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https://mathoverflow.net/questions/426409 | 0 | Let $X$ be a compact metric space, and fix an arbitrary point $x\_\ast \in X$. By the Kantorovich-Rubinstein duality theorem, the $1$-Wasserstein metric $W\_1$ on the set of Borel probability measures on $X$ can be expressed as
$$ W\_1(\mu,\nu) = \max\left\{ \int f \, d\mu - \int f \, d\nu \, : \, f \in \mathcal{K}\_0\... | https://mathoverflow.net/users/15570 | Is there a "smooth Kantorovich-Rubinstein duality" for Wasserstein distances on smooth/Euclidean space? | $\newcommand{\K}{\mathcal K}\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}$No, the formula
\begin{equation\*}
W\_1(\mu,\nu) = \max\Big\{ \int f \, d\mu - \int f \, d\nu \, : \, f \in \K\_r\Big\}, \tag{1}\label{1}
\end{equation\*}
with $\max$ rather than $\sup$, does not hold in general, for whatever choices of an ... | 3 | https://mathoverflow.net/users/36721 | 426415 | 173,066 |
https://mathoverflow.net/questions/426420 | 1 | A connected sum decomposition of a closed $n$-manifold $M^n$:
$M^n = M\_1^n \# M\_2^n$, is to view $M^n$ as two closed $M\_1^n$ and $M\_2^n$, joined
by a neck $I\times S^{n-1}$.
Similarly, a $k$-connected sum decomposition of a closed $n$-manifold $M^n$:
$M^n = M\_1^n \#\_k M\_2^n$, is to view $M^n$ as two closed $M\... | https://mathoverflow.net/users/17787 | Multi-connected sum decomposition of $n$-manifolds | I don't have a reference, but I think it's not too hard to see that $M\_1 \#\_k M\_2 \approx M\_1 \# X\_k \# M\_2$, where $X\_k = (S^1 \times S^{n-1})^{\# (k-1)}$. (Connected sum depends on choices of embedded disks, and so does $k$-connected sums; I'm assuming you want all $k$ pairs of disks to be isotopic.) So decomp... | 5 | https://mathoverflow.net/users/171227 | 426421 | 173,068 |
https://mathoverflow.net/questions/426423 | 3 | As is known to all, the Lie algebra $\frak{sl}\_2$ admits a very nice representation on
$$
\mathbb{K}[X,Y]
$$
the polynomials in two variables, given by
$$
E \mapsto X\frac{\partial }{\partial Y}, ~~ F \mapsto Y\frac{\partial }{\partial X}, ~~ H \mapsto X\frac{\partial }{\partial X} - Y\frac{\partial }{\partial Y}.
$$
... | https://mathoverflow.net/users/153228 | A representation of $\frak{sl}_n$ as partial derivatives on polynomials | The group $GL(V)$ acts on a vector space $V$ by linear automorphisms, and this induces the action of its Lie algebra $\mathfrak{gl}(V)$, i.e. a homomorphism to the Lie algebra of vector fields (differentiations). Explicitly
$$e\_{ij} \mapsto X\_i \partial\_j.$$
The composition of two differentiations is a second order ... | 5 | https://mathoverflow.net/users/6468 | 426424 | 173,070 |
https://mathoverflow.net/questions/424857 | 1 | I have a set of points $X=\{x\_1,\ldots,x\_N\}$ on the unit sphere in $\mathbb{R}^d$ ($N>d\ge 3$). What is an algorithm to find any facet of the polyhedron whose vertices are $X$?
The set $X$ has some features that may be helpful:
* It is highly symmetric in the sense that for each pair $x,x'\in X$ there is an isom... | https://mathoverflow.net/users/484293 | Algorithm to find a facet of a polyhedron given the vertices? | Inspired by Matt F's response, I figured out a solution (I think). Intuitively, a procedure that will find *some* facet starting from a given vertex $x\_1$ is the following: Start with a hyperplane containing the point $x\_1$ and orthogonal to the vector $x\_1$. Then tilt the hyperplane in some direction until it inter... | 1 | https://mathoverflow.net/users/484293 | 426428 | 173,072 |
https://mathoverflow.net/questions/254953 | 5 | This question is closely connected to the following paper and to a prior question posted to Mathoverflow titled "Can Cantor's theorem be proved in Paraemter Free Zermelo",
<https://wwwmath.uni-muenster.de/u/rds/ZFC_without_parameters.pdf>
That paper provided an answer to a question about whether $\text{ZFC}^o$ is e... | https://mathoverflow.net/users/95347 | Is Extreme Parameter Free ZFC equivalent to ZFC? | The answer is to the negative for both questions. A result proved by Levy. This can be seen in Kanamori's article: [Levy and set theory](http://math.bu.edu/people/aki/11.pdf). Annals of Pure and Applied Logic 140 (2006) 233–252
In page 247, he wrote:
>
> Levy’s main, negative results addressed another issue of se... | 5 | https://mathoverflow.net/users/95347 | 426433 | 173,074 |
https://mathoverflow.net/questions/426435 | 4 | Let $F: \mathcal{C} \to \mathcal{D}$ be a left Quillen functor between model categories. In Definition 2.16 of [Goerss–Schemmerhorn - Model Categories and Simplicial Methods](https://arxiv.org/abs/math/0609537), the left derived functor $LF: \mathcal{C} \to \operatorname{Ho}(\mathcal{D})$ is defined by $X \mapsto F(P)$... | https://mathoverflow.net/users/169035 | Can we define derived functors in model categories without functorial factorisations? | This depends on whether one insists on derived functors landing in the original model category (as is the case with modern approaches of Hinich and Dwyer–Hirschhorn–Kan–Smith), or in its homotopy category (as is the case with the classical approaches of Quillen, Grothendieck, etc.).
In the latter case it is indeed po... | 8 | https://mathoverflow.net/users/402 | 426439 | 173,076 |
https://mathoverflow.net/questions/426440 | 7 | Let $\mathcal A$ be an abelian category. In [this lecture](https://youtu.be/7RNh_cxDYmk), Thomas Nikolaus
1. Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\mathcal A)=N\_{\mathrm{dg}}(\operatorname{Ch}(\mathcal A))$ and $W$ is the set of quasi-i... | https://mathoverflow.net/users/37110 | Derived functors out of an unbounded derived $\infty$-category | An account of derived functors between ∞-categories equipped with weak equivalences and fibrations can be found in Section 7.5 of Cisinski's *Higher Categories and Homotopical Algebra*. This setting is sufficient to treat the case of the unbounded derived category of an abelian category.
A more general treatment of d... | 5 | https://mathoverflow.net/users/402 | 426444 | 173,077 |
https://mathoverflow.net/questions/426445 | 6 | Not thinking about $h$-cobordism, one usually defines a *cobordism* between manifolds, realizes it is an equivalence relation, chooses an appropriate class of *structured manifolds* (framed, unoriented, spin, ...), and dreams about the group of cobordism classes of these manifolds.
Thinking about $h$-cobordism, whose... | https://mathoverflow.net/users/472967 | Do $h$-cobordism groups arise from a 'Thom-like' spectrum? | This topic has a very different flavor from what is usually meant by cobordism. No, the correspondence between cobordism classes and homotopy classes (of a Thom space or Thom spectrum) has no analogue here; when manifolds, or cobordisms between manifolds, are created by transversality, we don't have much control over t... | 10 | https://mathoverflow.net/users/6666 | 426446 | 173,078 |
https://mathoverflow.net/questions/426453 | 8 | Let $\Lambda\subset \mathbb{R}^4$ be a lattice. We identify $\mathbb{R}^4$ with the space $M\_2(\mathbb{R})$ of $2\times 2$ matrices over $\mathbb R$. It then is is clear that the set
$$
\det(\Lambda)=\big\{\det(\lambda):\lambda\in\Lambda\big\}
$$
is not necessarily discrete in $\mathbb R$.
But now we additionally insi... | https://mathoverflow.net/users/473423 | Are the determinants of a lattice discrete? | This seems to be related to issues of Diophantine approximation. Fix such $\Lambda$ and $\Gamma$. Fix $\lambda \in \Lambda$ so that $\det(\lambda) \neq 0$. After rescaling $\Lambda$, we may assume that $\det(\lambda) = 1$. Fix some $\gamma \in \Gamma$ other than the identity. Conjugating $\Gamma$ (and transforming $\La... | 5 | https://mathoverflow.net/users/1650 | 426460 | 173,080 |
https://mathoverflow.net/questions/426172 | 22 | I am able to construct functions $\sin,\cos\colon \mathbb R \to \mathbb R$ satisfying the following properties:
1. $\sin^2 x + \cos^2 x = 1$,
2. $\sin(x+y)=\sin x \cos y + \sin y\cos x$, $\cos(x+y)=\cos x \cos y - \sin x \sin y$,
3. $\sin(0)=0$, $\cos(0)=1$
4. there exists $\tau>0$ such that $\sin$ and $\cos$ are $\t... | https://mathoverflow.net/users/36826 | Axiomatic construction of trigonometric functions | This is to complete the [answer](https://mathoverflow.net/a/426247) by Emanuele Paolini by showing that
\begin{equation\*}
a\_n:=\frac{\sin \frac xn}{\frac xn}
\end{equation\*}
is increasing in natural $n$ for each $x\in(0,h]$, where $h$ is any positive real number such that $\sin>0$, $\cos>0$, and $\tan<1$ on the int... | 4 | https://mathoverflow.net/users/36721 | 426488 | 173,088 |
https://mathoverflow.net/questions/426403 | 14 | Let $A$ be a set of $n$ integers and consider the quantity:
$$\int\_{0}^1 \left| \sum\_{a \in A} e^{2\pi i a x} \right|dx. $$
The (now solved) Littlewood conjecture is the claim that this quantity is lower bounded by $c \log n$ for some universal constant $c>0$.
The (still open) strong Littlewood conjecture is the cl... | https://mathoverflow.net/users/630 | The first case of the strong Littlewood conjecture | The following human-verifiable proof is in collaboration\* with Fedja.
**Lemma** $1$: We have the following for $0 \le x \le 3$: $$\frac{6204}{6750}x^2-\frac{8429}{60750}x^4+\frac{4475}{546750}x^6\le x.$$
>
> Proof: The above is equivalent to $$\frac{6204}{750}\left(\frac{x}{3}\right)^2-\frac{8429}{750}\left(\fra... | 8 | https://mathoverflow.net/users/129185 | 426498 | 173,092 |
https://mathoverflow.net/questions/426491 | 1 | Let $f\_n: \mathbb R\_+\to (0,1]$ be continuous and strictly decreasing for every $n\ge 1$. Assume that the pointwise limit of $(f\_n)\_{n\ge 1}$ exists, denoted by $f$, and is also strictly decreasing. Can we prove
$$\lim\_{n\to\infty}f\_n^{-1}(t)=f^{-1}(t), \quad \mbox{for almost every } t\in (0,1)?$$
The definit... | https://mathoverflow.net/users/nan | Identification of $\lim_{n\to\infty}f_n^{-1}$ with $f_n:\mathbb R_+\to (0,1]$ strictly decreasing and converging pointwise | $\newcommand{\de}{\delta}\newcommand{\ep}{\varepsilon}$Yes, this is true. Moreover, the continuity and the strictness of the decrease of the $f\_n$'s are not needed.
Indeed, for any nonincreasing function $g\colon[0,\infty)\to(0,1]$ and any $t\in(0,1)$, let
\begin{equation\*}
g^{-1}(t):=\sup\{x\in[0,\infty)\colon g(... | 1 | https://mathoverflow.net/users/36721 | 426504 | 173,094 |
https://mathoverflow.net/questions/426503 | 5 | I have a feeling the following is true.
