parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/434322 | 6 | $\DeclareMathOperator\Hom{Hom}$Let $X$ be a condensed set in the sense of Clausen-Scholze. If there is a universal anima $Y$ (or $\infty$ groupoid, or homotopy type) together with a map of condensed anima $X \to \pi^\*Y$ that induces an equivalence $$\Hom(Y,Z) \to^\sim \Hom(X,\pi^\* Z)$$ for every anima $Z$, we say tha... | https://mathoverflow.net/users/494541 | Domain of left adjoint from condensed sets to anima | Great question!
The answer is Yes. Let me elaborate a little. The question is more generally about the left adjoint to the inclusion $\mathrm{An}\to \mathrm{CondAn}$ from anima to condensed anima. This left adjoint is only partially defined; what exists in general is the functor $F: \mathrm{CondAn}\to \mathrm{Pro}(\m... | 5 | https://mathoverflow.net/users/6074 | 434418 | 175,679 |
https://mathoverflow.net/questions/434420 | 0 | Given a presheaf, in Angelo Vistoli's 2007 [Notes on Grothendieck topologies,
fibered categories
and descent theory](http://homepage.sns.it/vistoli/descent.pdf) there is a construction of the sheafification (Proof for theorem 2.64).
>
> **Note:** In this context, a Grothendieck topology is a Singleton Grothendieck ... | https://mathoverflow.net/users/485069 | Does the (Vistoli-)sheafification induce isomorphism? | You cannot prove this since it is not true. The map $F \to F^+$ is an isomorphism if and only if $F$ is already a sheaf. (Also notice that Vistoli, of course, does not claim that $F \to F^+$ is an isomorphism. In that case there would also be no need to construct $F^+$ in the first place.)
| 2 | https://mathoverflow.net/users/2841 | 434428 | 175,682 |
https://mathoverflow.net/questions/434410 | 6 | Is there a prime $p$ and a field $k$, not real closed, with $k^\times$ not $p$-divisible, such that there exists a finite extension $l/k$ such that $l^\times$ is $p$-divisible?
This question came up since I have proved a result in which I have the hypothesis that $l^\times$ is not $p$-divisible for any finite extension... | https://mathoverflow.net/users/127489 | For which fields $k$ with $k^\times$ not $p$-divisible, does there exist finite $l/k$ such that $l^\times$ is $p$-divisible? | For odd $p$ there are no such fields: If $a\in k$ is not a $p$-th power, then for every $n$ the polynomial $X^{p^n}-a\in k[X]$ is irreducible (see for example Theorem 9.1 in Chapter VI of Lang's Algebra), hence if $l/k$ is any finite extension of degree $d$, and $p^n>d$, then this $a$ has no $p^n$-th root in $l$.
For... | 6 | https://mathoverflow.net/users/50351 | 434439 | 175,685 |
https://mathoverflow.net/questions/434441 | 5 | My question is more or less related to basic set theory. But I don't know even that. Apologies if I added the wrong tags.
>
> **Motivation:** How many non-compact (planar) surfaces are there upto homeomorphism?
>
>
>
>
> **Question:** How many pairwise non-homeomorphic non-empty **closed** subsets of the Can... | https://mathoverflow.net/users/363264 | How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there? | There are $2^{\aleph\_0}$ different subsets of the Cantor set up to homeomorphism.
There can't be more than $2^{\aleph\_0}$ of them because any subset of the Cantor set is separable. To construct $2^{\aleph\_0}$ of them, consider first the ordinal spaces $A\_n=\omega^n+1$, with their order topologies. All of them [ca... | 6 | https://mathoverflow.net/users/172802 | 434443 | 175,686 |
https://mathoverflow.net/questions/434437 | 1 | Let $Y$ be a complemented subspace in a dual Banach space $X$. *Is it true that $Y$ is itself isomorphic to a dual?*
This is the case of a $w^\*$-closed subspace $Y$, but a complemented subspace of $X^\*$ need not be $w^\*$-closed (for instance $Z^\*\subset Z^{\*\*\*}$ is complemented but never $w^\*$-closed unless $... | https://mathoverflow.net/users/6101 | Complemented subspaces in a dual Banach space | $L^1$ is complemented in the measure space $M([0,1])$, $L^1$ is not a dual space.
| 4 | https://mathoverflow.net/users/164350 | 434444 | 175,687 |
https://mathoverflow.net/questions/434408 | 0 | I'm trying to find a mapping $f$ from the 2D real projective plane to $\mathbb{R}^3$ which
1. is smooth
2. has non-singular directional derivative. That is,
$\forall x, v, \quad v \ne 0 \implies D\_v f(x) \ne 0$.
This is different from an embedding, which is impossible because the surface would have to intersect ... | https://mathoverflow.net/users/494626 | Smooth mapping from $\mathrm{RP}^2$ to $\mathbb{R}^3$ with nonsingular derivative | This is a famous problem that was solved by a doctoral student of David Hilbert named Werner Boy in 1901. The kind of mapping you are looking for is called an "immersion" (of its domain — the real projective plane — into 3-space).
The surface Boy discovered is now called "Boy's surface" and there are plenty of refere... | 4 | https://mathoverflow.net/users/5484 | 434447 | 175,688 |
https://mathoverflow.net/questions/434424 | 0 | This question is on the same topic of [this one](https://mathoverflow.net/q/434157/136218), but simpler, and I have also included some numerical tests here.
Consider a $h \times (n-1-h)$ matrix $A$ with all entries $a\_{ij}$, $1 \le i \le h$, $1 \le j \le n-1-h$, equal to $0$ or $1$. We know that each row has $\lfloo... | https://mathoverflow.net/users/136218 | Number of couples of columns "connecting" top to bottom of a matrix | We can view (unordered) pairs of zeros in each row as covering the pairs of columns (and thus eliminating them from being counted by $c$). Since we want to minimize $c$, the more pairs are covered the better. Thus, we naturally arrive at partial $(n-1-h,\lfloor n/2\rfloor - h,2)$-[covering designs](https://www.dmgordon... | 1 | https://mathoverflow.net/users/7076 | 434465 | 175,693 |
https://mathoverflow.net/questions/434157 | 1 | (See also this similar [question](https://mathoverflow.net/q/434424/136218)).
Consider a $h \times 4n-h$ binary matrix (a matrix with all entries $a\_{ij}$, $1 \le i \le h$, $1 \le j \le 4n-h$, equal to $0$ or $1$). We know that each row has $2n$ entries equal to $1$ and $2n-h$ entries equal to $0$. All rows are diff... | https://mathoverflow.net/users/136218 | Number of sets of columns "connecting" top to bottom of a matrix | Quite similarly to [my answer](https://mathoverflow.net/q/434465) to your other question, we have
$$c(h,n) \geq \binom{4n-h}m - h\binom{2n-h}m.$$
I'm not sure how good is this bound.
| 1 | https://mathoverflow.net/users/7076 | 434467 | 175,694 |
https://mathoverflow.net/questions/434459 | 2 | Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion points of $E$. Then if $m=p\_1^{r\_1}\cdots p\_k^{r\_k}$ where $p\_i$, $i=1,2,\cdots,k$ are prime numbers, can we say t... | https://mathoverflow.net/users/483436 | Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups | It is not in general true that
$$ {\rm Gal}(K(E[m])/K) \cong \prod\_{i=1}^{k} {\rm Gal}(K(E[p\_{i}^{r\_{i}}])/K)$$
for elliptic curves $E/\mathbb{Q}$ which have CM by an order in $K$. One reason for this is the typical reason: the square root of the discriminant of $E$ is contained in $K(E[2])$, and the square root of ... | 7 | https://mathoverflow.net/users/48142 | 434469 | 175,695 |
https://mathoverflow.net/questions/434446 | 0 | Consider a $n\times n$ GOE random matrix. If we assume that $|\lambda\_1|>|\lambda\_2|\ge \dots \ge |\lambda\_n|$, can we get the order of $｜\lambda\_1｜/｜\lambda\_2｜$ or even $\lambda\_1/\lambda\_2$?
Any reference are appreciate!
---
If we change the condition that $\lambda\_1>\lambda\_2\ge \dots \ge \lambda\_n... | https://mathoverflow.net/users/168083 | The ratio of spectral edge of the GOE matrix | $|\lambda\_1|=2+\delta\_1$, $|\lambda\_2|=2+\delta\_2$, with $\delta\_1$ and $\delta\_2$ both of order $n^{-2/3}$, so
$$|\lambda\_1|/|\lambda\_2|=1+(\delta\_1-\delta\_2)/2+{\cal O}(n^{-4/3})=1+{\cal O}(n^{-2/3}).$$
Since you do not know the sign of $\lambda\_1$ and $\lambda\_2$, the ratio $\lambda\_1/\lambda\_2$ with... | 1 | https://mathoverflow.net/users/11260 | 434475 | 175,696 |
https://mathoverflow.net/questions/434453 | 4 | Let $G$ be a $p$-adic reductive group and $\pi$ an irreducible non-supercuspidal representation. Then there exist a parbaolic subgroup $P=MN$ and a supercuspidal representation of $M$ such that $\pi$ appears as a subrepresentation of $\operatorname{Ind}\_P^G\sigma$, namely $\sigma$ is the supercuspidal support of $\pi$... | https://mathoverflow.net/users/32746 | On a theorem of Bernstein-Zelevinsky regarding supercuspidal resentations | Higher multiplicities can occur. See Keys, *L-indistinguishability and R-groups for quasisplit groups: unitary groups in even dimension*, Ann. Sci. ENS, 4th series, vol 20, no. 1, 1987, pp. 31-64.
Given a minimal parabolic subgroup $B=TU$ and a unitary character $\lambda$ of $T$, the multiplicities of the components ... | 3 | https://mathoverflow.net/users/4494 | 434479 | 175,697 |
https://mathoverflow.net/questions/434391 | 2 | Let $\mathbb{B}\_1(0)\subseteq\mathbb{R}^n$ be the ball of radius $1$ in the Euclidean space, $n>1$. Suppose we have a cylinder $C=[0,1]\times \mathbb{B}\_1(0)$ and suppose we are given smooth functions
* $\rho\_0\colon [0,\varepsilon)\times \mathbb{B}\_1(0)\to\mathbb{R}$;
* $\rho\_1\colon (1-\varepsilon,1]\times \ma... | https://mathoverflow.net/users/494613 | Smooth extension of functions at corners | $\newcommand\ep\varepsilon$Take any $\ep\in(0,1/2)$. Let
$$A:=A\_0\cup A\_1\cup F,$$
where $A\_0:=[0,\ep)\times B$, $A\_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball (you did not say if your unit ball is closed or open; let us assume that it is closed).
