mathoverflow.net/questions/415339 | 3 | Let $k = \mathbb{F}\_q$ be a finite field with $q$ elements and $Q\subset\mathbb{P}^n\_k$ a quadric hypersurface defined over $k$.
By the Chevalley-Warning theorem if $n\geq 2$ then $Q$ has a point. Is there a formula for the number of points of $Q$ or at least a bound of the type $\#Q(k)\geq f(q,n)$ where $f$ is a f... | https://mathoverflow.net/users/14514 | Number of points of a quadric hypersurface over a finite field | Yes. The number of points of a smooth quadric hypersurface is $\frac{q^n-1}{q-1}$ if $n$ is even or $\frac{q^n-1}{q-1} \pm q^{ \frac{n-1}{2}}$ if $n$ is odd.
This can be proven in a number of ways. Here is a (perhaps extravagantly) geometric approach:
Fix a point $x$. A line through $x$ contains one other points of... | 8 | https://mathoverflow.net/users/18060 | 415348 | 169,334 |
https://mathoverflow.net/questions/415192 | 0 | Let $(\Omega, \mathcal F, \mu)$ be a standard probability space.
**Question:** For each $f \in L^\infty (\Omega)$, does there exist an ergodic measure preserving transformation $T: \Omega \to \Omega$ such that the following expression is maximised?
$$\lim\_{n \to \infty} \frac{1}{n} \sum\_{k = 1}^n\int\_\Omega |T^k... | https://mathoverflow.net/users/173490 | Entropy maximising ergodic transformation | I don't know if you're still interested now that the average collapses, but I think that there is such a maximizing invertible ergodic $T$ as long as you assume that your probability space is Lebesgue.
Under these assumptions, if you normalize so that the median of $f$ is $0$ (i.e. so that $\mu(f < 0) = \mu(f > 0)$),... | 2 | https://mathoverflow.net/users/116357 | 415351 | 169,335 |
https://mathoverflow.net/questions/415125 | 3 | Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.
**Question:** Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B\_t = A\_t \}) = 0$$
almost surely?
| https://mathoverflow.net/users/173490 | Intersection of Brownian motion and finite variation process | Fix a finite interval $I$.It suffices to show that almost surely,
$$\mathcal L(\{t \in I \, | \, B\_t = A\_t \}) = 0 \,.$$
Brownian motion restricted to $\{t \in I \, | \, B\_t = A\_t \}$ has bounded variation, so a positive answer is implied by the following stronger result, a special case of Theorem 1.3 of [1]:
**T... | 4 | https://mathoverflow.net/users/7691 | 415358 | 169,338 |
https://mathoverflow.net/questions/415381 | 5 | Let $A$ be a finite-dimensional algebra over a field $F$. A representation $M$ of $A$ is called **absolutely irreducible** if $M\otimes\_FE$ is irreducible as a representation of $A\otimes\_FE$ for all field extensions $E$ over $F$.
A field $F$ is said to be a **splitting field** of $A$ if any irreducible representat... | https://mathoverflow.net/users/56989 | Absolutely irreducible representation and splitting field | Yes, $F$ is a splitting field of $A$. Put $A\_E = A\otimes\_F E$ for convenience. Let $S'$ be a simple $A$-module. Then $S'\otimes\_F E$ has a simple $A\_E$-submodule, which by hypothesis is of the form $S\otimes\_F E$ for some simple $A$-module $S$. Note that $$0\neq \hom\_{A\_E}(S\otimes\_F E,S'\otimes\_F E)\cong \ho... | 3 | https://mathoverflow.net/users/15934 | 415386 | 169,343 |
https://mathoverflow.net/questions/415383 | 2 | MOTIVATION.
Let $f:[0,+\infty)\to \mathbb{R}$.
Solutions to $\partial\_tf(t) = -\lambda f(t)$, $f(0)\neq 0$ approach zero *exactly* as $e^{-\lambda t}$.
This property is preserved if we apply an exponentially decaying perturbation, i.e. we consider
$$\partial\_t f(t) = -\lambda f(t) +R(t) f(t)$$
where $|R(t)|< Ce^{-\... | https://mathoverflow.net/users/99042 | Decay rate for a small perturbation of a simple linear ODE | $\newcommand\R{\mathbb R}$The answer is no.
E.g., suppose that $H=\R^2$, with the standard basis $(e\_1,e\_2)$, $Ae\_1=3e\_1$, $Ae\_2=e\_2$, $R(t)e\_1=0$, $R(t)e\_2=e^{-t}e\_1$, and $f(0)=(0,1)$. So, $(e\_1,e\_2)$ is an ortho-eigenbasis of $A$, with the corresponding eigenvalues $3$ and $1$.
However, $f(t)\cdot e\_... | 1 | https://mathoverflow.net/users/36721 | 415398 | 169,346 |
https://mathoverflow.net/questions/415395 | 9 | There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones.
The six orientable ones are determined by their holonomy groups
$$
C\_1,C\_2,C\_3,C\_4,C\_6
$$
and
$$
C\_2 \times C\_2
$$
The five with cyclic holonomy... | https://mathoverflow.net/users/387190 | Compact flat orientable 3 manifolds and mapping tori | The remaining orientable manifold is called the Hantzsche-Wendt manifold $M^{HW}$, and is not a mapping torus over the Klein bottle. It has first homology $H\_1(M^{HW}, \mathbb{Z}) = \mathbb{Z}\_4 \times \mathbb{Z}\_4$. Any mapping torus $MT(M, f)$ of a manifold $M$ via the map $f: M \to M$ has $\pi\_1(MT(M, f)) \cong ... | 4 | https://mathoverflow.net/users/136267 | 415400 | 169,347 |
https://mathoverflow.net/questions/415170 | 2 | Let $R$ be a commutative Noetherian ring and $I$ is a proper ideal of $R$. suppose that $M$ is a f.g. $R$-module.