Assume that there are $n$ mutually disjoint closed disks $D\_i$ in the complex plane and $n$ complex polynomials $p\_i(z)$ of degree $n - 1$, with both types of objects indexed from $1$ to $n$, such that
condition: the polynomial $p\_i(z)$ has exactly 1 complex root in each d... | https://mathoverflow.net/users/81645 | A statement on complex polynomials | The conjecture is easily seen to be true for $n<3$.
We give a counterexample for $n=3$.
Let $p\_i = z^2 - \omega\_i$ where the $\omega\_i$ are the cube roots of unity.
These are linearly dependent because they're in the $2$-dimensional subspace
spanned by $1$ and $z^2$. But their zeros are the sixth roots of unity,
f... | 16 | https://mathoverflow.net/users/14830 | 426505 | 173,095 |
https://mathoverflow.net/questions/426499 | 6 | Let $X$ be a separable metric space which is *homogeneous*, i.e. for every two points $x,y\in X$ there is a homeomorphism $h$ of $X$ onto itself such that $h(x)=y$.
A compactification of $X$ is a compact metric space which contains a dense homeomorphic copy of $X$.
Does $X$ have a homogeneous compactification?
Ex... | https://mathoverflow.net/users/95718 | Do all homogeneous spaces have homogeneous compactifications? | Since you want a connected example:
A surface of infinite genus has no homogeneous compactification.
Indeed first observe a dense locally compact subset has to be open.
So the surface has to be open, and by homogeneity the compactification is a closed surface. But an open subset of a closed surface has (each comp... | 14 | https://mathoverflow.net/users/14094 | 426508 | 173,097 |
https://mathoverflow.net/questions/426511 | 1 | Let $\hat{\pi}^N$ be an AW-consistent estimator of $\pi$ (i.e., $\hat{\pi}^N$ is a strongly consistent estimator of $\pi$ under adapted (or called nested) Wasserstein distance $AW(\pi, \hat{\pi}^N)\to 0 $ a.s.).
How to prove that $W(\hat{\pi}^N)$ is a consistent estimator of $W(\pi)$?
| https://mathoverflow.net/users/168083 | How to prove that is a consistent estimator? | Take $\pi^N$ with $AW(\pi^N, \pi) \leq \frac{1}{N}$, where we denote by $\mu^N$ and $\nu^N$ the marginals of $\pi^N$.
Note that by the backward induction for $AW$ (cf. [here](https://epubs.siam.org/doi/pdf/10.1137/16M1080197)), it holds
$$
AW(\pi, \pi^N) = \inf\_{\kappa\_1 \in \Pi(\mu, \mu^N)} d\_{X\_1}(x\_1, y\_1) +... | 1 | https://mathoverflow.net/users/106046 | 426529 | 173,103 |
https://mathoverflow.net/questions/372903 | 5 | Let $C$ be a category with finite hom-sets.
Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition).
For which categories $C$ does it follow that $X \cong Y$?
(Of course, it is true for posets).
A somewhat related question is the following.
Let $C$ be a sy... | https://mathoverflow.net/users/166165 | When is an object determined by the number of maps from the other objects? | The first question is the main topic of two recent papers:
* Reggio's [Polyadic Sets and Homomorphism Counting](https://arxiv.org/abs/2110.11061), which is also a good reference for previous results in the literature (including that of Pultr mentioned in the comments). In particular, see Theorems 4.3, 5.10, and 5.12,... | 3 | https://mathoverflow.net/users/152679 | 426540 | 173,105 |
https://mathoverflow.net/questions/426537 | 1 | Let $X$ be an infinite moving average time series, i.e.
$$
X(t) = \sum\_{k = -\infty}^\infty a\_j \varepsilon\_{t-j}, \quad t \in \mathbb{Z},
$$
where $\varepsilon\_{j}$ are uncorrelated zero mean, finite variance and identically distributed random variables.
In my opinion, $\mathbb{E}X(t) = 0$ for all $t \in \mathbb... | https://mathoverflow.net/users/302666 | Expected value of long memory moving average | This is a slightly more elementary version of Anthony Quas's answer, without appealing to martingales.
By the condition $\sum\_j|a\_j|^2<\infty$, the sequence of random variables $X\_n:=\sum\_{j=-n}^n a\_j\epsilon\_{-j}$ is [Cauchy convergent](https://en.wikipedia.org/wiki/Cauchy%27s_convergence_test) in $L^2$ and he... | 1 | https://mathoverflow.net/users/36721 | 426544 | 173,108 |
https://mathoverflow.net/questions/426502 | 4 | We work in weakly predicatively constructive mathematics, in that we accept function sets but do not accept power sets or excluded middle. More specifically, we shall assume a sequential universe hierarchy in our foundations, where the elements of every universe are (codes for) sets, and every universe $\mathcal{V}$ em... | https://mathoverflow.net/users/483446 | Equivalence of real numbers in terms of Dedekind cuts and Cauchy nets of rational numbers | This seems like something that would be in the literature somewhere, but I couldn't find a reference, so here's a proof outline. I'll sketch how to get multivalued Cauchy real from Dedekind, generalised Cauchy from multivalued Cauchy and Dedekind from generalised Cauchy. It follows that all three are equivalent. In ord... | 6 | https://mathoverflow.net/users/30790 | 426553 | 173,111 |
https://mathoverflow.net/questions/426552 | 5 | In the paper by L.Clozel [in this book](https://www.jmilne.org/math/Books/AA1988a.pdf) (a French text), there is this conjecture (conjecture 4.5 p139)
**Conjecture**: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote by $E$ it's definition field. Then there exists a degree $n$ motiv... | https://mathoverflow.net/users/169282 | Motive associated to a cuspidal representation of $GSp_{4}$ | The formula you quote from Harris defines a Galois representation, not a motive. We expect that there is a motive whose etale realisation is Harris' space, but that is not immediate.
The problems are: firstly, the Siegel threefold is not proper; secondly, the projector cutting out M(pi\_f) might not be an idempotent ... | 3 | https://mathoverflow.net/users/2481 | 426570 | 173,115 |
https://mathoverflow.net/questions/426567 | 4 | Let $ \pi : X \rightarrow B $ be a family of compact complex manifolds parametrized by a connected base $ B $. (By this I mean $ \pi $ is a proper holomorphic submersion.) Let $ L $ be a holomorphic line bundle on $ X $ and $ n $ a positive integer. What can be said about the locus of points $ b \in B $ such that $ L|\... | https://mathoverflow.net/users/152391 | Roots of line bundles in a family | The locus you consider is either empty, or equal to $B$.
This can be seen as follows. Line bundles on a fiber $F$ of $\pi $ are parameterized by $H^1(F,\mathscr{O}^\*\_F)$. This group fits into an exact sequence
$$H^1(F,\mathscr{O}\_F) \rightarrow H^1(F,\mathscr{O}^\*\_F) \xrightarrow{\ c\_1\ }H^2(F,\mathbb{Z})\xrigh... | 7 | https://mathoverflow.net/users/40297 | 426577 | 173,116 |
https://mathoverflow.net/questions/236323 | 11 | Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix
$$A = \left(\begin{array}{}
1&g\_1 & x\_1&g\_1 x\_1 & x\_1^2&g\_1 x\_1^2 & \cdots & x\_1^{n/2-1} & g\_1 x\_1^{n/2-1} \\
1&g\_2 & x\_2&g\_2 x\_2 & x\_2^2&g\_2 x\_2^2 & \cdots & x\_2^{n/2-1} & g\_2 x\_2^{n/2-1} \\
\vdots &\... | https://mathoverflow.net/users/90413 | Determinant of a certain Vandermonde matrix | In Appendix B of <https://arxiv.org/abs/2103.10776> (J. Phys. A: Math. Theor. 54, 375201 (2021)), I derived a transformation of the block Vandermonde determinant above to a Hankel matrix, which in my case dramatically simplified the problem. This is a block-matrix generalization of the well known "Vandermonde factoriza... | 3 | https://mathoverflow.net/users/90413 | 426579 | 173,117 |
https://mathoverflow.net/questions/426585 | 3 | I have some doubts about what an abelian covering is, and I'll try my best to articulate them.
In Serre's *Algebraic groups and class fields* Chapter VI.2, he fixed a base field $k$ with algebraic closure $\bar{k}$, and $V$ a normal irreducible algebraic variety over $\bar{k}$ with function field $K = \bar{k}(V)$. Fo... | https://mathoverflow.net/users/172132 | The notion of abelian covers | (1) $G$-torsors are always unramified, because they are étale-locally trivial, and unramifiedness may be checked etale-locally.
(2) A $G$-torsor is a $G$-covering in the sense of Serre as long as it is irreducible. We simply take $L$ to be the function field $\overline{k}(Y)$. Since the morphism $Y \to V$ is finite, ... | 8 | https://mathoverflow.net/users/18060 | 426587 | 173,118 |
https://mathoverflow.net/questions/426304 | 1 | Language: Multi-sorted first order logic with equality and membership, where for each natural $n$ we have variables $x\_i^n$ of sort $n$, and for each decidable monotonic strictly increasing sequence of naturals $s$ [with 0 in its domain] ,we have binary relation symbols $=^s, \in^s$ with the following syntatical restr... | https://mathoverflow.net/users/95347 | Is this theory equivalent to Tangled Type Theory? | It is not. It is very important, very awful, and quite intended that the membership relation between the same two types is the same in different sequences of types.
| 4 | https://mathoverflow.net/users/485780 | 426599 | 173,120 |
https://mathoverflow.net/questions/426591 | 1 | Assume $A$ is an operator on a Hilbert space with discrete spectrum. Assume $B$ is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume $A$ and $B$ commute.
I'm looking at the small $t$ expansion of the expression,
$$W(t) = \mathrm{Tr} \left( A e^{-tB} \right) = \sum\_{n=0}^\inft... | https://mathoverflow.net/users/41312 | $\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$ | The answer by Carlo Beenakker is correct that any contribution to $D\_n A\_n \sim n^p$ with $p\ge 0$ does not generate any logarithmic asymptotic terms. Instead of numerical checks, one can also examine the [Taylor-Maclaurin](https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula#Asymptotic_expansion_of_sums) as... | 1 | https://mathoverflow.net/users/2622 | 426604 | 173,121 |
https://mathoverflow.net/questions/426605 | 4 | In his Tohoku article, Grothendieck says that the category $\mathbf{Ab}$ of abelian groups satisfies the axiom AB6, namely
"All small colimits exist in $\mathbf{Ab}$. Moreover for any index family $J$ and filtered categories $I\_j, j\in J$, with functors $I\_j\to \mathbf{Ab}, i\mapsto M\_i$, the natural map
$
\und... | https://mathoverflow.net/users/485722 | Why does the category of abelian groups satisfy the axiom AB6? | The objects we are taking the limit over in the left side are $(i\_j \in I\_j)\_j$, i.e. tuples of, for each $j\in J$, an element $i\_j$ of $I\_j$. These are the same as elements of $\prod\_{j\in J}$.
Thus, the left side is a limit over $\prod\_{j\in J} I\_j$.
On the other hand, you have defined $I\_j = I$ for all ... | 5 | https://mathoverflow.net/users/18060 | 426607 | 173,122 |
https://mathoverflow.net/questions/426600 | 3 | Consider the permutation group $\mathfrak{S}\_n$ on $n$ letters $\{1,2,\dots,n\}$. Let $\iota=(1,2,3,\dots,n)\in\mathfrak{S}\_n$ be the *identity* permutation in a $1$-line notation. Given $\pi, \rho\in\mathfrak{S}\_n$ define the *dot product* $\pi\cdot\rho=\pi\_1\rho\_1+\cdots+\pi\_n\rho\_n$.
