parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/390771 | 0 | Let $\mathcal{P}\_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \int\_{x \in X} I\_C(x) d\mathbb{P}(x),
$$
continuous on $P\_p(X)$ with respect to the Wasserstein distance?
| https://mathoverflow.net/users/36886 | Is integration against an indicator Wasserstein-Continuous | Assuming $C$ isn't also open, find a sequence $(x\_n)$ in $X\setminus C$ which converges to a point $x$ in $C$. Then the point measures $\delta\_{x\_n}$ converge to $\delta\_x$ but their integrals against $1\_C$ do not.
| 2 | https://mathoverflow.net/users/23141 | 390772 | 161,744 |
https://mathoverflow.net/questions/390774 | 3 | There is a result of Soundararajan on the upper bound of the partial sums of the Möbius function assuming GRH [here](https://arxiv.org/abs/0705.0723). [Suger and Halupczok](https://projecteuclid.org/journals/functiones-et-approximatio-commentarii-mathematici/volume-48/issue-1/Partial-sums-of-the-M%C3%B6bius-function-in... | https://mathoverflow.net/users/nan | Partial sums of the Möbius function on arithmetic progressions | A variant of the Siegel-Walfisz theorem states that there is a constant $c>0$ with the following property. For any $A>0$ and $q\leq(\log x)^A$, we have
$$\displaystyle \sum\_{\substack{n\leq x\\n \equiv a \bmod q}}\mu(n)\ll\_A x\exp\left(-c\sqrt{\log x}\right).$$
See Exercise 13 for Section 11.3 of Montgomery-Vaughan: ... | 12 | https://mathoverflow.net/users/11919 | 390778 | 161,745 |
https://mathoverflow.net/questions/389954 | 1 | Let $T$ be a compact Hausdorff space and $X$ be an infinite-dimensional complemented subspace of $C(T)$.
Question 1. Assume that $X$ has a subspace $U$ that is isomorphic to $c\_{0}$. Given any positive integer $n$ and a finite-dimensional subspace $M$ of $X$. Do there exist a constant $K$, depending only on the Bana... | https://mathoverflow.net/users/41619 | Every complemented subspace of a $C(T)$-space is an $\mathcal{L}_{\infty}$-space | Question 1 has an affirmative answer. You don't need $X$ complemented in a $C(K)$ space and for $U$ you only need that it fails cotype. Use the Mazur technique for constructing basic sequences: norm to $1+\epsilon$ the finite dimensional $M$ by finitely many linear functionals of norm one. The intersection of the kerne... | 1 | https://mathoverflow.net/users/2554 | 390779 | 161,746 |
https://mathoverflow.net/questions/390630 | 9 | Let $T$ be a first-order theory, and suppose we want to build a [saturated model](https://en.wikipedia.org/wiki/Saturated_model) $\mathbb U$ of $T$. That is, we want a model $\mathbb U$ of cardinality bigger than $|T|$, saturated in its own cardinality. In particular $\mathbb U$ will be universal and homogeneous for mo... | https://mathoverflow.net/users/2362 | Do saturated models require choice? | On request, I summarize here some of the results mentioned in the comments (now unfortunately deleted), even though they do not really answer the question.
For definiteness, I assume that *$M$ is saturated* means that for every $A\subseteq M$ that does not surject onto $M$, every partial $1$-type over $A$ which is fi... | 4 | https://mathoverflow.net/users/12705 | 390783 | 161,747 |
https://mathoverflow.net/questions/390788 | 6 | I have been a bit sloppy in the title, but let me be specific. I stepped again into the subtle difference between homotopy limit and limit in the homotopy category, in the following version.
Suppose you have two diagrams of spaces $X\_{\bullet}, Y\_{\bullet}$ and a homotopy equivalence $f\_i : X\_i \to Y\_i $ such th... | https://mathoverflow.net/users/140013 | Deformation of a diagram preserve the homotopy limit | This is false.
Consider the two $C\_2$-spaces $S^{2\sigma}$ and $S^2$, where $\sigma$ is the sign representation and $S^V$ denotes the one-point compactification. Then the two underlying spaces are the same and the action looks trivial in the homotopy category. However, the homotopy fixed points differ. (After 2-comp... | 14 | https://mathoverflow.net/users/6936 | 390797 | 161,752 |
https://mathoverflow.net/questions/390643 | 3 | In 1968, J. Lindenstrauss and A. Pe{\l}czy'{n}ski posed a problem: Is every complemented subspace $X$ of an $\mathcal{L}\_{p}$-space ($1\leq p\leq \infty$) either an $\mathcal{L}\_{p}$-space or isomorphic to a Hilbert space? If $p=1$ or $\infty$ and $X$ is infinite-dimensional, $X$ can not be a Hilbert space. In 1969, ... | https://mathoverflow.net/users/41619 | Complemented subspaces of $\mathcal{L}_{p}$-spaces | I don't know if there is a good reference for Theorem 1. The Lindenstrauss-Pelczynski proof shows that $Y$ is contractively complemented in $L\_p(\mu)$ for some measure $\mu$ (at least for $p$ in the reflexive range), and contractively complemented subspaces of $L\_p(\mu)$ are isometrically isomorphic to some $L\_p(\nu... | 1 | https://mathoverflow.net/users/2554 | 390800 | 161,753 |
https://mathoverflow.net/questions/390799 | 2 | Let $t>1$ and $X\_1,..., X\_t$ a set of real random variables from a discrete distribution, whose pmf is $p(x)$, supported on the points $1,...,k$.
Let $N\_t(x) = \sum\_{i = 1}^t \mathbb{1}\_{X\_i =\, x}.$ It is easy to show that
$$
P\left[\max\_x\left|\frac{1}{t} N\_t(x) - p(x)\right| <\varepsilon\right] \geq 1- 2k\... | https://mathoverflow.net/users/156139 | Concentration on discrete probability estimator | $\newcommand\ep\varepsilon$For $[t]:=\{1,\dots,t\}$, we have
$$N\_t-tp=\sum\_{i\in[t]}J\_i,$$
where $N\_t$ is the random vector in $\mathbb R^{[k]}$ with coordinates $N\_t(x)$ for $x\in[k]$, $p$ is the vector in $\mathbb R^{[k]}$ with coordinates $p(x)$ for $x\in[k],$ and
the $J\_i$'s are iid zero-mean random vectors i... | 2 | https://mathoverflow.net/users/36721 | 390810 | 161,756 |
https://mathoverflow.net/questions/390708 | 4 | I make the following observation:
Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.)
This one has eigenvalues and eigenfunctions
$$ \lambda\_j = -4 \sin^2\left(\frac{\pi j}{2(n + 1)}\right)$$
and
$$v\_{j,i} = \sqrt{\frac{2... | https://mathoverflow.net/users/119875 | Convergence of discrete Laplacian to continuous one | I would consider this a "homogenization problem," or at least it can be thought of as very similar to one. In fact, I think the analysis will be essentially the same as (or maybe easier than) the case of periodic homogenization for uniformly elliptic equations.
In the case of periodic elliptic homogenization, the qua... | 4 | https://mathoverflow.net/users/5678 | 390813 | 161,757 |
https://mathoverflow.net/questions/390816 | 15 | Is there an explicit finitely presented group $G$ and an element $g\in G$ such that the statement "$g$ is equal to the identity" is independent of ZFC?
| https://mathoverflow.net/users/nan | Element being trivial in a finitely presented group independent of ZFC | The answer is yes.
This is just an instance of the general phenomenon that every non-computable decision problem is saturated with logical independence. (See [this related MO answer](https://mathoverflow.net/a/130815/1946).)
**Theorem.** If $A$ is a computably enumerable undecidable decision problem, such as the wo... | 21 | https://mathoverflow.net/users/1946 | 390821 | 161,760 |
https://mathoverflow.net/questions/390747 | 2 | This question is about semigroup theory.
Let $E$ be a locally compact metric space, and $X=(X\_t,t\ge 0;\,P\_x,x\in E)$ be a Markov process on $E$. We assume that $X$ is symmetric with respect to $m$, a Radon measure on $E$. The semigroup $\{T\_t\}\_{t \ge 0}$ of $X$ is extended to a strongly continuous contraction s... | https://mathoverflow.net/users/68463 | On a property of resolvents associated with holomorphic semigroups | As pointed out in the comments, this is true for $Re\, \lambda >0$ but can fail in general. An example is the Ornstein-Uhlenbeck operator $L=D^2-xD$ in $L^2(e^{-x^2/2}\, dx)$. The $L^2$ spectrum consists of the negative integers but the $L^\infty$ spectrum equals the left half plane. If $Re\, \lambda <0$, $\lambda \not... | 4 | https://mathoverflow.net/users/150653 | 390822 | 161,761 |
https://mathoverflow.net/questions/390744 | 3 | There are three common ways to represent the hyperbolic plane. One usually starts with the hyperboloid $x\_1^2+x\_2^2-x\_3^2=-1$ embedded in a Minkowski space with metric $ds^2 = dx\_1^2+dx\_2^2-dx\_3^2$. If the hyperboloid is parameterized with the embedding
$$
x\_1(t,\phi)=\sinh t \cos\phi\\
x\_2(t,\phi)=\sinh t\s... | https://mathoverflow.net/users/89713 | Coordinate transformation for the hyperbolic plane to the pseudo sphere | In the *pseudosphere* representation you wrote down, $T\in (-\infty,0)$.
If you let $R = e^{-T}$ (which now takes values in $(1,\infty)$), your metric becomes $R^{-2} (dR^2 + d\Phi^2)$ and conects to the upper half plane model. From there you can invert your transformation to get to the Poincare disk model, [which ha... | 3 | https://mathoverflow.net/users/3948 | 390825 | 161,763 |
https://mathoverflow.net/questions/390811 | 5 | Is there a homomorphism of finitely presented groups $f:G\to H$ and an element $h\in H$ such that the statement "$f^{-1}(h)$ is empty" is independent of ZFC?
| https://mathoverflow.net/users/nan | Empty preimage under homomorphism of finitely presented groups independent of ZFC | The answer is yes, as a consequence of [my answer to your other question](https://mathoverflow.net/a/390821/1946).
Namely, in that answer, we have a finite group presentation $H$ and a word $h$ such that the question $h=1$ in $H$ is independent of ZFC. So if we take $G$ to be trivial and $f:G\to H$ the unique homomor... | 5 | https://mathoverflow.net/users/1946 | 390827 | 161,764 |
https://mathoverflow.net/questions/390818 | 4 | **Setup.** Let $k$ be an algebraically closed field of characteristic zero, and let $G/k$ be a semi-abelian variety i.e., $G$ is a commutative algebraic group which is an extension of an abelian variety $A/k$ by a torus $T/k$, so it admits the following presentation:
$$0 \to T \to G \to A \to 0$$
Let $Z/k$ be a smooth,... | https://mathoverflow.net/users/56667 | Extending rational maps to semi-abelian varieties | What you would like to say is that $G \cong A \times T$, which is obviously not quite true. But it is true smooth-locally, and that's enough to conclude.