You have a well-defined function $f$ on... | 0 | https://mathoverflow.net/users/36721 | 434483 | 175,699 |
https://mathoverflow.net/questions/434493 | 1 | Let $(\mu\_n)\_{n \in \mathbb{N}}$ be a sequence of a measures. We know, by the Portmanteau Theorem, that:
$$\int f d \mu\_n \to \int f d\mu, \quad \forall \, f \in C\_b \hbox{(class of continuous and bounded function)}$$ is equivalent to $\mu\_n(E) \to \mu(E)$, for all $E \in \mathcal{C}\_\mu$, class of continuity set... | https://mathoverflow.net/users/479236 | Show that a certain convergence of measures is equivalent to a certain convergence of integrals | It seems that this can be *deduced* from the Portmanteau theorem: Assume the convergence of the intergals $\int fd\mu\_n$ for all $f\in C\_b$ vanishing in a neighbourhood of $0$ and fix $E\in C\_\mu$ with $0\notin \overline E$. You may then choose a continuous function $g$ with values in $[0,1]$ which is $1$ in a neigh... | 4 | https://mathoverflow.net/users/21051 | 434496 | 175,701 |
https://mathoverflow.net/questions/433635 | 5 | Consider an elementary class, $K$, of some $\mathcal{L}$-theory, $T$ equipped with the usual $\mathcal{L}$-structure homomorphisms. (Not elementary embeddings, which elementary classes are more frequently equipped with.) Suppose we have $K' \subseteq K$, the elementary class of models of $T'$. When is $K'$ a [reflectiv... | https://mathoverflow.net/users/39865 | When is an elementary subclass reflective? | This is a great question, which I will only partially address. A complete, general, answer to the question goes beyond the energy I am happy to put into MathOverflow.
**Def**. Let $T \subset T'$ be first order theories (having models with at least two elements and quotients of definable equivalence relations), so tha... | 2 | https://mathoverflow.net/users/104432 | 434504 | 175,704 |
https://mathoverflow.net/questions/434438 | 1 | Let $M$ be a continuous martingale. Denote by $E$ the event that its total quadratic variation is finite, i.e.
$$E := \{\langle M, M \rangle\_\infty < \infty\}.$$
**Question:** Is it true that as $t \to \infty$, $M\_t$ converges almost surely on $E$?
| https://mathoverflow.net/users/173490 | Does a continuous martingale converge almost surely on the event that its quadratic variation is finite? | It is true even for local Martingales, see Proposition 1.26 page 124 in [1].
Here is an intuitive way to understand it:
The proof of the Dambis-Dubins-Schwarz theorem [2, 3] (see also [1,4]) implies that (on an enlarged probability space) we can write $M\_t=B\_{\langle M, M \rangle\_t}$ for some Brownian motion $B$, ... | 5 | https://mathoverflow.net/users/7691 | 434505 | 175,705 |
https://mathoverflow.net/questions/433997 | 11 | Consider an $O(N)$ invariant quadratic equation
$$
T\_{ijkl}= T\_{ijmn}T\_{klmn}+ T\_{ikmn}T\_{jlmn}+ T\_{ilmn}T\_{jkmn},
$$
where $T\_{ijkl}$ is a real, totally symmetric 4-tensor, and the indices run from 1 to $N$.
[Although this is not important for what follows, this equation appears in theoretical physics where ... | https://mathoverflow.net/users/38654 | A quadratic $O(N)$ invariant equation for 4-index tensors | Well, this is not actually an answer to either of the OP's questions; at most, it provides an easier way to classify the solutions for the $n=3$ case, and that *might* point a way towards an analysis for larger $n$, but I don't make any guarantees.
Much that I'll say to begin with works for all $n$, so I'll start wit... | 13 | https://mathoverflow.net/users/13972 | 434507 | 175,706 |
https://mathoverflow.net/questions/434511 | 5 | It was shown in
P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short
vectors in a lattice
that the construction of a shortest nonzero vector of a Euclidean lattice w.r.t. the $L^{\infty}$-norm is NP-hard.
But for the $L^2$ norm, is this question still open? Can anyone expl... | https://mathoverflow.net/users/489992 | Is it still not known whether the construction of shortest nonzero vector of a lattice w.r.t. $l^2$-norm is NP-hard? | The NP-hardness of the shortest vector problem in $L\_2$ norm is discussed in this 2015 [lecture](https://people.csail.mit.edu/vinodv/6876-Fall2015/L10.pdf) by Vinod Vaikuntanathan. An algorithm for this problem would give a *randomised* algorithm for any problem in NP.
The original reference is [The Shortest Vector ... | 10 | https://mathoverflow.net/users/11260 | 434513 | 175,707 |
https://mathoverflow.net/questions/434477 | 2 | For a given integer $n$, I am interested in the number of different numerical semigroups one can make with a generating set consisting only of integers in $[n]$.
I have done some small examples. For $n=1$ there is only $\langle 1\rangle$ and the same goes for $n=2$. For $n=3$ there are only $\langle 1\rangle$ and $\l... | https://mathoverflow.net/users/168142 | Counting numerical semigroups by largest element of minimal generating set | A key observation is that two sets of generators $g, g'\subseteq [n]$ produce the same semigroup if and only if $\langle g\rangle \cap [n] = \langle g'\rangle \cap [n]$. Hence, the number of different semigroups equals the number of different $\langle g\rangle \cap [n]$ for $g\subseteq [n]$ (and this is what is compute... | 3 | https://mathoverflow.net/users/7076 | 434514 | 175,708 |
https://mathoverflow.net/questions/434506 | 0 | I want to ask what advantage of using vorticity equations in fluid dynamics.
Does it help to find large curls? Does it have singularities connected to presence of curls?
| https://mathoverflow.net/users/493428 | Vorticity equation for incompressible 2D fluid dynamics | The vorticity equation for the Euler equation in 3D is, with $\omega=\text{curl } v$,
$$
\dot\omega + (v\cdot\nabla)\omega-(\omega\cdot\nabla)v=0,
$$
so that if $v$ is two-dimensional, i.e.
$
v=\begin{pmatrix}v\_1(x\_1, x\_2)\\
v\_2(x\_1, x\_2)\\
0\end{pmatrix}
$
you get that
$$
\text{curl } v=\begin{pmatrix}0\\
0\\
\p... | 4 | https://mathoverflow.net/users/21907 | 434516 | 175,709 |
https://mathoverflow.net/questions/434515 | 7 | Suppose $X$ is a normed linear space. If for every Banach space $Y$ and for every linear operator $T:X\to Y$, graph of $T$ is closed implies $T$ is continuous, then can we prove that $X$ is a Banach space?
| https://mathoverflow.net/users/41137 | Converse of closed graph theorem | No. The closed graph theorem in this form is equivalent to $X$ being a barreled space. See item 15 [here.](https://en.wikipedia.org/wiki/Barrelled_space#Characterizations_of_barreled_spaces)
There are incomplete normed spaces that are barreled. See [here.](https://link.springer.com/article/10.1007/BF03322742)
| 14 | https://mathoverflow.net/users/48839 | 434520 | 175,711 |
https://mathoverflow.net/questions/434527 | 1 | Let $M$ be a real square matrix of order $n\ge 3$.
Assume that for every *nonnegative* vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$.
Can we deduce that $\textbf{1}^T M \textbf{1}\ge 0$, where $\textbf{1}$ is the all one vector?
| https://mathoverflow.net/users/139975 | A question about the sign of quadratic forms on nonnegative vectors | I think $$M=\pmatrix{5&-3&-3\cr-3&5&-3\cr-3&-3&5\cr}$$ is a counterexample.
If $z=(a,b,0)$, then $z^tMz=5a^2-6ab+5b^2\ge0$ for all $a,b$ (and, by symmetry, the same should be true for $(a,0,b)$ and $(0,a,b)$), but if $z=(1,1,1)$, then $z^tMz=-3<0$.
| 1 | https://mathoverflow.net/users/3684 | 434528 | 175,714 |
https://mathoverflow.net/questions/434539 | 2 | Let $G$ be a $p$-adic reductive group, and $P=MN\subseteq G$ a parabolic subgroup. How do you know that the space of the induced representation $\operatorname{Ind}\_P^G\pi$ is non-zero? Namey, how do you know there even exists a nonzero map $f: G\to V\_\pi$ satisfying the defining property for $f\in\operatorname{Ind}\_... | https://mathoverflow.net/users/32746 | How do you construct elements in $\operatorname{Ind}_P^G\pi$? | To have it written explicitly, $\pi$ is a smooth representation of $M$, extended trivially across $N$ to $P$.
Choose an open subgroup $K\_M$ of $M$ that is so small that $\pi^{K\_M} \ne 0$, a $K\_M$-fixed vector $v \ne 0$, and a compact, open subgroup $K$ of $G$ such that $K \cap P$ is contained in $K\_M N$, and defi... | 4 | https://mathoverflow.net/users/2383 | 434541 | 175,718 |
https://mathoverflow.net/questions/434116 | 2 | Let $\mathcal{O}$ be a bounded open subset of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(\mathcal{O},\mathbb{R}^n)$. For $x\_0\in\mathcal{O}$, let $\big(x(t)\big)\_{t\geq 0}$ be the solution of
\begin{align\*}
& x(0)=x\_0 \\
& \dot{x}=v(x).
\end{align\*}
Let $V\subset\mathcal{O}$ a submanifold of dimension $... | https://mathoverflow.net/users/159940 | Continuity of a reaching time of a submanifold | If $V$ is closed in $\mathcal{O}$, then $\tau^V$ is continuous at $x\_0$. If $V$ is not closed, then it is not hard to find counterexamples (e.g. imagine $\mathcal{O}=\mathbb{R}^3$, $V$ is an open disk and $x(t)$ passes through the boundary of the disk and later intersects the disk).
To begin with, let $t\_0=\tau^V(x... | 1 | https://mathoverflow.net/users/172802 | 434544 | 175,720 |
https://mathoverflow.net/questions/434180 | 0 | Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup\_n \|X^n\|\_\alpha<\infty$. Define the standard Riemann-Stieltjes integrals $\mathbb X\_{s,t}^n=\int\_s^t (X^n(r)-X^n(s) )\otimes dX^n(r)$. Then is it true that $\sup\_n\|\m... | https://mathoverflow.net/users/479223 | Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms are uniformly bounded | This cannot hold, and in a sense rough path theory has to be developed precisely because of this reason; otherwise, rough path lifts would be defined uniquely for any curve of Hölder regularity $>1/3$.
For a simple example, take $Y\_n:t\mapsto n^{-1}\exp(in^2t)$, which converges to zero in every $(1/2-\varepsilon)$-H... | 2 | https://mathoverflow.net/users/129074 | 434548 | 175,721 |
https://mathoverflow.net/questions/433886 | 3 | Let $U$ be a bounded domain of $\mathbb{R}^d$, and write $m$ for the Lebesgue measure on $U$. For $k=1,2$, we denote by $H^k(U)$ the set of all locally $m$-integrable functions $u\colon U \to \mathbb{R}$
such that for any multi-index $\alpha$ with $|\alpha|\le k$, the weak derivative $D^\alpha u$ exists and belongs to ... | https://mathoverflow.net/users/68463 | On the domain of the Neumann Laplacian | This is a partial (positive) answer for the convex case only but not every detail has been worked out.
