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https://mathoverflow.net/questions/352114 | 5 | A $(p,q)$-shuffle is a permutation $\sigma$ of the set $\{1, \dots, p, p+1, \dots,p+q\}$ such that $$\sigma(1)<\dots<\sigma(p)$$ and $$\sigma(p+1) < \dots<\sigma(p+q)\,.$$
It is known that the [number of $(p,q)$-shuffles](https://en.wikipedia.org/wiki/Riffle_shuffle_permutation) is ${p+q \choose p}$.
Looking at the... | https://mathoverflow.net/users/152023 | Parity of shuffle permutations | As Philippe suggests in the comments, it is well know that
$$\sum\_{\sigma \text{ is a $(a,b)$-shuffle}} q^{\ell(\sigma)}={a+b \choose b}\_q$$
where the $q$-binomial coefficient (or Gaussian binomial coefficient) on the right side is defined by
$${n \choose k}\_q=\frac{[n]\_q! }{[k]\_q! [n-k]\_q!}$$
where $[n]\_q!=[1... | 3 | https://mathoverflow.net/users/33089 | 352147 | 148,876 |
https://mathoverflow.net/questions/352119 | 1 | Let $\Omega$ be a bounded and smooth domain. Suppose I have a sequence of non-negative functions $u\_n \in L^\infty((0,1)\times \Omega) \cap L^\infty((0,1);L^\infty(\Omega))$ with
$$0 \leq u\_n \leq 1 \quad \text{a.e}$$
and
$$u\_n \rightharpoonup^\* u \quad\text{in $L^\infty((0,1)\times \Omega)$}$$
to some $u$.
Is i... | https://mathoverflow.net/users/102901 | Weak-* convergence in $L^\infty((0,T)\times\Omega)$ implies weak-* convergence in $L^\infty(\Omega)$ for a.e. $t \in (0,T)$? | If I understood correctly the question, this is not true. If $u\_n(t,x)=sin(2πnt)f(x)$ for a fixed $f\neq 0$, then $u\_n \to 0$, $w^\*$ in $(t,x)$ by Riemann Lebesgue, but for fixed $t\neq 0,1/2,1$ does not converge $w^\*$ in $x$. Similarly, there is no way to find a sequence $n\_j$ such that $sin(n\_jt)$ converges a.e... | 2 | https://mathoverflow.net/users/150653 | 352155 | 148,879 |
https://mathoverflow.net/questions/352141 | 9 | Let $G= SU(n)$ and let $\mathbb{T}$ be the maximal torus in $G$ given by diagonal matrices. We have
$$
H^\*(G,\mathbb{Q}) \cong \Lambda\_{\mathbb{Q}}[x\_3, x\_5, \dots, x\_{2n-1}] \ .
$$
Now consider the Weyl map
$$
p \colon G/\mathbb{T} \times \mathbb{T} \to G \quad , \quad ([g],z) \mapsto gzg^{-1}\ .
$$
The induced ... | https://mathoverflow.net/users/3995 | Weyl map for $SU(n)$ | First, let me fix generators for $H^\*(SU(n))$ and $H^\*(SU(n)/\mathbb T)$: For the first, consider the vector bundle on $\Sigma SU(n)$ with clutching map $\operatorname{id}\_{SU(n)}$, i.e. with classifying map $f\_n\colon\Sigma SU(n)\simeq \Sigma\Omega BSU(n)\to BSU(n)$, and let $\Sigma x\_{2i-1} = f^\*c\_i$. For the ... | 7 | https://mathoverflow.net/users/35687 | 352157 | 148,880 |
https://mathoverflow.net/questions/351886 | 4 | Let $G$ be a topological group. It has a classifying space $BG$, which has homology groups $H\_{\*}BG$. Changing the topology of $G$ affects the space $BG$ and hence its homology groups.
For example the group $\mathbb{R}$ with its usual topology has $H\_{\*}B\mathbb{R}\simeq H\_{\*}pt$. Changing the topology to be mu... | https://mathoverflow.net/users/148857 | Shifting the group homology of a topological group? | Let $p$ be the projection map $\Bbb R \to S^1$. Let $T$ be $\Bbb R$, where we say that a subset $U$ is open if and only if $U = p^{-1} V$ for some open set $V$ of $S^1$. This makes $T$ a topological group, and the maps $\Bbb R \to T \to S^1$ are homomorphisms of topological groups. The space $T$ also has the property t... | 5 | https://mathoverflow.net/users/360 | 352161 | 148,882 |
https://mathoverflow.net/questions/352120 | 3 | Let $\mathsf{Stone}$ denote the category of Stone spaces (compact, totally disconnected Hausdorff spaces) and continuous maps. The forgetful functor $U : \mathsf{Stone} \to \mathsf{Set}$ has a left adjoint $F : \mathsf{Set} \to \mathsf{Stone}$ which maps a set $X$ to the space $F(X)$ of ultrafilters on $X$ (i.e. the St... | https://mathoverflow.net/users/2841 | Ultrafilter comonad on the category of Stone spaces | I believe the answer is *yes*: via the ultrafilter functor, $Set$ is comonadic over $Stone$ (and also over $CpctHaus$). Instead of the plain comonadicity theorem, let's use the crude comonadicity theorem, which says the following:
>
> Let $F: C \to D$ be a conservative left adjoint such that $C$ has and $F$ preserv... | 2 | https://mathoverflow.net/users/2362 | 352168 | 148,885 |
https://mathoverflow.net/questions/352152 | 9 | I am cross-posting [this question from MSE](https://math.stackexchange.com/questions/3528142/how-to-show-this-is-not-a-martingale) since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer.
The most surprising thing is that this exercise was t... | https://mathoverflow.net/users/132216 | Show that this process is not a martingale | Here's an approach that comes from
*Li, Xue-Mei*, [**Strict local martingales: examples**](http://dx.doi.org/10.1016/j.spl.2017.04.025), Stat. Probab. Lett. 129, 65-68 (2017). [ZBL1386.60159](https://zbmath.org/?q=an:1386.60159), <https://arxiv.org/abs/1609.00935>. Indeed, she mentions this very example after Corollar... | 7 | https://mathoverflow.net/users/4832 | 352171 | 148,887 |
https://mathoverflow.net/questions/352085 | 18 | I have this question for a moment now, so I think it is time that I sort it out.
I got into category theory and homotopy type theory at the same time, and so I have always read and been told that one should be careful about "evil" properties, i.e. the ones that are not invariant under equivalence of categories. So in... | https://mathoverflow.net/users/128382 | Are evil properties really evil | Before I try to answer the question itself, let me make a few preliminary remarks.
Firstly, a minor point: it's not really correct to say that categories are the objects of Ho(Cat), if by Ho(Cat) you mean the homotopy 1-category of Cat; the correct thing to say is that categories are the objects of the *2-category* C... | 15 | https://mathoverflow.net/users/49 | 352181 | 148,890 |
https://mathoverflow.net/questions/352196 | 4 | Let $G$ be a non-Abelian infinite group. Can $G$ admit more than one (inequivalent) non-compact locally compact metrizable second countable topologies that make it a topological group?
Thank you.
| https://mathoverflow.net/users/89313 | Different locally compact metrizable second countable topologies on the same group | Turning my comment into an aswer: yes, it is possible for such a $G$ to have more than one topology with your requirements.
Consider $(\Bbb R,+)$ with its usual topology. As a group it is isomorphic to $(\Bbb C,+)$, so there is also a topology in which $(\Bbb R,+)$ is a topological group homeomorphic to $\Bbb R^2$, b... | 7 | https://mathoverflow.net/users/49381 | 352202 | 148,894 |
https://mathoverflow.net/questions/327927 | 1 | Let $X$ be a Banach space and
$$\mathrm C\_0(\mathbb R,X):=\{f\colon\mathbb R\to X\colon f \text{ is continuous and } \lim\limits\_{|t|\to\infty}f(t)=0\}$$
normed by
$$\|f\|:=\sup\limits\_{t\in\mathbb R}\|f(t)\|, \quad f\in \mathrm C\_0(\mathbb R,X).$$
Let $(U(t,s))\_{t\ge s}$ denote an evolution family of bounded line... | https://mathoverflow.net/users/138304 | Does an evolution family commute with the right shift semigroup? | In general no; this happens only in very special cases.
To make this simpler, let's suppose $U$ is a semigroup (a time-homogeneous evolution family), so $U(s,t) =U(t-s)$. Then formally, this commutation would only occur if the generator $A$ of $U(t)$ were to commute with the generator of $R(t)$, which is $\frac{d}{dx... | 1 | https://mathoverflow.net/users/4832 | 352211 | 148,897 |
https://mathoverflow.net/questions/352186 | 3 | Are there any equivalent representations of the following (real valued) sum, in particular that are suitable for evaluation as $z\rightarrow0$ ?
$$ S=\sum\_{k=-\infty}^\infty \frac{i^k(z-2ik)}{(\rho^2+(z-2ik)^2)^{3/2}} $$
I am aware that $S$ resembles a coulomb force sum which can be rearranged using Lekner summation... | https://mathoverflow.net/users/94200 | Alternating 1D lattice sum | For $\rho$ equal to an even integer the sum $S$ diverges as $1/z^{3/2}$ when $z\rightarrow 0$. For $\rho$ unequal to an even integer and $z>0$, one has
$$S\_0=\lim\_{z\rightarrow 0}\sum\_{k=-\infty}^\infty \frac{i^k(z-2ik)}{(\rho^2+(z-2ik)^2)^{3/2}}=-4\,\Re\sum\_{k=1}^\infty\frac{ i^k k}{\left(4 k^2-\rho^2\right)^{3/2}... | 1 | https://mathoverflow.net/users/11260 | 352217 | 148,899 |
https://mathoverflow.net/questions/352200 | -1 | **Replacement:** if $\phi(x,z)$ is a formula in which all and only symbols $``x,z,x\_1,..,x\_n"$ occur free, and non of them occur as bound, and in which the symbol $``B"$ never occur; then: $$\forall x\_1,..,x\_n \\\forall A \exists B \forall y \ [y \in B \leftrightarrow \exists x \in A \forall z (\phi(x,z) \leftright... | https://mathoverflow.net/users/95347 | Is cyclic replacement inconsistent with ZFC-Foundation? | Unless I'm misunderstanding something here, Cyclic Replacement is inconsistent with just about any set theory. If $p$ and $q$ are two distinct sets, let $\phi(x,z,B)$ be $(z=p\land B\neq\{p\})\lor(z=q\land B=\{p\})$. Let $A$ be your favorite nonempty set. (Since you insist that $\phi$ contain $x$ free, add a clause "$\... | 2 | https://mathoverflow.net/users/6794 | 352223 | 148,901 |
https://mathoverflow.net/questions/352082 | 11 | Every $Π^1\_1$ formula $φ$ without free second order variables can be converted into a $Σ^1\_1$ $ψ$ such that $φ ⇔ ψ^\mathrm{HYP}$, and vice versa. ($\mathrm{HYP}$ is the hyperarithmetical universe, which can be identified with $L\_{ω\_1^\mathrm{CK}}$.) Essentially, arbitrary $Π^1\_1$ statement $⇔$ well-foundedness of ... | https://mathoverflow.net/users/113213 | Does $Π^1_{2n+1} = (Σ^1_{2n+1})^{M_{2n-1}}$? | Your conjecture is true. We assume PD throughout. The proof I see requires citing a number of facts from inner model theory and descriptive set theory. First, it uses Woodin's theorem characterizing the reals of $M\_{2n-1}$ as the set $Q\_{2n+1}$ of points in Baire space that are $\Delta^1\_{2n+1}$ definable from a cou... | 11 | https://mathoverflow.net/users/102684 | 352234 | 148,906 |
https://mathoverflow.net/questions/352097 | 6 | Consider an $n \times n$ irreducible and reversible (in the sense of a Markov chain) stochastic matrix $P$; assume that it has uniform stationary distribution (so, by reversibility, the matrix is symmetric and doubly stochastic).