$\DeclareMathOperator\Ext{Ext}$I'm looking for an example that has this property:
$$\Ext^i\_R(M,R/I)\neq 0$$ for all $i$ and there exists $n$ such that for all $i>n$
$$\Ext^i\_R(R/I,R/I)=0.$$
I don'... | https://mathoverflow.net/users/66837 | Example of non vanishing Ext | Take $R$ a local artinian non-Gorenstein ring, $I=0$, $M$ the residue field of $R$. The vanishing of $Ext^i\_R(R/I,R/I)=Ext^i\_R(R,R)$ for $i>0(=:n)$ is immediate, while for the non-vanishing of $Ext^i\_R(M,R/I)=Ext^i\_R(k,R)$ see e.g. Bourbaki, Algèbre Commutative, chap. X, $\S$ 3, n. 7, Lemme 2.
| 4 | https://mathoverflow.net/users/92322 | 415405 | 169,351 |
https://mathoverflow.net/questions/403152 | 4 | Let $M\_0^4$ and $M\_1^4$ be two closed smooth 4-manifolds and let $M$ be an $h$-cobordism between them (i.e., a compact smooth 5-manifold with boundary the disjoint union of $M\_0$ and $M\_1$ and with the inclusions of $M\_0$ and $M\_1$ into $M$ both being homotopy equivalences). In the case where $M\_0$ and $M\_1$ ar... | https://mathoverflow.net/users/99414 | h-cobordisms between non-simply-connected 4-manifolds | Let $X$ be a smooth, closed 4-manifold. Every element of $\operatorname{Wh}(\pi\_1(X))$ can be realised, for some $k \in \mathbb{N}$, as the Whitehead torsion $\tau(W,X \#^k S^2 \times S^2) \in \operatorname{Wh}(\pi\_1(X))$ of a smooth $h$-cobordism $(M;X\#^k S^2 \times S^2,Y)$, for some $Y$. The Whitehead group is non... | 4 | https://mathoverflow.net/users/102454 | 415407 | 169,352 |
https://mathoverflow.net/questions/415396 | 4 | I first asked this question at [math.stackexchange](https://math.stackexchange.com/questions/4373073/contact-variety-is-equidimensional) with no success, so I decided to repost it here.
I am reading the paper "Weakly Defective Varieties" by L. Chiantini and C. Ciliberto, available [here](https://www.ams.org/journals/... | https://mathoverflow.net/users/131868 | Contact variety to projective variety is equidimensional | I think that the monodromy argument they refer to is not entirely obvious. It is presumably related to a version of the *uniform position principle* but in the context of tangent hyperplanes, which I guess involves some intricate technical details. Let me instead sketch a proof of their statement that avoids monodromy.... | 4 | https://mathoverflow.net/users/37214 | 415408 | 169,353 |
https://mathoverflow.net/questions/415441 | 11 | Sheffer polynomials $\{P\_n(x)\}$ have generating function $P(x,t) = \sum\_{n=0}^{\infty}P\_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra correspondence. Is there any connection to Lie theory, obvious or not?
(I've deleted my reference to orthogonal polynomials, which was incorrect and i... | https://mathoverflow.net/users/94182 | How are Sheffer polynomials related to Lie theory? | First, a general set of Sheffer polynomials is not orthogonal with respect to some weight function; for example, the prototypical sequence $p\_n(x) = x^n$, which belongs to both special sub-groups of Sheffer polynomials—the Appell and binomial Sheffer polynomials—with the e.g.f. $e^{xt}$, is not an orthogonal set—neith... | 13 | https://mathoverflow.net/users/12178 | 415451 | 169,361 |
https://mathoverflow.net/questions/415401 | 1 | Given a (bounded) sequence $\{q\_n\}\_{n\geq 0}$ such that $\lvert q\_n\rvert \leq 1$ for all $n \geq 0$ and $\sum\_{n\geq 0} q\_n = 0$. We can impose the condition that $\sum\_{n\geq 0} \lvert q\_n\rvert \leq 2$ as well. I am wondering whether there exists a fixed constant, independent of the sequence $\{q\_n\}$, such... | https://mathoverflow.net/users/163454 | A discrete version of Poincaré's inequality | You can duplicate the usual proof of Hardy type inequalities to the discrete case.
Suppose $\{q\_n\}$ is an eventually 0 sequence (you can weaken this to $\lim\_{n\to \infty} n^{1/2} q\_n = 0$). Then by telescoping you have (all sums are over $n\geq 0$)
$$ \sum (n+1) q\_{n+1}^2 - n q\_{n}^2 = 0 $$
Rewrite as
$$... | 5 | https://mathoverflow.net/users/3948 | 415453 | 169,362 |
https://mathoverflow.net/questions/415476 | 5 | There is the following exercise in [Vershynin's book on High-Dimensional Probability](https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html).
**Exercise 3.1.6:**
Let $X = (X\_1, \dots, X\_n) \in \mathbb{R}^n$ be a random vector with independent coordinates $X\_i$ that satisfy $\mathbb{E}[X\_i^2] = 1$ and $... | https://mathoverflow.net/users/37267 | Variance of the norm of a random variable under finite-moment assumptions | $\newcommand\R{\mathbb R}
\renewcommand{\P}{\operatorname{\mathsf P}}\newcommand\E{\operatorname{\mathsf{E}}}\newcommand\Var{\operatorname{\mathsf{Var}}}$The best bound on $\Var \|X\|\_2$ is about $n$.