The $q$-factorial is gi... | https://mathoverflow.net/users/66131 | Sums over permutations relates to permutations? | **Question 1.** All entries except for the last $2$ positions remain constant in blocks of even size and, therefore, contribute $0$ to the signed sum. Now consider the last $2$ entries. Each pair of values $(a,b)$, where $1\le a<b\le n$, comes in consecutive pairs $\pi^{(2j-1)}=(\dots,a,b)$ and $\pi^{(2j)}=(\dots,b,a)$... | 5 | https://mathoverflow.net/users/113161 | 426608 | 173,123 |
https://mathoverflow.net/questions/426598 | 3 | Let $(X, T, \mathcal F, \mu)$ be an ergodic measure preserving system with finite measure.
Suppose $T$ is uniformly recurrent, in the following sense:
For every $A \in \mathcal F$, there exists an $M \in \mathbb N$ such that for almost every $x \in A$, we have $T^{n(x)} \, (x) \in A$ for some $n(x) \leq M$.
**Que... | https://mathoverflow.net/users/173490 | Does uniform recurrence imply uniform convergence of the Birkhoff sums? | This is just a comment on your definition of "uniformly recurrent". Can you give an example of a system you'd consider uniformly recurrent?
If this notion is standard, the following must be nonsense, but it seems to me that your notion does not make much sense.
>
> Lemma. If $T$ is ergodic and uniformly recurrent... | 3 | https://mathoverflow.net/users/123634 | 426610 | 173,125 |
https://mathoverflow.net/questions/426615 | 2 | Let $X$ be a smooth projective algebraic surface over $\mathbb C$, and let $L\_1, L\_2$ be ample line bundles satisfying $H^i(X,L\_j) = 0$ for $i > 0$.
Do we necessarily have $(h^0(L\_1) + h^0(L\_2))^2 \geq h^0(L\_1 \otimes L\_2)$, or are there examples where this inequality is violated?
| https://mathoverflow.net/users/485787 | Inequalities of dimensions of global sections for line bundles on surfaces | Here is a counterexample. Let $X = C \times \mathbb{P}^1$ where $C$ is a curve of genus $g \geq 2$ and let $L\_1 = L\_2 = p\_1^\*L \otimes p\_2^\*\mathcal{O}(1)$ where $L$ s a generic line bundle of degree $g-1$. Then $H^0(L) = 0$ since the map $\mathrm{Sym}^{g-1}(C) \to \mathrm{Pic}^{g-1}(C)$ is not surjective for dim... | 3 | https://mathoverflow.net/users/12402 | 426616 | 173,126 |
https://mathoverflow.net/questions/426575 | 1 | I've got a question to the Hoeffding Inequality which states, that for data points $X\_1, \dots, X\_n \in X$, which are i.i.d. according to a probability measure $P$ on $X$, we find an upper bound for:
$$P\left(\left|{\sum\_{I=1}^n}X\_i - \int\_XX\_1dP(x)\right| \ge \varepsilon\right) \le \alpha.$$
In machine learn... | https://mathoverflow.net/users/485752 | Using Hoeffding inequality for risk / loss function | I think this question is more on the definition side. Often times people don't distinguish $\mathbb{P}$ the probability measure in the underlying probability space, and $P$ the distribution or the probability measure induced by a random variable. Also people tend to use $P$ directly, without making any explicit referen... | 2 | https://mathoverflow.net/users/483494 | 426619 | 173,128 |
https://mathoverflow.net/questions/426569 | 2 | Let $K$ be a CM-field with totally real subfield $F$. Let $(V\_1, h\_1)$ and $(V\_2, h\_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.
>
> **Question 1** Does every $F$-group isomorphism $PU(V\_1, h\_1)\cong PU(V\_2, h\_2)$ arise from an isometry of $(V\_1, h\_1)\c... | https://mathoverflow.net/users/108424 | Does the F-unitary group isomorphism arises from a conformal isometry? | The answer to Question 1 is **No.** Indeed, take $V\_1=V\_2=V:=K^n$ and write
$$ h\_1(z)=h\_2(z)=h(z):=\lambda\_1 z\_1 \bar z\_1+\dots+\lambda\_n z\_n\bar z\_n\quad\text{with}\
\lambda\_i\in F^\times.$$
Write $\widetilde G=U(V,h),\ G={\rm PU}(V,h)$.
Define
$$\tilde\sigma\colon\,\widetilde G\to \widetilde G,\quad g\maps... | 3 | https://mathoverflow.net/users/4149 | 426620 | 173,129 |
https://mathoverflow.net/questions/426606 | 1 | Let $C^{m,\alpha}\_M([0,1])$ be a Holder ball consisting of real-valued functions $g$ on $[0,1]$ such that
$$ \|g\|\_{C^{m,\alpha}} := \max\_{0\leq j \leq m } \sup\_{x\in [0,1]} |g^{(j)}(x)| + \sup\_{x,y\in [0,1], x\neq y} \frac{ |g^{(m)}(x) -g^{(m)}(y)|}{|x-y|^\alpha} \leq M. $$
Let $f \in C^{m,\alpha}\_M([0,1])$ be f... | https://mathoverflow.net/users/483494 | Approximating a smooth function under some restrictions | The revised question, where you just want uniformly bounded Holder norm on the approximating polynomials, the answer is **yes**.
In fact, you have more.
**Theorem 1** If $f\in C^{m,\alpha}\_M([0,1])$ is such that the function $\tilde{f}(x) = \begin{cases} f(x) & x\in [0,1]\\ 0 & x\not\in [0,1]\end{cases}$ is in $C^... | 1 | https://mathoverflow.net/users/3948 | 426621 | 173,130 |
https://mathoverflow.net/questions/426623 | 2 | Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly?
What I mean by implicitly is that there exists a smooth function $f:\mathbb{R}^n\to \mathbb{R}^m$, such that $\Sigma$ is diffeomorphic to $f^{-1}(0)$, and the Euclidean metric on $\mathbb{R}^n$ pulled back to $f^{-1}(0)$ is eq... | https://mathoverflow.net/users/485792 | Under what conditions can an orientable Riemannian 3-manifold be defined implicitly? | By the Nash embedding theorem every Riemannian manifold $M$ embeds isometrically into some ${\Bbb R}^n$. You may then take $f(x)=dist(x,M)$ for $x\in{\Bbb R}^n$.
| 9 | https://mathoverflow.net/users/39082 | 426625 | 173,132 |
https://mathoverflow.net/questions/426346 | 1 | Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous non-linear map, and let $A$ be a connected subset of $\mathbb{R}^n$ with $\text{dim}(A)=d\leq n$. When can we say that the dimension of the image, $\text{dim}(F(A))$, is also $d$? In other words, when does the map $F$ preserve dimension?
A non-example would be some ... | https://mathoverflow.net/users/484618 | Dimension-preserving non-linear map | *Dimension Theory* by Ryszard Engelking provides the answer I was looking for. Specifically, theorem 1.12.8 which I have provided below, although there are other results in this same area that may be useful to someone.
**Alexandroff's Theorem**: *If $f:X\to Y$ is an open mapping of a locally compact seperable metric ... | 1 | https://mathoverflow.net/users/484618 | 426652 | 173,137 |
https://mathoverflow.net/questions/426655 | 3 | It started with a conjecture I had, see [A statement on complex polynomials](https://mathoverflow.net/questions/426503/a-statement-on-complex-polynomials?noredirect=1#comment1096888_426503), which was false for $n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage the mai... | https://mathoverflow.net/users/81645 | On well separated circular regions in the Riemann sphere and complex polynomials | Let $\{ z\_1,\ldots,z\_n\}$ be any finite set,
and
$$L\_k(z)=\prod\_{j\neq k}(z-z\_j).$$
Then $L\_k,\; 1\leq k\leq n$ are linearly independent since
the set of their linear combinations consists of all polynomials
of degree $\leq n-1$ (Lagrange interpolation formula says that),
and the space of all polynomials of degre... | 2 | https://mathoverflow.net/users/25510 | 426667 | 173,139 |
https://mathoverflow.net/questions/426601 | 6 | Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups
$$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$
are finitely generated? We know that they are finite-dimensional after tensoring with $\mathbb Q$, since they become id... | https://mathoverflow.net/users/126183 | Finite generation of motivic cohomology of number fields | Indeed, I believe it is known that these are finitely generated. First, the Gysin sequence shows that the map
$$H^1(\mathcal{O}\_F;\mathbb{Z}(n))\rightarrow H^1(F;\mathbb{Z}(n))$$
is injective with cokernel given by
$$\oplus\_\nu H^0(k\_\nu;\mathbb{Z}(n-1))$$
where $\nu$ runs over all maximal ideals of $\mathcal{O}\_F$... | 3 | https://mathoverflow.net/users/480363 | 426674 | 173,141 |
https://mathoverflow.net/questions/426668 | 3 | The [Tracy–Widom distributions](https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution) admit many interpretations.
[One of them](https://www.ggi.infn.it/sft/SFT_2018/LectureNotes/Majumdar_slides.pdf) is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) ... | https://mathoverflow.net/users/125498 | What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution? | This is not a complete answer, but more of an approach and an invitation to look at the relevant literature. As you write, you would like to insert the so-called Fermi factor into the Fredholm Pfaffian expression for $F\_1$ or $F\_4$ (by analogy with the Fredholm determinant for $F\_2$). Such Fermi factors are also pre... | 2 | https://mathoverflow.net/users/979 | 426680 | 173,142 |
https://mathoverflow.net/questions/426681 | 3 | Let $(G,+)$ be an abelian group and $A$, $B$ and $C$ be finite subsets of $G$ with $A+B=C$. One may conclude that $A\subset C-B$. However, $A$ need not be equal to $C-B$. What is a necessary and sufficient condition to have $A=C-B$?
| https://mathoverflow.net/users/165074 | Sumsets with the property "$A+B=C$ implies $A=C-B$" | Regardless of whether $A,\ B$, and $G$ are finite or infinite, the necessary and sufficient condition is that $B$ lies in a coset of the *period* (or *stabilizer*) of $A$, defined to be the subgroup of all those group elements $g$ with $A+g=A$.
To see this, notice that $A\subseteq (A+B)-B$ holds true in a trivial way... | 6 | https://mathoverflow.net/users/9924 | 426686 | 173,144 |
https://mathoverflow.net/questions/426279 | 5 | I have the $N$x$N$ matrix below where $N$ is a power of 2 (usually 64 or 256) and $\omega = 2\pi/N$. What is its largest eigenvalue?
$\begin{bmatrix}
2 & 1 & 0 & 0 & \cdots & 0 & 0 & 1\\
1 & 2\cos(\omega) & 1 & 0 & \cdots & 0 & 0 & 0\\
0 & 1 & 2\cos(2\omega) & 1 & \cdots & 0 & 0 & 0\\
0 & 0 & 1 & 2\cos(3\omega) & \cd... | https://mathoverflow.net/users/20757 | The discrete Fourier transform's Gaussian-like eigenvector | A quick numerical investigation for $N$ up to 512 leads to a conjectured exact large-$N$ expansion of the largest eigenvalue,
$$
\lambda\_0(N) = 4 - \frac{2\pi}{N} + \frac{\pi^2}{2 N^2} - \frac{\pi^3}{24 N^3} + \frac{\pi^4}{48 N^4}
- c\_5 \frac{\pi^5}{N^5} + \ldots.
$$
The coefficient $c\_5^{-1} = -114.63(1)$ is negati... | 6 | https://mathoverflow.net/users/90413 | 426691 | 173,146 |
https://mathoverflow.net/questions/426682 | 3 | Maps between real numbers are often defined by convergent series. For example, to define the exponential map, we can just prove that series
$$
\sum\_{n = 0}^{\infty} \frac{x^n}{n!}
$$
converges, which implies that there is a map to which it converges. Can we use this approach to define (constructively) localic maps bet... | https://mathoverflow.net/users/62782 | Localic maps given by series | Here is a fairly general methods for this sort of thing :
**Step 1)** We give a constructive proof that for each (Dedekind) real $x$, the serie $\sum \frac{x^n}{n!}$ converge. We define $exp(x)$ as the limit.