Indeed, note that the quotient map $G \to A$ is faithfully flat with smooth fibres, hence smooth [Tag [01V8](https://stacks.math.columbia.edu/tag/01V8)]. In fact, $... | 2 | https://mathoverflow.net/users/82179 | 390828 | 161,765 |
https://mathoverflow.net/questions/390794 | 3 | Let $\left\{a\_k\right\}\_{0\leqslant k\leqslant n}$ and $\left\{b\_k\right\}\_{0\leqslant k\leqslant n}$ be two positive decreasing sequences such that $c\_1a\_k\leqslant b\_k\leqslant c\_2 a\_k$ for all $k$ for some positive constants $c\_1,c\_2$. Let $\epsilon>0$ and $f>0$ a function such that $$\sum\_{k=0}^na\_k\si... | https://mathoverflow.net/users/192560 | Oscillating sums II | $\newcommand\ep\varepsilon$The answer is still no (assuming you wanted $c>0$). Indeed, let
$$s\_n:=\sum\_{k=0}^n a\_k\sin(k\ep),\quad t\_n:=\sum\_{k=0}^n b\_k\sin(k\ep),$$
where $\ep=3\pi/4$, $a\_0=b\_0=10$, $a\_1=1$, $b\_1=9$, and $a\_k=b\_k=2^{-k}$ for $k\ge2$.
Then $c\_1a\_k\le b\_k\le c\_2a\_k$ for $c\_1=1$, $c\_... | 4 | https://mathoverflow.net/users/36721 | 390829 | 161,766 |
https://mathoverflow.net/questions/390746 | 8 | I am a person living in a 3rd world country and completed my masters in mathematics in July 2020. Then I began to study some additional topics in Pure Mathematics as I was applying for Ph.D. abroad( In Number Theory) as I felt that I should study more topics before applying in December2020- January 2021.
Background i... | https://mathoverflow.net/users/151209 | Advice: What topics to study now in analytic number Theory( And if there are video lectures( Open Online course) / Course notes available on website) | (I'm posting this as an answer since I prefer to mantain my anonymity, thus I cannot comment.)
I'm also from a third-world country. At the age of 24, when I was finishing my master's thesis, I was diagnosed with bipolar disorder, and I was drinking way too much. It took me years to recover from alcoholism and to hand... | 19 | https://mathoverflow.net/users/193667 | 390840 | 161,769 |
https://mathoverflow.net/questions/390243 | 5 | Given a smooth regular curve $\gamma$ in $\mathbb{R}^{3}$, one defines the *tangent indicatrix* of $\gamma$ to be the spherical curve $\gamma'/\lVert \gamma'\rVert$. It is then natural to look for *spherical* curves $\gamma$ whose tangent indicatrix is the same as the original curve, modulo an isometry of $\mathbb{R}^{... | https://mathoverflow.net/users/74033 | When does a spherical curve equal its tangent indicatrix? | Your question is not very clearly phrased, which may explain why you didn't get any answers on MSE.
When you say, "spherical curve $\gamma$ whose tangent indicatrix is the same as the original curve, modulo an isometry of $\mathbb{R}^3$", the simple interpretation is this: Assume that $\gamma$ is unit speed, so that ... | 5 | https://mathoverflow.net/users/13972 | 390842 | 161,771 |
https://mathoverflow.net/questions/390659 | 15 | Let $A\_1, A\_2, \ldots, A\_n$ be $n$ sets such that:
(1) for each $i\in [n]$, $\frac{n}{3}\leq |A\_i|\leq n$;
(2) for any $1\leq i<j<k\leq n$, $|A\_i\cap A\_j\cap A\_k|\leq a$, where $a$ is a constant and $n$ is sufficiently large.
What is $\min |A\_1\cup A\_2\cup \cdots \cup A\_n|$? Is it $\Omega(n^2)$ or $o(n^... | https://mathoverflow.net/users/165069 | n sets, each is large, the intersection of every three is small, what is the size of the union? | Let $m$ be chosen later, and let $A\_1, A\_2, \dots, A\_n$ be independently chosen random subsets of $\{1,2,\dots m\}$, each having size $n$.
For a fixed $a+1$-tuple $(x\_1, x\_2, \dots, x\_{a+1})$ of distinct elements from $\{1,\dots,m\}$, and a fixed triple $(i,j,k)$, the probability that $\{x\_1, \dots, x\_a\} \su... | 6 | https://mathoverflow.net/users/405 | 390846 | 161,772 |
https://mathoverflow.net/questions/390343 | 2 | It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$.
>
> Can I assume that, if one uses polynomials of degree $p$ and the
> respective function to be interpolated $f\in C^p([a,b])$, that the
> interpolation error of this spline is ... | https://mathoverflow.net/users/182708 | Spline Interpolation error of higher degree | The following paper by de Boor suggests that this is the case, although he develops the proof only up to degree 6 splines.
*de Boor, C.*, [**On the convergence of odd-degree spline interpolation**](http://dx.doi.org/10.1016/0021-9045(68)90033-6), J. Approximation Theory 1, 452-463 (1968). [ZBL0174.09902](https://zbma... | 1 | https://mathoverflow.net/users/123142 | 390855 | 161,777 |
https://mathoverflow.net/questions/390814 | 0 | I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and Grassmanian of $k$-planes, which I don't understand. I know how to check the intersection a plane and a line, but I don't know how... | https://mathoverflow.net/users/127839 | Intersection Grassmanian planes | For example, if we take any basis $e\_1,e\_2,e\_3,e\_4$ of a 4-dimensional vector space, and if $F$ is the span of $e\_1,e\_2$, so $F\in Gr\_2$ and $F'$ is the span of $e\_3,e\_4$, so $F'\in Gr\_2$, then $F\cap F'$ is the set of vectors which are both of the form $ae\_1+be\_2$ and of the form $ce\_3+de\_4$. Every vecto... | 1 | https://mathoverflow.net/users/13268 | 390858 | 161,778 |
https://mathoverflow.net/questions/390856 | 12 | Let $f:G\to H$ be a surjective homomorphism of finitely presented groups. If the kernel of $f$ is finitely generated then is $G\times\_H G$ is a finitely presented group? Can one compute an explicit finite presentation?
| https://mathoverflow.net/users/nan | Is the diagonal of finitely presented groups computable? | The answer is "no". The hypotheses for the fibre product to be finitely presentable are given by the 1-2-3 theorem of Bridson--Baumslag--Miller--Short:
>
> **1-2-3 Theorem (BBMS):** The fibre product $G\times\_H G$ is guaranteed to be finitely presentable as long as:
>
>
> 1. the kernel of $G\to H$ is finitely ge... | 17 | https://mathoverflow.net/users/1463 | 390859 | 161,779 |
https://mathoverflow.net/questions/390866 | 1 | I am asking whether the Bochner spaces $L^\infty(a,b;L^p(c,d))$ and $L^p(c,d;L^\infty(a,b))$ are the same. Or, whether one is included/embedded in the other.
We have the norms
$$\|u\|\_{L^\infty L^p}=\sup\_{t \in (a,b)} \left(\int\_c^d |u(t,s)|^pds \right)^{1/p}$$
$$\|u\|\_{L^pL^\infty}= \left(\int\_c^d \left(\su... | https://mathoverflow.net/users/78193 | On the Bochner spaces $L^\infty(a,b;L^p(c,d))$ and $L^p(c,d;L^\infty(a,b))$, or: Interchange of supremum and integral | If you want a hint, then in such situations it is often advisable to look first at simple, non-trivial cases. Here you can start with a drastically simplified one—I looked at $p=1$ and two point measure spaces where it soon becomes clear what is going on.
| 2 | https://mathoverflow.net/users/194568 | 390871 | 161,781 |
https://mathoverflow.net/questions/390876 | -3 | Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?
| https://mathoverflow.net/users/110301 | Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $? | No. Identify $\mathbb{R}^2$ with $\mathbb{C}$ and consider $f(z) = z^2$. If $g$ is close enough to $f$ then $\alpha \mapsto g(e^{i \alpha})$ stays in an annulus and winds around the origin twice, which cannot be done injectively.
| 10 | https://mathoverflow.net/users/38566 | 390879 | 161,784 |
https://mathoverflow.net/questions/390878 | 8 | This is a follow up of my previous MO question "[Non orientable, closed manifold covered by two simply-connected charts](https://mathoverflow.net/questions/390739/non-orientable-closed-manifold-covered-by-two-simply-connected-charts)." Nick L's nice answer shows that such manifolds actually exist, examples being provid... | https://mathoverflow.net/users/7460 | Non orientable, closed manifold covered by two contractible charts | No. Recall that the *Lusternik-Schnirelmann category* of a space $X$, denoted $\operatorname{cat}(X)$, is the minumum $k$ such that $X$ may be covered by open sets $U\_0,U\_1,\ldots, U\_k$ such that each inclusion is null-homotopic. The standard lower bound for LS-category is the cup-length of reduced cohomology: if $R... | 16 | https://mathoverflow.net/users/8103 | 390880 | 161,785 |
https://mathoverflow.net/questions/390767 | 1 | The traditional knapsack problem is that: given a sequence of $i$ items with positive weights $w\_1,w\_2,...,w\_i$, positive values $v\_1,v\_2,...,v\_i$, and a bag with capacity $B$, we want to insert items into the bag without exceeding the capacity $B$ while maximising the total values (i.e., maximising $\sum\_{h=1}^... | https://mathoverflow.net/users/168850 | Knapsack problem with capacity constraint | I assume that your constraint should read $\sum\_{h=1}^i p\_h w\_h < B < 2 \sum\_{h=1}^i p\_h w\_h$, not $\sum\_{h=1}^i w\_h < B$ otherwise the trivial solution $p\_h \equiv 1$ always applies.
There is an reduction by adding a very heavy and expensive item. Take any Knapsack problem like you stated. Calculate $W := \... | 2 | https://mathoverflow.net/users/51695 | 390885 | 161,786 |
https://mathoverflow.net/questions/318786 | 0 | Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also.
Is there an expression for the Fourier transform of $g\circ f$ in terms of that of $f$?
| https://mathoverflow.net/users/36886 | Transformation of Fourier Transform | Too long for a comment. I find your question interesting for a reason linked to the proof of the Faà de Bruno formula. Let $f,g$ be two functions from $\mathbb R$ into itself. Let me assume that $f(0)=0$ and let me give a formal expression for $(g\circ f)(x)$ using the Fourier transform of $g$:
$$
g(f(x))=\int \hat g(\... | 1 | https://mathoverflow.net/users/21907 | 390893 | 161,787 |
https://mathoverflow.net/questions/390900 | 5 | Although this question might be formulated in higher generality, let me try to be concrete:
Let $(\mathbf{Top},\times,\*)$ be the monoidal category of compactly generated weak Hausdorff spaces; and let $\mathbf{C}$ be a small and $\mathbf{Top}$-enriched category (in the easiest case I have in mind: a topological grou... | https://mathoverflow.net/users/124042 | Enriched coends which preserve equivalences | One sufficient condition is that either $B$ or both $A$ and $A'$ are cofibrant in the projective model structure.
If $C$ is a Reedy category, possibly in the generalized sense defined by Berger and Moerdijk, and $A$, $A'$ and $B$ are cofibrant in the Reedy model structure, this is sufficient too.