Let first $U$ be convex and smooth and all functions be in $C^3$ up to the boundary. Integrating by parts one obtains
$$
\int\_U |\Delta u|^2=\int\_U |D^2 u|^2+\int\_{\partial U}\sum\_{i,j}(D\_{jj}uD\_i u-D\_{ij}uD\_j... | 1 | https://mathoverflow.net/users/150653 | 434552 | 175,722 |
https://mathoverflow.net/questions/434556 | 5 | Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?
| https://mathoverflow.net/users/1441 | Stable torus that is not a torus | Suppose $M\times S^1$ is homeomorphic to $T^{n+1}$. Then $\pi\_1(M\times S^1) \cong \pi\_1(T^{n+1})$, so $\pi\_1(M)\oplus\mathbb{Z} \cong \mathbb{Z}^{n+1}$, and hence $\pi\_1(M) \cong \mathbb{Z}^n$. Moreover, as $T^{n+1}$ is aspherical, so is $M$. Since $\pi\_1(M) \cong \pi\_1(T^n)$ and both are aspherical, we see that... | 13 | https://mathoverflow.net/users/21564 | 434559 | 175,724 |
https://mathoverflow.net/questions/434557 | 3 | Let $C,D$ be two non-compact complex algebraic smooth curves. Suppose that two unramified regular finite maps $p\_1, p\_2: C \rightarrow D$ are given and have the same degree. Is there always an automorphism $\varphi:C \rightarrow C$ such that $p\_1=p\_2\circ \varphi$?
| https://mathoverflow.net/users/494203 | Existence of covering isomorphism | I suppose that "non-compact complex algebraic curve" means complex affine curve.
The following counterexample was proposed by my friend Fedor Pakovich.
Let $D=\mathbf{C}\backslash\{-1,1\}$.
Consider the $4$-th Chebyshev polynomial
$$p\_1(z)=2(2z^2-1)^2-1=8z^4-8z^2+1.$$
It has critical points at $0,\pm1/\sqrt{2}$, w... | 2 | https://mathoverflow.net/users/25510 | 434567 | 175,728 |
https://mathoverflow.net/questions/434562 | 6 | Let $G$ be a vertex-transitive locally finite graph and $c\_n$ the number of self-avoiding walks in $G$ starting from some fixed vertex $v\_0$. One can easily see that $c\_{m+n} \leq c\_m c\_n$ and hence Fekete's subadditive lemma gives that there exists the limit $\mu := \lim\_{n\to\infty} c\_n^{1/n}$. This quantity i... | https://mathoverflow.net/users/318999 | Origin of the term "connective constant" | **Q:** Is there some application where $\mu$ plays a role in some kind of "connectedness" which would excuse the name?
**A:** The application is to crystalline structure. The name originates from Hammersley, who introduced [1,2] the "connective constant" to characterize the bonds in a crystal.
>
> In a certain se... | 4 | https://mathoverflow.net/users/11260 | 434582 | 175,733 |
https://mathoverflow.net/questions/434595 | 1 | Given the positive integers $n$ and $m$, consider the set of graphs $\mathcal{G} = \{G=(V,E): |V|=n \land |E|=m\}$.
For which values of $n$ and $m$ does the following requirement hold:
$\forall G \in \mathcal{G}$ there exist $V\_1 \subset V,V\_2 \subset V$ such that:
1. $(v\_1 \in V\_1 \land v\_2 \in V\_2) \impli... | https://mathoverflow.net/users/136218 | Do all graphs with $n$ vertices and $m$ edges have a special property? | For $n=53$ and $m=113$, you can't even get close in general. Take 7 copies of $K\_5$ and 3 copies of $K\_6$, all disjoint. Remove any two edges; now you have 53 vertices and 113 edges. No complete bipartite subgraph has more than 6 vertices. If you don't like disconnected graphs, delete 9 more edges without disconnecti... | 3 | https://mathoverflow.net/users/9025 | 434605 | 175,741 |
https://mathoverflow.net/questions/434470 | 0 | Suppose $\mathbf{A}\_{k\times n}$ ($k<n$) is a matrix whose entries are generated i.i.d. from Gaussian distribution and $\mathbf{s}\_{n\times 1}$ is a sparse vector with $m$ sparsity (i.e., $\|\mathbf{s}\|\_0=m$). Then what is the probability that the sparse solution of $\mathbf{A}\mathbf{s}=\mathbf{b}$ can be obtained... | https://mathoverflow.net/users/68835 | Probability of accurate sparse recovery | A good starting point is "Mathematics of sparsity (and a few other things)" by E. Candes, or a book on compressed sensing such as "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut.
Let me change the notation slightly as your choice of variable name is atypical. Let $N$ be the sample size, $d$... | 1 | https://mathoverflow.net/users/141760 | 434608 | 175,742 |
https://mathoverflow.net/questions/433600 | 2 | Odifreddi doesn't give a cite (at least in proposition XI.2.10) for the proposition that every non-zero r.e. degree computes a 1-generic. What paper should I cite for this proposition?
| https://mathoverflow.net/users/23648 | Cite for fact that every r.e. degree bounds a 1-generic | For the benefit of others, I emailed Shore and asked him about it and he told me that while he had assumed when he proved it that it wasn't a novel result he never actually found any earlier proof (much less a publication). So, absent contrary evidence turning up I think it's Shore who should get the citation for the c... | 0 | https://mathoverflow.net/users/23648 | 434609 | 175,743 |
https://mathoverflow.net/questions/434612 | 2 | Let $X$ be a scheme of finite type over $\mathrm{Spec}(A)$, where $A$ is a commutative ring. Let $U\subset X$ be any open subset, is $\Gamma(U,\mathscr{O}\_{X})$ a finitely generated $\Gamma(X,\mathscr{O}\_{X})$-algebra?
| https://mathoverflow.net/users/494530 | Is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{O}_{X})$-algebra? | No. The following counterexamples are part of the folklore:
* Even if $X$ is an affine variety over a field $k$ (so $\Gamma(X,\mathscr{O}\_X)$ is certainly of finite type over $k$) and $U\subseteq X$ open, then $\Gamma(U, \mathscr{O}\_X)$ can still fail to be of finite type over $k$ (and in particular, over $\Gamma(X... | 8 | https://mathoverflow.net/users/17064 | 434613 | 175,745 |
https://mathoverflow.net/questions/434537 | 7 | Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G\_c\rightarrow G\rightarrow G/G\_c\rightarrow 1$$
where $G\_c$ is the normal subgroup which contains all elements in the same connected component as the identity element, and $G/G\_c$ can be thought of as the "... | https://mathoverflow.net/users/314049 | Injectivity of the cohomology map induced by some projection map | Ok, I will follow Fernando's advice and post an answer. I learned the computation below from the beginning of [Pin(2)-equivariant Seiberg--Witten Floer homology and the triangulation conjecture](https://arxiv.org/abs/1303.2354).
The group $\text{Pin}(2) = S^1 \cup jS^1 \subset S^3$ is a subgroup of the unit quaternio... | 5 | https://mathoverflow.net/users/40804 | 434621 | 175,746 |
https://mathoverflow.net/questions/434565 | 3 | Consider the $9\times 9$ matrix
$$M = \begin{pmatrix} i e\_3 \times{} & i & 0 \\
-i & 0 & -a \times{} \\
0 & a \times{} & 0 \end{pmatrix}$$
for some vector $a \in \mathbb R^3$, where $\times$ is the cross product.
It is claimed in [Fu and Qin - Topological phases and bulk-edge correspondence of magnetized cold p... | https://mathoverflow.net/users/457901 | Reducing $9\times9$ determinant to $3\times3$ determinant | I think the simplest way to reduce $$A=M-\omega I$$ to a $3\times 3$ matrix is to use the Schur complement with respect to the $(\bar 2,\bar 2)$-elements of $A$,
\begin{align}
C = A/A\_{\bar 2,\bar 2} = A\_{2,2} - A\_{2,\bar 2} A\_{\bar 2,\bar 2}^{-1} A\_{\bar 2,2}.
\end{align}
Here, $\bar 2$ denotes the index compleme... | 4 | https://mathoverflow.net/users/90413 | 434627 | 175,747 |
https://mathoverflow.net/questions/434637 | 3 | I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book:
Wybourne, B.G.; "Symmetry Principles and Atomic Spectroscopy"; Wiley--Interscience, New York, 1970.
I ordered the book, but in the mean time, co... | https://mathoverflow.net/users/81645 | References for applications of Young diagrams/tableaux to Quantum Mechanics | Young diagrams or Young tableaux (the latter being diagrams with integers in each box) are used in particle physics to describe the states of indistinguishable fermions or bosons: $n$ indistinguishable particles, each of which can be in one of $m$ states form an irreducible representation of $U(m)$. A Young tableaux in... | 4 | https://mathoverflow.net/users/11260 | 434639 | 175,750 |
https://mathoverflow.net/questions/434614 | 1 | In graph theory, a Hamiltonian cycle is a cycle that visits each vertex exactly once. Hamiltonian cycle has a long history, and I have followed some articles.
We can find plenty of examples of Hamiltonian cycles by using google scholar.
* S. Špacapan, A counterexample to prism-hamiltonicity of 3-connected planar gr... | https://mathoverflow.net/users/171032 | Is there a monograph or review of Hamiltonian cycles of graphs (or long cycles of graphs)？ | **Q:** *Is there a monograph (or review) of Hamiltonian cycles of graphs (or long cycles of graphs)？*
One possible answer (from a specific perspective) is
[Hamiltonian Cycle Problem and Markov Chains](https://books.google.nl/books?id=lj6hDyx6asUC) (2012)
>
> This monograph summarizes a line of research that cas... | 2 | https://mathoverflow.net/users/11260 | 434643 | 175,752 |
https://mathoverflow.net/questions/434635 | 3 | I recently asked in [this thread](https://mathoverflow.net/questions/434320/lower-bounds-for-pattern-complexity-of-aperiodic-subshifts) about lower bounds on the complexity in the case where we have an aperiodic subshift. If I denote $c\_n(\Omega)$ as the number of possible patterns on $Q\_n= \big\{ 0,...,n-1 \big \}^d... | https://mathoverflow.net/users/143153 | 'Trivial' lower bounds for pattern complexity of aperiodic subshifts | I'll try to do three dimensions for simplicity. I am on the bus and have to be very quick.
>
> Theorem. Suppose $X \subset A^{\mathbb{Z}^3}$ is a subshift, such that $\liminf\_n \frac{P(n)}{n^2} = 0$. Then every point in $X$ is periodic.
>
>
>
Lou look at two-dimensional slices of your configuration, and use (... | 4 | https://mathoverflow.net/users/123634 | 434646 | 175,753 |
https://mathoverflow.net/questions/434649 | 1 | Let $p:X\to S$ be a morphism of schemes. Let $\mathcal F$ be an $\mathcal O\_X$-modules. Assume that:
* $\mathcal F$ is quasi-coherent of finite type;
* $\mathcal F$ is flat over $S$;
* the support of $\mathcal F$ is proper over $S$.
Let $\varphi\in\mathrm{Aut}\_{\mathcal O\_X}(\mathcal F)$ be an automorphism of $\... | https://mathoverflow.net/users/105537 | For an automorphism of a flat family of sheaves, is there a subscheme of the base where the automorphism is identity? | First, replacing $X$ by the support of $\mathcal{F}$ we may assume $X$ is proper over $S$. Second, using Chow's lemma we may assume $X$ is projective over $S$ (I assume here that $S$ is noetherian).