If, say the first row and column of the matrix was deleted (call this matrix $P'$), is t... | https://mathoverflow.net/users/148528 | Maximum eigenvalue of a doubly stochastic matrix with deleted row and column | The following proof shows that this is indeed the case. Let $\gamma$ denote the spectral gap of $P$ and $\lambda$ denote the absolute value of the eigenvalue of $P$ that achieves the spectral gap. Then,
\begin{equation}
\rho\_1 \le 1 - \frac{\gamma}{2n}
\end{equation}
for large $n$.
*Proof:*
Let $\{ \mathbf{u}\_i \... | 0 | https://mathoverflow.net/users/148528 | 352247 | 148,908 |
https://mathoverflow.net/questions/218911 | 1 | Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that $e<h<g$?
| https://mathoverflow.net/users/43954 | Name of the class of linearly ordered groups with no minimal positive element | I assume that the order is left-invariant. All orders here are total.
First, by homogeneity, the order is either dense or discrete (in the sense that the order topology is discrete). Beware that being discrete doesn't make it simpler: indeed for every left-ordered group $G$, the direct product with lexicographic orde... | 1 | https://mathoverflow.net/users/14094 | 352250 | 148,909 |
https://mathoverflow.net/questions/352209 | 1 | Let us make two small holes around points $0$ and $1$ on the complex plane and consider non-self-intersecting paths that start on the boundary of one hole and finish at the boundary of the another. It seems,
that all these paths are homotopic to the segment of the real line.
How to prove this? Is there any conceptual... | https://mathoverflow.net/users/152066 | Non-self-intersecting paths on $\mathbb{C}\setminus\{0,1\}$ | Such a homotopy exists and in fact you can assume that it is an isotopy. This is a "standard fact" in the theory of mapping class groups. See Proposition 2.2 of the "Primer" by Farb and Margalit.
| 1 | https://mathoverflow.net/users/1650 | 352272 | 148,918 |
https://mathoverflow.net/questions/352258 | 1 | The following content was based on [Minlos'](https://rads.stackoverflow.com/amzn/click/com/0821813374) book on statistical physics. Let $\Lambda \subset \mathbb{R}^{d}$ be fixed (Minlos takes $d=3$ but I think the ideas follow without change to $d \ge 1$). We denote by $\Lambda^{N}$ the $N$-fold cartesian product of $\... | https://mathoverflow.net/users/152094 | Measure, volume and cardinality on Minlos' book on statistical physics | If $\text{Vol}$ denotes the Lebesgue measure on $(\mathbb R^d)^N$, if $|\cdot|$ denotes the Lebesgue measure on $\mathbb R^d$, and if $\mu\_\Lambda=\mu\_\Lambda^{(N)}$, then indeed $\mu\_\Lambda(\Gamma\_{\Lambda,N})=\frac{|\Lambda|^N}{N!}$. This follows because (i) $\Pi^{-1}(\Gamma\_{\Lambda,N})=(\Lambda^N)'$ and (ii) ... | 0 | https://mathoverflow.net/users/36721 | 352283 | 148,923 |
https://mathoverflow.net/questions/352042 | 4 | Given an integer number $m>0$ and a real number $\alpha\in [1, 2]$, I am interested in finding a lower-bound for $\Pr[X\geq m]$ subject to $X \sim \text{Binomial}(n, m\alpha/n)$.
For large values of $m$, I use Chernoff bound to get my desired bound. However, for smaller values of $m$, I need a tighter bound. Based on m... | https://mathoverflow.net/users/130779 | Lower-bound for $\Pr[X \geq m]$ subject to $E[X]>m$ where $X$ is a binomial random variable | $\newcommand{\al}{\alpha}
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\De}{\Delta}$
The answer to your question is yes, and the restriction $\al\le2$ is not even needed.
Indeed, let $S\_n:=X\sim\text{Binomial}(n,m\al/n)$, where $m$ is any natural number, $\al$ is any real number $\ge1$, and $n$ ... | 5 | https://mathoverflow.net/users/36721 | 352285 | 148,924 |
https://mathoverflow.net/questions/352260 | 1 | I am looking for examples of transcendental entire functions $f:\mathbb C\to \mathbb C$ such that the set of non-escaping points in the Julia set of $f$ is not totally disconnected. I denote this set $J\_r(f)=\{z\in J(f):f^n(z)\not\to\infty\}$ because in some interesting cases it is the same as the "radial Julia set'' ... | https://mathoverflow.net/users/95718 | Is the set of non-escaping points in a Julia set always totally disconnected? | If $f$ has order $<1/2$ then there is a sequence $r\_k\to\infty$ with the property that
$$\min\_{|z|=r\_k}|f(z)|>r\_k.$$
Restricting $f$ on $\{ z:|z|<r\_k\}$ we obtain a polynomial-like map in the sense of Douady and Hubbard. If $J\_k$ is the Julia set of this map, then evidently $J\_k\subset J(f)$, and the points of $... | 1 | https://mathoverflow.net/users/25510 | 352287 | 148,925 |
https://mathoverflow.net/questions/352246 | 3 | Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance).
What is the distribution of the singular values and singular (left and right) vectors of $A$?
For a symmetric square matrix, the conc... | https://mathoverflow.net/users/16615 | Singular value decomposition of random rectangular matrices | To be specific, let me assume $m\leq n$. The $m\times n$ matrix $A$ then has $n-m$ singular values equal to zero. The remaining $m$ singular values $s\_i$, $i=1,2,\ldots m$ are the positive square roots of the eigenvalues $\sigma\_i$ of the symmetric matrix product $AA^t$. The distribution $P(\sigma\_1,\sigma\_2,\ldots... | 4 | https://mathoverflow.net/users/11260 | 352288 | 148,926 |
https://mathoverflow.net/questions/352294 | 4 | I was looking for an earliest reference or the name of the mathematician who showed calculating the derivatives is possible in the Fourier domain?
The Fourier transform of the derivative is ([Wikipedia](https://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships))
$$
\mathcal{F}(f')(\xi)=2\pi i\xi\cdot\m... | https://mathoverflow.net/users/142414 | Earliest reference on the calculation of derivatives by Fourier transform | Despite this being complicated by Fourier’s use of sine and cosine transforms, I think @PiyushGrover is right: Théorie analytique de la chaleur, [Art. 419](//archive.org/details/thorieanalytiq00four/page/557):
>
> $$
> \begin{align}
> \frac{d^{2i}}{dx^{2i}}fx
> =\pm&\int d\alpha f\alpha\int dp\,p^{2i}\cos.(px-p\alp... | 7 | https://mathoverflow.net/users/19276 | 352297 | 148,929 |
https://mathoverflow.net/questions/352305 | 10 | For $\mathcal{S}$ the $(\infty,1)$-category of spaces [its homotopy category $h\mathcal{S}$ does not have pushouts or pullbacks](https://mathoverflow.net/questions/239383/the-homotopy-category-is-not-complete-nor-cocomplete). Even if it does, they won't always agree with the (homotopy) pushouts or pullbacks in $\mathca... | https://mathoverflow.net/users/91925 | Can filtered colimits be computed in the homotopy category? | No, for a diagram $X: I \to \mathcal{S} \to h\mathcal{S}$ the colimit in $h\mathcal{S}$ would satisfy $[\mathrm{colim} X(i),Y] \cong \lim [X(i),Y]$ where brackets denote morphisms in $h\mathcal{S}$. Informally, the set of maps in $h\mathcal{S}$ from the (homotopy) colimit in $\mathcal{S}$ should involve also higher der... | 9 | https://mathoverflow.net/users/152131 | 352314 | 148,932 |
https://mathoverflow.net/questions/352255 | 14 | In his book [1], Paul Larson remarks (Remark 1.1.22) that in *L* there is a function $h:\omega\_1\rightarrow\omega\_1$ such that for any countable elementary submodel $X$ of $V\_\gamma$ (where $\gamma$ is the first strong limit cardinal), we have the order-type of $X\cap Ord$ is strictly less than $h(X\cap\omega\_1)$. ... | https://mathoverflow.net/users/18128 | Consequences of existence of a certain function from $\omega_1$ to $\omega_1$ | In the paper "[The consistency strength of the perfect set property for universally Baire sets of reals](https://arxiv.org/abs/1807.02213)", Ralf Schindler and I show that the negation of a very similar statement is equiconsistent with the existence of what we call a virtually Shelah cardinal.
We say that a cardinal ... | 6 | https://mathoverflow.net/users/1682 | 352319 | 148,933 |
https://mathoverflow.net/questions/352307 | 8 | Let's suppose that
$f:X\rightarrow X$ is a continuous map such that
1. $H\_{\ast}(f): H\_{\ast}(X)\rightarrow H\_{\ast}(X)$ is a homology isomorphism (with integral coefficients)
2. $X$ is a finite connected CW-complex.
3. $\pi\_{1}(f): \pi\_{1}(X)\rightarrow \pi\_{1}(X)$ is an isomorphism of fundamental groups.
... | https://mathoverflow.net/users/141114 | When homology isomorphism implies homotopy isomorphism | Here's a counterexample.
Set $X'=S^1\vee S^2$.
Consider the following map $F':S^2\vee S^2\vee S^2\rightarrow X'$: It maps the first $S^2$ summand to the $S^2$ summand of $X'$ via a map that represents $2\in\pi\_{2}S^2$; it maps the second summand once around the $S^1$ factor of $X'$, and maps the third $S^2$ summ... | 15 | https://mathoverflow.net/users/148857 | 352327 | 148,937 |
https://mathoverflow.net/questions/352291 | 2 | Let $R\subset S$ be commutative rings, $I\trianglelefteq R$ an ideal and $M$ be an $R$-module. Suppose that
1) $R$ is Noetherian and $I$-adically complete.
2) $M$ is a finite $R$-module (hence $M$ is $I$-adically complete)
3) $S$ is a flat $R$-algebra.
4) $S$ is $I$-adically complete
5) $M/IM$ is free module ... | https://mathoverflow.net/users/105386 | Completion and extension by scalars | This follows by adapting the proof of Tag [00MA](https://stacks.math.columbia.edu/tag/00MA), even without assumption 5. We also don't need $S$ to be an algebra; a complete $R$-module suffices. Finally, we never use that $R$ is $I$-adically complete!
Indeed, by assumption 2 there exists a short exact sequence $0 \to K... | 2 | https://mathoverflow.net/users/82179 | 352336 | 148,939 |
https://mathoverflow.net/questions/352309 | 1 | It is well known that changing variables from a symmetric matrix to its eigenvalue decomposition involves a Jacobian which is just the Vandermonde determinant of the eigenvalues.