The point is that ... | 8 | https://mathoverflow.net/users/6668 | 390905 | 161,792 |
https://mathoverflow.net/questions/390897 | 6 | In the "ordinary" operad category, it is known that there is a colored operad $Op$ with set of colors $\mathbb{N}$ corresponding to "degrees" of vertices and with operations indexed by trees, such that algebras over $Op$ in $\mathrm{Set}$ (or more generally any symmetric monoidal category) correspond to monochromatic o... | https://mathoverflow.net/users/7108 | Monochromatic infinity operads as algebras over the "operad operad" | Yes, combine Corollary 9.4.1 and Theorem 7.11 of [arXiv:1410.5675](https://arxiv.org/abs/1410.5675), for example.
This topic is also examined more explicitly
in the work of Chu and Haugseng, [arXiv:1707.08049](https://arxiv.org/abs/1707.08049).
Corollary 5.1.13 shows that enriched ∞-operads are equivalent
to ∞-alge... | 3 | https://mathoverflow.net/users/402 | 390914 | 161,795 |
https://mathoverflow.net/questions/390924 | 1 | $$F(m,n)= \begin{cases}
1, & \text{if $m n=0$ }; \\
\frac{1}{2} F(m ,n-1) + \frac{1}{3} F(m-1,n )+ \frac{1}{4} F(m-1,n-1), & \text{ if $m n>0$. }%
\end{cases}$$
Please a proof of:
$$\lim\_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\lim\_{n\rightarrow \infty}\frac{F(n,n-1)}{F(n-1,n-1)}=\frac{9}{8}$$
$$\lim\_... | https://mathoverflow.net/users/168671 | Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\frac{9}{8}$? | We will compute the generating function, and use the method described in section 2 of [this paper](https://arxiv.org/abs/math/0702595).
Let $F\_{m,n}=F(m,n)$. Consider the generating function
$$G(x,y)=\sum\_{m=0}^\infty\sum\_{n=0}^\infty F\_{m,n}x^my^n.$$
Then the recurrence gives
\begin{align\*}
&G(x,y)=\sum\_{m=0}^... | 11 | https://mathoverflow.net/users/95685 | 390936 | 161,798 |
https://mathoverflow.net/questions/390933 | 4 | Let $X\_1,X\_2,X\_3,...$ be iid non-negative random variables with $E[X\_i]=\infty$. I am looking for references on the growth in $n$ of the empirical average under assumptions on $X\_1.,..,X\_n$.
A more specific question is the following:
* Under the moment assumption $E[X\_i^p]=1$ for some $p\in (0,1)$, what are ... | https://mathoverflow.net/users/141760 | Large deviations: Growth of empirical average of iid non-negative random varialbes with infinite expectations? | Here it is more convenient to consider the order of magnitude of $S\_n:=\sum\_1^n X\_i$, rather than that of $S\_n/n$.
Take any real $c>0$. Let $x:=cn^{1/p}$, $Y\_i:=X\_i\,1(X\_i<x)$, $T\_n:=\sum\_1^n Y\_i$, $M\_n:=\max\_1^n X\_i$. Then
$$P(X\_1\ge x)\le EX\_1^p\,1(X\_i\ge x)/x^p<<1/x^p\tag{1}$$
as $n\to\infty$,
with... | 5 | https://mathoverflow.net/users/36721 | 390939 | 161,800 |
https://mathoverflow.net/questions/382881 | 1 | Consider a system $S$ of polynomial equations, $p\_1=0,...,p\_m=0$, for $p\_i\in K[x\_1,...,x\_n]$, for a field $K$: the system $S$ is *zero-dimensional* if it has finitely many solutions. It is well-known that zero-dimensionality can be decided by algorithms that rely on the construction of a Groebner basis for the id... | https://mathoverflow.net/users/61364 | Sufficient syntactic conditions for zero-dimensionality of polynomial systems | This is just an elaboration on my comment.
Here are two sufficient conditions:
* *Condition 1:* We have $m \geq n$, and each polynomial $p\_i$ with $i \leq n$ has the form $p\_i = x\_i^{m\_i} + \left(\text{some polynomial of degree $< m\_i$}\right)$ for some nonnegative integer $m\_i$.
* *Condition 2:* We have $m \... | 1 | https://mathoverflow.net/users/2530 | 390942 | 161,802 |
https://mathoverflow.net/questions/390889 | 13 | **Motivation.** A while ago I attended a party and I only knew some, but not all, of the attendees. There were 2 kinds of drinks: beer and soda. I noticed that amongst my acquaintances, more than half drank beer. So in order to go against what I perceived as the mainstream ("drink a beer"), I picked a soda. Then I wond... | https://mathoverflow.net/users/8628 | "Drinking number" of a graph | Known as unfriendly partition conjecture. Open for countable graphs: <http://www.openproblemgarden.org/op/unfriendly_partitions>.
| 12 | https://mathoverflow.net/users/195625 | 390945 | 161,803 |
https://mathoverflow.net/questions/390934 | 3 | $\DeclareMathOperator\rank{rank}$Let $G$ be a compact connected Lie group and $H$, $K$ be two closed connected subgroups. By Mikhail Borovoi's [answer](https://mathoverflow.net/a/62467) to [In a compact lie group, can two closed connected subgroups generate a non-closed subgroup?](https://mathoverflow.net/questions/624... | https://mathoverflow.net/users/38302 | Rank of a Lie subgroup generated by two Lie subgroups | No. A simple example is to let $G = \mathrm{SO}(6)$, which has rank $3$. Let $v\_1$ and $v\_2$ be unit orthogonal vectors in $\mathbb{R}^6$ and let $H\_1\simeq \mathrm{SO}(5)\subset G$ be the stabilizer of $v\_1$ and let $H\_2\simeq \mathrm{SO}(5)\subset G$ be the stabilizer of $v\_2$. Then $H\_1\cap H\_2\simeq\mathrm{... | 8 | https://mathoverflow.net/users/13972 | 390948 | 161,804 |
https://mathoverflow.net/questions/390815 | 10 | Consider a transitive set $M$. Let's call the *well-ordering number* for $M$ the smallest ordinal $\alpha$ so that $L\_\alpha(M)$ contains as an element a well-order of $M$, and denote it as $\upsilon = \upsilon(M)$. My question is which ordinals are the $\upsilon(M)$ for some $M$.
This is phrased in a general format... | https://mathoverflow.net/users/64676 | How far to find a well-order? | If there is a transitive set model $M$ of ZFC, then all finite successor ordinals $n+1$ and all $\omega+n+1$ are $\upsilon(M)$ for some such $M$, and also many other ordinals:
Claim: Let $\Omega,\alpha<\omega\_1^L$ be such that $L\_\Omega\models\mathrm{ZFC}$ and $0<\alpha$ and there is a surjection $f:\omega\to\Omega... | 10 | https://mathoverflow.net/users/160347 | 390953 | 161,808 |
https://mathoverflow.net/questions/390882 | 2 | I am working in geometry control field, fall last week on this exercice and I can't figure it out. I have a distribution $\mathscr{D}$ with $rank(\mathscr{D})=m+1$ in $\mathbb{R}^n$ with $n\leq 2m+1$. I know that there exists an involutive sub-distribution $\mathscr{L}\subset\mathscr{D}$ with rank $m$. I also know that... | https://mathoverflow.net/users/174936 | Finding a local normal form regarding distribution rank properties | In what follows I am assuming (but I'm pretty sure) that your integrable sub-distribution $\mathscr{L}$ is supposed to be the same distribution as $\mathscr{F}$ that you mention later in your question but please correct me if I've misunderstood. Additionally, I'll assume we're working locally, i.e. on sufficiently smal... | 2 | https://mathoverflow.net/users/103158 | 390954 | 161,809 |
https://mathoverflow.net/questions/390786 | 3 | I am reading the recent book by Kawamata, *Algebraic Varieties: Minimal Models and Finite Generation*. There is an English translation [here](http://homepage.fudan.edu.cn/chenjiang/files/2019/06/main.pdf) .
In the bottom of page 16 he says that an $\mathbb{R}$-Cartier divisor can be written as $D = \sum d\_i D\_i$ wi... | https://mathoverflow.net/users/133871 | Pullback of $\mathbb{R}$-Cartier divisors | 1. If you have morphism of free abelian groups $g : A \to B$ then $g \otimes Id\_{\mathbb R} : A \otimes \mathbb R \to B \otimes \mathbb R$ which sends $a \otimes r \mapsto g(a) \otimes r$ is well-defined (in the sense that it does not depend on how you present the element $\sum a\_i \otimes r\_i \in A \otimes \mathbb ... | 1 | https://mathoverflow.net/users/54337 | 390957 | 161,810 |
https://mathoverflow.net/questions/390863 | 7 | Let $X$ be an incomplete Alexandrov space with sec $\ge -1$ in the sense that for any point in $X$ there exists a small neighborhood in which the four-points criterion is satisfied.
Suppose $X$ is convex in the sense that any pair of points $p,q \in X$ there exists a minimizing geodesic connecting $p$ and $q$. Can we... | https://mathoverflow.net/users/105900 | Completion of an Alexandrov space | Yes, the conclusion is exactly the main result of Petrunin's paper "A globalization for non-complete but geodesic spaces", Mathematische Annalen volume 366, pages387–393(2016).
| 6 | https://mathoverflow.net/users/16323 | 390965 | 161,815 |
https://mathoverflow.net/questions/390973 | 12 | The Riemann hypothesis is equivalent to the assertion that the prime counting function $\pi(x) := \sum\_{p \le x} 1$ deviates from the logarithmic integral $Li(x) = \int\_2^x \frac{dx}{\log x}$ in the order $O(\sqrt{x} \log x)$. Since $\log x = O(x^\alpha)$ for any $\alpha>0$, the Riemann hypothesis implies that:
$$\fo... | https://mathoverflow.net/users/156792 | Optimality of the Riemann Hypothesis | See Chapter 15 ("Oscillation of error terms") in Montgomery-Vaughan: Multiplicative number theory I. See especially Theorems 15.2-15.3 and 15.11.
**Added by Steven Clark and GH from MO.** For convenience, we copied below the relevant theorems and some additional text from the book. As usual,
$$M(x):=\sum\limits\_{n\l... | 16 | https://mathoverflow.net/users/11919 | 390975 | 161,819 |
https://mathoverflow.net/questions/389750 | 8 | Reading Scholze's notes on Condensed Mathematics it is mentioned that when considered as $\infty$-categories,
$$ D(\operatorname{Cond(Ab)}) \cong \operatorname{Cond}(D(\operatorname{Ab}))$$
and that this is not a feature of condensed abelian groups but rather of the category of sheaves on a site.