Now, the condition $\varphi = \mathrm{id}$ is equivalent to
$$
\varphi - \mathrm{id} = 0.
$$
Let $L$ be a line bundle o... | 2 | https://mathoverflow.net/users/4428 | 434657 | 175,756 |
https://mathoverflow.net/questions/434659 | 2 | What is a reasonable axiomatization of S2S?
S2S is the monadic second order theory with two successors (Wikipedia [link](https://en.wikipedia.org/wiki/S2S_(mathematics))). It has finite binary strings, operations $s→s0$ and $s→s1$ on strings, and arbitrary sets of strings. It is one of the most expressive decidable t... | https://mathoverflow.net/users/113213 | Axiomatization of S2S | S2S can be axiomatized by:
1. $∃!s ∀t \, (t0≠s ∧ t1≠s)$ (empty string, denoted by $ε$)
2. $∀s,t \, ∀i∈\{0,1\} \, ∀j∈\{0,1\} \, (si=tj ⇒ s=t ∧ i=j)$ (tree successors; the use of $i$ and $j$ is an abbreviation; for $i=j$, 0 does not equal 1)
3. $∀S \, (S(ε) ∧ ∀s \, (S(s) ⇒ S(s0) ∧ S(s1)) \,⇒\, ∀s \, S(s))$ (induction)
... | 3 | https://mathoverflow.net/users/113213 | 434660 | 175,757 |
https://mathoverflow.net/questions/434662 | 1 | Let $X$ be an algebraic complex K3 surface, we know that $X$ is deformation equivalent to a smooth quartic surface or more generally a K3 surface with Picard number $1$ (a very general K3 surface in the family of K3 surfaces). It means that there exists a proper morphism $\mathcal{X}\rightarrow S$ with each fibre a K3 ... | https://mathoverflow.net/users/nan | One-dimensional family of complex algebraic K3 surfaces | Choose any ample class in $\mathrm{Pic}(X)$, assume its degree is $d$. Let $M\_d$ be the moduli space of polarized K3 surfaces of degree $d$ with appropriate level structure so that it has a universal family (alternatively one can use here the $\mathrm{Quot}$-scheme that is used to construct the moduli space). Let $x \... | 3 | https://mathoverflow.net/users/4428 | 434667 | 175,759 |
https://mathoverflow.net/questions/434671 | 1 | Let $\{0,1\}^{<\omega}$ be the collection of $x \in \{0,1\}^\omega$ such that there is $N\in\omega$ with $x(k) = 0$ for all $k\geq N$. We say that $ x, y\in \{0,1\}^{<\omega}$ form an edge if they have Hamilton distance $1$ (that is, there is a unique $k\in\omega$ such that $x(k) \neq y(k)$.
**Question.** Is there a ... | https://mathoverflow.net/users/8628 | Does $\{0,1\}^{<\omega}$ have a Hamiltonian path? | *I'm going to write $\mathcal{S}$ for what you call $\{0,1\}^{<\omega}$, since I'm used to the latter referring to the set of finite binary strings.*
Yes, and moreover there is a relatively simple process for building such a path. The key is the following lemma:
>
> For any finite set $A\subseteq\mathcal{S}$ and ... | 7 | https://mathoverflow.net/users/8133 | 434672 | 175,761 |
https://mathoverflow.net/questions/434663 | 1 | Let $P$ be a non-negatively curved (in the Alexandrov sense) polyhedral space (of dimension 3, say), $p,q\in P$ be vertices, and let $e$ be an edge connecting $p$ and $q$. Assume $e$ has cone angle $0< \alpha \leq 2\pi$. We know that the space of directions of $p,$ $\Sigma\_p P,$ is a polyhedral surface of curvature bo... | https://mathoverflow.net/users/100597 | Intersection of conical neighbourhoods on a polyhedral space | You say "The same is true for a tubular neighborhood of the edge".
This is not correct, but it is true if you stay away from the endpoints.
So $U$ should be defined as a tubular neighborhood of subarc of $e$.
In this case $B$ and $U$ might intersect, but the intersection does not contain $p$.
| 2 | https://mathoverflow.net/users/1441 | 434677 | 175,764 |
https://mathoverflow.net/questions/434655 | 4 | Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S\_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on $\mathscr{H}$.
Denote by $\rho(S\_r)$ the spectral radius of $S\_r$ and assume that $(\sup\_{r \in [0,1]}\rho(S\_r))... | https://mathoverflow.net/users/122635 | Uniform decay of operator norm for smooth family of operators | This works, essentially because $\|T^k\|^{1/k}$ for a given $k=n$ also controls this quantity for $k\ge n$.
More specifically, suppose that $\|T^n\|^{1/n}\le 1-\delta$. Clearly, $\|T^{kn}\|^{1/kn}\le \|T^n\|^{1/n}$, and for general $N\ge n$, write $N=kn+j$, $0\le j<n$, and estimate
$$
\|T^N\|^{1/N}\le \|T^{kn}\|^{1/N... | 3 | https://mathoverflow.net/users/48839 | 434679 | 175,766 |
https://mathoverflow.net/questions/434221 | 4 | By a good closed cover of a topological space $X$, I mean a collection of closed subspaces of $X$, such that the interior of them cover $X$, and any finite intersection of these closed subspaces is contractible.
Every triangulable space $X$ admits a good open cover: just fix a triangulation and take open stars at all... | https://mathoverflow.net/users/119189 | Closed good cover of a triangulable space | **Claim:** Let $Z$ be a simplicial complex. For each simplex $\sigma\in Z$, let $N\_2 (\sigma, Z)$
denote the simplicial neighborhood of $\sigma$ (or really, the second barycentric subdivision of $\sigma$)
inside the second barycentric subdivision of $Z$. Then
$\{|N\_2 (\sigma, Z)|: \sigma \in Z\}$ is a good cover of $... | 4 | https://mathoverflow.net/users/4042 | 434689 | 175,769 |
https://mathoverflow.net/questions/434630 | 1 | This question arises from a request for an algorithm to do such, [from 9 sets of 12 elements, arrange 12 groups of the 9 elements, selecting 1 element from each set](https://stackoverflow.com/questions/74334415/from-9-sets-of-12-elements-arrange-12-groups-of-the-9-elements-selecting-1-ele)
Given a set, $S$, of sets, ... | https://mathoverflow.net/users/494835 | Recombining set elements with no duplicated pairing of elements | Let $\Sigma$ be an set of cardinality $e$. Then this problem is equivalent to selecting $se$ vectors from $\Sigma^s$ such that the minimum Hamming distance is $s−1$. (To see this, take $S\_i = \{i\} \times \Sigma$). We could also phrase it as $\Sigma$ being an alphabet of cardinality $s$ and selecting $se$ words of len... | 2 | https://mathoverflow.net/users/46140 | 434702 | 175,773 |
https://mathoverflow.net/questions/434651 | 4 | This question arises as a variation of [this question](https://mathoverflow.net/questions/434183/quantitative-analytic-continuation-estimate-for-a-function-small-on-a-set-of-pos), which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails ... | https://mathoverflow.net/users/146531 | Quantitative analytic continuation estimate for functions small except on a small set | This conjecture is correct. Take $K=e$, and let $\gamma\leq 1/4$; we will fix $\gamma$ later.
First we give a crude estimate of $c\_0$.
Let $g(z)=\sum\_{1}^\infty c\_nz^n.$ Since $|c\_n|\leq e^n$,
we obtain $|g(z)|\leq e^n$ for $|z|\leq 1/2$ by the trivial estimate. Then by Cauchy, $|f'(z)|=|g'(z)|\leq 4\cdot e^n,\; ... | 3 | https://mathoverflow.net/users/25510 | 434715 | 175,775 |
https://mathoverflow.net/questions/434709 | 4 | The starting point of this question is the observation that if $\lambda$ is a countable ordinal, then there is an order-embedding $e:\lambda \hookrightarrow \mathbb{Q}$.
Given an infinite cardinal $\kappa$, is there a linearly ordered set $L\_\kappa$ of cardinality $\kappa$ such that for every ordinal $\alpha$ with $... | https://mathoverflow.net/users/8628 | Ordinal-universal linear order on $\kappa$ elements | Let $\lambda$ be an infinite cardinal. Give $[\lambda]^{<\omega}$ the linear ordering $\leq$ where we set $A>B$ if there is some $\alpha\in B$ where $\alpha\cap A=\alpha\cap B$ and $\alpha\not\in A$.
We shall show using what is essentially transfinite induction that every ordinal less than $\lambda^+$ embeds into $[\... | 7 | https://mathoverflow.net/users/22277 | 434717 | 175,777 |
https://mathoverflow.net/questions/434701 | 4 | Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and right eigenvector are never orthogonal to each other?
If true, could this generalize in a way to degenerate eigenvalues? -... | https://mathoverflow.net/users/457901 | Left and right eigenvectors are not orthogonal | Yes, this is always true if $\lambda \not= 0$. The subsequent theorem shows a more general result. To formulate it, we need the following terminology:
* For an eigenvector $\lambda$ of a bounded linear operator $T: X \to X$ on a complex Banach space $X$, the vector subspace $\bigcup\_{n \ge 1} \ker\big((\lambda-T)^n\... | 4 | https://mathoverflow.net/users/102946 | 434724 | 175,780 |
https://mathoverflow.net/questions/434680 | 15 | **Question 1**: Are there any contexts in which replacing the category of (non-symmetric or symmetric) operads (in some monoidal category or symmetric monoidal category, respectively) with the category of their corresponding monads leads to undesirable behavior? (I know that for non-symmetric operads [this functor is n... | https://mathoverflow.net/users/148161 | Why are operads sometimes better than algebraic theories? | First - yes, for symmetric set-operads this functor is "injective", though it is not fully faithful. It is faithful on general maps and fully faithful on isomorphisms. Its image can easily be characterized: symmetric set-operads are the so-called "[analytic monads](https://ncatlab.org/nlab/show/analytic+monad)" that is... | 22 | https://mathoverflow.net/users/22131 | 434728 | 175,782 |
https://mathoverflow.net/questions/434688 | 6 | Let's say that a formula in the language of set theory is *flexibly stratified* iff there exists a function $f$ from variable symbols to $\omega$ such that if $x=y$ appears in the formula, then $f(x)=f(y)$, and if $x\in y$ appears in the formula, then $f(x)<f(y)$ (this is in contrast to regular stratification which req... | https://mathoverflow.net/users/44115 | Strengthening Quine's New Foundations with a more flexible stratification criterion? | Flexibly Stratified Comprehension is inconsistent. By Flexibly Stratified Comprehension, there is an s such that
$$\forall x\:\bigl(x\in s\leftrightarrow\exists y\:\bigl(\forall t\:(t\in y\leftrightarrow t\in x)\land y\notin x\bigr)\bigr).$$
If $s\in s$, then there is an $S$ with the same members as $s$ such that $S\... | 9 | https://mathoverflow.net/users/133981 | 434731 | 175,784 |
https://mathoverflow.net/questions/434706 | 4 | It is well known how to derive the field operations from the construction of the real numbers as the Dedekind completion of the rational numbers and as the Cauchy completion of the rational numbers; see section 11.2.1 and 11.3.3 of [this textbook](https://hott.github.io/book/hott-online-1353-g16a4bfa.pdf) for an explic... | https://mathoverflow.net/users/483446 | The field structure on the locale of real numbers | There are several (equivalent) way to go about it:
You can start form the fields operation on $\mathbb{Q}$ and use that they are "locally uniformly continuous" to extend them by continuity to the localic completion (for the inverse map, you'll have to stay away of $0$ of course).