Indeed,
\begin{equation\*}
\Var \|X\|\_2\le \E \|X\|\_2^2=\sum\_{i=1}^n \E X\_i^2=n. \tag{1}\label{1}
\end{equation... | 5 | https://mathoverflow.net/users/36721 | 415478 | 169,371 |
https://mathoverflow.net/questions/415477 | 1 | Say we generate an $N \times N$ sparse random matrix $W$, where each element $W\_{ij}$ was independently chosen to be $1$ with probability $p=\frac{a}{N}$, and $0$ with probability $1-p$. We are interested in the case where $N \rightarrow \infty$.
I wonder what happens if we permute the rows and the columns of $W$ so... | https://mathoverflow.net/users/173974 | Permute a sparse random matrix to resemble a diagonal matrix as much as possible | When $a=1$, the probability to get a permutation matrix is
$$N! \left(\frac{1}{N}\right)^N \left(1-\frac{1}{N}\right)^{N^2-N} < \frac{N!}{N^N}e^{-(N-1)} < \sqrt{2\pi N}\,e^{-2N + 1 + \frac1{12N}}.$$
So, it's not high, but rather tends to 0 exponentially.
| 2 | https://mathoverflow.net/users/7076 | 415479 | 169,372 |
https://mathoverflow.net/questions/415448 | 4 | I'm interested in whether there exists a well known class of finite groups that generates all finite groups through quotienting.
I will be more formal: does there exist a class of ("nice") well known finite groups $\{G\_n\}$ such that for any finite groups $G$ we have that there exists $G\_n$ and a normal subgroup $N... | https://mathoverflow.net/users/155306 | Families of finite groups of which every finite group is a quotient | It obviously depends a lot on your definition of "nice", but one class defined by a reasonably nice property, while also relevant in theory, is the class of all complete groups (where complete means that both the center and the outer automorphism group are trivial). That every finite group is a quotient of a complete g... | 6 | https://mathoverflow.net/users/127660 | 415483 | 169,374 |
https://mathoverflow.net/questions/415482 | 3 | Let $k$ be a positive integer. Note that $a/b=ab^{k-1}/b^k$ for any integers $a$ and $b>0$.
If every $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x\_1^k+\cdots+x\_{s}^k$ with $x\_1,\ldots,x\_s\in\mathbb N$, then each $r\in\mathbb Q\_{\ge0}$ can be written as $x\_1^k+\cdots+x\_s^k$ with $x\_1,\ldots,x\_s\in\mathb... | https://mathoverflow.net/users/124654 | Waring's problem over $\mathbb Q_{\ge0}$ | I can answer the last question: $s(4)$ equals $15$, not $5$. Variables below will denote integers.
First we show that $s(4)\leq 15$. We use the result of [Davenport (1939)](http://djvu.org/) that there exists $m\geq 1$ such that every positive integer outside $\{16^h k:\text{$h\geq 0$ and $k\leq m$}\}$ is a sum of $1... | 13 | https://mathoverflow.net/users/11919 | 415489 | 169,375 |
https://mathoverflow.net/questions/415508 | 3 | Given 2-tangles $T\_1,T\_2\subset B^3$ with their endpoints at some fixed points NW, NE, SW, SE of $\partial B^3$ we can glue them along $\partial B^3$ to obtain a link $L=T\_1\cup T\_2\subset S^3.$
**Q:** Does that link $L$ together with $T\_1$ determine the homeomorphism class of $(B^3,T\_2)$?
Remarks
1. Note t... | https://mathoverflow.net/users/23935 | Glueing two 2-tangles | No, $L$ and $T\_1$ together do not determine $T\_2$. Take a non-trivial knot $K$ and suppose that $L$ is $K\#K$, and $T\_1$ is the trivial tangle. Then $T\_2$ can certainly be:
* a 2-tangle with one boundary-parallel component, and one with a boundary-parallel component connected-summed with $L$;
* a 2-tangle with tw... | 2 | https://mathoverflow.net/users/13119 | 415515 | 169,385 |
https://mathoverflow.net/questions/304644 | 9 | The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely
\begin{equation}
\chi=\textrm{#vertices}-\textrm{#edges}+\textrm{#faces}-\ldots
\end{equation}
Are there any other (independent) invariants t... | https://mathoverflow.net/users/115363 | Are there invariants of cell complexes similar to the Euler characteristic? | As pointed out in the comments, every characteristic class in $H^d(BO(d), G)$ provides a $G$-valued locally computable invariant of $d$-manifolds, by pulling back via the classifying map of the tangent bundle. As argued by [Levitt and Rourke](https://www.ams.org/journals/tran/1978-239-00/S0002-9947-1978-0494134-6/S0002... | 2 | https://mathoverflow.net/users/115363 | 415526 | 169,390 |
https://mathoverflow.net/questions/415469 | 18 | I am trying to understand some things about [Condensed Mathematics](https://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/condensed+mathematics) and the [Liquid Tensor Experiment](https://xenaproject.wordpress.com/2021/06/05/half-a-year-of-the-liquid-tensor-experiment-amazing-developments/). The aim of the LTE is to ... | https://mathoverflow.net/users/10366 | Breen-Deligne packages and the liquid tensor experiment | The comments have already given the answers, but let me assemble them here with my account of the story.
When Scholze first posted the [Liquid Tensor Experiment](https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-experiment/), it was quickly identified (by both Peter and Reid, somewhat independently I think) ... | 27 | https://mathoverflow.net/users/21815 | 415540 | 169,396 |
https://mathoverflow.net/questions/415514 | 8 | Cross posting from MSE, where [this question](https://math.stackexchange.com/q/4369960/643521) received no answers.