Now suppose I have a rectangular matrix and I want to change variables to its singular value decomposition. What is the Jacobian of this tr... | https://mathoverflow.net/users/16615 | Jacobian of changing of variables to singular value decomposition | For an $m\times n$ real matrix $A=U\Sigma V^t$ with $m\leq n$, diagonal matrix of singular values $\Sigma={\rm diag}\,(\sqrt\sigma\_1,\sqrt\sigma\_2,\ldots\sqrt\sigma\_m)$, orthonormal left and right eigenvector matrices $UU^t=VV^t=\mathbb{1}$, the Jacobian $J$ in the measure $dA=JdUdV\prod\_{i=1}^m\sigma\_i$ follows f... | 2 | https://mathoverflow.net/users/11260 | 352337 | 148,940 |
https://mathoverflow.net/questions/352242 | 1 | want to prove the duality relation:
$$||f||\_{L^{p,1}} =C\_{p} \cdot \sup\{\int\_X fg d\mu: \text{ for any } ||g||\_{L^{p',\infty}}\le 1 \} $$
where $\frac{1}{p}+\frac{1}{p'}=1, p>1, \mu$ is $\sigma$-finite meaure on $X$, $C\_{p}\in (0, \infty)$ and
$$||g||\_{L^{p,q}}:=(p \cdot \int\_{0}^{\infty} \lambda^{q-1}\mu(... | https://mathoverflow.net/users/124254 | Duality relation of Lorentz space $L^{p,1}$ | You may consult the book by Grafakos, Classical Fourier Analysis. A detailed study of Lorentz dual spaces is in Theorem 1.4.17, p.52
| 1 | https://mathoverflow.net/users/7294 | 352339 | 148,942 |
https://mathoverflow.net/questions/352299 | -1 | Let $p,q \in \mathbb{C}[x,y]$ be two polynomials such that $p\_xp\_yq\_xq\_y \neq 0$
(namely, each partial derivative is non-zero).
Assume that the following four conditions are satisfied:
**(1)** $\frac{\mathbb{C}[x][y]}{\langle p \rangle}$ is a flat $\mathbb{C}[x]$-module.
**(2)** $\frac{\mathbb{C}[y][x]}{\langle... | https://mathoverflow.net/users/72288 | Flatness of certain quotient rings | Think about the geometry: let $X = V(p)$ and $Y = V(q)$ in $\mathbf A^2$. The nonvanishing of partial derivatives of $p$ and $q$ is equivalent to the statement that $X$ and $Y$ project surjectively onto each coordinate axis.
Since $\mathbf C[x]$ and $\mathbf C[y]$ are principal ideal domains, flatness is equivalent t... | 1 | https://mathoverflow.net/users/82179 | 352346 | 148,945 |
https://mathoverflow.net/questions/352129 | 14 | A chiral algebra on a smooth curve $X$, in the sense of Beilinson-Drinfeld, is a right $D\_{X}$-module with a *chiral bracket*, which is a map $\mathcal{V}^{\boxtimes 2}(\infty\Delta)\rightarrow \Delta\_{\*}\mathcal{V}$ satisfying certain conditions making it look like a Lie bracket. In fact $\mathcal{V}$ is a Lie alge... | https://mathoverflow.net/users/nan | Factorization and vertex algebra cohomology |
>
> If $X$ is taken to a be a formal disc $D$, then we obtain a vertex algebra. I believe in this case the above construction produces the vertex algebra cohomology (with coefficients in the adjoint module) studied in the work of Bakalov, de Sole, Heluani and Kac. Is this true? If not what is the precise relation?
> ... | 8 | https://mathoverflow.net/users/17980 | 352364 | 148,952 |
https://mathoverflow.net/questions/156246 | 15 | The famous Higman embedding theorem says that every recursively presented group embeds in a finitely presented group. This is a convenient tool to construct finitely presented groups with bizarre properties from recursively presented ones, which are usually easier to construct.
One cannot hope for an exact analogue ... | https://mathoverflow.net/users/15934 | Is there a finitely generated residually finite group with solvable word problem that does not embed in a finitely presented residually finite group? | A preprint by E. Rauzy appeared today [on the arXiv](https://arxiv.org/abs/2002.02540), and gives a negative answer to this question. In other words (if the proof is correct), there exists a f.g. residually finite group with decidable word problem which does not embed in a f.p. residually finite group.
| 5 | https://mathoverflow.net/users/120914 | 352374 | 148,954 |
https://mathoverflow.net/questions/352344 | 0 | In [positive set theory](https://www.sciencedirect.com/science/article/pii/B9780444516213500086?via%3Dihub), the axiom scheme of generalized positive comprehension in $GPK^+\_\infty$ [of Olivier Esser] is stated in a manner as to forbid the symbol of the asserted set to occur in the defining formula.
To quote from t... | https://mathoverflow.net/users/95347 | Can we have cyclic generalized positive comprehension? | Yes, there is a clear inconsistency, take the formula:
$\exists A.\ (\forall x.\ (x\in A \leftrightarrow A=\emptyset))$.
Clearly, such a set $A$ cannot exist: if $A\neq \emptyset$, then any $x\in A$ would give a contradiction. If $A=\emptyset$, then this formula would imply that any $x$ satisfies $x\in A$, but then... | 4 | https://mathoverflow.net/users/44629 | 352397 | 148,961 |
https://mathoverflow.net/questions/352393 | 12 | I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds from the differentiable viewpoint; I wonder if a text introducing differential manifolds from an algebraic viewpoint exists... | https://mathoverflow.net/users/104710 | Book on manifolds from a sheaf-theoretic/locally ringed space PoV | I'm not very familiar with this book (in particular, I don't know how introductory or not it is), but I think
>
> Torsten Wedhorn, [Manifolds, Sheaves, and Cohomology](https://www.springer.com/gp/book/9783658106324). Springer Studium Mathematik—Master. Springer Spektrum, Wiesbaden, 2016. xvi+354 pp. ISBN: 978-3-658... | 7 | https://mathoverflow.net/users/130058 | 352402 | 148,965 |
https://mathoverflow.net/questions/352306 | 8 | **Motivation.** If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum\_{k=1}^n \lambda\_k \, h\_k \otimes h\_k,
$$
where $\lambda\_1,\dots,\lambda\_n$ are the eigenvalues of $A$ (counting multiplicities), $(h\_1,\dots,h\_n)$ is a corresponding orthonormal bas... | https://mathoverflow.net/users/102946 | Representation theorem for matrices (reference request) | My comments converted to an answer:
>
> **1st comment.** I know no reference but the proof need not be [as per OP] lengthy — one just computes the two scalars by which the map $\mathscr I:$
> $$
> \textstyle A\mapsto\mathscr I(A)=\int\_S h\langle h,Ah\rangle\langle h,\cdot\rangle d\lambda(h)
> \tag1
> $$
> acts on... | 6 | https://mathoverflow.net/users/19276 | 352407 | 148,969 |
https://mathoverflow.net/questions/352399 | 0 | Let $\prod\_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some concrete examples of:
* Locally-positive Borel probability measures on $(\prod\_{n \in \mathbb{N}} \mathbb{R},B)$,
* Local... | https://mathoverflow.net/users/36886 | Explicit examples of (probability) measures on $\prod \mathbb{R}$ | $\newcommand\R{\mathbb R}$ Let $T$ be any countable nonempty set. Let $B$ be the Borel $\sigma$-algebra over $\R^T$ generated by the Tikhonov product topology on $\R^T$. Take any $t\_0\in T$. For each natural $k$ and each $t\in T$, let
$$\nu\_{k,t}:=
\begin{cases}
N(0,1)&\text{ if } t\ne t\_0,\\
N(k,1)&\text{ if } t=t... | 4 | https://mathoverflow.net/users/36721 | 352408 | 148,970 |
https://mathoverflow.net/questions/352381 | 7 | Let $C$ be a small category and $M$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $Fun(C,M)$ of functors from $C$ to $M$, all with the same (levelwise) weak equivalences. The whole point of having such a model structure is that it should present the $\... | https://mathoverflow.net/users/2362 | When is the model structure on functors correct, i.e. when does localization commute with taking functor categories? | If your model structures are assumed to have small limits or colimits, the answer to the question of the title is: always. For any model category $M$ and any small category $C$, inverting levelwise weak equivalences in $Fun(C,M)$ is equivalent to considering the $\infty$-category of functors from $C$ to $M[W^{-1}]$. Th... | 10 | https://mathoverflow.net/users/1017 | 352411 | 148,973 |
https://mathoverflow.net/questions/352325 | 6 | I am looking at [this recent paper](https://arxiv.org/pdf/1912.09029.pdf) by Budney and Gabai and I am confused by a certain sentence in it. Theorem 4.7 states that if $\Delta\_1$ and $\Delta\_2$ are two 3-balls smoothly embedded in $S^4$ that are identical near there boundary, then there is a diffeomorphism $\Phi: S^4... | https://mathoverflow.net/users/99414 | 3-balls with the same boundary in $S^4$ differ up to diffeomorphism | The cited (early) work by Cerf proves that, given a submanifold Y in a manifold X, the obvious map Diff(X)->Emb(Y,X) is a locally trivial fibration.
I guess that Budney and Gabai mean the following. By Palais, all embeddings D^3->S^4 are isotopic. Hence, for i=0, 1, the complement C\_i of a small
open tubular neighbo... | 5 | https://mathoverflow.net/users/105095 | 352426 | 148,976 |
https://mathoverflow.net/questions/352369 | 7 | We say that $A\subseteq \omega$ is a *nullset* if $$\lim\sup\_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$
Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering relation $\leq^0$ on $\omega^\omega$ by saying that $f\leq^0 g$ if $f(x) \leq g(x)$ for all $x\in\omega\setminus N$ whe... | https://mathoverflow.net/users/8628 | Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets | The answer seems to be positive according to this paper: Barnabás Farkas, Lajos Soukup: *The zero density ideal, cardinal invariants and related forcing problems*.1
>
> **Theorem 2.3.** If $\mathcal I$ is a rare ideal on $\mathbb N$ then $\mathfrak b = \mathfrak b\_{\mathcal I}$ and $\mathfrak d = \mathfrak d\_{\ma... | 5 | https://mathoverflow.net/users/8250 | 352432 | 148,977 |
https://mathoverflow.net/questions/352253 | 7 | A Kodaira fibration is a compact complex surface X endowed with a holomorphic submersion onto a Riemann surface $\pi: X\to\Sigma$ which has connected fibers and is not isotrivial.
Is there an easy way to see why a compact complex surface that admits a Kodaira fibration is Kahler? I know for a complex compact surface ... | https://mathoverflow.net/users/88180 | Compact complex surface that admits a Kodaira fibration is Kahler | Let $f \colon S \longrightarrow B$ be a Kodaira fibration, and let $F$ be a general fibre. Then by [Kas68, Thm. 1.1] we have $g(B) \geq 2$ and $g(F) \geq 3$.
In particular, $S$ contains no rational or elliptic curves: in fact, such curves cannot neither dominate the base (because $g(B) \geq 2$) nor be contained in f... | 3 | https://mathoverflow.net/users/7460 | 352433 | 148,978 |
https://mathoverflow.net/questions/352430 | 3 | I have a question on the Schwartz-Christoffel formula.
The map is a conformal map from the unit disk to polygons. Let me give you a specific example. In fact,
\begin{align\*}
\phi(z)=\int\_{0}^z (1-u^n)^{-2/n}\,du
\end{align\*}
maps $\mathbb{D}$ onto the interior of a regular polygon with $n$ sides.