Regarding the existence of such con... | 4 | https://mathoverflow.net/users/22131 | 434732 | 175,785 |
https://mathoverflow.net/questions/434693 | 3 | Note added by YC: the definition below of the cyclic sub-complex is incorrect; and the "higher order derivations" referred to here are traditionally known (since the 1940s) as n-cocycles.
---
$\DeclareMathOperator{\Ker}{\mathrm{Ker}}\DeclareMathOperator{\Mod}{\mathrm{Mod}}\DeclareMathOperator{\Der}{\mathrm{Der}}\... | https://mathoverflow.net/users/130058 | Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses? | The answer is *yes* and it is very simple. It helps to understand the case $n=1$ first in the way I explained in my [thesis](https://arxiv.org/abs/1410.1716) in Prop. 4.5.3. Namely, $\Omega^1\_{S/R}$ can be constructed as the quotient of the right $S$-module $S \otimes\_R S$ by the $S$-submodule generated by those $ab ... | 2 | https://mathoverflow.net/users/2841 | 434742 | 175,788 |
https://mathoverflow.net/questions/434740 | 1 | Following to my [previous question](https://mathoverflow.net/q/434595/136218) on the same topic, I would like to have some opinions whether the present refinement have some chances to work or is doomed to fail.
Given the positive integers $n$ and $m$, consider the set of graphs $\mathcal{G} = \{G=(V,E): |V|=n \land |... | https://mathoverflow.net/users/136218 | Graphs with $n$ vertices and $m$ edges and more probable property | There is a $C\_4$-free bipartite graph $B$ with 19 vertices on one side, 20 vertices on the other side, and 92 edges. Its vertices have degree 4 or 5, so it is easy to find a path $P$ of 21 edges in $K\_{19,20}-B$. Now consider the 113-edge union $G=B\cup P$.
Since $G$ is bipartite, it has no complete multipartite su... | 1 | https://mathoverflow.net/users/9025 | 434747 | 175,791 |
https://mathoverflow.net/questions/434713 | 4 | Let $X$ be a complex smooth projective variety with trivial topological Euler characteristic $\chi\_{\text{top}}(X)=0$. We assume that $D$ is a smooth irreducible divisor in the linear system $|K\_X|$ of the canonical divisor $K\_X$ of $X$. Is $\chi\_{\text{top}}(D)=0$?
| https://mathoverflow.net/users/493291 | topological Euler characteristic of canonical divisor | This is not true. Consider, for instance a Calabi--Yau threefold $Y$ with $h^{2,1}(Y) = h^{1,1}(Y) + 1$ (an example of such can be found in <https://arxiv.org/abs/1602.06303>, see page 29) and let $X$ be the blowup of $Y$ in a point. Then $\chi\_{\mathrm{top}}(X) = 0$, but the canonical class of $X$ is equal to the exc... | 4 | https://mathoverflow.net/users/4428 | 434751 | 175,793 |
https://mathoverflow.net/questions/434707 | 25 | Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim\limits\_{n\to\infty}\prod\limits\_{k=1}^n f(\frac{k}{n})<\infty$ ?
I do not see any reason why such a function could not exist, but I have not been able to find an example of such a function.
Context: If such a function does not exist, the... | https://mathoverflow.net/users/494920 | Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$? | If you do not require monotonicity of $f$, the construction is pretty simple and is a combination of a few facts we normally (should) teach in elementary number theory and Fourier analysis classes. However, the monotonicity condition seems rather natural to impose and it seems to change the game completely, so look at ... | 17 | https://mathoverflow.net/users/1131 | 434762 | 175,794 |
https://mathoverflow.net/questions/434757 | 3 | Consider propositional logic.
Frege systems are textbook-style proof systems, with a finite set of logically sound axioms and rules. However you can generalise each axiom/rule with substitutions, replacing each variable simultaneously everywhere with a term. This is fine because if we take any propositional tautology... | https://mathoverflow.net/users/494987 | Does extended Frege p-simulate circuit Frege with substitutions? | The answer is yes. I’d say this is essentially folklore, but if you want a published reference, see Lemma 2.6 in my paper [1]. The lemma is stated more generally for proof systems for transitive modal logics; plain classical logic is just the special case for the modal logic axiomatized by $A\leftrightarrow\Box A$ (or ... | 2 | https://mathoverflow.net/users/12705 | 434763 | 175,795 |
https://mathoverflow.net/questions/434750 | 2 | **Question:**
what can be recommended for calculating $f(x)$ that solves $\frac{f(x)}{f(x)+f(1-x)}\approx g(x)$ for $x\in[0,1]$?
I have tried comparing Taylor series, but they look intimidating and I would appreciate suggestions for better solutions
| https://mathoverflow.net/users/31310 | Approximation with special partitions of unity | I don't know what significance the approximate equality has, so I'll just replace $\approx$ by $=$. Any function $g(x)$ on $[0,1]$ can be split into its even $g\_e(x) = (g(x)+g(1-x))/2$ and odd $g\_o(x) = (g(x)-g(1-x))/2$ parts, $g(x) = g\_e(x) + g\_o(x)$. So the desired equation takes the following form, with an immed... | 2 | https://mathoverflow.net/users/2622 | 434764 | 175,796 |
https://mathoverflow.net/questions/433638 | 2 | I've just checked that this is constructed to mimic the ordinary Hamiltonian equation in symplectic geometry. There are several literatures, and they use
$$
\eta(X\_H) = -H\\
\mathrm{d}\eta(X\_H,-) = \mathrm{d}H - R(H)\eta
$$ to descirbe the Hamiltonian equation, where $\eta$ is a contact form and $R$ is the associated... | https://mathoverflow.net/users/120948 | What is the motivation of contact Hamiltonian equation | One mathematical motivation is that it shows that the automorphism group of contact manifolds is huge. To give a bit of an idea of what I mean, compare this to Riemannian metrics.
The isometry group of a given metric is finite dimensional, and in fact, an isometry is uniquely determined by the information on how it a... | 2 | https://mathoverflow.net/users/67031 | 434783 | 175,803 |
https://mathoverflow.net/questions/434777 | 6 | I have been trying to understand projective objects in the derived category of chain complexes of modules over a ring.
If we stick to the category of chain complexes, the only projective objects are split exact complexes of projectives. These would all be trivial in the derived category.
What happens if we consider... | https://mathoverflow.net/users/170467 | Projective objects in the derived category of chain complexes | In a stable $\infty$-category, there are no nontrivial projectives. Of course, $0$ is always projective.
Now let $X$ be an arbitrary projective in some stable $C$, $X\simeq\Sigma \Omega X$ is a simplicial colimit of things of the form $\Omega X^n$ for some $n$'s, so that the identity $X\to X \simeq \Sigma \Omega X$ f... | 17 | https://mathoverflow.net/users/102343 | 434786 | 175,804 |
https://mathoverflow.net/questions/434802 | 7 | Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\tilde g: X \to \mathbb{R}$ be a measurable function and assume that there exists a norm dense vector subspace $D$ of $L^2$ ... | https://mathoverflow.net/users/102946 | Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace | The answer is **no** and the following result provides a quite interesting counterexample. This is a known result, but I am not sure where to find it in the literature.
>
> **Theorem.** If $f\in L^1\_{\rm loc}(\mathbb{R}^n)\setminus L^2(\mathbb{R}^n)$, then there is an orthonormal basis
> $\{\varphi\_k\}\_{k=1}^\in... | 7 | https://mathoverflow.net/users/121665 | 434803 | 175,807 |
https://mathoverflow.net/questions/434811 | 1 | I've been reading the chapter of Uniform Integrability on Probability Theory by Achim Klenke which has the following proposition.
**If $\mu$ is $\sigma$-finite, then there is a function $h \in L^{1}(\mu)$ such that $h > 0$ almost everywhere.**
In the proof, he constructed an increasing sequence of sets $A\_{n}$ w... | https://mathoverflow.net/users/494934 | On conditions for the existence of $h\in L^1$ such that $h>0$ a.e | It is false without assuming that the measure is $\sigma$-finite. Let $\mu$ be the counting measure on $\mathbb{R}$. Then $h>0$ a.e. means $h>0$ everywhere and clearly
$$
\int\_{\mathbb{R}}hd\mu=\sum\_{x\in\mathbb{R}} h(x)=+\infty.
$$
| 0 | https://mathoverflow.net/users/121665 | 434815 | 175,811 |
https://mathoverflow.net/questions/434257 | 2 | Given a general Banach space $B$ and a one-parameter family of contractions $F\_t:B\to B$ which is defined for all $t \in (a,b)$. $F\_t$ depends continuously on $t$ (in the sense $\lim\_{\varepsilon\to 0} \parallel F\_{t+\varepsilon}(x)-F\_t(x)\parallel = 0$). I want to study how the fixed points $x\_t$ of $F\_t$ depen... | https://mathoverflow.net/users/494474 | Differentiability of the fixed points of a family of contraction maps | I found the answer myself: One can simply apply the Banach space version of the implicit function theorem to the function $G(t,x) = x-F\_t(x)$. The implicit function theorem shows that, given $G$ is in $C^k$, that $x\_t$ is in $C^k$ as well. The precise statement can be found for instance in the book 'Analysis Tools wi... | 0 | https://mathoverflow.net/users/494474 | 434821 | 175,813 |
https://mathoverflow.net/questions/295655 | 17 | The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization ([wikipedia link](https://en.wikipedia.org/wiki/Stallings_theorem_about_ends_of_groups)) in terms of amalgam/HNN splittings over fi... | https://mathoverflow.net/users/14094 | Finitely generated groups with Hölder-exotic space of ends? | The question is solved positively in the paper *The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups* by Matthieu Dussaule, Wenyuan Yang, which appeared on arXiv on October of 2020 ([link](https://arxiv.org/abs/2010.07671)). It even says that this is almost always the case. This means that t... | 2 | https://mathoverflow.net/users/14094 | 434837 | 175,819 |
https://mathoverflow.net/questions/392625 | 10 | Recently I've been working with o-minimal expansions of $(\mathbb{R},\times,+)$, and I want to work "internally" to the language of o-minimal sets instead of working with "definable families".
This is for a simple practical reason: I can't explain ANYTHING in the definable family formalism to my combinatorialist coll... | https://mathoverflow.net/users/3404 | Definable constructions in o-minimal geometry | I will expand my comment into at least a partial (but for me quite illuminating answer). As I’ve mentioned in the comments, the question of complexity in o-minimality was recently adressed in an ICM talk by Gal Binyamini and Dmitri Novikov, and then formally introduced together with Benny Zack in the following preprint... | 0 | https://mathoverflow.net/users/45493 | 434843 | 175,821 |
https://mathoverflow.net/questions/434860 | 3 | Suggested by [this problem](https://mathoverflow.net/q/405252/9924):
>
> Do the sets of all odious / evel numbers meet every infinite arithmetic progression?