The following Latin square
$$\begin{bmatrix}
1&2&3&4&5&6&7&8\\
2&1&4&5&6&7&8&3\\
3&4&1&6&2&8&5&7\\
4&3&2&8&7&1&6&5\\
5&6&7&1&8&4&3&2\\
6&5&8&7&3&2&4&1\\
7&8&5&2&4&3&1&6\\
8&7&6&3&1&5&2&4
\end{bmatrix}... | https://mathoverflow.net/users/144261 | Latin squares with one cycle type? | One way to achieve the required property is to construct a Latin square whose autotopism group acts transitively on unordered pairs of rows. This can be achieved for orders that are a prime power congruent to 3 mod 4, by means of the quadratic orthomorphism method, described [in this paper](https://www.combinatorics.or... | 6 | https://mathoverflow.net/users/351290 | 415541 | 169,397 |
https://mathoverflow.net/questions/415532 | 14 | Let $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$. Then
$$A-A:=\{a-b:\ a,b\in A\}=\{u^4+v^4-x^4-y^4:\ u,v,x,y\in\mathbb Q\}.$$
Motivated by [Question 415482](https://mathoverflow.net/questions/415482), here I ask the following question.
**Question.** Is it true that $A-A=\mathbb Q$? Any effective way to approach it？
By my co... | https://mathoverflow.net/users/124654 | Does $A-A=\mathbb Q$ hold for $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$? | According to Tito Piezas's website [$x^4+y^4-(z^4+t^4) = N$](https://sites.google.com/site/tpiezas/001b),
There is an identity
$((2a+b)c^3d)^4 + (2ac^4-bd^4)^4 - (2ac^4+bd^4)^4 - ((2a-b)c^3d)^4 = a(2bcd)^4$
where $b = c^8-d^8$, for arbitrary {$a,c,d$}.
| 19 | https://mathoverflow.net/users/150249 | 415560 | 169,403 |
https://mathoverflow.net/questions/415522 | 16 | As the title suggests, I am a physicist and have a question about how to show certain superelliptic curves are also hyperelliptic. The superelliptic Riemann surfaces in question has the form $$w^n = \prod\_{\alpha=1}^N(z-u\_\alpha)(z-v\_\alpha)^{n-1},\quad u\_1<v\_1<\dotsb<u\_N<v\_N$$ and is of interest for certain que... | https://mathoverflow.net/users/476604 | From a physicist: How do I show certain superelliptic curves are also hyperelliptic? | This curve is not hyperelliptic unless $n=2$ or $N=2$.
First, note that it is more convenient to write the curve as
$$ w^n = \frac{ \prod\_{\alpha=1}^N (z- u\_\alpha )} {\prod\_{\alpha=1}^N (z- v\_\alpha)}$$ after a change of variables dividing $w$ by $\prod\_{\alpha=1}^N (z- v\_\alpha)$.
A straightforward calcul... | 14 | https://mathoverflow.net/users/18060 | 415570 | 169,404 |
https://mathoverflow.net/questions/415538 | 1 | I am reading a paper on multiple solutions for boundary value problems of fourth-order differential systems. In the paper, there is a nonlinear term $f\in C\left[(0,1)\times \mathbb{R}^+\times \mathbb{R}, \mathbb{R}\right]$, which is assumed to be semipositone. I found a definition of semipositione online, which says: ... | https://mathoverflow.net/users/244943 | What are semipositone functions? | The continuous function $f:[0,\infty)\rightarrow\mathbb{R}$ is called *semipositone* if it is monotonically increasing, $f(0)<0$ and $f(u)>0$ for some $u>0$. So indeed, $f$ is negative in a neighborhood of the origin. If instead $f(0)>0$ the function is called *positone*. The case $f(0)=0$ does not have a special name.... | 4 | https://mathoverflow.net/users/11260 | 415572 | 169,405 |
https://mathoverflow.net/questions/415565 | 17 | Étale cohomology of schemes $X$ is constructed as follows: one associates to $X$ the so-called étale topos of $X$, and then one just takes the sheaf cohomology of that topos.
Is it possible to associate to each smooth manifold $M$ a "de Rham topos of $M$", whose sheaf cohomology yields the de Rham cohomology of $M$?
... | https://mathoverflow.net/users/476516 | De Rham via topoi | One can define an analogue of the crystalline topos for smooth manifolds.
This is known as the [de Rham stack](https://ncatlab.org/nlab/show/de+Rham+space) of $M$.
One of the easiest constructions of the de Rham stack
embeds smooth manifolds fully faithfully (using the Yoneda embedding)
into the category of ∞-sheav... | 13 | https://mathoverflow.net/users/402 | 415581 | 169,406 |
https://mathoverflow.net/questions/415567 | 2 | I'm studying an article but I'm not able to understand one of his statements. I have the following hypotheses:
1. $G$ is a solvable group with trivial center, $J=\phi^{-1}(F(G/F(G)))$ and $J\_2=\phi^{-1}(Z(F (G/F(G)))$ where $\phi$ is the canonical homomorphism of $G$ to $G/F(G)$.
i) $\mathrm{mdc}(|\phi(J/F(G))|,|... | https://mathoverflow.net/users/334556 | Element that is in $\phi^{-1}(Z(F (G/F(G)))$ | For simplicity let $\bar G=G/F(G)$ and for any subset or element $x$ of $G$, let $\bar x$ be the image of $x$ in $\bar G$ (the "bar convention"). By (v), $\langle \bar x\_r\rangle\triangleleft \bar G$. In particular, $\bar x\_r\in F(\bar G)$.