We can know ... | https://mathoverflow.net/users/68463 | Inverse of the Schwartz-Christoffel map and the continuity | The inverse map satisfies $|\phi^{-1}(z)-\phi^{-1}(a)|\leq c|z-a|^{n/(n-2)}$
at any vertex $a$ (reciprocal to the exponent of the direct map). Since this exponent is $>1$ the inverse is just Lipschitz
everywhere. You do not need any general theorems for this. At the vertex the inverse map behaves like a power. At all o... | 3 | https://mathoverflow.net/users/25510 | 352446 | 148,980 |
https://mathoverflow.net/questions/352429 | 3 | This is a sort of follow-up question to [this old post](https://mathoverflow.net/questions/54923/is-there-a-disjoint-union-sigma-algebra) I came across.
**Setup:**
---
Let $\{X\_n\}\_{n \in \mathbb{N}}$ be a collection of Hausdorff topological spaces and let $\{\Sigma\_n\}\_{n \in \mathbb{N}}$ be their respecti... | https://mathoverflow.net/users/36886 | Disjoint union of measures | 1. As written your definition doesn't quite make sense because $\bigsqcup\_{n \in \mathbb{N}} \Sigma\_n$ doesn't contain nearly enough sets (e.g., $X \mathrel{:=} \bigsqcup\_n X\_n$ itself is not in it). It seems better to me just to set
$\Sigma = \left\{ \bigsqcup\_n A\_n : A\_n \in \Sigma\_n \right\}.$ Equivalently, ... | 2 | https://mathoverflow.net/users/4832 | 352450 | 148,981 |
https://mathoverflow.net/questions/352149 | 2 | Suppose $f\_1\colon K\to [0,\infty)$ and $f\_2\colon K\to[0,\infty)$ are two upper semi-continuous affine functions,
$$
f\_i(\lambda x+(1-\lambda)y)=\lambda f\_i(x)+(1-\lambda)f\_i(y)\ \mbox{ for all }\ x,y\in K\ \mbox{and}\ 0<\lambda<1,
$$
on a compact convex set $K$ in a locally convex Hausdorff space $E$. If $f\_... | https://mathoverflow.net/users/58366 | Inequalities for upper semi-continuous affine functions on compact sets by using extreme points | First, let us assume that $f\_2$ is continuous, and let $h=f\_1-f\_2$, which is an upper semi-continuous affine function, negative on the extreme points of $K$.
Let $c=\sup\_K h$. Since an upper semi-continuous function attains its maximal value on the compact set, $L=h^{-1}(c)$ is non-empty and convex. Since $c$ is ... | 2 | https://mathoverflow.net/users/53155 | 352456 | 148,986 |
https://mathoverflow.net/questions/352390 | 4 | Let $S^+$ be the upper hemisphere of the standard sphere in $\mathbb R^3$ and $b$ -- the boundary of $S^+$ (the equator).
Let $b'$ be a small isometric deformation of $b$ (a nearby curve of the same length). Is there a smooth (or of regularity higher than $C^2$) isometric immersion of $S^+$ which bounds $b'$?
I'm al... | https://mathoverflow.net/users/13842 | Bending the hemisphere | The answer to the infinitesimal version is 'no', which makes it very unlikely that the answer to the isometric deformation version is 'yes'. Here is how one can see this:
One can parametrize the upper hemisphere by the unit disk $x^2+y^2\le 1$ conformally by the well-known formula
$$
F(x,y) = \left(\frac{2x}{1+x^2+y... | 9 | https://mathoverflow.net/users/13972 | 352458 | 148,987 |
https://mathoverflow.net/questions/352476 | 2 | Let $R$ be a commutative ring, $p >0$ prime and $G$ a finite, locally free group scheme over $R$ of rank $p^n$; $n \in \mathbb{N}\_{\ge 1}$. Assume $p \in R^\*$ (i.e. is a unit in $R$).
Question: Why this condition on the rank implies that $G$ is étale?
By definition etale is equivalent to flat & unramified. As $G... | https://mathoverflow.net/users/108274 | Locally free group scheme étale | This is not so easy, but relies on a well-known structure theorem for connected group schemes over a perfect field.
**Lemma 1.** *A finitely presented morphism $Y \to X$ of schemes is unramified if and only if $Y\_x \to x$ is unramified for all $x \in X$.*
*Proof.* See [EGA IV$\_4$, Cor. 17.4.2]. $\square$
Thus, ... | 8 | https://mathoverflow.net/users/82179 | 352478 | 148,991 |
https://mathoverflow.net/questions/352490 | 1 | Let $\gamma : [0,1] \rightarrow \mathcal{M}$ be a continuous map so that $[0,1]$ is homeomorphic to $\gamma([0,1])$, where $\mathcal{M}$ is a manifold (Hausdorff, second countable, and locally Euclidean). Using a chart containing $\gamma(0)$, I think it is always possible to find a circle centered at $\gamma(0)$ that i... | https://mathoverflow.net/users/151406 | Find a circle that intersects the image of $[0,1]$ in a manifold $\mathcal{M}$ at only 1 point | My understanding is that $\operatorname{dim}\mathcal{M}=2$ since we are taking about circles. Also that the precise formulation of the question is:
>
> There is a coordinate system $\phi:U\subset\mathcal{M}\to\mathbb{R}^2$ and $r>0$ such $\gamma(0)\in U$ and the set
> $$
> \phi(\gamma([0,1])\cap U)\cap \underbrac... | 1 | https://mathoverflow.net/users/121665 | 352493 | 148,996 |
https://mathoverflow.net/questions/352443 | 3 | Let $0<a \le 1, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \ln(x)))},$$
is decreasing for $\beta <1$ and increasing for $\beta>1$.
By drawing the graph for some values with mathemat... | https://mathoverflow.net/users/149793 | How to prove monotonicity of such function? | $\newcommand{\Ga}{\Gamma}$The desired result actually holds for all real $a>0$ and, moreover, $f$ is monotonic on the entire interval $(0,\infty)$.
Indeed, let $c:=-\alpha>0$ and $b:=\beta>0$. Note that for any real $u\ne0$ we have
\begin{equation}
u^{-a}(\Ga(a)-\Ga(a,u))=u^{-a}\int\_0^u t^{a-1}e^{-t}\,dt
=\int\_... | 5 | https://mathoverflow.net/users/36721 | 352500 | 149,002 |
https://mathoverflow.net/questions/352504 | 2 | I am stuck on the following problem. I have a discrete distribution $\mu\_0$ (it is actually an empirical distribution). I have some $\mu\_i$ (again discrete, some empirical distribution). I have some bound on the Wasserstein distance $W\_2(\mu\_0, \mu\_i).$ I now want to consider a simple mixture of $\mu\_i,$ that is,... | https://mathoverflow.net/users/69849 | $2$-Wasserstein distance between mixtures | $\newcommand\Ga{\Gamma}$ $\newcommand\ga{\gamma}$ $\newcommand\la{\lambda}$
Let $\Ga(\mu,\rho)$ denote the set of all measures with marginals $\mu$ and $\rho$. For each $i$, take any real $c\_i>W\_2(\mu\_0,\mu\_i)^2$, so that
$$\int d(x,y)^2\ga\_i(dx\times dy)<c\_i$$
for some $\ga\_i\in\Ga(\mu\_0,\mu\_i)$. Let
$$\ga... | 5 | https://mathoverflow.net/users/36721 | 352505 | 149,004 |
https://mathoverflow.net/questions/352057 | 2 | Let $\Omega$ be a bounded domain (smooth if necessary) and let $J:H^1(\Omega) \times H^1\_0(\Omega) \to \mathbb{R}$ be defined by
$$J(u,v) = \int\_\Omega f(u)|\nabla v|^2$$
where $f\colon \mathbb{R} \to \mathbb{R}$ is a smooth function, bounded above and below away from zero, which can make as nice as necessary.
Un... | https://mathoverflow.net/users/137958 | Weak lower semicontinuity of functional with two arguments | This functional is sequentially weakly lower semicontinuous under fairly mild assumptions on $f$.
We need that $f$ is non-negative, continuous and bounded from above.
Let $u\_n \rightharpoonup u$ in $H^1(\Omega)$ and $v\_n \rightharpoonup v$ in $H\_0^1(\Omega)$.
Rellich-Kondrachov implies that $u\_n \to u$ in $L^2(\O... | 2 | https://mathoverflow.net/users/32507 | 352535 | 149,009 |
https://mathoverflow.net/questions/352510 | 4 | I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity' and reducing a system of elliptic PDE down to one PDE, does anyone know of a recent reference which explains this theo... | https://mathoverflow.net/users/119114 | References for systems of elliptic PDEs | In matrix analysis, the **Schur complement** is an object that you obtain after eliminating a part of the unknowns. It works that way: you have to solve $Mx=b$ where $M$ is a square, invertible matrix. You write the system in block form
$$\begin{pmatrix} A & B \\ C & D \end{pmatrix}\binom{y}{z}=\binom{c}{d},$$
where yo... | 5 | https://mathoverflow.net/users/8799 | 352542 | 149,012 |
https://mathoverflow.net/questions/352540 | 10 | I'm going by the maxim
>
> Groups, like men, are known by their actions
>
>
>
This naturally leads one to ask "given groups $G, H$ which act on sets $S, T$ and the semidirect product $G \rtimes H$, how does one *visualize* the action of $G \rtimes H$? What does it act on? Some combination of $S$ and $T$? ($S \... | https://mathoverflow.net/users/123769 | The geometry of the action of the semidirect product | This is a bit of a special example in a couple ways, but maybe it might be useful for building intuition.
Suppose $1 \longrightarrow H \longrightarrow G \longrightarrow \mathbb{Z} \longrightarrow 1$ is a short exact sequence of groups. The algebraic situation you describe at the end of your question is sometimes desc... | 5 | https://mathoverflow.net/users/135175 | 352543 | 149,013 |
https://mathoverflow.net/questions/352552 | 6 | The Section 5 of the book:
**Billingsley, P., Convergence of Probability Measures, 1999**,
studies Prohorov's theorem. A short reminder is given below.