**Step 2)** We show that $x \mapsto exp(x)$ is a geometric construction. That is if $f : \mathcal{E} \to \m... | 3 | https://mathoverflow.net/users/22131 | 426692 | 173,147 |
https://mathoverflow.net/questions/426684 | 14 | This question is motivated by the superficial observation that [Birkhoff's representation theorem](https://en.wikipedia.org/wiki/Birkhoff%27s_representation_theorem) and the [cryptomorphism between matroids and geometric lattices](https://en.wikipedia.org/wiki/Geometric_lattice#Cryptomorphism) are sort of similar. The ... | https://mathoverflow.net/users/19864 | Birkhoff's representation theorem vs matroid-geometric lattice correspondence | [Antimatroids](https://en.wikipedia.org/wiki/Antimatroid)
are a good example. We have the syllogism "Antimatroids are to
matroids as join-distributive lattices are to geometric lattices."
Two other examples are the characterizations of complemented modular
lattices of finite length $n\geq 4$ and primary modular lattice... | 7 | https://mathoverflow.net/users/2807 | 426695 | 173,148 |
https://mathoverflow.net/questions/426696 | 3 | This might be related to an open problem.
For odd natural $n$ define the Euler quotient:
$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$
>
> Q1 Are there infinitely many $n$ with unbounded smallest prime factor
> for which $a(n)$ is non-zero?
>
>
>
>
> Q2 Are there infinitel... | https://mathoverflow.net/users/12481 | Are there infinitely many nonzero Euler quotients $a(n)=\frac{2^{\phi(n)}-1}{n} \bmod n$? | Let $p$ be any odd prime. Suppose that $2^{p-1}-1=p^ku$ for some integer $u$ with $(p,u)=1$. Then by Lifting The Exponent Lemma for all integer $K\geq 0$ we have
$$
2^{p^K(p-1)}-1=p^{K+k}u\_K
$$
For some integer $u\_K$ such that $(p,u\_K)=1$. Therefore, for all $K>k$ the number $n=p^{K+1}$ satisfies
$$
2^{\varphi(n)}-1... | 8 | https://mathoverflow.net/users/101078 | 426700 | 173,150 |
https://mathoverflow.net/questions/426701 | 0 | If $G = (V, E)$ is a simple, undirected graph and $T \subseteq V$, let $$N(T) = \{v \in V: \{v, t\}\in E \text{ for some }t\in T\}.$$
Given $v\in V$ we let $N\_0(v) = \{v\}$ and $N\_{k+1}(v) = N\_k(v) \cup N(N\_k(v))$ for all $k\geq 1$. The *iterated degree sequence* of $v$, denoted by $(\text{deg}\_k(v))\_{k\in\omeg... | https://mathoverflow.net/users/8628 | Non-isomorphic graphs with identical iterated degree matrix | Yes. Consider all graphs with $V=\{1,\ldots,n\}$ for which the vertex $n$ has degree $n-1$. There are $2^{n^2/2+o(n^2)}$ isomorphism classes of such graphs. But $\deg\_k(v)=n$ for all $v$ and all $k\geqslant 2$, thus there exist only at most $n^{n-1}$ distinct matrices.
| 4 | https://mathoverflow.net/users/4312 | 426702 | 173,151 |
https://mathoverflow.net/questions/426554 | 1 | Let $\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$ be a faithful representation of the commutative Lie algebra $\mathbb{R}^n$ into the Lie algebra of skew-symmetric matrices. There is an orthonormal basis $d\_1,\ldots,d\_{2m}$ of $\mathbb{R}^{2m}$ such that for all $i=1,\ldots,n$ and $j=1,\ldots,m$
$$\rho(e\_i)(d\_{2j-1})=... | https://mathoverflow.net/users/56980 | Degenerate representation | Suppose $V = \mathbb{R}^n$ has a basis $(e\_1,\dots,e\_n)$. Your assumption is that you have a family of linear maps $\lambda\_1,\dots,\lambda\_m \in V^\*$ which are defined such that $\lambda\_r(e\_i) = \lambda\_{ir}$ for any $1 \leqslant r \leqslant m$. Working inside the algebra of all functions $\mathrm{Fun}(V,\mat... | 1 | https://mathoverflow.net/users/22846 | 426718 | 173,157 |
https://mathoverflow.net/questions/426727 | 1 | I am looking for preferably a closed form (or series solution if not possible) for the following integral:
$$\int\_0^a x^{3/2} J\_\nu (bx) dx$$
where $\nu$ is an integer. This 1D integral appears when taking the polar Fourier transform of a separable radially symmetric function in 2D that I would like to propagate ... | https://mathoverflow.net/users/483438 | Definite integral of Bessel function of the first kind times $x^{3/2}$ | The integral requires $\nu>-5/2$ for convergence, and then becomes a hypergeometric function:
$$\int\_0^a x^{3/2} J\_\nu (bx) dx=\frac{2^{1-\nu} a^{\nu+\frac{5}{2}} b^{\nu}}{(2 \nu+5) \Gamma (\nu+1)}\, \_1F\_2\left(\frac{\nu}{2}+\frac{5}{4};\frac{\nu}{2}+\frac{9}{4},\nu+1;-\frac{1}{4} a^2 b^2\right).$$
| 4 | https://mathoverflow.net/users/11260 | 426729 | 173,161 |
https://mathoverflow.net/questions/426732 | 2 | Suppose that $X,X\_1,X\_2,X\_3\dots$ is a sequence of $\mathbb{P}$-i.i.d. random variables **supported in the interval $[0,1]$**. Let $F$ be the cumulative distribution of $X$, i.e. $F(x):=\mathbb{P}[X \le x]$ and let $\hat{F}\_n$ be the empirical cumulative distribution given $X\_1,\dots,X\_n$, i.e, $\hat{F}\_n(x) := ... | https://mathoverflow.net/users/126216 | DKW inequality for $L^1$-norm | The exponent 2 on $\epsilon$ cannot be improved by passing to the $L^1$ norm. Consider $X\_i$ i.i.d. uniform in $[0,1]$. Let $C$ be a constant, and denote $\varepsilon\_n= C/\sqrt{n}$. Let $A\_n $ be the event that $\hat{F}\_n(1/2)<1/2-9\varepsilon\_n$ and let $B\_n$ be the event that $\|\hat{F}\_n-F\|\_{L^1([0,1])} \g... | 2 | https://mathoverflow.net/users/7691 | 426741 | 173,164 |
https://mathoverflow.net/questions/426534 | 1 | Let $(W\_t)\_{t\ge 0}$ be a standard Brownian motion and $\tau$ be a stopping time lying in $[1,2]$. For $x, y>0$, can we show
$$\mathbb E\big[{\bf 1}\_{\{x+\inf\_{0\le t\le 2}W\_t>0\}}(W\_{\tau}-y)^+\big]>0?$$
| https://mathoverflow.net/users/261243 | Is this expectation $\mathbb E\big[{\bf 1}_{\{x+\inf_{0\le t\le 2}W_t>0\}}(W_{\tau}-y)^+\big]$ strictly positive? | Let $A= \{W\_1>y+2, \, \inf\_{0 \le t \le 1} W\_t >-x\}.$
By the reflection principle,
$$P(A)=P(W\_1>y+2)-P(W\_1<-2x-y-2)$$ $$=P(W\_1>y+2)-P(W\_1>2x+y+2)>0\,.$$
Let $D= \{ \inf\_{1 \le t \le 2} W\_t >y\}.$ Then by the Markov property and the reflection principle,
$$P(D|A) \ge P(\inf\_{0 \le t \le 1} W\_t >-1)=P(W\_1>-1... | 2 | https://mathoverflow.net/users/7691 | 426744 | 173,165 |
https://mathoverflow.net/questions/426227 | 14 | In their celebrated paper "[A new approach to the representation theory of the symmetric group. II](https://doi.org/10.1007/s10958-005-0421-7)", Okounkov and Vershik prove that $Z(n-1,1)$, the centralizer of $\mathbb{C}[S\_{n-1}]$ in $\mathbb{C}[S\_n]$ is the (surprisingly commutative!) algebra generated by $Z(\mathbb{... | https://mathoverflow.net/users/138150 | What is the centralizer of a Young subgroup of $S_n$? | This answer is largely inspired by the wonderful paper of Samuel Creedon, [The Farahat-Higman Algebra of Centralizers of Symmetric Group Algebras](https://arxiv.org/abs/2206.02939), which studies in detail the case of $\mathbb{C}S\_n^{S\_{n-m}}$, where $m$ is considered fixed and $n$ varies. Let $\lambda = (\lambda\_1,... | 10 | https://mathoverflow.net/users/159272 | 426751 | 173,167 |
https://mathoverflow.net/questions/426664 | 6 | Let me recall the following definition. Let $F: C \to D$ be a functor between homotopical categories. Denote by $\gamma\_C: C \to \mathrm{Ho} C$ the localization and similary for $D$. A **total left derived functor** $\mathbf{L}F: \mathrm{Ho}C \to \mathrm{Ho}D$ for $F$ is a right Kan extension of $\gamma\_D \circ F$ al... | https://mathoverflow.net/users/482564 | Is the composite of absolute derived functors a derived functor? | Here is a somewhat degenerate example that illustrates what can go wrong.
Let $\textbf{Ab}$ be the category of abelian groups, considered as a homotopical category where the weak equivalences are the isomorphisms, and let $\textbf{Ch}$ be the category of chain complexes of abelian groups, considered as a homotopical ... | 6 | https://mathoverflow.net/users/11640 | 426752 | 173,168 |
https://mathoverflow.net/questions/426762 | 17 | Are there uncountably many $A\_\alpha $ of subsets of $\mathbb{N}$ with the following two properties:
1. Each $A\_\alpha$ has positive upper natural density
2. $A\_\alpha \cap A\_\beta$ is a finite set for $\alpha \neq \beta$
If the answer is no then the next question:
Are there uncountably many $A\_\alpha $ of s... | https://mathoverflow.net/users/36688 | Uncountably many subsets of the natural numbers with certain natural density condition | Amazingly, the answer to the main question is yes. For each $n$ let $I\_n = [2^{2^n}, 2^{2^{n+1}})$. Then let $\mathcal{C}$ be an uncountable family of infinite subsets of $\mathbb{N}$ any two of which have finite intersection. For each $B\in \mathcal{C}$ let $B' = \bigcup \{I\_n: n \in B\}$. Then $\{B': B \in \mathcal... | 25 | https://mathoverflow.net/users/23141 | 426766 | 173,171 |
https://mathoverflow.net/questions/426772 | 0 | I want to plot the two surfaces which are defined in $ \mathbb{ R }^3 \ni ( x, y, z ) $ via the equations $ 0 = y^2 - x\*(x^2 + 1) $ and $ 0 = z^2 - y\*(y^2 + 1) $, respectively.
Moreover, I want also to highlight the intersection curve of these surfaces.
Any tool is ok for me, but I know that sage has some functions... | https://mathoverflow.net/users/132492 | Plot two implicit surfaces in 3D and highlight their intersection | Something like this?
Blue curve is the intersection of the two surfaces $ 0 = y^2 - x(x^2 + 1) $ and $ 0 = z^2 - y(y^2 + 1) $.
The 3D plots were generated with Mathematica, as documented [here.](https://reference.wolfram.com/language/example/HighlightTheIntersectionOfTwoSur... | 2 | https://mathoverflow.net/users/11260 | 426774 | 173,173 |
https://mathoverflow.net/questions/426759 | 6 | Let $f \in L^1 (\mathbb R)$. Suppose $g\_n \in L^1 (\mathbb R)$ are a sequence of positive functions.