I have been look... | https://mathoverflow.net/users/170467 | Derived category of abelian sheaves on a site equivalent to sheaves on the derived category of abelian groups | It's true in any $1$-topos for hypercomplete sheaves, see Theorem 2.1.2.2 in [Spectral Algebraic Geometry](https://www.math.ias.edu/%7Elurie/papers/SAG-rootfile.pdf).
| 6 | https://mathoverflow.net/users/6074 | 390988 | 161,826 |
https://mathoverflow.net/questions/390956 | 8 | I am aware that a quasiconformal map satifies the formula
$$
\frac{\partial f}{\partial \overline{z}} = \mu(z) \frac{\partial f}{\partial z}
$$
where $\sup\{\mu(z):z \in \text{Domain}\{f\}\}<1$ imposes a bound on the eccentricity of the ellipses in the image of $f$. For a multicomplex function $F(z\_1, z\_2, \dots, z\_... | https://mathoverflow.net/users/170939 | Quasiconformal maps in arbitrary dimensions | The correct definition of higher-dimensional quasiconformal maps does not use complex variables. The correct condition is
$$
|Df(x)|\le K |J\_f(x)|
$$
where $J\_f$ is the Jacobian determinant. An orientation-preserving homeomorphism $f$ between two domains $U, V$ in $R^n$ is called quasiconformal if it belongs to the S... | 5 | https://mathoverflow.net/users/39654 | 391002 | 161,828 |
https://mathoverflow.net/questions/390995 | 5 | Let $X$ be a compact complex manifold of complex dimension $n$ and let $\omega$ be a smooth Kahler form on it. Let $Y \subset X$ be a complex (possibly singular) hypersurface and let $u: X \setminus Y \to
\mathbb{R}$ be a smooth function.
**Question:** If $ \sup\_{X \setminus Y} |u| < \infty$ and $\omega + i \partia... | https://mathoverflow.net/users/195890 | Volume of singular Kahler metric | Yes, this is true. By the boundedness assumption, $u$ extends to a (bounded) $\omega$-psh function on $X$.
Bedford and Taylor defined in '82 the Monge-Ampère operator of a bounded psh function, which has later been extended to the global quasi-psh case.
Almost by definition of that operator, you have $\int\_X(\omeg... | 5 | https://mathoverflow.net/users/5659 | 391003 | 161,829 |
https://mathoverflow.net/questions/390958 | 5 | Let's say you have a set of first order differential equations with known Jacobian $J$. Let $x\_0, x\_1, ..., x\_n$ be sampled points on the trajectory near the attractor.
Let $T\_n = J(x\_{n-1})J(x\_{n-2})...J(x\_0)$.
Oseledet says that
$$\frac{1}{n} \log(\mbox{the singular values of } T\_n)$$
converge to the ... | https://mathoverflow.net/users/152429 | Can you use Oseledet's theorem to numerically approximate the Lyapunov spectra? | Once you have a method to estimate the top Lyapunov exponent, you can use the action on wedge products to estimate the other exponents (This should be clear if you examine the proof of Oseledet's theorem.) Estimating the top exponent is difficult, in general. Doing it via the definition is very slow as you observed. Mo... | 4 | https://mathoverflow.net/users/7691 | 391008 | 161,830 |
https://mathoverflow.net/questions/355488 | 5 | Consider $f\in H^{\sigma}(S^1)=W^{\sigma, 2}$ (the usual Sobolev space on the circle) and let $S\_Nf$ be its truncated Fourier series $S\_Nf = \sum\_{|n|\leq N} \hat{f}(n)e^{2\pi i n x}$. I am looking for the theory for inequalities of the following type:
$$\|f-S\_Nf\|\_{H^s} \leq CN^{c(s,\sigma)} \|f\|\_{H^{\sigma}} \... | https://mathoverflow.net/users/42864 | Sobolev convergence of Fourier series | Let us start with pointing out that $f\in H^\sigma$ is equivalent to
$$
(\langle n\rangle^\sigma\hat f(n))\_{n\in \mathbb Z}\in \ell^2(\mathbb Z),
\quad \text{with $\langle n\rangle=\sqrt{1+n^2}$.}
$$
Then you have for $s<\sigma$
$$
\Vert f-S\_N(f)\Vert\_{H^s}^2=\sum\_{\vert n\vert> N}\langle n\rangle^{2s}\vert\hat f(... | 3 | https://mathoverflow.net/users/21907 | 391011 | 161,832 |
https://mathoverflow.net/questions/390963 | 2 | Consider a probability density function $\phi(\cdot)$ that is positive everywhere. For any a>0, $\phi(x)/\phi(x+a)$ is strictly increasing in x. I think a function like this must be continuous, first strictly increasing, and then strictly decreasing. Could anyone tell me how to prove it? Thanks.
| https://mathoverflow.net/users/140169 | An implication of SMLRP | If $\phi(x)/\phi(x+a)$ is strictly increasing, then you get
$$
\frac{\phi(x)}{\phi(x+a)} < \frac{\phi(x+a)}{\phi(x+2a)}
$$
for any $x$.
From this you get that the function $-\log(\phi)$ is strictly "midpoint-convex", and any Lebesgue-measurable midpoint-convex function is in fact convex; see for example these [two](htt... | 4 | https://mathoverflow.net/users/5784 | 391015 | 161,834 |
https://mathoverflow.net/questions/391016 | 4 | Messing around, I noticed that
$$\sqrt{2}=e^{i\pi /4}+e^{-i\pi /4}$$
$$\sqrt{3}=e^{i\pi /6}+e^{-i\pi /6}$$
and (even more surprisingly)
$$\sqrt{5}=e^{2\pi i/5}-e^{4\pi i /5}-e^{6\pi i /5}+e^{8 \pi i /5}$$
>
> **Question.** Is this always possible? Stated more succinctly, can every square root be expressed a... | https://mathoverflow.net/users/196113 | Can every square root be represented as a linear combination on roots of unity? | Yes, this is always the case.
If $\alpha=\sqrt{n}$, then $|\mathrm{Gal}(\mathbb{Q}(\alpha)/\mathbb{Q})|=2$ (since there are no intermediate fields) so $\mathrm{Gal}(\mathbb{Q}(\alpha)/\mathbb{Q})=\mathbb{Z}/2\mathbb{Z}$ since that is the only group with two elements, thus it is abelian.
By the Kronecker-Weber theor... | 10 | https://mathoverflow.net/users/159298 | 391017 | 161,835 |
https://mathoverflow.net/questions/391035 | 0 | Suppose that we have the following expression $$p^{-2}|E|^6+16p^{-1}|E|^4+p^{2}|E|^2-8p^{-\frac{3}{2}}|E|^5-2|E|^4+8p^{\frac{1}{2}}|E|^3,$$ where $p$ is a prime number.
1. I was wondering is it possible to find some $C>0$ such that if $|E|\geq Cp$ then the above inequality is positive?
2. If the above is not possible... | https://mathoverflow.net/users/121924 | Finding the optimal size of $|E|$ for specific inequality | Let's handle the lowest-order terms, treating $p$ and $|E|$ as degree $1$, using $p\geq 2$
$$p^{-2}|E|^6+16p^{-1}|E|^4+p^{2}|E|^2-8p^{-\frac{3}{2}}|E|^5-2|E|^4+8p^{\frac{1}{2}}|E|^3 $$ $$\geq p^{-2}|E|^6+0+p^{2}|E|^2-4 \sqrt{2} p^{-1}|E|^5-2|E|^4+0 $$
$$ = p^4 ( (|E|/p)^6 - 4\sqrt{2} (|E|/p)^5 - 2 (|E|/p)^4 + (|E|/p)... | 2 | https://mathoverflow.net/users/18060 | 391037 | 161,843 |
https://mathoverflow.net/questions/391023 | 7 | Given a potential function $U: \mathbb{R}^n \to \mathbb{R}$, Langevin diffusion is gradient descent plus a Brownian motion term: $X' = -\nabla U(X) + \sqrt{2} \text{ }dW$.
It happens that the stationary distribution of Langevin diffusion is very nice: it is proportional to $\exp(-U)$. I can verify this by plugging $\... | https://mathoverflow.net/users/39142 | Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$? | The reason a Langevin diffusion leaves $\nu(x)=e^{-U(x)}$ invariant is because it is *[symmetric or reversible with respect to $\nu$](https://www.cambridge.org/core/journals/advances-in-applied-probability/article/abs/time-reversible-diffusions/6CEF3A3069A15D58C55FBB06D6F467D8)*. In comparison to general diffusion proc... | 8 | https://mathoverflow.net/users/64449 | 391044 | 161,846 |
https://mathoverflow.net/questions/391039 | 8 | Let $X$ be a normal variety over a field of characteristic zero with rational singularities.
>
> If $\pi:Y \to X$ is a birational proper morphism with $Y$ also normal, then does $Y$ also have rational singularities?
>
>
>
It is easy to see that this is true if $\dim(X) = 2$, but the higher dimensional case see... | https://mathoverflow.net/users/519 | An invariance property of rational singularities | No, $Y$ need not have rational singularities. See Section III of the paper for an example in dimension three: Cutkosky, *A characterization of rational surface singularities*, Inventiones Mathematicae, 1990. In that example, $Y$ ($Z$ in the notation of the paper) is normal but not Cohen-Macaulay, so it is not a rationa... | 6 | https://mathoverflow.net/users/14895 | 391061 | 161,850 |
https://mathoverflow.net/questions/390305 | 3 | Let $H = H\_1 \times H\_2$ be a closed subgroup of a second-countable locally compact Hausdorff group $G = G\_1 \times G\_2$, with $H\_i \leq G\_i$. Let $\chi = \chi\_1 \otimes \chi\_2$ be a unitary character of $H$, for $\chi\_i$ a character of $H\_i$. The induced representation
$$I(\chi) = \operatorname{Ind}\_H^G\c... | https://mathoverflow.net/users/38145 | Dense subspace of $\operatorname{Ind}_{H_1 \times H_2}^{G_1 \times G_2} \chi$ | Yes, $V$ is always norm dense.
You have $V=V\_1\otimes V\_2$ for obvious choices of dense subspaces $V\_i<I(\chi\_i)$
and you ask whether $V$ is dense in $I(\chi)$. Equivalently, you may ask directly whether
$I(\chi\_i) \otimes I(\chi\_i)$ is dense in $I(\chi)$.
Why reformulating in this way? Because now one can us... | 2 | https://mathoverflow.net/users/89334 | 391063 | 161,852 |
https://mathoverflow.net/questions/391055 | 7 | 1. **Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group?**
2. **Has this membership problem been implemented in GAP/Magma?**
More precisely, I have a finitely generated free group $F$, two finitely generated subgroups $A,B$, given by their fre... | https://mathoverflow.net/users/7644 | Membership to double cosets in free groups | Here is a second answer that is just rephrasing @DerekHolt’s answer based on the comments. So upvote his answer first! Let $X$ be a finite alphabet. An inverse automaton is a finite directed graph labeled over the alphabet $X$ which is folded in the sense of Stallings, meaning you cannot find two edges entering or leav... | 6 | https://mathoverflow.net/users/15934 | 391069 | 161,856 |
https://mathoverflow.net/questions/391086 | 3 | Given a measurable space, the vector space of signed measures is a Banach space. Does it have the Radon-Nikodym property? What if the space is of a special type, such as a nice topological space with the Borel $\sigma$-algebra?