Let $\Pi$ be a family of probability measures on $(S,\mathcal{F})$. We call $\Pi$ *relatively compact* if every sequence of elements of $\Pi$ contains a weakly c... | https://mathoverflow.net/users/117762 | Is the separability of the space needed in the proof of the Prohorov's theorem? | You are correct that separability is not needed. However, there is also not really any loss of generality in assuming it. For suppose that $\Pi$ is tight. Then for every $n$ there exists a compact set $K\_n$ such that $\mu(K\_n) > 1-\frac{1}{n}$ for all $\mu \in \Pi$. So if we set $S\_0 = \bigcup\_{n=1}^\infty K\_n$, t... | 3 | https://mathoverflow.net/users/4832 | 352567 | 149,022 |
https://mathoverflow.net/questions/352517 | 2 | Let $u$ be a bounded function and let a closed set $E$ be given. The compliment of $E$ can be covered with a Whitney type covering $B\_i$ such that the following are satisfied:
1) $E^c \subset \bigcup 4B\_i$
2) $16B\_i \cap E \neq \emptyset$
3) $\sum \chi\_{4B\_i}(x) \leq C(n)$
4) If $r\_i$ is the radius of $B... | https://mathoverflow.net/users/100801 | Joining Hölder continuous functions on Whitney covering | **Your function need not be Hölder continuous.** Let $\Omega$ be the union of two exponential cusps with a common vertex and let $E$ be the complement of these cusps. Let $u=1$ in the upper cusp and $u=0$ in the lower cusp. Because the cusps are "sharp", if $B\_i$ is on one cusp, $8B\_i$ will not intersect the oper cus... | 4 | https://mathoverflow.net/users/121665 | 352573 | 149,023 |
https://mathoverflow.net/questions/352298 | 37 | Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The idea is that every statement that mathematicians care about is equivalent to some question about sets. An example of su... | https://mathoverflow.net/users/4613 | Could groups be used instead of sets as a foundation of mathematics? | The answer is yes, in fact one has a lot better than bi-interpretability, as shown by the corollary at the end. It follows by mixing the comments by Martin Brandenburg and mine (and a few additional details I found on MO). The key observation is the following:
**Theorem:** The category of co-group objects in the cate... | 37 | https://mathoverflow.net/users/22131 | 352577 | 149,025 |
https://mathoverflow.net/questions/352558 | 1 | Is there a non-constant continous function $f:\mathbb{R}\rightarrow \mathbb{R}$ and matrices $A=\begin{pmatrix}
a\_1 & 0\\
0 & a\_2\\
\end{pmatrix}$ and
$B=\begin{pmatrix}
b\_1 & 0\\
0 & b\_2\\
\end{pmatrix}$ for which there does not exist any matrices $C\in \mathrm{Mat}\_{d\times 2},D \in \mathrm{Mat}\_{2\times d}$ a... | https://mathoverflow.net/users/36886 | Non-existent matrices with "essential zeros" | **Yes**, we can find such a triple $(f, A, B)$.
Let us first observe that OP's question can be rephrased as follows.
>
> **Question.** Find a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$
> such that $f$ does not lie in the $\mathbb{R}$-linear span $L(f)$ of $\{ x \mapsto f(cx + c') \}\_{(c, c') \i... | 1 | https://mathoverflow.net/users/84349 | 352581 | 149,026 |
https://mathoverflow.net/questions/352533 | 0 | I am trying to calculate the following function in floating-point arithmetic.
$$f(c,z)=\frac{(c-1)z}{(z-1)^2}\left( \sum\_{k=2}^{c-1}\frac{1}{c-k}\left(\frac{z-1}{z}\right)^k-\left(\frac{z-1}{z}\right)^c\log(1-z)\right)$$
where $z\in(0,1)$ and $c \in \mathbb{N}$, and $c>1$.
The following implementation, which I e... | https://mathoverflow.net/users/76238 | Numerical problems floating-point arithmetic | It is actually pretty simple if you are comfortable with Taylor series (definitely not MO level, so ask on MSE next time). Let $w=\frac{z-1}{z}$. If $|w|<1$, you are in no trouble computing the expression as it is. So let's consider the case $|w|>1$. Then $\frac1{1-z}=1-\frac 1w$, so your expression in parentheses (the... | 5 | https://mathoverflow.net/users/1131 | 352582 | 149,027 |
https://mathoverflow.net/questions/352541 | 2 | Let $\Omega = \left\lbrace x : |x| = r \, \text{for} \, x \in \mathbb{Z}\_2^n \right\rbrace$ for $r \in \mathbb{N}$. We want to find the the biggest subset of $\Omega$, $\Gamma = \left\lbrace x \in \Omega : |x - y| \geq m ,\, \forall x, y \in \Gamma \right\rbrace$ for a given value of $m \in \mathbb{N}$.
| https://mathoverflow.net/users/152265 | Find the largest subset of all binary arrays of length $n$ with $r$ ones which have pairwise distance greater than $m$ | What you are looking for is precisely the optimal (largest cardinality) constant weight (this weight is $r$ in your case) binary code with length $n$ and distance $m$, which is a well-researched and very difficult problem in general.
Let this quantity be denoted $A(n,m,r)$ in your terminology. In fact normally, this ... | 3 | https://mathoverflow.net/users/17773 | 352585 | 149,029 |
https://mathoverflow.net/questions/352570 | 7 | Suppose $\mathfrak{U}$ is a variety of groups. Let's define $F\_n(\mathfrak{U})$ as [relatively free groups in $\mathfrak{U}$](https://en.wikipedia.org/wiki/Free_object).
Suppose $G \in \mathfrak{U}$ is a finitely generated group. We call $G$ finitely presented in $\mathfrak{U}$ iff $\exists n \in \mathbb{N}$ and fin... | https://mathoverflow.net/users/110691 | Does Higman's embedding theorem hold inside group varieties? | Kharlampovich proved that there exist two finitely based varieties of solvable groups $ {\mathfrak A} \subset {\mathfrak B}$ such that the word problem is not solvable in the groups f.p. in the smaller variety but solvable in the groups that are f.p. in the bigger variety (the result can be found in our joint survey "A... | 10 | https://mathoverflow.net/users/nan | 352586 | 149,030 |
https://mathoverflow.net/questions/352579 | 6 | For each $n\in\mathbb{N}$, let:
* $\chi\_n\pmod{q\_n}$ a real non-principal Dirichlet character ($q\_1 < q\_2 < \cdots$),
* $\beta\_n$ the largest real zero of $L(s,\chi\_n)$,
* $\delta\_n := (1-\beta\_n)\log(q\_n)$.
Let $\chi\pmod{q}$ be a Dirichlet character, and consider $s = \sigma + it$ with $|t| < 1$. In p. 2... | https://mathoverflow.net/users/74026 | "Sub-logarithmic" zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem | Let $\chi$ be a non-principal real Dirichlet character modulo $q$. Let
$$\beta\_0=1-\frac{1}{\eta\log q}$$
be a real zero of $L(s,\chi)$ satisfying $\eta\geq 100$ for convenience (Heath-Brown's condition is $\eta\geq 3$). Let $\rho=\beta+i\gamma$ be any zero of $L(s,\chi)$ such that $\rho\neq\beta\_0$ and $|\gamma|\le... | 4 | https://mathoverflow.net/users/11919 | 352591 | 149,031 |
https://mathoverflow.net/questions/352484 | 1 | Consider a random walk $X\_t = \sum\_{s=1}^t D\_s$ with i.i.d. increments $D\_t \in \mathbb{R}^n$,such that $X$ is a martingale $\mathbb{E}[D\_t]=\vec{0} \in \mathbb{R}^n$, the support of $D\_t$ is bounded, and $D\_{t,i}$ has strictly positive variance for all $i \in \{1,\ldots,n\}$.
Is it true that the probability t... | https://mathoverflow.net/users/82510 | Multidimensional random walk falling pointwise below some threshold | An extended comment, answering the question asked by fedja in one of the comments. *(Edited: In order to simplify the example I chose to work in dimension two, but this example requires dimension at least three.)*
My bet was incorrect: I am very surprised to find that the answer is negative! In order to construct a c... | 1 | https://mathoverflow.net/users/108637 | 352610 | 149,038 |
https://mathoverflow.net/questions/352604 | 14 | Let $\mathbb T=\mathbb R/\mathbb Z$ be the circle group and $\mathbb T^\omega$ be the infinite-dimensional torus, considered as an abelian compact topological group.
>
> **Problem 1.** Is it true that for any finite set $F\subset\mathbb T^\omega$ and any neighborhood $U\subseteq \mathbb T^\omega$ of zero there exi... | https://mathoverflow.net/users/61536 | How flexible is the infinite-dimensional torus? | This is a draft proof of an affirmative answer to Problem 3.
**Proposition.** For any $n\in\mathbb N$, $\varepsilon>0$ and vectors $x\_1,\dots,x\_n\in\mathbb R^{2n}$ there exists a linear transformation $A\in SL(2n,\mathbb Z)$ such that $A(\{x\_1,\dots,x\_n\})\subset[-\varepsilon,\varepsilon]^n\times\mathbb R^n$.
... | 4 | https://mathoverflow.net/users/43954 | 352614 | 149,040 |
https://mathoverflow.net/questions/352621 | 0 | Is there any way to find the following limit
$$R(n,m)=\lim\_{N\to\infty}\frac{H\_{nN,m}}{H\_{N,m}}$$
which involves harmonic numbers (generalized if $m\neq 1$)
$$H\_{N,m}=\sum\_{k=1}^N k^{-m}\qquad ?$$
I am more specifically looking for a convenient way to compute it numerically for $m<1$ (if it converges to so... | https://mathoverflow.net/users/111000 | Limit of a ratio of harmonic numbers? | Suppose first that $0\le m<1$. Then, using the inequality $k^{-m}\ge\int\_k^{k+1} x^{-m}\,dx$ for $k>0$, we have
$$H\_{N,m}\ge\int\_1^{N+1}x^{-m}\,dx=\frac{(N+1)^{1-m}-1}{1-m}\sim\frac{N^{1-m}}{1-m}\tag{1}$$
(as $N\to\infty$). Similarly, using the inequality $k^{-m}\le\int\_{k-1}^k x^{-m}\,dx$ for $k>1$, we have
$$H\... | 4 | https://mathoverflow.net/users/36721 | 352625 | 149,042 |
https://mathoverflow.net/questions/352623 | 1 | Let $T:[0, 1]\rightarrow [0, 1]$ be map such that $T(x)=4x(1-x)$. For any $\alpha \in \mathbb{R}$, we define the level set as follows
$$F(\alpha)=\{x\in [0,1]: \lim\_{n\rightarrow \infty}\frac{1}{n}\log |(T^{n})^{'}(x)|=\alpha\}.$$
>
> Is not $\alpha \mapsto h\_{top}(F(\alpha))$ concave? $h\_{top}(.)$ defines in [t... | https://mathoverflow.net/users/127839 | Entropy spectrum is not concave | The only values of $\alpha$ for which $F(\alpha)$ is non-empty are $-\infty$ and $\log 2$. This is because the map is conjugate (by the conjugacy $H\colon x\mapsto \sin^2(\pi x/2)$ to the full tent map $S$ (that is, $T\circ H=H\circ S$).
In particular, $T^n\circ H=H\circ S^n$, so that for any $x$, $|(T^n)'(H(x))|H'(x... | 1 | https://mathoverflow.net/users/11054 | 352627 | 149,043 |
https://mathoverflow.net/questions/352618 | 2 | I've recently read about the notion of (Rademacher) type and cotype of a Banach space in some article. In the books I checked afterwards, typical examples studied were $L^p$-spaces or the Schatten classes but nothing was said about spaces of continuous functions. As these are arguably one of the most important examples... | https://mathoverflow.net/users/49733 | type and cotype of spaces of continuous functions | It is known that $C(K)$, for infinite $K$, contains a copy of $c\_0$, hence it does not have nontrivial type (meaning $>1$) or nontrivial cotype (meaning $<\infty$).
| 5 | https://mathoverflow.net/users/127871 | 352630 | 149,045 |
https://mathoverflow.net/questions/352603 | 10 | Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using [modular forms](https://www.quantamagazine.org/sphere-packing-solved-in-higher-dimensions-20160330/). I understand some sphere pac... | https://mathoverflow.net/users/136553 | Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$? | This is a tough question, and I don’t think there’s a definitive answer yet. For some mathematical details, see the following survey articles:
<https://arxiv.org/abs/1611.01685>
<https://arxiv.org/abs/1603.05202>
Instead, I’ll focus on the big picture here. Why modular forms? I can see a couple of potential answers... | 13 | https://mathoverflow.net/users/4720 | 352634 | 149,046 |
https://mathoverflow.net/questions/352633 | 3 | Let $\phi:\Sigma^2\to M^3$ an conformal isometric immersion into a Riemannian 3-manifold $(M,g)$.