>
>
>
A number is [odious](https://oeis.org/A000069) if it contains an odd number of digits $1$ in its binary representation; otherwise, it is [evel](htt... | https://mathoverflow.net/users/9924 | Where odious numbers meet arithmetic progressions | Yes, we even know that the density of such $n$ is the expected one.
A. O. Gelfond proved in 1968, in a short paper ("Sur les nombres qui ont des propriétés additives et multiplicatives données", Acta Arithmetica 13, pages 259-265) that, for all $b \bmod a$ and $j \bmod m$,
$$(\star) \, \lim\_{N \to \infty}\frac{1}{N/... | 8 | https://mathoverflow.net/users/31469 | 434861 | 175,826 |
https://mathoverflow.net/questions/434710 | 9 | It is well known that the principal block $\mathcal{O}\_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain Koszul algebra.
Namely, if we let $W$ denote the Weil group and take $P=\bigoplus\_{w\in W}P\_w$ to be a projective generato... | https://mathoverflow.net/users/492515 | Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request | You can find quiver and relations (not sure if they are admissible always) here: <http://www.math.uni-bonn.de/ag/stroppel/Quivers.pdf>
In particular the explicit algebra is only fully understoof for $sl\_4$ and below. See also <https://link.springer.com/article/10.1007/s00222-006-0005-2> which gives results on the calc... | 2 | https://mathoverflow.net/users/61949 | 434862 | 175,827 |
https://mathoverflow.net/questions/434725 | 3 | Is there any better than a brute force method for finding the maximum
$$\max\limits\_{ (d\_{1},\dots,d\_{n}) \in \mathbb Z\_{m}^{n}} \sum\_{j=0}^{m-1} \left(\sum\_{i=1}^{n}v\_{i,(j+d\_{i})\bmod m}\right)^{2}$$
for $m,n \in \mathbb N^{+}$ and $v\_{i,j} \in \mathbb Z$?
| https://mathoverflow.net/users/nan | How to find the maximum of a sum of squares of sums? | You can solve the problem via binary quadratic programming as follows. Let binary decision variable $x\_{id}$ indicate whether row $i$ is rotated $d$ places. The problem is to maximize
$$\sum\_{j=0}^{m-1} \left(\sum\_{i=1}^n \sum\_{d=0}^{m-1} v\_{i,(j+d) \bmod m} x\_{id} \right)^2$$
subject to
$$\sum\_{d=0}^{m-1} x\_{i... | 0 | https://mathoverflow.net/users/141766 | 434880 | 175,834 |
https://mathoverflow.net/questions/434867 | 13 | Let $M$ be a connected $n$ dimensional boundary-less smooth manifold with the property that for any connected boundary-less $n$ dimensional manifold $\overline{M}$ and any embedding $i:M\rightarrow \overline{M}$, we must have $\overline{M}=i[M]$.
Question: Must $M$ be compact ?
| https://mathoverflow.net/users/32135 | Is an inextensible manifold necessarily compact? | Yes, $M$ must be compact. In fact, if $M$ is non-compact, it admits a non-surjective self embedding $f:M\rightarrow M$.
When $n=1$, the only non-compact manifold is $\mathbb{R}$, which obviously admits non-surjective self-embeddings. So we assume $n=2$.
We will first construct $f$ for the special case where $M = \m... | 15 | https://mathoverflow.net/users/1708 | 434881 | 175,835 |
https://mathoverflow.net/questions/434870 | 0 | Suppose $f$ is a matrix convex function over symmetric, positive semidefinite matrices with spectra in some interval $I$ [1]. That is, for $A,B\succeq 0$ with spectra in $I$, and any $\theta\in[0,1]$,
$$
\theta f(A)+(1-\theta)f(B)\preceq f(\theta A + (1-\theta) B)\,\,.
$$
$f(X)=X^{-1/2}$ is one such example.
If t... | https://mathoverflow.net/users/100164 | Do subgradient inequalities hold for matrix convex functions? | Some further research has pointed out the answer as affirmative (though I will leave the question unanswered for now, as I'm still perplexed by Ando's rank restriction, which now seems unnecessary).
**Theorem V.3.3** of [1]. Suppose the matrix map $f$ is induced by a scalar function on $I\subset \mathbb{R}$ applied t... | 0 | https://mathoverflow.net/users/100164 | 434883 | 175,836 |
https://mathoverflow.net/questions/434854 | 4 | $\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for the algebra generated by the set of all the product sets $A\times B\in\B\times\B$.
Let $L^2$ be the Lebesgue measure on $... | https://mathoverflow.net/users/36721 | On partial absolute continuity | The notation $L^2$ for Lebesgue measure is confusing. I denote $\lambda$ and $\lambda\_2$ the Lebesgue measure on $\mathbb{R}$ and $\mathbb{R}^2$, respectively.
The answer is no.
Fix a discrete measure $\mu$ which gives a positive mass to every rational number in $[0,2]$. Call $\nu$ the image of $\lambda \otimes \m... | 2 | https://mathoverflow.net/users/169474 | 434884 | 175,837 |
https://mathoverflow.net/questions/434886 | 0 | Let $(y\_n)\_{n\ge 1}$ be a sequence with values in $(0,1)$ such that $\lim\_n y\_n=1$. Let also $f: [0,1]\to \mathbb{R}$ be a bounded function such that $f(0)=0$ and satisfies
$$
\forall n\ge 1, \forall x \in [0,1-y\_n], \quad
f(x+y\_n)=f(x)+f(y\_n)
\quad \quad (\star)
$$
and
$$
\forall x \in [0,1], \quad f(x)+f(1-x)... | https://mathoverflow.net/users/32898 | Symmetric and nearly additive bounded functions | Let $C$ be the usual Cantor set in $[0,1]$, and split the complement $I\setminus C$ as a disjoint union of intervals $I\_1,I\_2,\dots$. Let $L\_i$ and $R\_i$ denote the left and right half of $I\_i$, and let $m\_i$ be the midpoint of $I\_i$.
Then the function
$$
f(x)=
\begin{cases}
0, & x\in C;\\
1, &x\in L\_i; \... | 1 | https://mathoverflow.net/users/17581 | 434893 | 175,840 |
https://mathoverflow.net/questions/434855 | 0 | Let $\varphi$ denote the Euler's totient function. Is there any reference in literature for the value of sum $$\sum\_{\substack{r\le x\\ d\mid r}}\gcd(\phi(d),r)$$ where $d$ is some fixed positive integer?
Thanks in advance.
| https://mathoverflow.net/users/160943 | Asymptotic for a sum involving GCD and Euler totient function | Let $c:=\gcd(\phi(d),d)$. then setting $r:=dk$ the sum can be rewritten as
$$c\sum\_{k\leq x/d} \gcd(\tfrac{\phi(d)}c,k) \approx \frac{c^2x}{d\phi(d)}f(\tfrac{\phi(d)}c),$$
where $f(m) := \sum\_{k=1}^m \gcd(k,m)$ is the multiplicative function given by its values on prime powers $f(p^s) = p^{s-1}((p-1)s+p)$. See also [... | 1 | https://mathoverflow.net/users/7076 | 434894 | 175,841 |
https://mathoverflow.net/questions/434903 | 3 | I would like to know if in some book or how could I compute the quadratic variation of the supremum of the bronian motion $S\_t=\sup\_{s\in[0,t]}W\_s$ where $W$ is a Brownian motion. I was thinking about using the fact that $S\_t\overset{d}{=}|W\_t|$ but also I don't know if two martingales with the same distribution h... | https://mathoverflow.net/users/495104 | Quadratic variation of supremum of brownian motion | The quadratic variation is identically $0$, i.e.
$$\langle S, S \rangle\_t = 0$$
for all $t$, almost surely.
To see this, note that $S$ is almost surely increasing, hence has bounded variation almost surely. Thus the quadratic variation is simply equal to the sum of the squared jumps of the process. However, $S$ ... | 7 | https://mathoverflow.net/users/173490 | 434905 | 175,842 |
https://mathoverflow.net/questions/434897 | 2 | Let's say that a limit diagram $\bar p:K^\triangleleft\to\def\sC{\mathscr{C}}\sC$ is a *weakly contractible* limit if the simplicial set $K$ is weakly contractible (in that $K\to\*$ is a weak homotopy equivalence).
I want to say for an $\infty$-category $\sC$ and an object $x\in\sC\_0$ that the forgetful functor $\sC... | https://mathoverflow.net/users/160838 | Does the forgetful functor from an over-$(\infty,1)$-category create weakly contractible limits? | You can also use Quillen's theorem A : to prove that $C \to D$ is initial, it suffices to show that for every $d\in D$, $C \times\_D D\_{/d}$ is weakly contractible.
In the case where $C\to D$ is fully faithful, this is always the case for $d$ in the image of $C$ as this pullback has a terminal object. So here we are... | 5 | https://mathoverflow.net/users/102343 | 434906 | 175,843 |
https://mathoverflow.net/questions/434600 | 1 | Suppose $k$ is a field. I wonder when the Witt ring of the quadratic forms $\textbf{W}(k)$ has a projective fundamental ideal, which is the kernel of the rank modulo 2 morphism. Here I want a sufficient condition on $k$.
| https://mathoverflow.net/users/149491 | Projectivity of the fundamental ideal of Witt groups | You have two easy families of examples.
* $k$ is a quadratically closed field, that is a field in which every element is a square. In this case , $I(k)=0$, and is free.
* $k$ is a Euclidean field, that is a field in which squares form an ordering. Real closed fields are euclidean, but there exist euclidean fields whi... | 0 | https://mathoverflow.net/users/36683 | 434908 | 175,844 |
https://mathoverflow.net/questions/434846 | 1 | Disclaimer: I first tried to ask this question on stackexchange <https://math.stackexchange.com/questions/4577320/sufficient-conditions-for-a-right-exact-functor-to-be-a-left-adjoint> but I did not get an answer there.
I know that by the adjoint functor theorem, for a right exact functor between Abelian categories to... | https://mathoverflow.net/users/161063 | Sufficient condition for right exact functor to be a left adjoint | Any additive functor between additive categories preserves finite coproducts and hence finite products, since they can be characterized as biproducts.
A right exact functor preserves all colimits iff it preserves coproducts.
Coproducts can be described as filtered colimits of finite coproducts. Hence, a right exact... | 1 | https://mathoverflow.net/users/2841 | 434909 | 175,845 |
https://mathoverflow.net/questions/434910 | 12 | Are there models of ZFC in which $\mathfrak{r}$ is strictly less than $\mathfrak{s}$? I've not been able to find any forcings that end up with this result.