It suffices to show that for any $\bar y\in F(\bar G)$ of prime power order... | 4 | https://mathoverflow.net/users/99221 | 415588 | 169,409 |
https://mathoverflow.net/questions/415556 | 6 | Suppose that $f \in S\_k(\Gamma\_0(N)) $ be a Hecke eigenform whose Fourier expansion at $ i\infty $ is given by
$$
f(z) = \sum\_{n=1}^{\infty} \lambda(n) n^{\frac{k-1}{2}} \exp(2\pi i n z),
$$
normalized so that $\lambda(1)=1$. In this setting the Ramanujan-Petersson conjecture states that $ |\lambda(n)| \leq d(n)... | https://mathoverflow.net/users/148866 | Ramanujan-Petersson conjecture at various cusps | The estimate $|\lambda(n)| \leq C\_N d(n)$ remains valid at all cusps, but $C\_N$ cannot in general be taken independent of $N$. See Remark 3.14 of [this paper](https://mathscinet.ams.org/mathscinet-getitem?mr=3110797) ([arxiv link](https://arxiv.org/abs/1205.5534)), where it is noted that for certain $(N,f,n)$, with $... | 9 | https://mathoverflow.net/users/29622 | 415589 | 169,410 |
https://mathoverflow.net/questions/415485 | 7 | How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
| https://mathoverflow.net/users/475670 | On eigenfunctions of the Laplace Beltrami operator | For $\mathrm{SU}(2)$, with the scale for the biïnvariant metric so that it becomes isometric to the unit $3$-sphere $S^3$ in Euclidean $4$-space, it is well-known what the eigenvalues of the Laplace-Beltrami operator are and, indeed, what the eigenfunctions are as well. The eigenvalues are $\lambda\_k = k(k{+}2)$ for $... | 11 | https://mathoverflow.net/users/13972 | 415595 | 169,411 |
https://mathoverflow.net/questions/415585 | 8 | 1. What is Deligne's motivation in Appendice 9 of Exposé VI to prove that every coherent topos has enough points? For instance, does that have applications in étale cohomology (or other parts of algebraic geometry)?
2. Which topics are discussed in the Exposés Vbis, VI, VIII, and IX? An answer to the question should pr... | https://mathoverflow.net/users/476516 | Questions about SGA 4 | 1. I'm not aware of any applications of Deligne's result to algebraic geometry at the time. It does formally imply that certain topologies (e.g. fppf) have enough points, but this wasn't very useful because the actual points in many well-known topologies were only described by [Gabber–Kelly](https://arxiv.org/abs/1407.... | 10 | https://mathoverflow.net/users/82179 | 415598 | 169,413 |
https://mathoverflow.net/questions/415425 | 3 | Recall that the $i$*-weight* of a Tychonoff space $X$, $iw(X)$, denotes the minimal weight of all Tychonoff spaces onto which $X$ can be condensed. A standard fact about this cardinal number is that the relations $\text{log}(X)\leq iw(X) \leq nw(X)$ always hold. Furthermore, if $X$ is an infinite discrete space, then $... | https://mathoverflow.net/users/146942 | $i$-weight of a metrizable space | Countable powers of copies of $H(\kappa)$, the hedghog space of spininess $\kappa$, are universal for metrizable spaces of weight $\kappa$ (i.e., any metrizable space of weight $\kappa$ embeds into such a countable power as a subspace). Note first that $H(2^\kappa)$ condenses onto a subspace of $[0,1]^\kappa$ (it is ea... | 4 | https://mathoverflow.net/users/121994 | 415600 | 169,414 |
https://mathoverflow.net/questions/415610 | 2 | Let $X$ be random vector on the unit-sphere $S\_{n-1}$ in $\mathbb R^n$. We don't assume that the distribution of $X$ is uniform on $S\_{n-1}$
>
> I'm interested in proving the existence of a (deterministic) direction $v \in S\_{n-1}$ such that
> $$
> \mathbb P(|\langle X,v\rangle | \ge \alpha) \ge \beta,
> \tag{1}... | https://mathoverflow.net/users/78539 | Existence of preferred direction for a random vector with arbitrary distribution on sphere, under a condition on its covariance matrix | $\newcommand\R{\mathbb R}\newcommand{\Si}{\Sigma}\newcommand{\si}{\sigma}$The answer is no: in general (and usually) there are no positive absolute constants $a$ and $b$ such that for some unit vector $v$ one has
$$P(|X\cdot v|\ge a)\ge b.$$
Indeed, otherwise one would have $E(X\cdot v)^2\ge c:=ba^2>0$. However, if $... | 1 | https://mathoverflow.net/users/36721 | 415619 | 169,419 |
https://mathoverflow.net/questions/415045 | 0 | I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the Jacobian of the fixed point $(1,0)$, which is
$$A\_{(1,0)} =
\begin{pmatrix}
-1 & -a\\
0 & 0
\end{pmatrix}$$
It is clear... | https://mathoverflow.net/users/471464 | Why all the coefficients of the center manifold of this system are zeros? | The center manifold for the first problem is not $y = O(x^2)$. It is
tangent to the center subspace of the linearized operator, which is
not $x$-axis in this case.