I would like to know what kind of informations is preserved (about the immersion) when we change $g$ by $e^f g$.
For example, since $g$ is in the same conformal class of $e^fg$ then $\phi:\Sigma^2\to M^3$ is also an ... | https://mathoverflow.net/users/74747 | Isometric immersions and metrics in the same conformal class | In general, having constant mean curvature is not preserved. This can be seen by a direct computation, or by looking at examples: The round metric on the 3-sphere without a point is conformal to flat 3-space. The Clifford torus in the 3-sphere (even without a point) is minimal, hence of constant mean curvature, and emb... | 5 | https://mathoverflow.net/users/4572 | 352635 | 149,047 |
https://mathoverflow.net/questions/352595 | 5 | I want to construct a $SU(N)$ matrix $V$, with the following property:
1. All the elements of the first row are given, i.e. $V\_{1,j}=a\_i$ (with $\sum\_i a\_i^2=1$)
2. All matrix elements are real, i.e. $V\_{i,j} \in \mathbb{R}$
>
> How can I find a matrix $V$ that satifies the criteria? Specifically, how can I ... | https://mathoverflow.net/users/63938 | Parametrization of real-valued SU(N) | In addition to the comments I made above about continuous solutions, I thought I'd point out a solution that works for all $n$ with only one point of discontinuity, namely
$$
(a\_1\ a\_2\ \ldots\ a\_n) = (1\ 0\ \ldots\ 0).
$$
Away from this point, one can start with the following formulae:
$$
V\_{i,1}= V\_{1,i} = a\_... | 3 | https://mathoverflow.net/users/13972 | 352642 | 149,049 |
https://mathoverflow.net/questions/352304 | 2 | In a quantum mechanics problem I encountered the following integral
\begin{equation\*}
\int\_0^\infty t^{\nu+1}J\_\nu(\beta t)L\_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,,
\end{equation\*}
where $L$ denotes the [associated Laguerre polynomial](https://en.wikipedia.org/wiki/Laguerre_polynomials), $J$ a [Bessel function of the fi... | https://mathoverflow.net/users/152124 | Integral involving associated Laguerre polynomial and Bessel function | Let $n=\mu - \nu$ be an integer, as specified in the problem. Then I'll indicate how to prove that
$$I:=\int\_0^\infty e^{-a\,t} t^{\nu+1} J\_\nu(b\,t) L\_n^{2v}(t) \,dt =\frac{(2b)^\nu \Gamma(\nu+1/2)}{\sqrt{\pi}}\sum\_{k=0}^n \frac{(-1)^k}{k!}\binom{n+2\nu}{n-k} $$
$$
(a^2+b^2)^{-(\nu+k+3/2)}a^{k+1}{}\_2F\_1(-k/2... | 5 | https://mathoverflow.net/users/121836 | 352644 | 149,050 |
https://mathoverflow.net/questions/352648 | 0 | Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This [Wikipedia article](https://en.wikipedia.org/wiki/Quotient_stack) (or also this related [MO question](https://mathoverflow.net/questions/159279/understanding-the-definition-of-the-quotient-stack-x-g)) defines a quotient stack $... | https://mathoverflow.net/users/143571 | basic question on quotient stacks | The map $D\to X$ induces a map $D/G\to X/G$. Since $D\to T$ is a principal $G$-bundle the induced map $D/G\to T$ is an isomorphism (e.g. by checking locally over $T$ and reduce to the case of the trivial bundle) and so we get a map $T \to X/G$. This is the map they refer to.
For the second, in general the map $X\to ... | 2 | https://mathoverflow.net/users/115052 | 352650 | 149,052 |
https://mathoverflow.net/questions/352649 | 1 | Let $X$ be a Banach space and $f:X\rightarrow \mathbb{R}$ be continuous. Suppose that $\{X\_n\}\_{n \in \mathbb{N}}$ is a strictly nested sequence of sub-Banach spaces, for which $\cup\_{n \in \mathbb{N}} X\_n$ dense in $X$. Then the colimit is an LF-space which is not metrizable.
Since $f|\_{X\_n}$ is continuous to... | https://mathoverflow.net/users/36886 | Continuous function on colimit | I did not understand fully your setting (for example, are the $X\_n$ given the topology of a subspace or simply the embedding map $X\_n \to X$ is required to be continuous... Or is $f$ linear... etc.). But the colimit topology on $X$ will be finer than the original topology on $X$, and therefore the functional $f$, bei... | 2 | https://mathoverflow.net/users/2095 | 352655 | 149,053 |
https://mathoverflow.net/questions/352034 | 6 | This question is motivated by discussion surrounding [this MO question](https://mathoverflow.net/questions/351827/a-new-cardinal-characteristic-related-to-partitions).
An ultrafilter $U$ on $\omega$ is a simple $P\_{\aleph\_2}$-point if it is generated by a sequence $\langle X\_\alpha:\alpha<\omega\_2\rangle$ such th... | https://mathoverflow.net/users/18128 | A question on simple $P_{\aleph_2}$-points | The answer is "yes". Alan Dow pointed out in correspondence that the construction of Blass and Shelah referenced below can be modified to yield such a model. The particular model arises by using finite support to add an $\omega\_1$-sequence of Cohen reals, followed by a finite support iteration adding the generating se... | 3 | https://mathoverflow.net/users/18128 | 352661 | 149,055 |
https://mathoverflow.net/questions/352656 | 1 | Say we have an affine variety $V \subset \mathbb{C}^n$, and suppose we intersect $V$ with a hyperplane $H$, possibly not in general position. Is it possible for the degree of $V \cap H$ to be larger than the degree of $V$?
| https://mathoverflow.net/users/152336 | Can the degree of an affine variety increase after intersecting with a hyperplane? | Yes, it is possible for the degree to increase. Say $V \subset \mathbb{C}^3$ is reducible: a union of a curve of degree $d$ plus one more point that doesn't lie on the curve. Then $V$ has degree $d$. But if $H$ is any plane through that extra point, then the intersection of $V$ with $H$ has degree $d+1$: the one point,... | 4 | https://mathoverflow.net/users/88133 | 352664 | 149,056 |
https://mathoverflow.net/questions/352613 | 0 | **Update:** It is almost sure that the expression of $\kappa$ in coordinates given in the book, namely
>
> $$\kappa(u\_t)=h\_{\alpha\beta}\frac{\partial u\_t^\alpha}{\partial x^i}\frac{\partial u\_t^\beta}{\partial x^j}$$
>
>
>
is a typo. I have edited the question and given my version of this below.
---
... | https://mathoverflow.net/users/143284 | How can I prove this Weitzenbock formula | $\newcommand{\pa}{\partial}$Edit: The answer is now LaTeXified.
I happen to have written notes on this. :)
\begin{aligned}
\frac{\pa\kappa(u\_t)}{\pa t}&=h\_{\alpha\beta}(u\_t)\frac{\pa^2 u\_t^\alpha}{\pa t^2}\frac{\pa u\_t^\beta}{\pa t}\\&=\Big\langle\frac{\pa^2u\_t}{\pa t^2},\frac{\pa u\_t}{\pa t}\Big\rangle.
\en... | 2 | https://mathoverflow.net/users/90076 | 352666 | 149,057 |
https://mathoverflow.net/questions/352643 | 4 | **Background:**
I am reading the book *Variational Problems in Geometry* by Seiki Nishikawa. The main purpose of this book is to prove the existence of harmonic maps $M\to N$ between two compact Riemannian manifolds $M$ and $N$ with the target manifold being nonpositively curved. Since a harmonic map is defined as a ... | https://mathoverflow.net/users/143284 | Possible flaw in the proof of the Eells-Sampson theorem on harmonic maps in Nishikawa's book | $\newcommand{\R}{\mathbb{R}}\newcommand{\pa}{\partial}$Edit: The answer is now LaTeXified.
Below are my notes on this. I reworked the proof:
*Proof*. Let $S:=\big\{T\in[0,\infty):$ the equation has a solution in $C^{2+\alpha,1+\alpha/2}(M\times[0,T],N)\big\}$. Let $T\_0:=\sup S$. By existence of local solution, $T\... | 6 | https://mathoverflow.net/users/90076 | 352667 | 149,058 |
https://mathoverflow.net/questions/352668 | 2 | Cocktail party graphs and $k\_{n,n,n}$ (a tripartite graph) are determined by the spectra of their adjacency matrices? I think thay are DS ( determined by the adjacency spectrum) but I can't find a reference for that. Does anyone know if there is a reference for this?
Thanks to everyone for the help.
| https://mathoverflow.net/users/152342 | Cocktail party and tripartite graphs are DS? | Yes, the cocktail party graphs and $K\_{n,n,n}$ are determined by the spectra of their adjacency matrix. See for example, Proposition 6 of the paper [Which graphs are determined by their spectrum?](https://www.sciencedirect.com/science/article/pii/S002437950300483X) by van Dam and Haemers. Proposition 6 shows that the ... | 4 | https://mathoverflow.net/users/2233 | 352676 | 149,060 |
https://mathoverflow.net/questions/352692 | -3 | Is this proposition established?
Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series
$$p(x)=\sum\_{n=0}^\infty a\_nx^n,$$
$$P(x)=\sum\_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+1+\nu)}a\_nx^{n+\nu}.$$
Suppose that $p'(x),P'(x)$ don't exist. For any $a\in[0,1)$ and any sufficiently small $\delta>0$... | https://mathoverflow.net/users/146060 | A proposition about power series | If $P'(a)=0\neq p'(a)$, then there is no such constant. Indeed, in this situation, we have for sufficiently small $\delta$,
\begin{align\*}
\sup\_{x,y\in[a,a+\delta]}|P(x)-P(y)|&\ \ll\_a\ \delta^2\\
\sup\_{x,y\in[a,a+\delta]}|p(x)-p(y)|\delta^{\nu}&\ \gg\_a\ \delta^{1+\nu}.
\end{align\*}
These bounds follow readily fro... | 1 | https://mathoverflow.net/users/11919 | 352696 | 149,066 |
https://mathoverflow.net/questions/352687 | 7 | A finite-dimensional associative $\mathbf{k}$-algebra $\mathbf{k}Q/I$ is of tame representation type if for each dimension vector $d\geq 0$, with the exception of maybe finitely many dimension vectors $d$ representations**\***, the indecomposable representations of $\mathbf{k}Q/I$ with that dimension vector can be des... | https://mathoverflow.net/users/64073 | What's an illustrative example of a tame algebra? | I think the following is an example of a tame algebra where there is more than one component to a moduli space of fixed dimension. I don't know any examples where there are dimension vectors with moduli of dimension $>1$.
Take a quiver with two vertices $1$ and $2$, two arrows $x\_1$ and $x\_2$ from $1$ to $2$ and tw... | 6 | https://mathoverflow.net/users/297 | 352701 | 149,068 |
https://mathoverflow.net/questions/352678 | 2 | One of the beneficial properties of persistent homology is its stability results (so called robustness to noise).
Usually the referenced paper is [this paper](http://pub.ist.ac.at/~edels/Papers/2010-J-01-LpStablePersistence.pdf)
titled *"Lipschitz functions have $L\_p$-stable persistence"*.
According to the author... | https://mathoverflow.net/users/83274 | Persistent homology stability results (query about Lipschitz functions) | The Lipschitz property (together with a bound on the dimension of the domain, roughly speaking) allows to prove stability of persistence diagrams in the Earth mover distance (aka Wasserstein distance).