Define, for each $n$, the function $f\_n$ by
$$f\_n (x) := \frac{1}{2g\_n (x)} \int\_{x - g\_n (x)}^{x + g\_n (x)} f(y) \, dy.$$
**Question:** Is it true that if $g\_n \to 0$ in weak $L^1$ norm, then $f\_n \to f$... | https://mathoverflow.net/users/173490 | Convergence of integral averages in $L^1$ | The answer is negative even if $f$ is supported in $[0,1]$ and $g\_n \to 0$ uniformly in $[0,1]$. The reason is that $f \in L^1$ does not suffice for the Hardy-Littlewood maximal function to be in $L^1$, so choosing $g\_n$ to capture that maximal function will yield a counterexample. It is still useful to see an explic... | 7 | https://mathoverflow.net/users/7691 | 426784 | 173,178 |
https://mathoverflow.net/questions/426780 | 1 | $S\_5$ is a set consisting of the following 5-length sequences $s$: (1) each digit of $s$ is $a$, $b$, or $c$; (2) $s$ has and only has one digit that is $c$.
$T\_5$ is a set consisting of the following 5-length sequences $t$: (1) each digit of $t$ is $a$, $b$, or $c$; (2) $t$ has two digits that are $c$.
Is there ... | https://mathoverflow.net/users/57534 | 3-partition of a special set | Here is one such partition of $S\_5$, obtained via integer linear programming:
\begin{align}
A\_0 &= \{aaaac,aaacb,aabca,aacba,ababc,abbac,abcab,abcba,acaaa,acbbb,baabc,
babac,bacaa,bacbb,bbaca,bbbbc,bbbcb,bbcaa,bcaab,bcbba,caaba,cabab,cabbb,
cbaab,cbabb,cbbaa\} \\
A\_1 &= \{aaaca,aabbc,aacab,abaac,abbca,abbcb,abcbb,ac... | 3 | https://mathoverflow.net/users/141766 | 426788 | 173,180 |
https://mathoverflow.net/questions/426793 | 2 | Define a coupling $\pi\in \Pi(\mu,\nu)$ on the product space $(X\times X,\mathcal{F}\times\mathcal{F})$. let $\pi\_x$ be the disintegration of $\pi$ with respect to the $\mu$, i.e. there exists a Borel measurable function $x\mapsto\pi\_x$ such that $$\pi(dx,dy)=\mu(dx)K(x,dy)$$ where $K(\cdot,\cdot)$ is a probability t... | https://mathoverflow.net/users/168083 | Can we say that there exists a measurable function $f$ such that $ \nu=f_{\#}\mu$? | Yes: if $\pi\_x=\delta\_{f(x)}$, then
$$\nu(A)=\pi(X\times A)=\int\_X\mu(dx)\pi\_x(A) \\
=\int\_X\mu(dx)\,1(f(x)\in A)=\int\_X\mu(dx)\,1(x\in f^{-1}(A))
=\mu(f^{-1}(A))$$
for all Borel subsets $A$ of $X$, as desired.
| 2 | https://mathoverflow.net/users/36721 | 426795 | 173,182 |
https://mathoverflow.net/questions/426796 | 3 | It is known that
$$
\cos(\frac{x}{2})\cos(\frac{x}{4})\cos(\frac{x}{8})\dots = \frac{\sin x}{x} = O\_{x \rightarrow \infty}(x^{-1})
$$
Is it true that
$$
f(x) = \cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\frac{x}{27})\dots = o\_{x \rightarrow \infty} (1) ?
$$
If so, what is the rate of convergence?
It seems to me that ... | https://mathoverflow.net/users/97209 | $\cos(\frac{x}{3})\cos(\frac{x}{9})\cos(\frac{x}{27})\dots$ as $x \rightarrow \infty$ | No. If $x = 3^n \pi$ then $|f(x)| = f(\pi) \neq 0$
(numerically it's about $0.466$), and $3^n \pi$ can be arbitrarily large.
Such a construction fails for $\prod\_{k=1}^\infty \cos(x/2^k)$
because if some $x/2^k$ is nearly $\pi$,
or more generally some *odd* multiple of $\pi$, then
$x/2^{k+1}$ is nearly a half-integr... | 9 | https://mathoverflow.net/users/14830 | 426797 | 173,183 |
https://mathoverflow.net/questions/426709 | -1 | Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N\_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving that there is an $H$ such that for $T\geq H$, $$N\_0(T+1)-N\_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$$ and $$N(T+1)-N(... | https://mathoverflow.net/users/nan | Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$ | The first two displays (together) are in between the Lindelöf hypothesis and the Riemann hypothesis. That is, they imply the Lindelöf hypothesis, while they follow from the Riemann hypothesis. They are not known unconditionally. See Sections 13.5-13.6 in Titchmarsh: The theory of the Riemann zeta-function.
It is stra... | 2 | https://mathoverflow.net/users/11919 | 426800 | 173,184 |
https://mathoverflow.net/questions/426773 | 2 | It is mentioned in multiple occasions [here](https://terrytao.wordpress.com/tag/mobius-function/) that the bound
$$
\mathop{\sum\_{n=1}^{N}}\_{n\equiv a\mod l} \mu(n) = o(N)
$$
is equivalent to the prime number theorem in arithmetic progressions. But I am not able to find a proof of this equivalence (or the proof of th... | https://mathoverflow.net/users/148866 | Averages of Möbius function in arithmetic progressions | When $(a,m) = 1$, let $\pi(x;a \bmod m)$ be the number of primes $p\leq x$ such that $p \equiv a \bmod m$.
The prime number theorem for arithmetic progressions
mod $m$ says for all $a \in (\mathbf Z/m\mathbf Z)^\times$ that $\pi(x;a \bmod m) \sim (1/\varphi(m))x/\log x$.
Harold Shapiro, in the paper *Some assertions ... | 5 | https://mathoverflow.net/users/3272 | 426811 | 173,185 |
https://mathoverflow.net/questions/426354 | 4 | Let $G$ be a countable abelian group and let $H \le G$ be a subgroup. Let $G \curvearrowright (X,\mu)$ be an ergodic measure preserving action on some probability space $(X,\mu)$. Now we know that the action $H \curvearrowright (X,\mu)$ may not be ergodic, but it has an ergodic decomposition $\mu = \int\_Y\mu\_y d\nu\_... | https://mathoverflow.net/users/47704 | Ergodic decomposition of the action of a subgroup | The answer to both questions is "no" - a counterexample (labeled "folklore") appears immediately prior to Question 6.6 in *Austin, Tim*, [**Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications**](http://dx.doi.org/10.4064/fm210-2-3), Fundam. Math. 210, No. 2, 133-206 (2... | 4 | https://mathoverflow.net/users/10457 | 426818 | 173,186 |
https://mathoverflow.net/questions/426776 | 4 | A well known result of A. Zuk states that for $\frac{1}{3} < d < \frac{1}{2}$, a random group $\Gamma$ with respect to Gromov's density model with density $d$ has Kazhdan's property (T) with overwhelming probability.
On the other hand, [C. J. Ashcroft](https://arxiv.org/pdf/2206.14616.pdf) has recently proved that th... | https://mathoverflow.net/users/136187 | Residual finiteness of random groups with property (T) | This is an open question: there are no densities $1/2>d\geq 1/6$ where a random group is known to be residually finite. Any progress would be a major step forward.
As mentioned in the question, at density $<1/6$, [Ollivier--Wise](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKE... | 4 | https://mathoverflow.net/users/1463 | 426831 | 173,189 |
https://mathoverflow.net/questions/426638 | -1 |
>
> On the other hand, if $\mu$ and $\nu$ are completely dependent then $\pi\_{x\_1}=\delta\_{f(x\_1)}$ for some function $f$. Then $W(\pi)=1$.
>
>
>
---
Note that
$$
\pi(dx, dy)=\pi\_x(dy)\mu(dx).
$$
If $\pi\_x(dy)=\delta\_{f(x)}(dy)$, then
$$
\pi(dx, dy)=\pi\_x(dy)\delta\_{f(x)}(dy).
$$
| https://mathoverflow.net/users/168083 | What does $\mu$ and $\nu$ "dependent" mean? | It makes no sense to say that two probability measures $\mu$ and $\nu$ are "completely dependent". Dependence (or lack thereof) is a property of random elements, not of probability measures.
Also, a disintegration $(\pi\_x)$ is an attribute, not of probability measures $\mu$ and $\nu$, but of a coupling $\pi$ of $\mu... | 1 | https://mathoverflow.net/users/36721 | 426837 | 173,191 |
https://mathoverflow.net/questions/426833 | 3 | *Note: This is a refinement of a [previous problem](https://mathoverflow.net/questions/426759/convergence-of-integral-averages-in-l1).*
Let $f \in L^1 (\mathbb R)$. Suppose $g\_n \in L^1 (\mathbb R)$ are a sequence of positive functions.
Define, for each $n$, the function $f\_n$ by
$$f\_n (x) := \frac{1}{2g\_n (x... | https://mathoverflow.net/users/173490 | Weak convergence of integral averages | This is true, because we can approximate $f$ in $L^1$ by continuous functions. For them the statement clearly holds, and for the difference we can apply the classical
Hardy-Littlewood maximal inequality. Next, I will add some details.
Define the operator $T\_n$ on functions in $L^1(\mathbb R)$ by $$(T\_nf)(x) := \fra... | 3 | https://mathoverflow.net/users/7691 | 426843 | 173,193 |
https://mathoverflow.net/questions/426849 | 1 | I am trying a find a reference to/proof of the following result:
>
> Let $(M, g)$ be a compact Riemannian manifold. Then there is $b$ so that the following holds: for any $r>0$, there is a covering $\mathscr U$ of $M$ by open geodesic balls of radius $r$ so that (i) every member of $\mathscr U$ intersects at most $... | https://mathoverflow.net/users/41094 | A different version of Besicovitch Covering Theorem involving balls of half radius | Are you allowing $b$ to depend on the manifold (as it appears to me from your statement)? In that case, Besicovitch is overkill, and this statement holds in much more generality than Besicovitch. One only needs to know that a compact Riemannian manifold is a [doubling metric space](https://en.wikipedia.org/wiki/Doublin... | 4 | https://mathoverflow.net/users/486022 | 426858 | 173,196 |
https://mathoverflow.net/questions/426850 | 3 | Say that a clone (in the sense of universal algebra) $\mathfrak{C}$ has *almost abelian symmetry groups* (= **aasg**) iff for each function $f(x\_1,...,x\_n)\in\mathfrak{C}$ there is an abelian subgroup $A\subseteq S\_n$ and a set $X\subseteq \{1,...,n\}$ such that $$\langle A\cup \mathsf{Fix}(X)\rangle=\{\sigma: f=\la... | https://mathoverflow.net/users/8133 | When does a clone on a two-element set have almost abelian symmetry groups? | **Question 1**:
Is the set of clones on $\{1,2\}$ which have aasg computable?
The answer is Yes. The aasg clones are just the essentially unary clones
on $\{1,2\}$, and there are six of them.