Please find an eigenvector associated with zero eigenvalue and change
the coordinates so that the center subspace corresponds one of the
coordinate axes. In... | 0 | https://mathoverflow.net/users/471464 | 415623 | 169,420 |
https://mathoverflow.net/questions/415361 | 3 | Every continuous vector bundle on a closed smooth manifold $M$ has a smooth structure. On the other hand, every vector bundle $E$ is the image of a trivial bundle $M\times\mathbb{C}^n$ under some projection $p$ in a matrix algebra $M\_n(C(M))$. I am wondering if it possible to approximate this projection by a smooth id... | https://mathoverflow.net/users/78729 | Approximation of continuous projections on a manifold by smooth idempotents | Well, I learned this was Rieffel's question and was already solved by Hanfeng Li. <https://mathscinet.ams.org/mathscinet-getitem?mr=2994680>
**Obsolete:**
The following is not an answer but maybe useful (see the comments above).
**Lemma.** Let $a,b$ be self-adjoint operators and $0<\epsilon<0.01$. Assume that $\|b-... | 2 | https://mathoverflow.net/users/7591 | 415625 | 169,421 |
https://mathoverflow.net/questions/415615 | 10 | Littlewood-Richardson coefficients $c\_{\nu\mu}^{\lambda}$ (where $\nu$,$\mu$ and $\lambda$ are integer partitions such that $|\nu| + |\mu| = |\lambda|$) are well-known coefficients appearing in various contexts.
In terms of Schur functions, which form a basis of the symmetric functions, the coefficients are the mult... | https://mathoverflow.net/users/73667 | Littlewood-Richardson coefficients in terms of Specht modules | The answer is Yes. You don't need $GL(n, \mathbb C)$ or symmetric functions to define/understand LR-coefficients. There are numerous references to choose from, which wary how much Young tableau combinatorics you want to see.
1. For more combinatorial explanations, I recommend two similarly titled papers by [Kerov](ht... | 10 | https://mathoverflow.net/users/4040 | 415636 | 169,422 |
https://mathoverflow.net/questions/415630 | 2 | The Gamma function $\Gamma$ is defined by
\begin{equation\*}
\Gamma(x)=\int\_{0}^\infty t^{x-1}e^{-t} \,\mathrm{d}t,
\end{equation\*}
for $x>0$. It satisfies the well-known functional equation
$$\Gamma(x+1)=x\Gamma(x)\label{1}\tag{i}$$
for all $x>0$. Apparently, the definition of Gamma function can be extended to $\ma... | https://mathoverflow.net/users/476697 | Gamma function and the somewhat extended version of Bohr-Mollerup theorem | $\newcommand\R{\mathbb R}\newcommand{\Ga}{\Gamma}\newcommand\Z{\mathbb Z}$The answer is yes. Indeed, the conditions $f\colon A\to\R$ and $f(x+1)=xf(x)$ for all $x\in A$ imply that $x+1\in A$ for any $x\in A$ and hence, by induction, $x+k\in A$ for all natural $k$. We also have $(0,\infty)\subseteq A\subseteq\R$ and $f(... | 2 | https://mathoverflow.net/users/36721 | 415656 | 169,429 |
https://mathoverflow.net/questions/415384 | 1 | One of the consequences of the well-known Motzkin-Taussky theorem (<https://www.jstor.org/stable/1990825>) is the following : if two complex matrices $A, B$ generate a vector space of diagonalisable matrices, then $A$ and $B$ commutes and in particular are simultaneously diagonalisable.
Does the result hold for nilpo... | https://mathoverflow.net/users/476498 | Nilpotent matrices with (Motzkin-Taussky) property L | The matrices $$A = \begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&0 \end{pmatrix}, B = \begin{pmatrix} 0&0&0 \\ 1&0&0 \\ 0&-1&0 \end{pmatrix}$$ satisfy $(sA+tB)^3=0$ for all $s,t$, but $$AB = \begin{pmatrix} 1&0&0 \\ 0&-1&0 \\ 0&0&0 \end{pmatrix},$$ which is not nilpotent, so $A$ and $B$ are not simultaneously triangularisable ... | 1 | https://mathoverflow.net/users/425351 | 415674 | 169,437 |
https://mathoverflow.net/questions/415643 | 15 | There is a folklore in the empirical computer-science literature that, given a tree $(X,d)$, one can find a bi-Lipschitz embedding into a hyperbolic space $\mathbb{H}^n$ and that $n$ is "much smaller" than the smallest dimension of a Euclidean space in which $(X,d)$ can be bi-Lipschitz embedded with similar distortion.... | https://mathoverflow.net/users/469470 | Are hyperbolic spaces actually better for embedding trees than Euclidean spaces? | I am not sure if the following paper answers your question. The abstract suggests so, but it is written in a computer science style that is less transparent to me in terms of stating a precise theorem. Also: (a) I am not an expert, (b) I am confused by the way that $n$ seems to play two different roles in your question... | 7 | https://mathoverflow.net/users/476736 | 415679 | 169,439 |
https://mathoverflow.net/questions/415617 | 16 | Let $f(x)$ be sufficiently regular (e.g. a smooth function or a formal power series in characteristic 0 etc.). In my research the following recursion made a surprising entrance
$$
f\_1(x) = f(x),\ f\_{n+1}(x) = f(x) f'\_{n}(x)
$$
Thus I would like to understand the sequence
$$
\left(f(x) \frac{d}{dx} \right)^n f(x)
$$
... | https://mathoverflow.net/users/1849 | A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$? | Revamped Feb. 12, 2022:
I posted an answer to this (perennial) question in detail in the old MO-Q "[Formula for n-th iteration of dx/dt=B(x)](https://mathoverflow.net/questions/41039/formula-for-n-th-iteration-of-dx-dt-bx)" and pointed out a common conflation of related but distinct number arrays, all related to 'nat... | 8 | https://mathoverflow.net/users/12178 | 415681 | 169,440 |
https://mathoverflow.net/questions/415678 | 1 | $\newcommand{\Perf}{\operatorname{Perf}}$This is a toy example that I want to understand, I will be grateful for any help. Given a ring $R$ and $A=\Perf(R)$ the category of perfect complexes over $R$ . Suppose that $m\in\Perf(R)$ and $B$ the smallest thick subcategory generated by
$ m $. Let $C$ be the verdier quotient... | https://mathoverflow.net/users/141114 | Verdier localisation | No, not in general. This is true if you pass to "big" categories, so $\mathrm{Mod}(R)$ instead of Perf, and you take the smallest localizing subcategory containing $m$, but in general wrong at the level of small categories.