The general stability result (for bottleneck distance) does hold even if the function is not Lipschitz.
| 3 | https://mathoverflow.net/users/112954 | 352709 | 149,071 |
https://mathoverflow.net/questions/352702 | 1 | For functions $f, g:\omega\to\omega$ we write $f \leq^\* g$ if $\{x\in\omega: f(x)> g(x)\}$ is finite.
Let $S\_\omega$ denote the collection of bijections $\varphi:\omega\to\omega$ Similarly to the [bounding number](https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum#Bounding_number_'"<code>UNIQ--p... | https://mathoverflow.net/users/8628 | Bounding and dominating numbers for preordering $\leq^*$ on bijections $f:\omega\to\omega$ | Of course $\mathfrak{b}^\mathrm{bij} \geq 2$ and it is not hard to see that $\{id\_\omega,f\}$ is an unbounded family if $f$ is the function that permutes each even number with its successor. So $\mathfrak{b}^\mathrm{bij} =2$.
I suspect $\mathfrak{d}^\mathrm{bij}= \mathfrak{c}$ but I don´t have time to check this.
| 3 | https://mathoverflow.net/users/17836 | 352714 | 149,073 |
https://mathoverflow.net/questions/352682 | 1 | Let $X$ be a very general hypersurface of degree $6$ in $\mathbb{P}^3$. Fix an integer $d$.
Define $Y:= \{ C \in \mathbb{P}(H^0(\mathcal{O}(3))) \text{ such that } \text{dim}(\text{ Hilb}^d(X \cap C)) > d \}$, where $\text{Hilb}^d(X \cap C)$ denotes the Hilbert scheme of zero-dimensional subschemes of length $d$. Note ... | https://mathoverflow.net/users/130022 | complete intersection curves with large Hilbert scheme of points | I believe that in characteristic zero for smooth $X$ we always have $\dim \text{Hilb}^d(X\cap C) = d$.
Sketch of proof: Let $Y = X\cap C$. For a point $p\in Y$ and any $e\in \mathbb{N}$ denote by $\text{Hilb}^e(Y, p) \subset \text{Hilb}^e(Y)$ the locus consisting of subschemes supported only at $p$. Since $X\subset \... | 1 | https://mathoverflow.net/users/98919 | 352716 | 149,075 |
https://mathoverflow.net/questions/352726 | 3 | What is the "correct" definition of a free [augmented commutative algebra](https://en.wikipedia.org/wiki/Augmentation_%28algebra%29)?
At least two definitions come to my mind:
Fix a commutative ring $k$. We need elements $\lambda\_1,\dotsc,\lambda\_n \in k$. They define an augmentation on the polynomial algebra $k[... | https://mathoverflow.net/users/2841 | Free augmented algebras | For any choice of $\lambda\_1,\dots,\lambda\_n$ there is an isomorphism:
$$ k[X\_1^{[\lambda\_1]},\dots,X\_n^{[\lambda\_n]} ] \simeq k[Y\_1^{[0]},\dots,Y\_n^{[0]} ] $$
Which is given by $X\_i \leftrightarrow Y\_i+e\lambda\_i$ where $e$ is the unit.
So the two constructions actually give you the same objects.
| 6 | https://mathoverflow.net/users/22131 | 352728 | 149,076 |
https://mathoverflow.net/questions/352724 | 2 | Consider a discrete state space $\mathcal{X}$. The expander Chernoff inequality gives subgaussian concentration for the sample mean $\frac1n \sum\_{t=1}^n f(X\_t)$ for some function $f : \mathcal{X} \to [0,1]$, where $X\_t$ follows a discrete time stationary Markov chain. The variance parameter of the subgaussian conce... | https://mathoverflow.net/users/148528 | Concentration in Markov chains | Your ``extreme case'' is not a good indication of what happens in a slightly less extreme case. If the chain has a narrow bottleneck between two strongly connected components, for some time it will stay on one side of the bottleneck and you will see the sample mean approach the average of $f$ over this side; but at som... | 4 | https://mathoverflow.net/users/4961 | 352733 | 149,079 |
https://mathoverflow.net/questions/352720 | 17 | I have heard that there exists the following conjecture (if I am not mistaken).
Let $u\_1,\dots,u\_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector $x$ such that
$$\sum\_{i=1}^n |( x,u\_i)|\geq \sqrt{n}.$$
>
> I am looking for a reference for this conjecture.... | https://mathoverflow.net/users/16183 | Reference to a conjecture on unit vectors in Euclidean space | That isn't a conjecture but a routine exercise assigned after the students learn about Bang's solution of the Tarski plank problem. The proof goes in 2 steps:
1) Consider all sums $\sum\_j \varepsilon\_i u\_i$ with $\varepsilon\_i=\pm 1$ and choose the longest one. Replacing some $u\_j$ with $-u\_j$ if necessary, we ... | 28 | https://mathoverflow.net/users/1131 | 352737 | 149,081 |
https://mathoverflow.net/questions/352703 | 14 | I have found, by accident, an identity that relates a sum of Chebyshev polynomials of the first kind to a Chebyshev polynomial of the second kind. It goes as follows:
Given an integer partition of $n$, let $g\_a$ be the number of times $a$ appears in said partition.${}^1$ Then the following identity holds for all $n ... | https://mathoverflow.net/users/144413 | Curious identity between the two kinds of Chebyshev polynomials | Here is how to prove it with more standard methods. First of all, let me
restate your identity:
>
> **Definition.** Let $\mathbb{N}=\left\{ 0,1,2,\ldots\right\} $. A
> *partition* shall mean [an integer
> partition](https://en.wikipedia.org/wiki/Partition_(number_theory)), i.e., a
> weakly decreasing finite list... | 8 | https://mathoverflow.net/users/2530 | 352747 | 149,084 |
https://mathoverflow.net/questions/352749 | 3 | The Diophantine equation
$$x^3 + y^3 + z^3 = 42$$
was recently solved by
Booker and Sutherland:
[Sum of three cubes for 42 finally solved](https://www.eurekalert.org/pub_releases/2019-09/uob-sot090619.php).
Is there a clean partition of the form of those
polynomial equations all of which *do* have integer solutions,
... | https://mathoverflow.net/users/6094 | Which Hilbert's 10th polynomials are known to have solutions? | The negative solution of [Hilbert's tenth problem](https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem) means that there is no algorithm that would recognize the solvable polynomial equations. Hence the "clean partition" you are looking for does not exist.
Of course, one can partition the polynomials any way one ... | 7 | https://mathoverflow.net/users/11919 | 352754 | 149,086 |
https://mathoverflow.net/questions/352760 | 1 | Let $\mathcal{M}^{+}(\mathbb{R}\_{+})$ be space of non-negative [Radon measures](https://en.wikipedia.org/wiki/Radon_measure) on $\mathbb{R}\_{+}$ with bounded total variation and define the metric $\rho$ on $\mathcal{M}^{+} (\mathbb{R}\_{+})$ as $$ \rho(\mu,\nu)= \sup \left \{ \int\_{\mathbb{R}\_{+}} \psi d (\mu - \nu... | https://mathoverflow.net/users/151864 | How to prove space of non-negative Radon measures is complete? | Assuming that the above equivalence is valid, all you have to show that $\lim\_{m,n \to \infty} \rho(\mu\_m,\mu\_n) = 0$ implies uniform tightness of the sequence $(\mu\_n)$. So let $\epsilon > 0$ be arbitrary. Then there is $n\_0 \in \mathbb{N}$ with $\rho(\mu\_m,\mu\_n) \leq \epsilon$ for $m,n \geq n\_0$, in particul... | 1 | https://mathoverflow.net/users/100904 | 352770 | 149,092 |
https://mathoverflow.net/questions/352751 | 2 | I've read books which have this statement, without explanation : 'every finite group is an algebraic group'. I'm trying to understand what exactly they mean. The definition I have in my mind of a group scheme $ G $ over a base field $ k $ is a representable functor $$ G : (\text{Sch}/k)^{op} \rightarrow (\text{Grp}) $$... | https://mathoverflow.net/users/152391 | Viewing a finite group as a group scheme | I don't think that there is a different natural construction of a group scheme associated to a finite group. You can assume that authors mean this construction unless otherwise stated.
Notice, however, that this construction can be also done in the functorial setup, and a little bit more general and elegant as well:
... | 7 | https://mathoverflow.net/users/2841 | 352777 | 149,093 |
https://mathoverflow.net/questions/352590 | 4 | The Mehta integral is the following expression:
$$\frac{1}{(2\pi)^{n/2}}\int\_{-\infty}^{\infty} \cdots \int\_{-\infty}^{\infty} \prod\_{i=1}^n e^{-t\_i^2/2}
\prod\_{1 \le i < j \le n} |t\_i - t\_j |^{2 \gamma} dt\_1 \cdots dt\_n =\prod\_{j=1}^n\frac{\Gamma(1+j\gamma)}{\Gamma(1+\gamma)}.$$
To simplify the notation,... | https://mathoverflow.net/users/150564 | Mehta integral and orthogonality | **Case 3:**
Let me define $t=x\sqrt{2\gamma}$, then it is known from random-matrix theory (see, for example, [Forrester's book](https://press.princeton.edu/books/hardcover/9780691128290/log-gases-and-random-matrices-lms-34)) that for a fixed $\gamma$ the probability distribution $P(x\_1)$ of a single eigenvalue $x\_1... | 2 | https://mathoverflow.net/users/11260 | 352779 | 149,094 |
https://mathoverflow.net/questions/352725 | 2 | I have a possibly elementary question. Let $\mathcal{M}$ be a manifold with $\text{dim} \; \mathcal{M} = 2$. Let $U \subseteq \mathcal{M}$ be homeomorphic to $\overline{\mathcal{B}(0,1)}$, and let $\partial U = U \backslash \text{int} \; U$. Construct the topological space $\mathcal{N}$ by removing $\text{int} \; U$ an... | https://mathoverflow.net/users/151406 | Cut out an open ball from a 2-manifold and glue the boundary | The following is theorem A1 in the paper by David Epstein, "Curves in 2-manifolds and isotopies", Acta Math, 1966.
Theorem. Let $M$ is a surface equipped with a PL structure. Then every topological embedding $f: S^1\to M$ is isotopic to a PL embedding. Moreover, isotopy takes place in an arbitrarily small neighborho... | 3 | https://mathoverflow.net/users/39654 | 352787 | 149,097 |
https://mathoverflow.net/questions/352786 | 6 | Let's say we have an automorphic form $f$ on $GL(2)$ that is self-dual. In particular, the associated L-function $L(s,f)$ satisfies a functional equation with sign $\varepsilon\_F = \pm 1$.
Is it known that the proportion of such automorphic forms with given sign (say $-1$) is exactly $1/2$?