Here $\mathfrak{r}$ is the reaping number $\lvert\lvert([\omega]^\omega,[\omega]^\omega,\text{does not split})\rvert\rvert$ and $\mathfrak{s}$ is the splitting nu... | https://mathoverflow.net/users/478588 | Can we force $\mathfrak{r}<\mathfrak{s}$? | The inequality $\mathfrak{r} \leq \mathfrak{u}$ is provable in ZFC (because every base for an ultrafilter is a reaping family). Blass and Shelah proved the consistency of $\mathfrak{u} < \mathfrak{s}$ in
>
> *Blass, Andreas; Shelah, Saharon*, [**There may be simple $P\_{\aleph\_1}$- and $P\_{\aleph\_2}$-points and ... | 11 | https://mathoverflow.net/users/70618 | 434920 | 175,848 |
https://mathoverflow.net/questions/434914 | 2 | In his paper [Constructive Renormalization Theory](https://arxiv.org/pdf/math-ph/9902023.pdf), V. Rivasseau describes the idea of Wilson's approach of solving path integrals step by step. In section 1.4, page 5, however, there is a statement which I do not follow. He discusses a $\phi^{4}$ theory and decomposes the cov... | https://mathoverflow.net/users/150264 | The ultraviolet limit as a limiting case of the renormalization group flow? | The telescopic argument you mentioned is incorrect. If you carefully look at Rivasseau's notations, you will see that the first term $C^{0}(p)$ in the sum
$$
\sum\_{j=0}^{\rho}C^{j}(p)
$$
is defined as
$$
\int\_{1}^{\infty} e^{-\alpha(p^2+m^2)}d\alpha\ .
$$
So the sum is just the integral over the range $[M^{-2\rho},\i... | 1 | https://mathoverflow.net/users/7410 | 434923 | 175,849 |
https://mathoverflow.net/questions/434924 | 5 | The Waring rank of a degree-$d$ homogeneous polynomial $p$ is the least integer $r$ such that you can write $p$ as a linear combination of $r$ $d$-th powers of linear forms $\{\ell\_k\}$:
$$
p = \sum\_{k=1}^r c\_k \ell\_k^d
$$
for some scalars $\{c\_k\}$. Lets use $\operatorname{rank}(p)$ to denote the Waring rank of $... | https://mathoverflow.net/users/11236 | Waring rank of monomials, and how it depends on the ground field | The answer to question 1 is affirmative. There are several lower bounds in various papers. I'll take the idea from <https://arxiv.org/abs/1503.08253> (Buczyński and myself, "Some examples of forms of high rank"). For any homogeneous form $F$, let $\operatorname{Derivs}(F)$ be the vector space spanned by all the partial... | 7 | https://mathoverflow.net/users/88133 | 434935 | 175,851 |
https://mathoverflow.net/questions/434853 | 1 | *This is a follow-up to [this previous question](https://mathoverflow.net/q/434802/102946), but under stronger assumptions.*
Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real
scalar field). Let $\tilde g: X \to \mathbb{R}$ be a measurable function and assume that there exis... | https://mathoverflow.net/users/102946 | Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II | Gro-Tsen's answer to your previous question provides a counterexample if you define $D$ to be all vectors in $\ell\_2$ that are of the form $\sum\_n a\_n f\_n$, where
$f\_n = e\_n + e\_{n+1}$, $(e\_n)$ is the unit vector basis for $\ell\_2$, and $\sum |a\_n| < \infty.$ $D$ is the range of a bounded linear operator from... | 2 | https://mathoverflow.net/users/2554 | 434936 | 175,852 |
https://mathoverflow.net/questions/434787 | 0 | I'm reading *Theorem 1* at page 98 of *Vector Measures* by Joseph Diestel, John Jerry Uhl. Here we use the [Bochner integral](https://math.stackexchange.com/questions/4298588/dominated-convergence-theorem-for-banach-space).
>
> **Theorem 1** Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, $1 \leq p<... | https://mathoverflow.net/users/99469 | If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all $f \in L_{p}(\mu, X)$, then $H=0$ $\mu$-a.e | It is always true that $L\_q(\mu,X^\*)\hookrightarrow L\_p(\mu,X)^\*$ isometrically (without the assumption that $X$ has the Radon-Nikodym property). We need the Radon-Nikodym property to guarantee that this isometric embedding is a surjection.
From this it follows that for any $T\in L\_q(\mu,X^\*)$, $$\|T\|=\sup\Big... | 3 | https://mathoverflow.net/users/469053 | 434937 | 175,853 |
https://mathoverflow.net/questions/434941 | 2 | In type theory, proving a statement means to exhibit an instance/element of a type corresponding to the statement. But if the statement is undecidable, no element of the type A nor its negation A → ⊥ can be generated. How can be proven that the statement A is undecidable?
| https://mathoverflow.net/users/495133 | Undecidable statements in type theory | "A is unprovable" is a shortcut of "A is unprovable in the theory T": provability is always relative to a specified theory.
The statement "A is unprovable in the theory T" cannot be a statement of the theory T itself, as the rules and axioms that define T are expressed in a meta-language "outside" T.
A standard way... | 3 | https://mathoverflow.net/users/110166 | 434945 | 175,855 |
https://mathoverflow.net/questions/434944 | 1 | Let $(a\_0,a\_1,\dotsc)$ be an infinite sequence of natural numbers containing all the natural numbers. Assume that $a\_x\equiv a\_y\pmod{m}$ if and only if $x\equiv y\pmod{m}$, for all $x,y\in\mathbb{N}$ and $m\in\mathbb{N}\_{\geq 1}$. Prove that $a\_n=n$ for all $n\in\mathbb{N}$.
I have tried to prove using inducti... | https://mathoverflow.net/users/495134 | Let $(a_0,a_1,\dotsc)$ be an infinite sequence of natural numbers such that $a_x\equiv a_y\pmod{m}$ iff $x\equiv y\pmod{m}$. Prove that $a_n=n$ | If $a\_x=a\_y$, then $a\_x\equiv a\_y\pmod{m}$ for all positive integers $m$, hence $x\equiv y\pmod{m}$ for all positive integers, so that $x=y$. Therefore $n\mapsto a\_n$ is a permutation of the natural numbers. Now the positive integer $m:=|a\_{n+1}-a\_n|$ satisfies $a\_{n+1}\equiv a\_n\pmod{m}$, whence $n+1\equiv n\... | 6 | https://mathoverflow.net/users/11919 | 434946 | 175,856 |
https://mathoverflow.net/questions/434927 | 3 | Let $\kappa\geq \aleph\_0$ be a cardinal. If $X\neq \emptyset$ is a set, we say that a family ${\cal C}\subseteq {\cal P}(X)$ has *property ${\bf B}$* if there is $S\subseteq X$ such that for all $C\in {\cal C}$ we have $S \cap C \neq \emptyset \neq C\setminus S$. (In other words, $S$ intersects every $C$ but contains ... | https://mathoverflow.net/users/8628 | Property ${\bf B}$ for families of large sets with small intersection | EDIT: I'll leave my previous answer up for now (at the end of this one), but here's an easier answer that doesn't need assumptions like CH that go beyond ZFC.
It's well-known that there is a family of continuum many infinite subsets of $\omega$ such that every two have finite intersection. List such a family as $\{A\... | 6 | https://mathoverflow.net/users/6794 | 434949 | 175,858 |
https://mathoverflow.net/questions/434943 | 3 | Let $k\_{1},\dots, k\_{d}>1$ be integers and consider the integral
$$J\_{\lambda }=\int\_{\mathbb{S}^{d-1}}e^{-\lambda \left(x^{2k\_{1}}\_{1}+\dots+ x^{2k\_{d}}\_{d}\right)} d\sigma(x)$$
where $d\sigma$ denotes the standard surface measure on $\mathbb{S}^{d-1}$, the unit sphere in $\mathbb{R}^{d}$, $d\geq 2$.
I can n... | https://mathoverflow.net/users/116555 | What is the optimal asymptotic behavior of this integral over the sphere? | $\newcommand\la\lambda\renewcommand{\S}{\mathbb S}\newcommand{\si}{\sigma}$Let us show that
\begin{equation\*}
J\_\la=e^{-\la(m+o(1))} \tag{1}\label{1}
\end{equation\*}
(as $\la\to\infty$), where
\begin{equation\*}
m:=\min\_{x\in\S^{d-1}}s(x),\quad s(x):=\sum\_1^d x\_j^{2k\_j}
\end{equation\*}
for $x=(x\_1,\dots,x\_... | 4 | https://mathoverflow.net/users/36721 | 434955 | 175,860 |
https://mathoverflow.net/questions/434845 | 1 | I have 2 normal distributions $\mathcal{N}(\mu\_1, \mathbb{I}\_d)$ where $\mu\_1$ is a fixed vector in $\mathbb{R}^d$ and $\mathcal{N}(\mu\_2, \mathbb{I}\_d)$, where $\mu\_2$ is $\mu\_1 + V$, where $V$ is uniformly distributed and orthogonal to $\mu\_1$, that is $V^\top \mu\_1 = 0$. (In 3 dimensions $V$ would be unifor... | https://mathoverflow.net/users/325572 | KL divergence between gaussian with uniform prior | $\newcommand{\R}{\mathbb R}\newcommand{\KL}{{\operatorname{KL}}}$For $j=1,2$, let $P\_j:=N(\mu\_j,I\_d)$, where $\mu\_2=\mu\_1+v$ and $v$ is a unit vector. So, for the pdf's $p\_j$ of $P\_j$ we have
\begin{equation\*}
p\_j(x)=(2\pi)^{-d/2} e^{-|x-\mu\_j|^2/2}
\end{equation\*}
for all $x\in\R^d$, where $|\cdot|$ is the... | 2 | https://mathoverflow.net/users/36721 | 434957 | 175,861 |
https://mathoverflow.net/questions/434172 | 2 | $\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\pr{pr}$I have some problems to understand the proof of *Proposition 1.5* from Mumford's [Geometric Invariant Theory](https://link.springer.com/book/9783540569633), p 34:
*Corollary 1.5*
Let $G$ be a conne... | https://mathoverflow.net/users/108274 | Proposition 1.5 in Mumford's Geometric Invariant Theory | I think the fist part of the proof of 1.5 may be rephrased as follows. The hypothesis that $[\mathcal{L}^n]$ is fixed by the $G$-action exactly means that for all $g \in G$, there is an isomorphism:
$$ \sigma^\* \mathcal{L}^n \big|\_{\{g\} \times X} \simeq p\_2^\* \mathcal{L}^n|\_{\{g\} \times X}.$$ Put differently, fo... | 1 | https://mathoverflow.net/users/37214 | 434966 | 175,862 |
https://mathoverflow.net/questions/434463 | 0 | I'm reading *Theorem 1* at page 98 of *Vector Measures* by Joseph Diestel, John Jerry Uhl.
>
> **THEOREM 1.** Let $(\Omega, \Sigma, \mu)$ be a **finite** measure space, $1 \leq p<\infty$, and $X$ be a Banach space. Then $L\_{p}(\mu, X)^\*=L\_{q} (\mu, X^\*)$ where $p^{-1}+q^{-1}=1$, if and only if $X^{\*}$ has the ... | https://mathoverflow.net/users/99469 | Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces? | Below is my formalization of @Nik's hints to finish the proof.