So what you can do is look at the full subcategory of $\mathrm{Mod}(R)$ of these objects, and ... | 3 | https://mathoverflow.net/users/102343 | 415695 | 169,446 |
https://mathoverflow.net/questions/415680 | 2 | My setting is the following : let $G$ be a topological group and $X$ be a topological space. I have the head filled with two possible definitions for a continuous action of $G$ on $X$.
The first could be called the separately continuous action and means that every element $g\in G$ acts via a continuous map of $X$ int... | https://mathoverflow.net/users/242651 | Difference between definitions of continuous action, profinite case | Both definitions are equivalent in much greater generality. I believe the profinite case has as more direct proof, perhaps it can be found in the book of Ribes and Zalesski.
There is a very general result by Robert Ellis (Ellis, Robert
Locally compact transformation groups.
Duke Math. J. 24 (1957), 119–125) that cove... | 2 | https://mathoverflow.net/users/15934 | 415698 | 169,448 |
https://mathoverflow.net/questions/415693 | 0 | Let $a(n)$ be the sequence of composite numbers (starting from $4$). Let
$$b(n)=a(n-1)a(n-2) \operatorname{mod} a(n)$$
Obviously, $b(1)=b(2)=0$.
I conjecture that with the only exception for the $b(3)=0$ all terms are belong to $\left\lbrace2,3,6,8\right\rbrace$.
My second conjecture is that the sequence $b(n)$ can... | https://mathoverflow.net/users/231922 | Property of composite numbers | For any $m$ we have that either $m+1$ or $m+2$ is composite. Therefore a tuple of three consecutive composites must be of one of the following forms:
$$(k-4,k-2,k),(k-3,k-2,k),(k-3,k-1,k),(k-2,k-1,k).$$
Since we have $(k-a)(k-b)\equiv ab\pmod k$, we get the four cases you ask for with $k=a(n)$. The second conjecture fo... | 3 | https://mathoverflow.net/users/30186 | 415699 | 169,449 |
https://mathoverflow.net/questions/415703 | 28 | I am thinking about the Axiom of Choice and I am trying to understand the Axiom with some but a little progress. Many questions are arising in my head. So, I know that there exists a model of ZF set theory in which the set of real numbers, which is provably uncountable, is a countable union of countable sets.
**Quest... | https://mathoverflow.net/users/48157 | Can a countable union of two-element sets be uncountable? | Yes, it is possible. This phenomenon is sometimes called Russell's socks, named after an analogy due to Russell about how one can pick out a shoe from an infinite set of pairs of shoes, but not for socks since socks in a pair are indistinguishable.
[Horst Herrlich, Eleftherios Tachtsis, On the number of Russell’s soc... | 39 | https://mathoverflow.net/users/30186 | 415707 | 169,454 |
https://mathoverflow.net/questions/415611 | 4 | Let $d\ge 3$ be a constant. Is there an explicit construction of an infinite family of $d$-regular graphs such that for $G$ in this family with $n$ vertices, every subgraph $H$ of on at most $\alpha n$ vertices where $\alpha$ is a constant depending on $d$ has average degree bounded by, say, $2.1$?
Note that a random... | https://mathoverflow.net/users/91432 | Explicit constructions of regular graphs with very sparse induced subgraphs | I think that for Ramanujan graphs the situation is similar to vertex expansion, which got more publicity.
Namely, a variation of the combinatorial construction of Kahale about vertex expansion (<https://www.researchgate.net/publication/2782658_Eigenvalues_and_Expansion_of_Regular_Graphs>, Theorem 7, the same paper whic... | 1 | https://mathoverflow.net/users/450073 | 415712 | 169,456 |
https://mathoverflow.net/questions/414250 | 2 | I know that the weak lower semi-continuity of the KL divergence was proved in [1]. If I remember well, the same property is true for any $f$ divergence (with suitable assumptions on the probability space). I am looking for some reference about it.
[1] Posner, Random Coding Strategies for Minimum Entropy, 1975.
**Ed... | https://mathoverflow.net/users/475450 | Looking for a reference: $f$-divergences are lower semicontinuous | I think the most general reference is:
M. Liero, A. Mielke, and G. Savaré. [Optimal entropy-transport problems and a newhellinger–kantorovich distance between positive measures](https://link.springer.com/article/10.1007/s00222-017-0759-8). *Inventiones mathematicae*, **211**, 2018.
[arxiv:1508.07941](https://arxiv... | 0 | https://mathoverflow.net/users/475450 | 415713 | 169,457 |
https://mathoverflow.net/questions/415618 | 3 | I'll first describe my problem in layman's terms. I have a map with $m$ countries and I want to color each country with a different color (this has nothing to do with the 4-color theorem). How do I choose the colors so as to maximize contrasts?