I know many results abo... | https://mathoverflow.net/users/116092 | Distribution of signs of automorphic forms | For simplicity, let's consider the case of holomorphic modular forms over $\mathbb Q$ of squarefree level and trivial nebentypus. Then one knows from
>
> Iwaniec, Henryk; Luo, Wenzhi; Sarnak, Peter. [Low lying zeros of families of $L$-functions.](http://www.numdam.org/item/?id=PMIHES_2000__91__55_0) Publications M... | 8 | https://mathoverflow.net/users/6518 | 352798 | 149,100 |
https://mathoverflow.net/questions/352796 | 7 | Does there exist a smooth nontrivial fiber bundle $p: F \hookrightarrow \mathbb R^n \to B$ such that $F$ and $B$ are connected manifolds with $F$ compact? "Nontrivial" here means the fiber $F$ is not a point.
| https://mathoverflow.net/users/105900 | Foliation of $\mathbb R^n$ by connected compact manifolds | There does not, even if you don’t require the fiber and base to be manifolds (or even connected, just that $F$ is not a single point). See
Borel, Armand; Serre, Jean-Pierre,
Impossibilité de fibrer un espace euclidien par des fibres compactes,
C. R. Acad. Sci. Paris 230 (1950), 2258–2260.
| 17 | https://mathoverflow.net/users/317 | 352800 | 149,101 |
https://mathoverflow.net/questions/352801 | 1 | Suppose we have the following data:
1) A group ring $\mathbb{Z}[G]$, where $G$ is a torsion free group.
2) $M\_{\bullet}$ a bounded (above and below) chain complex of $\mathbb{Z}[G]$-modules such that each $M\_{i}$ is a finitely generated free $\mathbb{Z}[G]$-module.
My question is the following:
If the homology... | https://mathoverflow.net/users/141953 | Faithfully flat modules over a group algebra | No way. Let $G={\mathbb Z}$ so that ${\mathbb Z}[G]={\mathbb Z}[x,x^{-1}]$. Then use the complex
$$\ldots \rightarrow 0 \rightarrow 0 \rightarrow {\mathbb Z}[x,x^{-1}] \xrightarrow{1-x+x^2} {\mathbb Z}[x,x^{-1}] \rightarrow 0 \rightarrow 0 \rightarrow \ldots$$
| 5 | https://mathoverflow.net/users/5301 | 352804 | 149,102 |
https://mathoverflow.net/questions/352809 | 4 | Let $C$ be a small poset category. I'm coming across a category where objects are (certain) functors valued in groups $F\colon C\to\operatorname{Grp}$, and morphisms between two objects $F$ and $G$
are *almost* natural transformations $\eta\colon F \Rightarrow G$.
The difference is that if $f\colon e \to v$ is an arrow... | https://mathoverflow.net/users/135175 | Naturality up to (inner) automorphism? | Assuming the inner automorphisms you get are assumed to satisfy some further coherency conditions, your morphisms should amount to [pseudonatural transformations](https://ncatlab.org/nlab/show/pseudonatural+transformation). This requires thinking of the category of groups as a 2-category.
| 11 | https://mathoverflow.net/users/136562 | 352811 | 149,104 |
https://mathoverflow.net/questions/352799 | 2 | I am looking for references on eigenfunctions with Neumann boundary condition.
In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it have critical points at vertices. But, I could not find the source. I think this is probably a well-known fact. But I am... | https://mathoverflow.net/users/68463 | References for Neumann eigenfunctions | From this recent [paper](https://arxiv.org/abs/1802.01800) I would conclude the statement is false: the second Neumann eigenfunction of an acute triangle has one non-vertex critical point.
This was a [Polymath problem.](http://michaelnielsen.org/polymath1/index.php?title=The_hot_spots_conjecture)
| 3 | https://mathoverflow.net/users/11260 | 352812 | 149,105 |
https://mathoverflow.net/questions/352660 | 0 | Suppose $K$ is a compact subset of $\mathbb R^n$ , $V\_0$ and $V\_1$ the complements of $K$ in $\mathbb R^n$ a and $\mathbb R^n\_\infty$ (one point compactification), respectively. Let $u$ be subharmonic on $V\_0$ and $H$ be the generalized solution of Dirichlet problem of $u$ on $V\_1$. So in particular $H$ is harmoni... | https://mathoverflow.net/users/100746 | Dirichlet problem for a subharmonic function | The examples given in Helms's book already answer your question in the negative: if $K$ is the unit ball and we prescribe zero boundary values, then we have $$H(x) = 0,$$ but we can have $$u(x) = c (1 - |x|^{2 - n})$$ for any $c \in \mathbb{R}$. (If $n = 2$, set $u(x) = c \log |x|$ instead.)
| 1 | https://mathoverflow.net/users/108637 | 352816 | 149,108 |
https://mathoverflow.net/questions/352837 | 6 | I have asked these questions as comments [here](https://mathoverflow.net/questions/352105/a-sufficient-condition-for-a-set-of-primes-to-be-the-set-of-reducibility-of-an-i/352109#352109) (these are related to the question there). The questions are: Let $S$ be one of the following sets of primes:
1. All primes of the f... | https://mathoverflow.net/users/nan | Primes mod 4 and integer polynomials | Here is a way to argue without showing directly that the polynomial must have degree $2$. It was explained to me by Borys Kadets (all further mistakes are, of course, my contribution).
**Lemma.** If a set of primes $S$ of density $\frac{1}{2}$ admits such polynomial then some subset $S'\subset S$ with $\#(S\setminus ... | 5 | https://mathoverflow.net/users/39304 | 352839 | 149,118 |
https://mathoverflow.net/questions/352792 | 0 | Let $H\_1$ and $H\_2$ be two planes in $\mathbb{P}^3$.Let $P$ be a set of $9$ points such that no three lie on a line. Suppose $H\_1$ contains 4 of them and $H\_2$ contains remaining $5$ points. Is it true that $P$ imposes independent conditions on quadrics ?
| https://mathoverflow.net/users/130022 | Independent conditions imposed by points in different planes | Let $C'' \subset H\_2$ be the unique conic on $H\_2$ containing the 5 points and let
$$
\{C'\_t\}\_{t \in \mathbb{P}^1} \subset H\_1
$$
be the pencil of conics containing the 4 points. Let
$$
L = H\_1 \cap H\_2.
$$
The points impose independent conditions on quadrics if and only if there is no $t$ such that
$$
C'\_t... | 1 | https://mathoverflow.net/users/4428 | 352849 | 149,119 |
https://mathoverflow.net/questions/352802 | 1 | $\DeclareMathOperator\Gr{Gr}$ Let $\Gr(k,n)$ be the set of $k$-dimensional subspaces in affine space $\mathbb{A}^n$ over an algebraically closed field. If $U\subseteq (\mathbb{A}^n)^{\times k}$ is an open subset of $n \times k$ matrices of rank $k$, is $\{\text{Im}(u): u \in U\} \subseteq \Gr(k,n)$ open?
Here is my c... | https://mathoverflow.net/users/150898 | Is the set of images of an open subset of full-rank matrices an open subset of the Grassmannian? | The map $U \to \mathbb{G}(k,n)$ that you are interested in is flat, and a flat finitely presented map is well known to be open.
Flatness can be checked in several ways: for example, the map is generically flat, by the generic flatness theorem. But it is also $\mathrm{GL}\_n$-equivariant, and the action of $\mathrm{GL... | 1 | https://mathoverflow.net/users/4790 | 352850 | 149,120 |
https://mathoverflow.net/questions/352845 | 0 | Suppose you replace the usual success conditions for a supermartingale (lim sup is infinite) with the requirement that the actual limit is infinite, e.g. a supermartingale $B$ succeeds on $X \in 2^\omega$ just if
$$\lim\_{n \to \infty} B(X\restriction\_n) = \infty$$
I'm 90% sure that you don't get a 'valid' randomn... | https://mathoverflow.net/users/23648 | Kurtz randomness and supermartingales with infinite *limit* | Assuming you mean a c.e. (super)martingale, this is just Martin-Löf randomness. What needs to be shown is that, for every non-random real, there is a martingale which limit-succeeds on the real. You can see this from the proof of the equivalence of the martingale definition and the Kolmogorov complexity definition:
L... | 1 | https://mathoverflow.net/users/32178 | 352857 | 149,121 |
https://mathoverflow.net/questions/352854 | 1 | Riemann and Slaman have some great work classifying what reals are 1-random with respect to a measure $\mu$ *relative* to $\mu$. In that paper they cite Levin and Kautz (but not to refs I can find) for the claim that if $X$ is 1-random with respect to a computable measure $\mu$ then $X$ is of 1-random degree. They expl... | https://mathoverflow.net/users/23648 | Non-relativized, Computable and Schnor randomness w.r.t a measure | 1. It's theorem 6.12.9 in Downey & Hirschfeldt. The answer should be yes, with the caveat that it won't hold if $X$ is an atom of $\mu$.
2. Levin proved the existence of a $\mu$ such that every real passes all $\mu$-c.e. ML tests, which is stronger than what you're requiring. For what you're asking, I believe we can us... | 1 | https://mathoverflow.net/users/32178 | 352860 | 149,122 |
https://mathoverflow.net/questions/352853 | 6 | Fix a parameter $\alpha\in(0,1)$ and take an i.i.d. sequence $X\_0,X\_1,\ldots$ of $\mathbb{R}^n$ valued random variables. Construct the limiting random variable
$X\_\infty = (1-\alpha)\sum\_{k=0}^\infty \alpha^k X\_k.$
Is any general result known about this kind of limit? What if the $X\_i$ follow a well known dis... | https://mathoverflow.net/users/58787 | The "Chaos Game" as a particular series of i.i.d. random variables | $\newcommand\al{\alpha}$Let us drop the factor $1-\al$, by considering
$$Y:=X\_\infty/(1-\al)=\sum\_{k=0}^\infty\al^k X\_k.$$
By [Kolmogorov's three-series theorem](https://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem), this series will converge almost surely (a.s.) unless at least one of the tails of the... | 10 | https://mathoverflow.net/users/36721 | 352864 | 149,124 |
https://mathoverflow.net/questions/352844 | 8 | $\require{AMScd}$I am in the following situation: the diagram
$$
\begin{CD}
\cal M @>r>> [{\cal B},Set] \\
@VuVV @VVf^\*V \\
\cal D @>>N\_g> [{\cal A}^\text{op},Set]
\end{CD}
$$
is a (strict) pullback in $\bf Cat$; moreover, $f : \cal A^\text{op}\to B$ is bijective on objects (and $f^\*$ is the "inverse image" functor ... | https://mathoverflow.net/users/7952 | Pulling back a functor, it becomes monadic | If $A$ and $B$ are small and $D$ is locally presentable then $u$ is a monadic right adjoint. More generally, what is non-trivial is the construction of a left adjoint of $u$. As soon as $u$ has a left adjoint, $u$ is monadic.
The proof is relatively simple: the forgetful functor $f^\*$ is monadic, so it satisfies all... | 10 | https://mathoverflow.net/users/22131 | 352866 | 149,125 |
https://mathoverflow.net/questions/352863 | 1 | I'm curious about the following argument in Morrey ("Multiple integrals in the calculus of variations", Lemma 2.3.1). Suppose $f\in L^1[0,1]$ and $$\int\_0^1 fg\,dx=0$$ for every test function $g\in C^\infty\_c[0,1]$. Then, "by approximations," the same is true if $g$ is merely bounded and measurable, in which case we ... | https://mathoverflow.net/users/90154 | Proof of the du Bois-Reymond lemma "by approximation" | One way to run this argument is to note that you can approximate a bounded measurable function by $C^\infty\_c$ functions, almost everywhere and boundedly. That is, given a bounded measurable $h$, you can find a uniformly bounded sequence $g\_n \in C^\infty\_c([0,1])$ with $g\_n \to h$ almost everywhere.
There are m... | 5 | https://mathoverflow.net/users/4832 | 352869 | 149,127 |
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