---
Let's prove that
$$
\sum\_{m=1}^M \|H\_m\|^q\_{L\_{p}(\mu\_m, X)^\*} \le \|H\|^q\_{L\_{p}(\mu, X)^\*} \quad \forall M \in \mathbb N^\*.
$$
Let $\Omega' := \bigcup\_{m=1}^M \Omega\_m$. We define a measure $\mu'$ on $\Omega$ by
$$
\mu' (B) := \mu(... | 1 | https://mathoverflow.net/users/99469 | 434973 | 175,863 |
https://mathoverflow.net/questions/434968 | 1 | Let $X,Y$ be Polish spaces and $c:X \times Y \to [0, \infty]$ lower semi-continuous. There is a sequence $(c\_\ell)\_{\ell \in \mathbb N}$ with $c\_\ell:X \times Y \to [0, \infty)$ of bounded Lipschitz continuous functions such that $c\_\ell \uparrow c$ pointwise. Fix $\varphi:X \to \mathbb R$. The following is taken f... | https://mathoverflow.net/users/99469 | Optimal transport: how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$? | Are you sure there aren’t additional conditions on $\varphi$? Because otherwise taking $X = \mathbb R$ and $Y$ to be a one point space, the following gives a counterexample:
$c(x, y) = 0$ if $x = 0$; $c(x, y) = 1$ otherwise, and
$\varphi(x) = 0$ if $x = 0$, $\varphi(x)= 2$ otherwise.
Indeed, $\varphi^c = -1$, whi... | 1 | https://mathoverflow.net/users/173490 | 434975 | 175,864 |
https://mathoverflow.net/questions/434967 | 1 | Let $X$ be a multivariate normal random variable with mean $\boldsymbol{\mu}$ and variance matrix $\mathrm{\Sigma}$. Next, define
Suppose that $Y = AX$ where $A$ is appropriate matrix. Can we say that the distribution of $Y$ is same as $X$ if and only if $A$ is an orthogonal symmetric matrix?
Thank you Iosif for po... | https://mathoverflow.net/users/120111 | Multivariate normal distribution and orthonormal transformation | $\newcommand\Si\Sigma$First here, there is no such thing as an orthonormal matrix. So, let us assume that you meant an orthogonal matrix instead.
Then we have this question:
>
> Suppose that $X\sim N(\mu,\Si)$ and $Y = AX$, where $A$ is appropriate matrix. Can we say that the distribution of $Y$ is same as $X$ if... | 1 | https://mathoverflow.net/users/36721 | 434980 | 175,866 |
https://mathoverflow.net/questions/434984 | 1 | Let $F$ be a nonarchimedean local field, say, charactersitic $0$. Is there any general theorem that tells when $\sqrt{-1}$ exists in $F$? How often does it happen?
| https://mathoverflow.net/users/32746 | How often does $-1$ have a square root in a local field? | For a $p$—adic field $K$ (a finite extension of $\mathbf Q\_p$), Hensel’s lemma for $f(x) = x^2+1$ with initial guess $x=a$ in $\mathcal O\_K^\times$ says a sufficient condition for $f(x)$ to have a root near $a$ in $K$ is $|f(a)| < |f’(a)|^2$, which says $|a^2+1| < |4|$. That is equivalent to $a^2+1 \equiv 0 \bmod 4\p... | 3 | https://mathoverflow.net/users/3272 | 434987 | 175,868 |
https://mathoverflow.net/questions/434922 | 1 | I want to solve this PDE with boundary conditions
$$
{u\_{xy}} + y{u\_y} = 0\,\,\,\,\,x > 0,y > 0\,,\,u\left( {x,0} \right) = {e^x},u\left( {0,y} \right) = \cos y
$$
I did the following
\begin{array}{l}
{u\_{xy}} + y{u\_y} = 0\,\,\,\,\,x > 0,y > 0\,,\,u\left( {x,0} \right) = {e^x},u\left( {0,y} \right) = \cos y\\
{u\_y... | https://mathoverflow.net/users/495120 | A PDE with boundary condition | If you write the equation as $(e^{xy}u\_y)\_x = 0$, then the boundary conditions tell you that $e^{xy} u\_y = e^0u\_y(0,y) = -\sin y$, so
$$
u\_y(x,y) = -\sin y\,e^{-xy},
$$
so the solution is
$$
u(x,y) = e^x - \int\_0^y\sin t\,e^{-xt}\,dt = e^x + {\frac {{{\rm e}^{-xy}} \left( x\,\sin y +\cos y \right) - 1 }{{x}^{2}+1... | 2 | https://mathoverflow.net/users/13972 | 434988 | 175,869 |
https://mathoverflow.net/questions/434932 | 2 | **Context**. The question arises from my former [question on the remainder of a power series](https://mathoverflow.net/questions/396814/on-the-remainder-of-a-power-series-evaluated-on-the-boundary-of-its-convergence). Precisely, I was trying to understand if the boundary behavior of power series considered by Ricci in ... | https://mathoverflow.net/users/113756 | On a lemma of Łojasiewicz in complex analysis of one variable | The assumption of the Lemma is that $f$ has a finite limit as $z\to \zeta\_0$. This assumption does not hold in any of the two examples that you mention. In these examples, $f$ has a limit only when $z\to 1$ AND $z$ is real.
The Lemma is also true for non-tangential limits, with essentially the same proof but again n... | 4 | https://mathoverflow.net/users/25510 | 434989 | 175,870 |
https://mathoverflow.net/questions/434981 | 5 | Any $z \in \widehat{\mathbb{Z}} = \lim\_{n} \mathbb{Z}/n\mathbb{Z}$ defines an operation on all finite groups: if $G$ is a finite group and $g \in G$, say $g^n=1$, then map it to $g^{z\_n}$. This defines a map of sets $G \to G$, which is natural with respect to all homomorphisms of finite groups. The same works for tor... | https://mathoverflow.net/users/2841 | Classification of natural endomorphisms on finite groups | It suffices to look at symmetric groups, as these are generated by transpositions. Indeed, let $z \in \widehat{\mathbf Z}$ and suppose it induces a homomorphism $\phi\_z$ on $S\_n$ for all $n$ given by $g \mapsto g^{z\_n}$ if $g$ has order $n$. We will show that $z$ is either $0$ or $1$.
* If $z\_2 = 0 \in \mathbf Z/... | 11 | https://mathoverflow.net/users/82179 | 434991 | 175,871 |
https://mathoverflow.net/questions/434658 | 9 | We know that an infinite dimensional Banach space has an uncountable Hamel basis. Now if $X$ is a vector space with an uncountable Hamel basis, does there exist a norm on $X$ for which $X$ is a Banach space. I could not proceed. Any help is appreciated.
| https://mathoverflow.net/users/41137 | Banach space with uncountable basis | I will pull together the comments into a community wiki answer with some of my own remarks so that the question isn't left on the unanswered questions list.
If you're willing to accept that it is consistent that $\aleph\_1 < 2^{\aleph\_0}$, you can get a relatively small example of an impossible uncountable Hamel dim... | 8 | https://mathoverflow.net/users/61785 | 434995 | 175,872 |
https://mathoverflow.net/questions/413071 | 5 | Let $M$ be a continuous martingale such that almost surely, the sample paths of $M$ are not constant.
**Question:** Is it true that $M$ is almost surely not differentiable?
| https://mathoverflow.net/users/173490 | Can an a.s. non constant continuous martingale be differentiable with nonzero probability? | Almost surely, we can write for every $t \ge 0$, $M\_t=M\_0+\beta\_{\langle M,M \rangle\_t}$, where $\beta$ is some Brownian motion.
By Kahane's theorem, almost surely, for every $s \ge 0$, $\limsup\_{\delta \to 0+} \delta^{-1/2}|\beta\_{t+\delta}-\beta\_t| \ge 1$.
<https://www.ams.org/journals/tran/1986-296-02/S0002... | 1 | https://mathoverflow.net/users/169474 | 434996 | 175,873 |
https://mathoverflow.net/questions/434992 | 0 | Consider $f:\mathbb{R}^{2}\_{0} \to \mathbb{R}\_{0}$ such that $f(x,y)$ is a continuous function and satisfies the following properties:
1. $f(x,y) = f(y,x)$
2. $f(tx,ty) = tf(x,y) \ \forall \ t > 0 $
3. $f(1,1) = 1$
**Can we show that if $g(x,y) := 3f(x,y) - 2(x+y)$, then $\underset{x,y}{\text{argmax}}[g(x,y)] = (... | https://mathoverflow.net/users/495167 | Find an optimizer for $g(x,y)$ if it exists | By conditions 2 and 3, $f(t,t)=t$ for all $t>0$. By continuity, $f(0,0)=\lim\_{t\to 0^+} f(t,t)=\lim\_{t\to 0^+}t=0$. From this it follows that $g(0,0)=0$.
As you said, if $g$ has a maximizer, then the maximum value must be $0$. We also know that $g(0,0)=0$. So if we're interested in showing that $(0,0)$ is a maximiz... | 0 | https://mathoverflow.net/users/469053 | 435001 | 175,874 |
https://mathoverflow.net/questions/435010 | 9 | Re-posted from math.stackexchange as I did not get any answers [there](https://math.stackexchange.com/questions/4575919/why-do-almost-all-points-in-the-unit-interval-have-kolmogorov-complexity-1).
I am reading
* Jin-yi Cai, Juris Hartmanis, [On Hausdorff and topological dimensions of the Kolmogorov complexity of th... | https://mathoverflow.net/users/485890 | Why do almost all points in the unit interval have Kolmogorov complexity 1? | I'm not an expert on Kolmogorov complexity, but this does seem like a counting argument: for any fixed $\epsilon > 0$, there are only $\sum\_{i = 1}^{(1-\epsilon)n} 2^i < 2^{(1-\epsilon)n+1}$ programs of length less than or equal to $(1-\epsilon)n$, and so fewer than $2^{(1-\epsilon)n+1}$ strings of $n$ bits can be des... | 21 | https://mathoverflow.net/users/116357 | 435011 | 175,877 |
https://mathoverflow.net/questions/434901 | 0 | Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n) $ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ be an ideal of $R$ such that $\text{depth}\_R(R/J)=0$. Then, is it true that $\text{depth}\_S\left(\dfrac {S}{JS+xS}\rig... | https://mathoverflow.net/users/493381 | On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings | Let $R=k[[u,v]]/(u^2,uv)$ and $S=R[x]/(x^2-u)$. Let $J=0$. You can check that $S/xS$ has depth one but $R$ has depth zero.
| 2 | https://mathoverflow.net/users/9502 | 435012 | 175,878 |
https://mathoverflow.net/questions/434261 | 5 | The following question is motivated from Chapter 2 (Generalized Hodge Systems in 2D), particularly Section 2.3 ($L^p$ theory for Hodge systems in 2D) of Christodoulou and Klainerman's book, *The global nonlinear stability of the Minkowski space.*
In this section (page 43, in my copy), the authors state that the Calde... | https://mathoverflow.net/users/147016 | Reference for Calderon-Zygmund $L^p$ inequalities on the sphere | I don't know of an exact reference, and in general this sort of result (transfering a "classical" result from the analysis of PDE in $\mathbb{R}^n$ to Riemannian manifolds) is often quite hard to track down. Instead, I can explain how you derive the inequalities you want from the standard Calderon-Zygmund estimates for... | 7 | https://mathoverflow.net/users/380 | 435021 | 175,883 |
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