If there is any related information, such as the Radon-Nikodym property relative to some s... | https://mathoverflow.net/users/156492 | Radon-Nikodym property for space of signed measures | The spaces you are interested in are abstractly AL-spaces and by [Kakutani's representation theorem](https://www.jstor.org/stable/pdf/1968915.pdf?casa_token=x9kvIzsjxHUAAAAA:btJzFUxpBLfAZcTE-3USjRKjmUMMNF-uGO24KQdIaD6F0fTqt3pijagbQptR5XaVx3uv1SD2NfUb0WE5GEW5eQHfXawLS61OLV0Nyv2NnW2teq3Plg_P), they can be represented as ... | 9 | https://mathoverflow.net/users/15129 | 391088 | 161,864 |
https://mathoverflow.net/questions/391065 | 2 | **Question.** Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}\_g X = 0$, $g(X,X) = 1$, and tangent to the boundary?
* Here an open domain $\Omega \subset \mathbf{R}^2$ is cal... | https://mathoverflow.net/users/103792 | Existence of divergence-free unit vector field in conformally rescaled euclidean metric | Now that the question has been changed so extensively, the remarks that I made for the old version are no longer of any interest. Here is what I understand the problem to look like now:
First, $\Omega\subset\mathbb{R}^2$ is a convex open set in the plane whose boundary is *polygonal*, i.e., a union of line segements ... | 4 | https://mathoverflow.net/users/13972 | 391090 | 161,865 |
https://mathoverflow.net/questions/391082 | 5 | $\newcommand{\C}{\mathcal{C}}$ $\newcommand{\g}{\mathfrak{g}}$
If $\C$ is a monoidal category (not necessarily a symmetric monoidal category), it's possible to define the notion of an algebra object $A$ in $\C$, with multiplication operations $$A^{\otimes n} (:= A\otimes\_\C A\otimes\_\C \cdots\otimes\_\C A)\to A.$$
... | https://mathoverflow.net/users/7108 | Uses for (Framed) E2 algebras twisted by braided monoidal structure | I don't know specific references (the papers - in reverse chronological order - of Liang Kong, Hao Zheng, Ingo Runkel, Christoph Schweigert and Jurgen Fuchs is where I'd start), but the notion is certainly very natural in TFT, in at least two (closely related) ways:
* if you think of your braided tensor category $C$ ... | 4 | https://mathoverflow.net/users/582 | 391098 | 161,867 |
https://mathoverflow.net/questions/391108 | 10 | I have an elementary question on modular forms, but which I don't know how to solve.
>
> a) Is there a congruence subgroup $\Gamma \leq \mathrm{SL}\_2(\Bbb Z)$, an integer $k \in \Bbb Z$ and a non-constant modular form $f \in M\_k(\Gamma)$ such that $f$ has only finitely many non-zero Fourier coefficients $a\_n(f)$... | https://mathoverflow.net/users/196880 | Modular forms with finitely many or very few non-zero Fourier coefficients | For simplicity, I will assume that $f$ is a cusp form (hence $k\geq 12$ and $k$ is even).
The answer to question (a) is negative. It was proved independently by [Rankin (1939)](https://doi.org/10.1017/S0305004100021095) and Selberg (1940) that the Dirichlet series
$$\sum\_n\frac{|a\_n(f)|^2}{n^s}$$
has a simple pole ... | 17 | https://mathoverflow.net/users/11919 | 391110 | 161,872 |
https://mathoverflow.net/questions/391113 | 1 | Let $F$ be a field. Call a unital subring $R\subset F$ dense if there is no subfield $K\subsetneq F$ such that $R\subset K$.
Is there a characteristic zero field $F$ such that the only regular Noetherian dense unital subring $R\subset F$ is $F$ itself?
| https://mathoverflow.net/users/196887 | Characteristic zero field with unique regular Noetherian dense unital subring | Welcome new contributor!
You are essentially asking if there is a field $F$ such that the only regular noetherian subring of $F$ with fraction field $F$ is $F$ itself.
Every algebraically closed field is such an example, as well as $\mathbb{R}$.
Indeed, suppose that $F$ is the fraction field of a regular noetheri... | 1 | https://mathoverflow.net/users/86006 | 391127 | 161,876 |
https://mathoverflow.net/questions/391130 | 3 | Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups.
Can $L$ contain a Noetherian unital subring $R\subsetneq L$ not contained in any subfield $F\subsetneq L$?
| https://mathoverflow.net/users/196887 | Non-abelian isomorphic absolute Galois groups of fields of different characteristic | I'm not sure I understand the point of the question, but the answer is yes: For example, $K=\overline{\mathbb{F}\_p}(t)$ and $L=\overline{\mathbb{Q}}(t)$ are both known to have absolute Galois group free profinite on countably many generators, but $R=\overline{\mathbb{Q}}[t]$ is a noetherian proper subring of $L$ with ... | 6 | https://mathoverflow.net/users/50351 | 391132 | 161,880 |
https://mathoverflow.net/questions/391105 | 4 | In stochastic analysis, for an Ito diffusion $X\_t$ such that $dX\_t=\mu(X\_t)dt+\sigma(X\_t)dB\_t$, we can exlpicitly compute a "natural scale function"
$$S(x)=\int^x\exp\left(-\int^y\frac{2\mu(z)}{\sigma^2(z)}dz\right)dy$$
with suitable conditions on $\mu$ and $\sigma$, making $S(X\_t)$ a martingale(or at least a loc... | https://mathoverflow.net/users/174600 | How to find the "natural scale function" for more general stochastic processes? | Just figured this out with the help of @Nawaf Bou-Rabee. Thank you!
The idea is to calculate the infinitesimal generator $L$ for the markov process $X\_n$ or $X\_t$, and a function $f$ making $f(X\_n)$ or $f(X\_t)$ a martingale(or local martingale for continuous time) if and only if $Lf=0$. See reference [here](https... | 3 | https://mathoverflow.net/users/174600 | 391143 | 161,883 |
https://mathoverflow.net/questions/391136 | 3 | Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such as we have for simple Lie algebras/groups.
| https://mathoverflow.net/users/167165 | Is the representation of finite simple groups fully understood? | A term that may fit in the scope of this problem is "generic character table", character tables of a whole family of groups of Lie type.
Example: Generic character table of $SL\_2(q)$, $q = 2^f$
| Representations | $I$ | $U$ | $S(a)$ | $T(b)$ |
| --- | --- | --- | --- | --- |
| Trivial | $1$ | $1$ | $1$ | $1$ |
... | 6 | https://mathoverflow.net/users/125498 | 391154 | 161,885 |
https://mathoverflow.net/questions/389935 | 1 | Let $S$ be a finite set of points in $\mathbb{R}^{d}$, $c(s) \in [0,1]$ such that $\sum\_{s \in S} c(s) = 1$, $\rho$ continuous and non-vanishing probability distribution on $[0,1]^{d}$ and $\mu $ measure with respect to $\rho$, i.e. $ \mu (X) = \int\_{X} \rho (x) \ dx$. Let us suppose that there exists an assignment $... | https://mathoverflow.net/users/nan | Least square assignment and hyperplanes | There is always a minimizing assignment with this hyperplane property. So if the minimizing assignment is unique, then the hyperplane property always holds.
It is enough to show that $L^{-1}(s)$ and $L^{-1}(t)$ can always be replaced by regions which are divided by an appropriate hyperplane and which do not increase ... | 1 | https://mathoverflow.net/users/nan | 391161 | 161,889 |
https://mathoverflow.net/questions/390978 | -2 | Define a ranking function $\cal R$ as:
$\mathcal{R}: V \to ON; \,\mathcal {R}(x)= \min \alpha \, \forall y \in x: \alpha > \mathcal {R}(y) $
Now the constructible rank $\mathcal R^c$ of a set $X$ in $L$ is the ordinal index of the first constructible stage $X$ appears as an element of.
Accordingly for some set $X... | https://mathoverflow.net/users/95347 | What does the Concordant constructible universe model? | We assume we are working in ZF. We assume that the definition of concordant is $concordant(X) \equiv\_{df} \mathcal {R^c}(X) = \mathcal {R}(X)$+1(without the "+1", no set is concordant).
We will show that union does not hold in ‘[]. For simplicity we will use Quine ordered pairs so that if x and y are in ‘[], and is ... | 1 | https://mathoverflow.net/users/133981 | 391164 | 161,891 |
https://mathoverflow.net/questions/391107 | 16 | (In this question, $p$ can be $0$.)
I'm curious if theorems on étale cohomology can be proved by easier way.
For example, proper base change theorem. This theorem can be stated as the following way.
>
> **Theorem.** Let $k$ be a field of charateristic $p$ and $p\ne \ell$. Let's consider that Cartesian diagram
> $... | https://mathoverflow.net/users/nan | Proof of main theorems in étale cohomology theory | You seem to mean two slightly different things by the $p=\ell$ case. Your first theorem is about a field of characteristic $p$, and the second is about $\mathbb C\_p$, which is a field of characteristic zero. Let's talk first about the significance of this.
For a field of actual characteristic $p$, the proof of "Arti... | 10 | https://mathoverflow.net/users/18060 | 391166 | 161,892 |
https://mathoverflow.net/questions/391172 | 13 | **Question.** Is there a large [colimit-sketch](https://ncatlab.org/nlab/show/sketch) $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$?
In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}$ in $\mathcal{E}$ such that topological spaces can be seen as those func... | https://mathoverflow.net/users/2841 | Is there a large colimit-sketch for topological spaces? | The answer is **no**. I think this example illustrates why pure colimit sketches are rarely studied; one generally allows limits into the sketch *before* generalizing to allow colimits into the sketch. That is, I think this example illustrates the point that pure colimit-sketches are just not that useful. (This is afte... | 15 | https://mathoverflow.net/users/2362 | 391174 | 161,897 |
https://mathoverflow.net/questions/391158 | 5 | I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$)
\begin{equation}\label{kg}
\left\lbrace
\begin{array}{ll}
(\square+m^2)F(x)=0\\
F(0,\vec{x})=g(\vec{x}) \\
\frac{\partial F}{\partial x^{0}}(0,\vec{x})=f(\vec{x})
\end{array}
\right.
\end{equation}
I could... | https://mathoverflow.net/users/96878 | Initial conditions in the Klein-Gordon equation | One must remark that derivatives in Sobolev spaces are usually taken in the sense of distributions: given $k\in\mathbb{N}\_0=\{0,1,2,\ldots\}$, $H^k(\mathbb{R}^n)$ is the space of tempered distributions $u$ on $\mathbb{R}^n$ such that all partial derivatives of order $\leq k$ belong to $L^2(\mathbb{R}^n)$, where these ... | 9 | https://mathoverflow.net/users/11211 | 391175 | 161,898 |
https://mathoverflow.net/questions/391120 | 7 | Consider the (quantum) Hamiltonian on the real line
$$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$
Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate minima,
and $\lim\_{|x|\to +\infty}V(x)=+\infty$. Such a $V$ is called double well potential.