Each color is a 3D vector in the unit cube. I want the minimum distance b... | https://mathoverflow.net/users/140356 | Lattice-like structure with maximum spacing between vertices | No proof of optimality, but here are conjectured values for $m \le 10$ based on numerical optimization:
\begin{matrix}
m & \text{maximin?} \\
\hline
2 & \sqrt{3} \\
3 & \sqrt{2} \\
4 & \sqrt{2} \\
5 & \sqrt{5}/2 \\
6 & 3\sqrt{2}/4 \\
7 & 1.0010898245 \\
8 & 1 \\
9 & \sqrt{3}/2 \\
10 & 3/4 \\
\end{matrix}
```
m = 2:
... | 3 | https://mathoverflow.net/users/141766 | 415716 | 169,458 |
https://mathoverflow.net/questions/415719 | 3 | I'm reading some notes(\*) about arithmetic lattices in $\operatorname{SU}(n,1)$. I'm trying to understand the data that classifies (up to commensurability) the arithmetic lattices of the "first type" in this group. Fix a totally real number field $F$ with a totally imaginary quadratic extension $E$. Then these lattice... | https://mathoverflow.net/users/151664 | Multiplicative group of number field mod field norms of quadratic extension | The index is definitely not finite. If $F$ is an arbitrary number field and $E$ is a finite extension with $[E:F] > 1$, the norm subgroup ${\rm N}\_{E/F}(E^\times)$ has infinite index in $F^\times$. A proof in general is in a recent stackexchange post [here](https://math.stackexchange.com/questions/4368481/is-f-times-n... | 5 | https://mathoverflow.net/users/3272 | 415721 | 169,459 |
https://mathoverflow.net/questions/346487 | 11 | Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^\*$-algebras is an equivalence of categories. In other words, all topological properties of such a topological space $X$ are encoded in the algebraic properties (and of course the norm) of $C(X)$.... | https://mathoverflow.net/users/2622 | Fundamental group under Gelfand duality | It seems that this question has been investigated in the literature, where the fundamental group is approached through the notion of a regular covering. For instance, the following reference establishes an equivalence between the categories of regular coverings $\tilde{X} \to X$ and Galois extensions of $C(X)$, meaning... | 2 | https://mathoverflow.net/users/2622 | 415729 | 169,461 |
https://mathoverflow.net/questions/415728 | 8 | Given an random variable $Y:\Omega \to \mathbb{R}$ with finite mean $\mu$ and finite, positive variance $\sigma^2$, let $X = \frac{Y-\mu}{\sigma}$ be the renormalization with mean $0$ and variance $1$. what are some general techniques for showing that $Y$ has a normal distribution? That is,
$$P(X\leqslant a) = \frac{1}... | https://mathoverflow.net/users/37327 | Ways of proving normal distribution (with a view towards Selberg's central limit theorem) | There are multiple books about ways to characterize the normal distribution. For instance, Bryc’s [book](https://homepages.uc.edu/%7Ebrycwz/probab/charakt/charakt.pdf) starts with Herschel-Maxwell’s theorem:
>
> If $X$ and $Y$ are independent variables whose joint distribution is rotationally invariant, then $X$ an... | 5 | https://mathoverflow.net/users/nan | 415741 | 169,467 |
https://mathoverflow.net/questions/415687 | 6 | Maybe the answer to the following question is known but I am unable to find it in the literature. Anyway, let me begin my question by fixing some notations and terms.
Let $G = (A, \Delta)$ be a compact quantum group in the sense of Woronowicz. For a finite dimensional unitary representation $U \in B(H) \otimes A$ of ... | https://mathoverflow.net/users/40789 | Norm of contragredient of unitary representations of compact quantum groups | Yes, $c(G) < +\infty$ implies that $G$ is of Kac type. As far as I know, this result is not in the literature, but can be proven as follows.
Let $h$ be the Haar state. For every irreducible unitary representation $u \in M\_n(\mathbb{C}) \otimes A$, there is a unique positive invertible $Q\_u \in M\_n(\mathbb{C})$ wit... | 6 | https://mathoverflow.net/users/159170 | 415746 | 169,468 |
https://mathoverflow.net/questions/415688 | 11 | What is a reference for ["the equivalence between geometric theories and frames internal to the free topos"](https://indico.math.cnrs.fr/event/747/contributions/2523/attachments/1818/1962/introductionJoyal.pdf)? [1] This seems to be an extremely interesting theorem.
[1] André Joyal, “A crash course in topos theory: t... | https://mathoverflow.net/users/476516 | Equivalence between geometric theories and frames internal to the free topos | Simon essentially answered the question already, but I will expand some of the parts that may not be clear to the experts. Sketches of an elephant is a good reference for everything I am going to say.
I will take a bit of a different narrative to Simon's, implicitly assuming a good understanding of classifying topoi, b... | 10 | https://mathoverflow.net/users/104432 | 415753 | 169,470 |
https://mathoverflow.net/questions/415584 | 1 | I'm looking for a solution to $$\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx.$$
Mathematica gives me the following solution, but I'd like to know/understand the steps involved in finding it.
$$\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx = \frac{x^{1-a} \left(\left(c x^{-d}\right)^{\frac{1-a}{d}} \Gamma
\left... | https://mathoverflow.net/users/103291 | Solution to $\int x^{-a} \Gamma\left( b, c x^{-d} \right) dx$ | To remove this question from the "unanswered" list; given the indefinite integral
$$\int z^{\alpha-1}\Gamma(b,z)\,\mathrm{d}z=\alpha^{-1}[ z^\alpha \Gamma(b, z)-\Gamma(b + \alpha, z)]$$
change variables to $z=cx^{-d}$, $\mathrm{d}z=-dcx^{-d-1}\mathrm{d}x$, hence
$$-\int dc^\alpha x^{-\alpha d-1}\Gamma(b,cx^{-d})\,\math... | 1 | https://mathoverflow.net/users/11260 | 415761 | 169,474 |
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