**Easy direction:**
Assume that $\mathfrak C$ is essentially unary
and $f(x\_1,\ldots,x\_n)\in \mathfrak C$
does not depend on i... | 4 | https://mathoverflow.net/users/75735 | 426859 | 173,197 |
https://mathoverflow.net/questions/426863 | 3 | Let $X$ be a smooth projective surface and $f:X\to\mathbb{P}^1$ be a $\mathbb{P}^1$-fibration with a singular fiber consisting of a tree with three irreducible rational ($-2$)-curves $D\_1$, $D\_2$, $D\_3$ and a ($-1$)-curve $L$ intersecting one of $D\_i$'s. I want to know is it possible to calculate multiplicity of $L... | https://mathoverflow.net/users/211682 | Multiplicity of irreducible component of a singular fiber of a $\mathbb{P}^1$-fibration | If we assume that the rational curves intersect in nodes, then we can compute it from the data of the dual graph. Since $L$ intersects only one of the $D\_i$, then the $D\_i$ necessarily form a chain, say $D\_1 \cup D\_2 \cup D\_3$ and without loss of generality, there are two possibilities: 1) $L$ intersects $D\_1$, o... | 7 | https://mathoverflow.net/users/12402 | 426870 | 173,200 |
https://mathoverflow.net/questions/426828 | 2 | Here I am dealing with the difference of triangulated hull and thick hull. Let $\mathcal{D}$ be a triangulated category and $\mathcal{E}\subset\mathcal{D}$ be a collection of objects. The triangulated hull $\mathcal{E}^{\star}$ in $\mathcal{D}$ is the smallest triangulated subcategory of $\mathcal{D}$ containing $\math... | https://mathoverflow.net/users/nan | triangulated hull and thick hull of $\mathcal{O}_X,\dots,\mathcal{O}_X(n)$ | No. The simplest counterexample is a smooth quadric surface $X = \mathbb{P}^1 \times \mathbb{P}^1 \subset \mathbb{P}^3$. If $F$ belongs to the triangulated hull of $\mathcal{O}\_X$, $\mathcal{O}\_X(1)$, $\mathcal{O}\_X(2)$, then its first Chen class is proportional to
$$
c\_1(\mathcal{O}\_X(1)) = h\_1 + h\_2,
$$
where ... | 3 | https://mathoverflow.net/users/4428 | 426875 | 173,202 |
https://mathoverflow.net/questions/426872 | 4 | Are all Frey elliptic curves semi-stable? If so, where exactly is this needed in the modularity approach, now that we know modularity for all rational elliptic curves?
Thank you!
| https://mathoverflow.net/users/478525 | Are Frey elliptic curves semi-stable? | See Proposition 1 in [http://www.fen.bilkent.edu.tr/~franz/ta/ta-flt.pdf](http://www.fen.bilkent.edu.tr/%7Efranz/ta/ta-flt.pdf) and the paragraph following it. You have to adjust the terms in a potential counterexample to Fermat's Last Theorem for prime exponent $p \geq 5$ to make the Frey curve semistable. Whether or ... | 9 | https://mathoverflow.net/users/3272 | 426876 | 173,203 |
https://mathoverflow.net/questions/426864 | 6 | I am interested in existing algorithms to compute whether a given non-cyclic, non-pure cubic extension $K/\mathbb{Q}$ is monogenic or not, and if so, to give me a defining polynomial for the integral power basis $\mathbb{Z}[\alpha]$. I am also interested in general about the following statement: If we are to fix an int... | https://mathoverflow.net/users/138669 | Algorithm for computing whether a cubic field is monogenic? | In the paper "[Computing all power integral bases of cubic fields](https://doi.org/10.2307/2008731)" (by Gaal and Schulte, published in Mathematics of Computation in 1989) the authors give an algorithm to determine if a cubic field $K/\mathbb{Q}$ is monogenic or not. It boils down to solving a cubic Thue equation, whic... | 9 | https://mathoverflow.net/users/48142 | 426880 | 173,206 |
https://mathoverflow.net/questions/157630 | 8 | This question is [cross-posted](https://math.stackexchange.com/questions/661801/infinite-matrix-leading-eigenvalue-problem) at Math.StackExchange.com.
I'm trying to find the leading eigenvalue and corresponding left and right eigenvectors of the following infinite matrix, for $\lambda>0$:
$$
\mathrm{A}=\left(
\begi... | https://mathoverflow.net/users/46551 | Infinite matrix leading eigenvector problem | The characteristic polynomial of $\mathrm A\_n$ can be calculated exactly in the limit $n\to\infty$: As pointed out by @Gottfried, the characteristic polynomial $c\_n(x)=\det(\mathrm A\_n-x 1)$ can be written in terms of the $q$-binomial as
$$
c\_n(x)=\sum\_{k=0}^{n/2+1}q^{k(k-1)}\binom{n-k+1}{k}\_{\!q} \, (-x)^{n-k},
... | 5 | https://mathoverflow.net/users/90413 | 426882 | 173,207 |
https://mathoverflow.net/questions/426881 | 4 | I was reading Brown's Cohomology Theory of Finite groups and was wondering whether there's an example of the following.
Let $n\in \mathbb{N}$. Does there exist a $\mathbb{Z}\_n$ module $M$ which is cohomologically trivial (i.e. $\smash{\hat{H}}^i(\mathbb{Z}\_n,M)$ is trivial for each $i$) but such that there exists s... | https://mathoverflow.net/users/161454 | Cohomologically trivial module $M$ such that $M/pM$ is not cohomologically trivial for some $p\in\mathbb{N}$ | For $n=p$ an odd prime, $M=\mathbb Z/p^2$, where a generator of $\mathbb Z/p$ acts by multiplication by $1+p$, is an example.
$M/pM$ is $\mathbb Z/p$ with the trivial action of $\mathbb Z/p$, which has cohomology groups $\mathbb Z/p$ in each degree.
But for $M$ itself, the invariants are generated by $p$ and the co... | 7 | https://mathoverflow.net/users/18060 | 426883 | 173,208 |
https://mathoverflow.net/questions/426905 | 1 | I was wondering if anything could be said at all about the well-psedness of the following time-inhomogeneous singular diffusion SDE:
\begin{align}d X\_t&=\sigma(X\_t,t ) d W\_t , \qquad t\geq 0, \, X\_0 \in \mathbb R \\
\sigma(x,t )&=t^{-\alpha} b(x) \qquad \, \alpha>0,
\end{align}
with $b$ as nice as required (say... | https://mathoverflow.net/users/97651 | Solution of SDE with time power law singular diffusion | It seems natural to view the Itô process on a different time scale where it becomes a Itô diffusion, and in turn, invoke existence/uniqueness theory for Itô diffusion.
In particular, under the (deterministic) time change given by $\tau(t) = (t-2 \alpha t )^{\frac{1}{1-2 \alpha}}$ (so that $\tau'(t) = \tau(t)^{2 \alph... | 3 | https://mathoverflow.net/users/64449 | 426908 | 173,213 |
https://mathoverflow.net/questions/426910 | -2 | Let $E:y^2＝x^3-x/ \Bbb{Q}(i)$ be elliptic curve and $L(E,1)$ be a special value of $L$ function of $E$ at $1$.
Let $L(ψ,1)$ be value at $1$ of Hecke $L$ function with respect to Hecke character $ψ$, It is known that $L(E,1)＝L( \bar{ψ},1)L(ψ,1)$.
In this case, why $L(ψ,1)＝1$ ?
I may forget some trivial fact about ... | https://mathoverflow.net/users/144623 | Special value of Hecke $L$ function | Not clear what you mean by $\varphi$ or $\psi$. At any rate, $E$ is a CM curve, hence $L(s,E)$ equals $L(s,\psi)$ for a suitable Hecke character $\psi$ of the number field $\mathbb{Q}(i)$. You can find the details (e.g. the definition of $\psi$) in Sections 8.3-8.4 of Iwaniec's book "Topics in classical automorphic for... | 3 | https://mathoverflow.net/users/11919 | 426911 | 173,214 |
https://mathoverflow.net/questions/426915 | 4 | Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, and $u\in\mathbb{R}^n\setminus\{0\}$ such that $u$ does not belong to the asymptotic cone of $C$ and is nowhere tangent to $\partial C$. For $x\in C$, let $p(x)$ be the projection of $x$ on $\partial C$ toward the direction $u$:
$$p(x)=x+\inf\{\lambda\geq ... | https://mathoverflow.net/users/159940 | Is this projection on the boundary of a convex Lipschitz? | No. E.g., let $n=2$ and
$$C=\{(s,t)\in\mathbb R^2\colon t\ge s^2\}.$$
Then the asymptotic cone of $C$ is $K:=\{(0,t)\colon t\ge0\}$. Let $u=(1,0)\notin K$. Then
$$p((0,t))=(\sqrt t,t)$$
for real $t\ge0$. So, $p$ is not Lipschitz.
---
Somehow I missed the condition that the direction $u$ be nowhere tangent to $\pa... | 4 | https://mathoverflow.net/users/36721 | 426918 | 173,218 |
https://mathoverflow.net/questions/426906 | 3 | While playing around with random matrices and I arrived at a different formula for the mean of the limiting normal distribution for a spectral CLT for sample covariance matrices. More precisely I have the formula
\begin{align\*}
& \sum\limits\_{b=1}^{r-1} c^{b} \left(A(r,b) - {r \choose b}^2 \, b\right)
\end{align\*}
f... | https://mathoverflow.net/users/409412 | Is this combinatorial identity known? (of interest for random matrix theory) | Firstly, exploit the finite support to simplify the limits of the sums. Secondly, split the second sum. We get
$$\begin{align\*}A(r,b) =& \sum\limits\_{m=1}^{r} (2m-1) {r \choose b-m}{r \choose b+m-1} \\
& + \sum\limits\_{m=1}^{r} 2m {r \choose b-m}{r \choose b+m} \\
& + \sum\limits\_{m=1}^{r} {r \choose b-m}{r \choo... | 7 | https://mathoverflow.net/users/46140 | 426919 | 173,219 |
https://mathoverflow.net/questions/426903 | 2 | Let $M$ be a closed oriented real manifold with a free smooth circle action. Denote $BS^1$ to be the classifying space of principal circle bundles and $ES^1\rightarrow BS^1$ to be the universal principal circle bundle.
Now we consider the fibre product $M\_{S^1}:=M\times\_{S^1}ES^1$. When the fundamental group $\pi\_... | https://mathoverflow.net/users/169860 | on second cohomology of $S^1$-manifold | Yes. This follows from the Leray-Serre spectral sequence of the fibre bundle
$$
M\to M\_{S^1} \to BS^1
$$
which has $E\_2^{p,q}=H^p(BS^1;H^q(M;\mathbb{Z}))$ and converges to (the associated graded of the filtration on) $H^\*(M\_{S^1};\mathbb{Z})$. Note that $BS^1\simeq \mathbb{C}P^\infty$ has cohomology concentrated in... | 7 | https://mathoverflow.net/users/8103 | 426920 | 173,220 |
https://mathoverflow.net/questions/426712 | 9 | I have done a course in Sieve Theory from [the notes of Prof. Rudnick](http://www.math.tau.ac.il/%7Erudnick/courses/sieves2015.html). Before this, I did 2 courses in Number Theory from the 2 volumes of Apostol.
I don't have any guidance by professor as I am living in a very poor country in Europe and I study mathemat... | https://mathoverflow.net/users/151209 | Status of current research in Sieve Theory | Sieve theory is not saturated. It is alive and thriving. The ICM just awarded the Fields medal to James Maynard in no small part due to his work in sieve theory (see [here](https://arxiv.org/abs/1311.4600) and [here](https://arxiv.org/abs/1408.5110)). Because of the natural role of multiplicative structure in sieve the... | 15 | https://mathoverflow.net/users/111215 | 426932 | 173,223 |
https://mathoverflow.net/questions/426666 | 4 | Consider a $1$-periodic Hamiltonian $H:S^{1}\times M\rightarrow \mathbb{R}$ defined on a compact symplectic manifold $M$. Let's suppose $M$ is nice enough so that we can develop Floer theory on it. Now when defining Floer's equation I have seen people considering time-dependent compatible almost complex structures $J\_... | https://mathoverflow.net/users/155363 | Choice of a family of almost complex structures when defining Floer Homology | For a lot of things, you can work with a generic time-independent $J\_0$ as you suggest; for instance, Audien & Damien work in this context in their book (so for most of the "fundamental" constructions in Hamiltonian Floer theory, you don't need time-dependence in your almost complex structure).