Let $E\_1,E\_2$ be the fi... | https://mathoverflow.net/users/16183 | Energy levels of double well potential | This is either
Helffer-Sjostrand
<https://www.tandfonline.com/doi/abs/10.1080/03605308408820335>
or Barry Simon
<https://www.jstor.org/stable/pdf/2007072.pdf?refreqid=excelsior%3A258084917fff9e0c10088abbb2679c55>
PS: I cannot resist pointing out that the last paper is in Annals of Mathematics proving, as you ... | 9 | https://mathoverflow.net/users/119875 | 391181 | 161,900 |
https://mathoverflow.net/questions/391188 | 8 |
>
> Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper
> map $f\colon \Sigma\to \Sigma$ of degree $1$ and **not** homotopic to
> any self-homotopy equivalence of $\Sigma$ ?
>
>
>
$\bullet$ Here $H\_\mathbf{c}^2(\Sigma;\Bbb Z)=\Bbb Z$, so for any *proper*... | https://mathoverflow.net/users/172285 | Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence | We can produce such a map by *folding*.
We define $f(x,y)$ to be $(x,y)$ if $x < 0$, to be $(-x,y)$ if $0 \leq x \leq 1$, and to be $(x-2,y)$ if $x > 1$.
This map is proper and degree one, but is not injective at the level of fundamental groups.
Edit: Beaten by 45 seconds! I’ll leave this here as (perhaps) it is ... | 7 | https://mathoverflow.net/users/1650 | 391190 | 161,903 |
https://mathoverflow.net/questions/391163 | 7 | Let $G$ be a group with generators $a, b\in G$.
Define $\mathrm{len}:G\to\mathbb{Z}\_{\geq 0}$ by sending $g$ to the minimum length of a word in $a, b, a^{-1}, b^{-1}$ equal to $g$.
Assume that for all $g\neq e\in G$ there is infinitely many $n\in \mathbb{Z}\_{\geq 0}$ such that $\mathrm{len}(g^{n+1})<\mathrm{len}(... | https://mathoverflow.net/users/nan | Infinite oscillation of minimum word length in 2-generated group | This answer is partly inspired by HJRW's comment.
**Definition 1.** Let $G$ be a finitely generated group and $g\in G$, and word length $|\cdot|$ with respect to some finite generating subset. Say that $g$ is very distorted if $\liminf |g^n|/\log(n)=0$, or equivalently if $u\_g(n)=\sup\{m:|g^m|\le n\}$ satisfies $\li... | 6 | https://mathoverflow.net/users/14094 | 391198 | 161,905 |
https://mathoverflow.net/questions/391178 | 2 | Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as
$$W(\sigma,t)=\sum\_{k=1}^\infty \frac{\cos(t\log p\_k)}{p\_k^\sigma},$$
where $p\_1, p\_2,\dots$ are the prime numbers, with $p\_1=2$.... | https://mathoverflow.net/users/140356 | Truncated Euler products, Dirichlet eta function, and convergence issues | Take some large $n$ and consider the summands in $W(\sigma,t)$ for which $t\log p\_k$ lie between $2n\pi$ and $2n\pi+\pi/3$. These are the primes $p\_k$ which lie in the interval $[e^{2n\pi/t},e^{2n\pi/t+\pi/3t}]$. Writing $N=e^{2n\pi/t}$, prime number theorem implies that this interval contains $\gg\frac{N}{\log N}$ p... | 2 | https://mathoverflow.net/users/30186 | 391199 | 161,906 |
https://mathoverflow.net/questions/391195 | 2 | Let $x\in S^{d-1}$ be chosen uniformly at random from the $d$-dimensional unit sphere.
I want to show that there exists a universal constant $c\in\mathbb R$ such that $\mathbb E\left[\frac{d}{||x||\_1^2}\right]\le c$ for all dimensions $d$.
Any thoughts?
---
Using numerical simulations, it seems that $\mathbb... | https://mathoverflow.net/users/197231 | Is $\mathbb E\left[\frac{d}{||x||_1^2}\right]=O(1)$ for all $d\in\mathbb R^+$, where $x\in S^{d-1}$ is a random $d$-dimensional unit vector? | We may sample $x$ as follows: choose i.i.d. standard Gaussian $\xi\_1,\ldots,\xi\_d$ and put $$x\_i=\frac{\xi\_i}{\sqrt{\sum\_{j=1}^d \xi\_j^2}},\quad i=1,2,\ldots,d.$$
Then
$$\frac1{\|x\|\_1^2}=\frac{\sum \xi\_j^2}{(\sum |\xi\_j|)^2}.
$$
By law of large numbers, the numerator is usually of order $d$ and the denominato... | 3 | https://mathoverflow.net/users/4312 | 391201 | 161,907 |
https://mathoverflow.net/questions/391211 | 3 | I am considering the family $\mathcal{F}$ of functions $f \colon \mathbb{R} \to \mathbb{R}$ which have at most linear growth at infinity, that is there exists a constant $M\_f$ such that
\begin{equation} |f(x)| \leq M\_f (1+|x|) . \end{equation}
In particular, for each $f \in \mathcal{F}$ we define $M\_f$ as the stri... | https://mathoverflow.net/users/176090 | Functions with at most linear growth at infinity: is the constant itself continuous? | The answer is no to both your hopes: it can happen that neither $M\_{f\_n}\to M\_f$ nor $\sup\_n M\_{f\_n}<+\infty$ hold, although $M\_f<\infty$.
As a counter-example take
$$
f\_n(x)=\max(0,n(x-n)).
$$
(I'm too lazy to include a picture: $f\_n$ is zero for $x\leq n$, and then starts growing with slope $n$ for $x\geq ... | 6 | https://mathoverflow.net/users/33741 | 391213 | 161,911 |
https://mathoverflow.net/questions/391196 | 3 | Let $U$ be an ultrafilter on $\mathcal{P}(\omega)$ and $\langle \sigma \_\alpha \mid \alpha < \omega\_1 \rangle$ be a sequence of elements of $U$.
I know that the limit sup of $\sigma \_\alpha$'s ($= \{i \in \omega \mid (\forall \alpha \in \omega\_1 )(\exists \beta \in \omega\_1 \setminus \alpha )\ i \in \sigma\_\bet... | https://mathoverflow.net/users/172604 | "Good limit" of an uncountable sequence of elements of an ultrafilter | Let $U$ be an ultrafilter on $\omega$ (or any family of subsets of $\omega$) and let $\langle\sigma\_\alpha:\alpha\lt\omega\_1\rangle$ be a sequence of elements of $U$.
Call a set $F\subset\omega$ *bad* if $F$ is finite and $\{\alpha\in\omega\_1:F\subset\sigma\_\alpha\}$ is countable. Let $B$ be the collection of all... | 3 | https://mathoverflow.net/users/43266 | 391215 | 161,912 |
https://mathoverflow.net/questions/391197 | 4 | Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.
Then, given an exact sequence $\eta: 0\to X \to Z\_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $\eta$ by $f$ gives an exact sequence $f\eta: 0\to X \to Z'\_{\eta} \to M \to 0$ in $Ext^1(M,Y)$.
Similarly, given ... | https://mathoverflow.net/users/135253 | Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences | An explicit reference for the additivity is
*Mac Lane, Saunders*, Homology, Classics in Mathematics. Berlin: Springer-Verlag. x, 422 p. (1995). [ZBL0818.18001](https://zbmath.org/?q=an:0818.18001)
in particular Chapter III, Theorem 2.1. The book seems to be treating rings $R$ which need not be commutative, hence th... | 3 | https://mathoverflow.net/users/8103 | 391217 | 161,913 |
https://mathoverflow.net/questions/391214 | 7 | Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs).
Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ are the respective vertex sets) such that:
* Any vertex of $Y$ is at most at distance $K$ from the image of a vertex ... | https://mathoverflow.net/users/148575 | Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic? | Questions of this flavour have indeed been studied before. There may be earlier references, but I know at least a body of work that was initiated by Benjamini, Ellis and Georgakopoulos (see [here](http://arxiv.org/abs/1508.02247) for precise references). They were interested in a question that is not exactly yours, but... | 9 | https://mathoverflow.net/users/10265 | 391220 | 161,915 |
https://mathoverflow.net/questions/390990 | 4 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\Tr{Tr}$Consider the diagonal action of $\GL(n,\mathbb{C})$ on the variety of $k$-tuples of matrices, $\M\_{n\times n}(\mathbb{C})^k$
through conjugation. Then it is known that the ring of invariants
$\mathbb{C}[\M\_{n\times n}(\mathbb{C})^k]^{\GL... | https://mathoverflow.net/users/85702 | Ring of invariants for $n$-tuples of Lie algebras | For some Lie groups you will need additional invariants. For example, for the even orthogonals you will need the Pfaffian in addition to the Trace.
See for example:
[*Invariant theory of special orthogonal groups*](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-168/issue-2/Invariant-theory... | 3 | https://mathoverflow.net/users/12218 | 391222 | 161,916 |
https://mathoverflow.net/questions/391147 | 4 | Let $k$ be either the field $\Bbb C$ of complex numbers or the field $\Bbb R$ of real numbers.
Let $X$ be an algebraic variety over $k$, say, quasi-projective and smooth (but not necessarily projective).
We consider the set of $k$-points $X(k)$ with the usual topology.
>
> **Question.** Is $X(k)$ homotopically equi... | https://mathoverflow.net/users/4149 | Is a complex or real algebraic variety homotopically equivalent to a CW complex? | This is same answer I left at the question linked above in the comments. The one benefit it offers is that it directly addresses the class of varieties in question.
In *[Triangulation of Locally Semi-Algebraic Spaces.](https://deepblue.lib.umich.edu/handle/2027.42/63851)* by K.R. Hofmann, necessary and sufficient con... | 4 | https://mathoverflow.net/users/12218 | 391225 | 161,917 |
https://mathoverflow.net/questions/391060 | 12 | Let $(\mathcal V,\otimes,I)$ be a closed symmetric monoidal category, and let $\mathcal C$ be a $\mathcal V$-enriched category. The (weak) enriched Yoneda Lemma gives us a nice description of the set $Hom(F,G)$ of natural transformations between two $\mathcal V$-enriched functors $F,G\colon\mathcal C\to\mathcal V$ when... | https://mathoverflow.net/users/126183 | Yoneda Lemma for monoidal functors | For your suspicion to work you need $G$ to be pseudomonoidal I would think, otherwise I don't see how to obtain a comonoid structure on $G(Y)$ from that on $Y$?
With both $F$ and $G$ lax monoidal considering Day-convolution $\hat\oslash$ on $\hat A = \mathcal V^{A^\text{op}}$ gives you a "monoidal yoneda lemma", not ... | 2 | https://mathoverflow.net/users/133974 | 391233 | 161,920 |
https://mathoverflow.net/questions/391237 | 1 | In [arXiv:math/0605371, Theorem 4 on p.8](https://arxiv.org/pdf/math/0605371v1.pdf), there is the following statement:
Let $X$ be a unit Killing vector field on a $n$-dimensional Riemannian manifold $M$. Then the Ricci curvature $\operatorname{Ric}$ of the manifold $M$ satisfies the condition $\operatorname{Ric}(X,X)... | https://mathoverflow.net/users/163410 | Unit Killing vector fields on pseudo Riemannian manifolds | All vector fields in Riemannian geometry papers are assumed real and smooth, unless otherwise stated. All vector fields in complex algebraic geometry papers are assumed complex and holomorphic, unless otherwise stated.
| 3 | https://mathoverflow.net/users/13268 | 391238 | 161,922 |
https://mathoverflow.net/questions/391005 | 1 | this paper is only available in Russian: <http://www.mathnet.ru/links/9f144b1d16e600dac49acbfe5acf938f/ppi797.pdf> According to MathSciNet, there is no link to the English article or journal publication yet :(. I saw [this entry](https://mathoverflow.net/questions/245163/need-help-with-paper-written-in-russian-yorgovs-... | https://mathoverflow.net/users/166974 | paper only available in Russian (Kozachenko Leonenko entropy estimator) | This journal is translated into English cover-to-cover
as *Problems of Information Transmission*.