But for other things,... | 5 | https://mathoverflow.net/users/134697 | 426943 | 173,226 |
https://mathoverflow.net/questions/425095 | 2 | When developing floer theory for an Hamiltonian $H:M\times S^{1}\rightarrow \mathbb{R}$ we will want $H$ to satisfy a non-degenerancy condition, that is, for every $x\in \mathcal{P}(H)$, a periodic solution of the hamiltonian system, $1$ is not an eigenvalue of $d\phi^{1}\_{H}(x(0))\in GL(T\_{x(0)}T^\*M)$. From all the... | https://mathoverflow.net/users/155363 | Generic choice of non-degenerate Hamiltonians $H$ in Floer theory | You can find a statement (and proof) of such a theorem in Hofer-Salamon's *Floer homology and Novikov rings*, where it appears as Theorem $3.1$. They require also that no holomorphic spheres with first Chern number $\leq 1$ lie on the periodic orbit (which becomes necessary when one deals with bubbling off in the Floer... | 1 | https://mathoverflow.net/users/134697 | 426950 | 173,229 |
https://mathoverflow.net/questions/426948 | 0 | Let's take a simple random walk on $\mathbb{Z}$, $(S\_n)\_{n\geq0}$, started at zero. If $\tau^+\_0 = \inf\{n \geq 1: S\_n = 0\}$ is the first time the walk returns on zero, we know that $\mathbb{E}[\tau^+\_0] = +\infty$, since the walk is recurrent null. Now I need to know the tail, $\mathbb{P}[\tau^+\_0> K]$, when $K... | https://mathoverflow.net/users/486100 | Simple random walk return time | This can be done by the reflection principle. Also, one can use [Theorem 0.6](http://cgm.cs.mcgill.ca/%7Ebreed/MATH671/lecture2corrected.pdf), which implies
$$P(\tau\_0^+>k)=\tfrac1k\,E|S\_k|.$$
By the central limit theorem and uniform integrability, for $k\to\infty$,
$$E|S\_k|\sim\sqrt k\,E|Z|=\sqrt k\,\sqrt{\frac2... | 1 | https://mathoverflow.net/users/36721 | 426952 | 173,230 |
https://mathoverflow.net/questions/426947 | 1 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\Tor{Tor}$Let's suppose $V$ is a $k$-vector space equipped with its standard (left) $\GL (V)$-action. The shuffle algebra is the graded dual of the tensor algebra with the standard Hopf algebra structure, and for ease I'm just going to denote this shuffle algebra by $T^\b... | https://mathoverflow.net/users/73780 | Equivariant description of indecomposable elements in shuffle algebra | There is a duality on covariant functors from $k$--vector spaces to itself: $DF(V) = F(V^{\vee})^{\vee}$, where $V^{\vee}$ is the dual of the vector space $V$. $D$ is known to preserve irreducible functors, so in the semisimple case (e.g. if $k$ has characteristic 0), $D$ acts as the identity on appropriately finite ob... | 5 | https://mathoverflow.net/users/102519 | 426958 | 173,232 |
https://mathoverflow.net/questions/426896 | 2 | Let $u(x, t)$ be a (non-negative, bounded) function on $\mathbb{R}^{n}\times [0, +\infty)$ and suppose that $u$ satisfies some time-independent PDE, e.g. $\partial\_{t}u=\Delta\_{p}u$. Let us assume that $u$ has the following property: if there exists a ball $B(x\_{0}, r\_{0})$ in $\mathbb{R}^{n}$ such that $$B(x\_{0},... | https://mathoverflow.net/users/163368 | Property implies finite propagation speed | As noted by the OP in a comment, $u(x, t)$ is assumed to be a (non-negative, bounded) function on $\mathbb{R}^{n}\times [0, +\infty)$ which solves a time-homogenous PDE.
``Time homogenous'' means that if $u(x,t)$ is a solution, then $\tilde{u}(x,t):=u(x,s+t)$ is also a solution for every $s \ge 0$. In particular, this ... | 5 | https://mathoverflow.net/users/7691 | 426959 | 173,233 |
https://mathoverflow.net/questions/426957 | 9 | Let $k$ be a field. By the Quillen-Suslin theorem, all vector bundles on $\mathbb{A}^n\_k$ are trivial for all $n \geq 0$. If $U \subset \mathbb{A}^n\_k$ is an affine open subset, then vector bundles on $U$ (of small rank) need not be trivial in general, but there do exist (non-trivial) such $U$, e.g., (split) algebrai... | https://mathoverflow.net/users/519 | Triviality of vector bundles on affine open subsets of affine space | For your final question, the answer is that all vector bundles over $U$ are trivial.
Sketch of proof: Let $R=k[x\_1,\ldots,x\_n]$.
Let $L\_1, L\_2,\ldots, L\_m$ be the equations of hyperplanes in $R$. Without loss of generality, we can assume $L\_1=x\_1$.
Set $A=R[L\_1^{-1},\ldots, L\_m^{-1}]$ and set $B=R[L\_2^{... | 11 | https://mathoverflow.net/users/10503 | 426961 | 173,234 |
https://mathoverflow.net/questions/426954 | 3 | In symbolic method, one often considers two operators on ordinary generating functions, namely
$$
\operatorname{PSET}F(x) = \exp\left(F(x)-\frac{F(x^2)}{2}+\frac{F(x^3)}{3}-\dots\right),
$$
and
$$
\operatorname{MSET}F(x) = \exp\left(F(x)+\frac{F(x^2)}{2}+\frac{F(x^3)}{3}+\dots\right).
$$
These operators allowed... | https://mathoverflow.net/users/116776 | Representing PSET as species | See Gilbert Labelle, *[On asymmetric structures](https://doi.org/10.1016/0012-365X(92)90371-L)*, Discrete Math. 99 (1992), 141–164.
| 5 | https://mathoverflow.net/users/10744 | 426963 | 173,235 |
https://mathoverflow.net/questions/425847 | 3 | Let $A$ and $B$ be two (non commuting) self-adjoint bounded operator acting on a Hilbert space and let $p,q>1$ such that $\frac1p+\frac1q=1$
Do we have a Young-type inequality such as $ \frac12|AB+BA| \leq \frac{|A|^p}{p}+ \frac{|B|^q}{q}$ in the sense of quadratic form ? It is of course obvious for $p=q=2$.
| https://mathoverflow.net/users/107004 | Young-type inequality for bounded operator | As a general criterion, these kind of inequalities hold at the level of singular values (which implies a version of the inequality where one of the sides is conjugated by a unitary/partial isometry), but they tend to fail at the plain operator level.
Here is a counterexample for $p=3$, $q=3/2$. Take
$$
A=\begin{bmatr... | 5 | https://mathoverflow.net/users/3698 | 426964 | 173,236 |
https://mathoverflow.net/questions/426942 | 3 | Say we have a power series of two variables, with an associated function $f$ defined as
$$
\begin{split}
f(x, y) =\, & \sum\_{n,m} a\_{n,m}x^ny^m,\\
& a\_{n,m} \geq 0 \quad \forall n, m \in\mathbb{N},
\end{split}
$$ which is known to converge in each point of a compact set $\mathcal{C} \subset \mathbb{R\_+}^2$. We can... | https://mathoverflow.net/users/102634 | Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set? | The series $f(x,y)=y+xy+x^2y+x^3y+\dots$ converges to $0$ when $y=0$, and converges to $y/(1-x)$ when $|x|<1$. This function is not continuous at $(x,y)=(1,0)$.
| 7 | https://mathoverflow.net/users/6666 | 426980 | 173,240 |
https://mathoverflow.net/questions/426972 | 6 | Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper [Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves](https://doi.org/10.24033/asens.2423) by Yifeng Liu, the notation $\underline{H}^p(X, \mathscr{F})$ is used. (More generally, $\mathscr{... | https://mathoverflow.net/users/112369 | Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$? | As indicated in the comments, the notation $\def\HH{\underline{\rm H}}\def\H{{\rm H}}\HH^p(X,F)$ is defined (for example) by Milne in *Étale Cohomology* as the $p$th right derived functor of the inclusion functor from the category of sheaves of abelian groups to the category of presheaves of abelian groups on the same ... | 6 | https://mathoverflow.net/users/402 | 426989 | 173,243 |
https://mathoverflow.net/questions/426984 | 2 | An integer partition is a sequence $\lambda=(\lambda\_1\geq\lambda\_2\geq\dotsb\geq\lambda\_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a positive integer $s>1$.
Let $\mathcal{O}\_{n,s}=\{\lambda\vdash n: \lambda\_i\in\{1,3,5,\dots,2s-1\}\}$. Parts are odd in... | https://mathoverflow.net/users/66131 | Seeking a bijective proof enumerating two partition sets: Part I | Take a partition (Young diagram) in $O\_{n,s}$.
Number of elements in first column is $a\_1$.
Removing the first column now gives a partition with only even parts.
Divide all parts by $2$, and take conjugate. This gives $a\_2 \geq a\_3 \geq \dotsc \geq a\_s$, and you now have an element in the second set.
*It is a ni... | 4 | https://mathoverflow.net/users/1056 | 426990 | 173,244 |
https://mathoverflow.net/questions/426956 | 8 | Since [this question](https://mathoverflow.net/questions/41310/) is on the front page again, a generalization.
>
> Let $p$ be prime, and let $a$ and $b$ be positive integers with $a+b=p-1$. Is it possible to have two loaded dice, one with sides labeled $\{ 0,1,\ldots, a \}$ and the other with sides labeled $\{ 0,1,... | https://mathoverflow.net/users/297 | Two dice yielding uniform distribution, part 2 | Suppose that you can factor $x^{p-1} + x^{p-2} + \cdots + 1$ as $f(x)g(x)$ with $\deg(f)=a,\deg(g) = b$ and $f,g$ having nonnegative coefficients.
Lemma: It must be the case that $f = \sum\_{i=0}^a c\_i x^i$ and $g = \sum\_{i=0}^b d\_i x^i$, where $c\_i, d\_j > 0$ for all $0 \le i \le a$ and $0 \le j \le b$.
Proof:... | 1 | https://mathoverflow.net/users/141277 | 426994 | 173,245 |
https://mathoverflow.net/questions/427000 | 1 | In [the proof that Martin's maximum implies (\*)](https://ivv5hpp.uni-muenster.de/u/rds/MM_implies_star_final_version_March_18_2021.pdf), the introduction gives the following theorem:
>
> Assume there is a proper class of Woodin cardinals. Then, the following are equivalent:
>
>
> * $(^\*)$
> * For any $\Pi\_2$ $... | https://mathoverflow.net/users/473200 | An extension of Woodin's star axiom | This principle is inconsistent, even if we just look at $(H\_{\omega\_2};\in)$. This is because - for example - the continuum hypothesis is expressible as a sentence in this structure ("There is an $\omega\_1$-sequence of reals such that every real appears in the sequence"), but both $\mathsf{CH}$ and $\neg\mathsf{CH}$... | 3 | https://mathoverflow.net/users/8133 | 427003 | 173,247 |
https://mathoverflow.net/questions/426998 | 3 | Let $T$ be a ring with involution $s:T\rightarrow T$. And let
$$h:T\otimes T^\text{op} \rightarrow T\otimes T^\text{op}$$ be the ring automorphism given by $h(a\otimes b)=s(b)\otimes s(a)$.