Here is a PDF version of the indicated article for your convenience:
<https://dmitripavlov.org/scans/kozachenko-leonenko.pdf>
| 1 | https://mathoverflow.net/users/402 | 391240 | 161,923 |
https://mathoverflow.net/questions/391253 | 2 | This question is ***almost*** a duplicate of [that question](https://mathoverflow.net/q/375368/4149),
which has a [good answer](https://mathoverflow.net/a/375415/4149).
The difference is that I ask for ***references*** rather than proofs.
By a reference I mean a reference to a book, or to a paper, or to an arXiv prepri... | https://mathoverflow.net/users/4149 | Smooth morphisms vs. submersions | Assertion 2 is stated as Proposition 10.4 page 270 in Hartshorne's book. In fact, assertions 1 is equally proved in Hartshorne proposition 10.4 (though it is not in the statement of the proposition). It suffices to notice that the kernel of:
$$ \Omega\_{Y/\mathbb{C}} \longrightarrow \Omega\_{X/ \mathbb{C}} \longrightar... | 3 | https://mathoverflow.net/users/37214 | 391256 | 161,926 |
https://mathoverflow.net/questions/391246 | 4 | Let $a>0$ be a real cyclotomic number. Is it always possible to solve in cyclotomics the equation $X\overline{X}=a$ ?
Equivalently, one might want to express $a$ as a sum of squares of two real cyclotomics. It is well-known that one square is not always enough.
(If two squares are not enough, then, is there an uppe... | https://mathoverflow.net/users/11100 | is every positive real cyclotomic number the norm of a cyclotomic? | If $a$ is a totally positive real cyclotomic number, then it is a sum of two squares of real cyclotomic numbers.
It suffices to check that the equation $x^2+ y^2 - a z^2=0$ has solutions in real cyclotomic numbers. It has solutions in a particular real cyclotomic number field $F$ if it has solutions everywhere locall... | 6 | https://mathoverflow.net/users/18060 | 391269 | 161,932 |
https://mathoverflow.net/questions/391153 | 2 | Consider sum:
\begin{equation}
S\_q(n) = \sum\_{x=1}^{n}\Big\{ \frac{x^q}{n} \Big\}
\end{equation}
where $\{x\}$ is fractional part of $x$. It's easy to see that $S\_{1}(n) = \frac{1}{2}(n-1)$, but already $S\_{2}(n)$ looks more complicated. I have some observations about the $S\_q(n),\; q > 1$. It is likely that they... | https://mathoverflow.net/users/37289 | Formulas for $\sum_{x=1}^{n}\Big\{\frac{x^q}{n}\Big\}$ | I have a solution when $-1$ is a power of $q$ mod $n$ (which generalizes your observations) and when $n$ is prime and $q$ is 2.
We'll show that if there is a $z$ such that $z^q\equiv -1 \mod n$ where $n=\prod\_i p\_i^{q\_i}$ then:
$$S\_q(n)= \frac{1}{2} ( n - \prod\_i p\_i^{q\_i-\lceil{ q\_i/q }\rceil}) $$
In parti... | 2 | https://mathoverflow.net/users/178682 | 391277 | 161,934 |
https://mathoverflow.net/questions/391278 | 0 | **Input**: System of $\Omega(t)$ independent polynomials in $\mathbb F\_2[x\_1,\dots,x\_{t}]$ of degree $O(t)$.
1. Can we output a common solution of the system in polynomial time?
2. Can we output parity of the number of common solutions of the system in polynomial time?
| https://mathoverflow.net/users/10035 | On a system of equations in $\mathbb F_2$ | 1 - No, the problem is NP-hard even if degree of all polynomials is 2.
2 - See, for example, paper [Solving Systems of Polynomial Equations over
GF(2) by a Parity-Counting Self-Reduction](https://acris.aalto.fi/ws/portalfiles/portal/35681013/LIPIcs_ICALP_2019_26.pdf) by Björklund et al.
| 5 | https://mathoverflow.net/users/7076 | 391279 | 161,935 |
https://mathoverflow.net/questions/391001 | 11 | *Note : this is a [crosspost](https://math.stackexchange.com/q/3907570/259363) from the Mathematics StackExchange, as suggested by [this](https://math.meta.stackexchange.com/a/23092/259363) meta post.*
Let $X$ be an $n$-connected ($n\geqslant1$) CW-complex and $Y$ be a $k$-connected ($k\geqslant1$) CW-complex. My goa... | https://mathoverflow.net/users/137265 | Whitehead product and a homotopy group of a wedge sum | Here are some details which are related to Tyler's comment.
I recommend looking at the paper "Induced Fibrations and Cofibrations" by Tudor Ganea (1967). For connected based spaces $X$ and $Y$, there is a fibration up to homotopy
$$
\Sigma (\Omega X) \wedge (\Omega Y) \to X\vee Y \to X\times Y
$$
where the first map ... | 10 | https://mathoverflow.net/users/8032 | 391281 | 161,936 |
https://mathoverflow.net/questions/391280 | 4 | According to Stack project, when referring to them I should use the Tag system
<https://stacks.math.columbia.edu/tags> since
>
> “The tag system provides stable references to definitions, lemmas, propositions, theorems”
>
>
>
and
>
> “The place of the lemma in the document may change, the lemma may be move... | https://mathoverflow.net/users/12770 | Can/should I cite some Stack projects Tag in a paper | Yes, you should cite Stack Project Tags! The conclusion of the Theorem/Lemma/etc shouldn't weaken over time.
Compare to that old reliable citation "personal communication", or, worse, to a paper that is "to appear", and is never written. At the very least, the proof is there to be examined, and indeed the entire proo... | 11 | https://mathoverflow.net/users/4177 | 391282 | 161,937 |
https://mathoverflow.net/questions/391283 | 4 | Let $(M,g)$ be a compact Riemannian manifold with boundary and assume it has positive scalar curvature.
>
> **Question.** Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature?
>
>
>
| https://mathoverflow.net/users/85934 | Positive scalar curvature on the double of a manifold | The answer is negative at least if you do not add some sort of convexity hypothesis for the boundary, and at least in dimension $2$. Take a round $2$-sphere with $h\ge 2$ round holes. It has positive curvature, but its double has genus $h-1\ge 1$ and cannot have a positively curved metric. Of course, in dimension $2$ s... | 9 | https://mathoverflow.net/users/4961 | 391290 | 161,939 |
https://mathoverflow.net/questions/391219 | 4 | Let $G$ be an infinite finitely generated group. Fix a finite generating set for $G$.
Define $\mathrm{len}\_G:G\to\mathbb{Z}\_{\geq 0}$ by sending $g$ to the minimum length of a word in the generators and their inverses equal to $g$.
Let $H\subset G$ is an infinite finitely generated subgroup. Fix a finite generati... | https://mathoverflow.net/users/nan | Computable change in minimum word length of subgroup elements | I'll put my comment here, so that the question has an answer.
The function $\operatorname{len}\_H$ is what is called an *actual distortion function* (for $H$ in $G$) by Margolis, Meakin & Šuniḱ (see [1]). This is a notion that has been studied before in various forms in some papers by e.g. Gromov and Gersten (as @YCo... | 4 | https://mathoverflow.net/users/120914 | 391298 | 161,941 |
https://mathoverflow.net/questions/391287 | 3 | According to [NIST](https://dlmf.nist.gov/13.16#E5), the integral representation of Whittaker $W$ functions
$$
W\_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(%
\frac{1}{2}+\mu-\kappa\right)}\int\_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-%
\frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa} ... | https://mathoverflow.net/users/156601 | On integral representation of Whittaker $W$ functions | When $2\mu$ is not an integer one has this [identity](https://dlmf.nist.gov/13.14.E33) between the two Whittaker functions $W$ and $M$,
$$W\_{\kappa,\mu}\left(z\right)=\frac{\Gamma\left(-2\mu\right)}{\Gamma\left(\frac%
{1}{2}-\mu-\kappa\right)}M\_{\kappa,\mu}\left(z\right)+\frac{\Gamma\left(2\mu%
\right)}{\Gamma\left(\... | 3 | https://mathoverflow.net/users/11260 | 391300 | 161,942 |
https://mathoverflow.net/questions/391291 | 6 | Let $S=(s\_{ij})$ be a skew-symmetric integral matrix of order $n$. We only consider the case that $n$ is even. Let $e$ be the all-one vector in $\mathbb{R}^n$. Define the walk matrix $$W(S)=[e,Se,\cdots,S^{n-1}e].$$ (the name "walk matrix" comes from graph theory, where $S$ is the adjacency matrix of an undirected gra... | https://mathoverflow.net/users/120597 | Determinant of walk matrix for a skew-symmetric matrix of even order | Surely, there is nothing special in the all-ones vector: the claim holds for any integer-valued $e$.
Notice that
$$
\det W^TW
=\det\bigl[e^T (-1)^iS^{i+j}e\bigr],
$$
Since $S$ is skew-symmetric, we have $e^TS^ie=0$ for all odd $i$. Permuting now the rows and columns of $W^TW$ we get
$$
\det W^TW=\det\begin{bmatrix... | 7 | https://mathoverflow.net/users/17581 | 391301 | 161,943 |
https://mathoverflow.net/questions/120173 | 14 | This question is inspired by the excellent question by Douglas Ulrich [When is $L$-Rank definable in inner models of $V=L$?](https://mathoverflow.net/questions/119882/when-is-mathbbl-rank-definable-in-inner-models-of-mathbbv-mathbbl)
Suppose $M \in L$ is a countable model of $ZFC$, and furthermore suppose $M \vDash V... | https://mathoverflow.net/users/10671 | Can a model of $V\neq L$ contain a class giving the $L$-ordering on all its sets? | This answer just considers the version of the question for transitive models $M$.
Under a reasonable interpretation of the order $<\_L$ of constructibility, there is no such transitive model $M$. However, that interpretation is fine structurally motivated, and with a more obvious interpretation, I’m not sure how to p... | 5 | https://mathoverflow.net/users/160347 | 391303 | 161,944 |
https://mathoverflow.net/questions/391274 | 7 | Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only been able to find a handful of papers about filtered 2-colimits, notably Dubuc and Street's [A construction of 2-filtered bi... | https://mathoverflow.net/users/49 | Filtered 2-colimits commute with finite 2-limits | Two relevant papers are:
* Dupont's [Interchange of filtered 2-colimits and finite 2-limits](https://arxiv.org/abs/0904.1553).
* Canevali's [2-filtered bicolimits and finite weighted bilimits commute in Cat](http://cms.dm.uba.ar/academico/carreras/licenciatura/tesis/2016).
The former proves that finite conical pseu... | 8 | https://mathoverflow.net/users/152679 | 391310 | 161,945 |
https://mathoverflow.net/questions/391257 | -4 | ***Update**: I added $exp[i\theta\_k(s)]$ in the definition of $\eta^\*(s)$ to address some critical convergence issues. Thanks for the contributors who pointed to these issues.*
Prime numbers are denoted as $p\_1,p\_2,\dots$ with $p\_1=2$. The modulus of a complex number $s$ is denoted as $|s|$. Finally, $S$ denotes... | https://mathoverflow.net/users/140356 | Scaled Riemann zeta function with no zero in the critical strip | The conditions presented are contradictory - for any $s=\sigma+it\in S$ with $t\neq 0$ it is impossible to choose positive real numbers $\tau\_k(s)$ such that $\prod\_{k=1}^\infty\frac{\tau\_k(s)}{1-p\_k^{-s}}$ converges. This is because the argument of the partial product cannot converge, and this is something that po... | 2 | https://mathoverflow.net/users/30186 | 391313 | 161,946 |
https://mathoverflow.net/questions/391318 | 2 | I have a question about the following example from the [Algebraic spaces and quotients by equivalence relation of schemes](https://people.kth.se/%7Eskjelnes/Pdffiler/algspaces.pdf) by Roy Mikael Skjelnes (page 12)
of a presheaf quotient, which
has associated sheaf which can not be a scheme.
It's originally an example f... | https://mathoverflow.net/users/108274 | Example of an Algebraic Space ("false" affine line with different tangents at origin) | You are confusing two completely different notions of "separated".
A presheaf of sets $F$ on a site is called "separated" if whenever $\{U\_i \to U\}$ is a covering family, $F(U) \to \prod F(U\_i)$ is injective. This is the notion that Skjelnes is talking about. The sheafification of a presheaf of sets is often const... | 5 | https://mathoverflow.net/users/1310 | 391322 | 161,948 |
https://mathoverflow.net/questions/391324 | 2 | So this question is probably not "research level", although, for what it is worth, it is coming up in a research paper I am presently writing.
Let $X,Y$ be irreducible affine varieties over $\mathbb{C}$. Suppose $f:X\to Y$ is a bijective birational morphism. Assume that $X$ is normal, but $Y$ is not and that both $X,... | https://mathoverflow.net/users/12218 | When is a birational bijection étale? | This map is étale if and only if $Y$ is normal.
You mentioned the proof of one direction. In the other, probably the most straightforward proof uses [the invariance of normality under étale morphisms of local rings](https://stacks.math.columbia.edu/tag/025P). If $f: X \to Y$ is étale, this result implies $X$ is norma... | 6 | https://mathoverflow.net/users/18060 | 391325 | 161,949 |
https://mathoverflow.net/questions/391309 | 2 | The Gaussian Poincare inequality says that if $q:R^n\to R$ is Lipschitz (for simplicity you may additionally assume smooth with compact support), then $Var[f(X)] \le L^2$
for $X\sim N(0,I\_q)$.
Now instead, $X\in R^{n\times d}$ is a matrix with iid entries and $f:R^{n\times d}\to R$ is again $L$-Lipschitz, but with r... | https://mathoverflow.net/users/141760 | Lipschitz condition with respect to operator norm of a Gaussian matrix with iid entries. Improved Gaussian Poincare Inequality? | You cannot improve the bound $L^2$ on $Var\,f(X)$ unless an additional condition on $f$ is assumed.
Indeed, let $f(x)\equiv Lx\_{11}$, where $x\_{11}$ is the first diagonal entry of a matrix $x\in R^{n\times d}$. Then $L$ is the Lipschitz constant for $f$ with respect to the operator norm, but $Var\,f(X)=L^2$ if $X\_... | 2 | https://mathoverflow.net/users/36721 | 391326 | 161,950 |
https://mathoverflow.net/questions/391305 | 9 | I'm struggling with the definition of an arithmetic subgroup of an algebraic group defined over $\mathbb{Q}$.
In Wikipedia you can read:
>
> If $\mathrm G$ is an algebraic subgroup of $\mathrm{GL}\_n(\mathbb{Q})$ for some $n$ then we can define an arithmetic subgroup of $\mathrm G(\mathbb{Q})$ as the group of integ... | https://mathoverflow.net/users/197544 | Definition of an arithmetic subgroup of an algebraic group | Write the coordinates $a\_{ij}$ of one embedding into $GL\_n$ as polynomial functions, defined over $\mathbb Q$, in the coordinates $b\_{ij}$ of a different embedding into $GL\_n$. We can do this because, by definition, embeddings give an isomorphism of algebraic varieties. (I guess we should allow the inverse of the d... | 14 | https://mathoverflow.net/users/18060 | 391327 | 161,951 |
https://mathoverflow.net/questions/390997 | 5 | **Definitions and notation:**
Let $\Omega$ be a open, convex, bounded subset of $\mathbb R^n$ with Lipschitz boundary, and $f \in W^{1,1}(\Omega)$ a Sobolev function. Given $x \in Ω$, we denote by $x’$ the point $(x, f(x)) \in \Omega \times \mathbb R$.
Consider the graph $G\_f$ of $f$ as a subset of $\Omega \times ... | https://mathoverflow.net/users/173490 | Is the graph of a Sobolev function “almost geodesically complete”? | The following is not a full proof, as I skip on some calculation details, just an extension of the remark I made in a comment but I am pretty certain it is correct.
**The answer is no, at least not wrt. to minimal geodesics**
**Boundary case:** The first counterexample shows that we can force the curve to touch the... | 1 | https://mathoverflow.net/users/51695 | 391334 | 161,953 |
https://mathoverflow.net/questions/391330 | 8 | Question: What is the relationship between provability in $RCA\_0$ and effectively true?
In other words: Given a problem, if a statement asserting the existence of a solution of the problem is provable in $RCA\_0$, does it follow that given a computable instance of the problem we can compute a solution to the problem... | https://mathoverflow.net/users/178532 | Relationship between provable in $RCA_0$ and effectively true | Provability in $\mathsf{RCA\_0}$ (or even truth in all $\omega$-models of $\mathsf{RCA\_0}$) guarantees a *version of* computable truth of a $\Pi^1\_2$ sentence, namely that for every instance of the corresponding problem $X$ there is a solution $Y$ which is computable from $X$ (note that this applies even when $X$ is ... | 5 | https://mathoverflow.net/users/8133 | 391338 | 161,955 |
https://mathoverflow.net/questions/390461 | 6 | Let $X$ be a random variable with values in a closed compact $\Omega \subset \mathbb{R}^m$. Assume $\Omega$ is has a Lipschitz boundary. Let $p(x)$ be the probability density function of $X$ and assume $p(x)>0\ \forall x\in\Omega$.
Now we draw two independent sets of $n$ iid copies of $X$. We denote them $A\_n = \{p\... | https://mathoverflow.net/users/14414 | A problem on rate of decay of fill distance? | $\newcommand\ep\varepsilon$Let $X:=\Omega$. Suppose that $|B\_x(\ep)\cap X|>0$ for all $x\in X$ and all real $\ep>0$, where $|\cdot|$ is the Lebesgue measure and $B\_x(\ep)$ is the open ball in $\mathbb R^m$ of radius $\ep$ centered at $x$. Then the condition $p(x)>0$ for all $x\in X$ implies that $P(p\_1\in B\_x(\ep)\... | 2 | https://mathoverflow.net/users/36721 | 391340 | 161,956 |
https://mathoverflow.net/questions/391306 | 3 | I just learned that there is a model structure on the category $Op\_{Top}$ of topological operads, due to Berger-Moerdjik [1], obtained by right transfer of the Quillen model structure on $Top$.
Since $Top$ is a proper model category, I was wondering if $Op\_{Top}$ is proper aswell. It should be right proper since al... | https://mathoverflow.net/users/169319 | Is the category of topological operads left proper? | It is not left proper. You can find a counterexample (at least for operads in simplicial sets but you can just apply realization to this counterexample) in section 4 of <https://arxiv.org/abs/1411.4668>.
| 6 | https://mathoverflow.net/users/10707 | 391343 | 161,958 |
https://mathoverflow.net/questions/391335 | 2 | Is there a function $f$ on $\mathbb{R}$ such that as $x \to 0$,
$$
f(x) = \sum\_{j=0}^N x^{1 - \frac{1}{j}} + o(x^{1- \frac{1}{N}}),
$$
for every $N \in \mathbb{N}$?
Heuristically there shouldn't be such a function, since the exponents accumulate at $1$ and thus higher terms are not "asymptotically independent", gi... | https://mathoverflow.net/users/117393 | Existence of function $f$ such that $f(x) \sim \sum_{j \in \mathbb{N}} x^{1 - \frac{1}{j}}$ | First, $\sum\_{j=0}^N x^{1-1/j}$ is undefined, since $1/0$ is undefined. So, let us use $\sum\_{j=1}^N x^{1-1/j}$ instead.
For $x>0$, let
$$f(x):=\sum\_{j=1}^\infty x^{1-1/j}1(x<e^{-j})=\sum\_{1\le j<\ln(1/x)} x^{1-1/j}.$$
Then for each natural $N$ and small enough $x\in(0,1)$
$$0\le f(x)-\sum\_{j=1}^N x^{1-1/j}=\sum... | 5 | https://mathoverflow.net/users/36721 | 391345 | 161,959 |
https://mathoverflow.net/questions/389790 | 14 | Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}\_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. [Lachaud and Martin-Descamps](http://dx.doi.org/10.4064/aa-56-4-329-340) established the bound $h \geq \tfrac{q^g}{g+1} \tfrac{(q-1)^2}{q... | https://mathoverflow.net/users/297 | Lower bounds for class number of function fields with fixed $q$, growing $g$ | $\def\FF{\mathbb{F}}$Easy results are usually not original, and this wasn't an exeption. What I was doing was almost exactly the same as [On the number of rational points of Jacobians over finite fields](https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/169/4/83945/on-the-number-of-... | 5 | https://mathoverflow.net/users/297 | 391350 | 161,961 |
https://mathoverflow.net/questions/391348 | 1 | Let $X$ be a set of continuum cardinality. The group of permutations of $X$ acts on the set of topologies on $X$.
What can be said about the fixed points of this action? Are manifolds fixed points?
What can be said about the finite orbits?
| https://mathoverflow.net/users/197683 | Action of the permutation group on the set of topologies on a continuum | Let $S(X)$ be the group of permutations of $X$, $X$ infinite. It is a classical consequence of the Baer theorem (Onofri for $X$ countable) that $S(X)$ has no nontrivial finite quotient [actually every nontrivial quotient has cardinal $2^{|X|}$]. Hence every finite orbit of $S(X)$ on $T(X)$, the set of topologies on $X$... | 3 | https://mathoverflow.net/users/14094 | 391353 | 161,962 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.