I was wondering if the induced homorphism
$$ K\_{\ast}(h): K\_{\ast}(T\otimes T^\text{op})\rightarrow K\_{\ast}(T\otimes T^\text... | https://mathoverflow.net/users/165456 | Involution map, and induced morphism in K-theory | No. Let $R$ be commutative, $T=R\times R$, $s(x,y)=(y,x)$. Then $T\otimes T^{op}=T\otimes T$ is the product of four copies of $R\otimes R$, so that $K(T\otimes T^{op})$ is the product of four copies of $K(R\otimes R)$ and $h$ permutes the copies in some nontrivial way.
| 6 | https://mathoverflow.net/users/6666 | 427006 | 173,250 |
https://mathoverflow.net/questions/426861 | 1 | Let $k$ be an integer with $k>1$ and let $$a\_k=\frac{2 k(k-1)}{\Gamma[k-1]} \int\_0^\infty \frac{e^{-r^2} r^{2k-3}}{2+4 r^2+r^4} dr.$$
How to prove that $a\_k<1$?
| https://mathoverflow.net/users/486023 | A bounded sequence $a_k=\frac{2 k(k-1)}{\Gamma[k-1]} \int_0^\infty \frac{e^{-r^2} r^{2k-3}}{2+4 r^2+r^4} dr$ | $\newcommand\Ga\Gamma$We have
$$a\_k=\frac{2 k(k-1)}{\Ga(k-1)} J\_k,\quad J\_k:=\int\_0^\infty e^{-r^2} r^{2k-3}g(r)\,dr,$$
$$g(r):=\frac1{2+4 r^2+r^4}.$$
Next (for $r>0$),
$$g(r)=h(r)-\frac{8 \left(70 r^2+41\right)}{r^{12} \left(r^4+4 r^2+2\right)}<h(r),$$
where
$$h(r):=\frac{1}{r^4}-\frac{4}{r^6}+\frac{14}{r^8}-\frac... | 3 | https://mathoverflow.net/users/36721 | 427010 | 173,251 |
https://mathoverflow.net/questions/426969 | 6 | Consider simple random walks that stop when reaching a given node $x$ in an undirected, unweighted and connected graph on $n$ nodes.
Let
* $H(i,x)$ denote the (expected) hitting time from $i$ to $x$, with $H(x,x)=0$.
* $H\_{\max} = \max\_{i \in V} H(i,x)$ and $H\_{\text{avg}} = \frac{1}{n}\sum\_{i \in V} H(i,x)$.
I... | https://mathoverflow.net/users/486114 | Average and max. hitting time to a specific vertex | **Notation:** Let $G=(V,E)$ be an undirected simple graph of $n$ nodes. If $\tau\_x$ is the (random) time it takes the walk to reach the node $x$,
then write $H(v,x)=E\_v(\tau\_x)$. Denote $H\_{\max}(x):=\max\_{u \in V} H(u,x)$ and
$ H\_{\rm avg}(x) =\frac1n\sum\_{v \in V} H(v,x)$.
**Claim:** $ H\_{\max}(x) \le 2n^{3... | 9 | https://mathoverflow.net/users/7691 | 427013 | 173,254 |
https://mathoverflow.net/questions/427007 | 1 | I am currently reading "Smoothing of Multivariate Data" by Klemela. It contains Lemma 11.6, which upper and lower bounds the KL-divergence of two densities in terms of the $L\_{2}$-metric. The Lemma specifically states the following 2 bounds ***without*** proof:
**Upper bound**
>
> Let $f, f\_{0}$ be densities. I... | https://mathoverflow.net/users/486152 | Lower bound for KL divergence of bounded densities and $L_{2}$ metric | $\newcommand{\ep}{\varepsilon}$As in your post and comments, suppose that $f$ and $f\_0$ are supported on a compact set $S$, and
\begin{equation\*}
a\le f\le b,\quad a\le f\_0\le b
\end{equation\*}
on $S$ for some real $a,b$ such that $0<a<b$.
Then
\begin{equation\*}
\int\_{\left\{x \in \mathbf{R}^{d}: f\_{0}(x)>0\... | 1 | https://mathoverflow.net/users/36721 | 427017 | 173,255 |
https://mathoverflow.net/questions/427025 | 1 | I have a problem in understanding the concept of trianguline representation. Maybe someone can enlighten me.
Let $K$ be a finite extension of $\mathbb{Q}\_p$ and $V$ be a $p$-adic representation of $G\_K$ (absolute Galois group of $K$) of dimension $n$. By $p$-adic Hodge theory $V$ corresponds to some étale $(\varphi... | https://mathoverflow.net/users/118220 | Trianguline representation | No, triangulations are not in general unique.
A simple way of seeing this is to consider the case when $K = \mathbf{Q}\_p$, $V$ is 2-dimensional and crystalline with distinct Hodge–Tate weights, say $\{0, -r\}$ and the eigenvalues of $\varphi$ on $D\_{cris}(V)$ are distinct, say $\alpha$ and $\beta$. Weak admissibili... | 3 | https://mathoverflow.net/users/2481 | 427031 | 173,258 |
https://mathoverflow.net/questions/427023 | 0 | Consider the scalar conservation law
$$u\_t+f(u)\_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$
where $f \in C^{2}(\mathbb{R})$ is a strictly convex function ($f''>c>0$).
The solution can be shown to satisfy so-called "Oleinik's entropy condition":
$$ \frac{u(x+a,t)-u(x,t)}{a} \leq... | https://mathoverflow.net/users/nan | Oleinik inequality (one-sided Lipschitz condition) implies $BV_{\mathrm{loc}}$ for solution of conservation law | The function $f(x):=u(x,t)-cx/t$ decreases, thus $u(x,t)=f(x)+cx/t$ is a sum of two monotone functions, so belongs to local BV.
| 3 | https://mathoverflow.net/users/4312 | 427033 | 173,259 |
https://mathoverflow.net/questions/426974 | 6 | Suppose the dynamical system $(X,T)$ has only proper factors (i.e. not $(X,T)$ itself) of zero topological entropy. Does the system $(X,T)$ also have zero entropy?
| https://mathoverflow.net/users/45092 | Topological dynamical systems with only zero-entropy factors | This question is very related to the question of **lowering topological entropy**, introduced in ``Can one always lower topological entropy?'' by Shub and Weiss and then very nearly solved by Lindenstrauss in "Lowering topological entropy" and "Mean Dimension, Small Entropy Factors, and an Embedding Theorem." There's t... | 8 | https://mathoverflow.net/users/116357 | 427036 | 173,260 |
https://mathoverflow.net/questions/427026 | 1 | Let $W$ be a standard one dimensional Brownian motion, and $\mathcal F\_t$ its natural filtration. Consider the SDE
$$dX\_t = \mu\_X (t, \omega) \, dt + \sigma\_X (t, \omega) \, dW\_t$$
$$dY\_t = \mu\_Y (t, \omega) \, dt + \sigma\_Y (t, \omega) \, dW\_t$$
$$X\_0 = x\_0, Y\_0 = y\_0 \text{ a.s.}$$
where $\mu\_X,... | https://mathoverflow.net/users/173490 | A comparison principle for SDE | Don't think so. Take both to satisfy something like $dZ = \sigma |Z| dW$, start one from -1, and one from 1. regardless of the exact parameters, one stays positive and one stays negative.
| 2 | https://mathoverflow.net/users/143907 | 427038 | 173,261 |
https://mathoverflow.net/questions/427035 | 15 | I feel that the following problem should have a clean and simple solution, but so far I couldn't find one.
>
> Suppose that $p$ is a prime, and that $A\subseteq\mathbb Z/p\mathbb Z$ is a set such that
>
>
> * for any $z\in\mathbb Z/p\mathbb Z$, either $z\notin A$ or $z+1\notin A$;
> * for any $z\in\mathbb Z/p\mat... | https://mathoverflow.net/users/9924 | Sets with both additive and multiplicative gaps | I can give an upper bound of $2p/5$ and a lower bound of $(2/5-o(1))p$ for a conjecturally infinite set of $p$.
For the upper bound, the numbers $z, z+1, 2z+2, 2z+1, 2z$ form a pentagon in your graph so at most $2$ out of the $5$ may be in the set $A$. As $z$ runs over $\mathbb Z/p\mathbb Z$, each of $z, z+1, 2z+2, 2... | 23 | https://mathoverflow.net/users/18060 | 427042 | 173,262 |
https://mathoverflow.net/questions/426995 | 0 | An integer partition is a sequence $\lambda=(\lambda\_1\geq\lambda\_2\geq\dotsb\geq\lambda\_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a positive integer $s>1$.
Let $\mathcal{O}\_{n,s}=\{\lambda\vdash n: \lambda\_i\in\{1,3,5,\dotsc,2s-1\}\}$. Parts are odd i... | https://mathoverflow.net/users/66131 | Seeking a bijective proof enumerating two partition sets: Part II | Just to mark this question as answered, let me convert my comments into an answer.
The set of partitions $\mathcal{B}\_{n,s}$ you define are called the "lecture hall partitions" and they were introduced by Mireille Bousquet-Mélou and Kimmo Eriksson in [1]. See also the survey on the mathematics of lecture hall partit... | 3 | https://mathoverflow.net/users/25028 | 427048 | 173,263 |
https://mathoverflow.net/questions/427053 | 0 | Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
~~Under what conditions does $X=h(Y)$, where $Y\sim g(y)$ and $h$ is some function?~~
Under what conditions does there exist a function $h$ such $h(Y)$ has the s... | https://mathoverflow.net/users/486206 | When can a convolution be written as a change of variables? | Suppose that $Y,U$ are independent random variables (r.v.'s) such that $h(Y)=U+Y$ for some (Borel-measurable) function $h$. Then $U=Z:=h(Y)-Y$ and $U,Z$ are independent. So, $U$ is independent of itself. So, $U$ is constant almost surely (a.s.): $P(U=u)=1$ for some real $u$. (Indeed, if $U$ is not constant a.s., then t... | 2 | https://mathoverflow.net/users/36721 | 427055 | 173,264 |
https://mathoverflow.net/questions/427060 | 6 | I am interested in knowing which natural categories of its representations the étale fundamental group of a scheme can be recovered from.
Suppose $X$ is a scheme. Let $\pi\_1^\text{ét}(X)$ be its étale fundamental group. Thinking along the lines of Tannakian reconstruction of a pro-algebraic group from the category o... | https://mathoverflow.net/users/486214 | Tannakian-type reconstruction of étale fundamental group | Yes, this is possible. Since finite groups are algebraic groups, pro-finite groups are pro-algebraic groups. So one can recover $\pi\_1$ in exactly the way you say from the category of algebraic representations of $\pi\_1$, i.e. finite-dimensional representations that are continuous for the discrete topology of $k$. (T... | 13 | https://mathoverflow.net/users/18060 | 427061 | 173,266 |
https://mathoverflow.net/questions/426917 | 5 | I am looking for a local integral domain $(D, m)$ with $Spec(D)=\{0,m\}\cup\{ P\_i\}\_i$ such that $P\_i's$ are incomparable (that is, $P\_i\not\subseteq P\_j$ and $P\_j\not\subseteq P\_i$ for $i\not= j$) and $\cap\_i P\_i=0$ and $\cup\_i P\_i\not=m$, where $Spec(D)$ is the set of all prime ideals of $D$ ($D$ is not No... | https://mathoverflow.net/users/338309 | An example of a local integral domain with special spectrum | Let $k$ be a field. Put $R=k[[x,y]]$ and let $D\subset R$ be the subring $k+xR$. It consists of power series without any term $ay^n$ ($a\in k^\times$, $n>0$), or (equivalently) series $f(x,y)$ such that $f(0,y)\in k$. It is easy to see that $D$ is local with maximal ideal $m=\{f\in D\mid f(0,0)=0\}$.
I claim that $D$... | 7 | https://mathoverflow.net/users/7666 | 427067 | 173,267 |
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