parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/392675 | 6 | [Sharkovskii's theorem](https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem) is a fascinating result and ready to stand on its own. But is it also used somewhere? Are there other theorems that rely on it or somehow use the Sharkovskii ordering?
| https://mathoverflow.net/users/175280 | Uses for Sharkovskii's theorem | Quoting from Thomas D Rogers, Remarks on Sharkovsky's Theorem, Rocky Mountain J. Math. 15 (1985) 565-569:
In regard to applications of Sharkovsky's theorem (difference equation
models) it is interesting that the result is sturdy to perturbations in $f$.
Block [5] shows that if $f$ has a point of period $n$, then ther... | 2 | https://mathoverflow.net/users/3684 | 392736 | 162,460 |
https://mathoverflow.net/questions/392667 | 3 | **Problem set up:**
Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space.
We say that a measure preserving transformation $T$ on $\mathbf X$ is *$\varepsilon$-almost mixing* if for every $\delta > \varepsilon$, and every pair of non-null measurable sets $A, B \in \mathcal A$, there exists an $N >... | https://mathoverflow.net/users/173490 | Does an “almost mixing” transformation admit a non-null ergodic component? | A paper of Martin and England (<https://www.ams.org/journals/bull/1968-74-03/S0002-9904-1968-11982-2/S0002-9904-1968-11982-2.pdf>) shows (in your language) that if $T$ is $\epsilon$-almost mixing for any $\epsilon<1$, then $T$ is weak-mixing (and hence ergodic).
| 4 | https://mathoverflow.net/users/11054 | 392740 | 162,461 |
https://mathoverflow.net/questions/392734 | 5 | In the paper [Reconstruction of hidden symmetries](https://www.mathematik.uni-muenchen.de/%7Epareigis/Papers/Hidsym.pdf) of Bodo Pareigis in the subsection "3.1 Reconstruction of coalgebras" there is the following proposition (3.3.).
Let $\mathcal{C}$ be a braided monoidal category, $\mathcal{A}$ a $\mathcal{C}$-mono... | https://mathoverflow.net/users/140716 | Reconstruction of coalgebras | The diagram
$\require{AMScd}$
\begin{CD}
\omega @>{\delta}>> \omega\otimes C\\
@VV{\delta}V @VV{1\_{\omega}\otimes\Delta}V \\
\omega\otimes C @>{\delta\otimes 1\_{C}}>> \omega\otimes C\otimes C \\
@V{1\_{\omega}\otimes\epsilon}VV @VV{1\_{\omega\otimes C}\otimes\epsilon}V \\
\omega\otimes1 @>\delta\otimes1\_1>> \omega\o... | 2 | https://mathoverflow.net/users/41291 | 392751 | 162,464 |
https://mathoverflow.net/questions/392739 | 1 | Formally, let $K$ be a field, $a \in K[x]$, $D \subseteq K$, and $E = \{(x\_i, a(x\_i)) \mid x\_i \in D\}$ be distinct evaluations of $a$ where $\lvert E\rvert > \operatorname{deg}(a)$ (so $E$ uniquely determines $a$).
Suppose that $D$, $E$ are partitioned into disjoint subsets $D = \bigcup D\_j$, $E = \bigcup E\_j$,... | https://mathoverflow.net/users/8846 | Can a polynomial be evaluated from evaluations of partial interpolations? (Or: can the unique solution of congruences be written in a certain way?) | No. The interpolating polynomial is a weighted sum $a(x) = \sum\_{x\_i} a(x\_i) P\_i(x)$ and the independence of the $a(x\_i)$ from each other imposes the independence of the $P\_i$ from each other, which isn't consistent with your desired decomposition.
A short counter-example will give the intuition. Take $D\_1 = \... | 4 | https://mathoverflow.net/users/46140 | 392752 | 162,465 |
https://mathoverflow.net/questions/392753 | 14 | In all that follows, let $k$ be a field and $G$ be a finite group.
It is well-known that the order of $G$ is invertible in $k$ iff the group ring $k[G]$ is semisimple, which is equivalent, *inter alia*, to the fact that $\operatorname{Ext}^1\_{k[G]}(V,W)$ vanish for all $V,W$ left $k[G]$-modules (= $k$-linear represe... | https://mathoverflow.net/users/17064 | Global homological dimension of group rings | Here is another proof that the global dimension is infinite that is specific to groups and explicitly identifies a module of infinite projective dimension. Let $G$ be a finite group and suppose that the characteristic $p$ of $k$ divides the order of $G$. Then I claim that the trivial $kG$-module has infinite projective... | 14 | https://mathoverflow.net/users/15934 | 392759 | 162,469 |
https://mathoverflow.net/questions/392427 | 2 | Let $\mathbf{U}$ be a random unitary matrix and $\mathbf{z}$ be a random i.i.d complex Gaussian vector (unitary invariant). Assume that the following relation is satisfied:
\begin{align}
\mathbf{y}=\mathbf{U}\mathbf{s}+\mathbf{z}.
\end{align}
Are the following relations hold for the mutual information between $\mathbf{... | https://mathoverflow.net/users/68835 | Mutual Information after Applying Random Unitary Matrix | No, this is not correct. Consider as a counterexample the case that $s$ can take only two values, a unit vector $e$ or minus $e$. Since $s$ is rotated randomly to construct $y=Us+z$, knowledge of $s$ gives you no information on $y$, so the mutual information $I(s,y)=0$.
On the other hand, knowledge of $s$ does give y... | 2 | https://mathoverflow.net/users/11260 | 392761 | 162,470 |
https://mathoverflow.net/questions/392757 | 2 | Consider the *word* a.k.a. *list monad* $W:\mathrm{Set}\to\mathrm{Set}$ assigning to a given set $M$ the set of words over $M$. Now, given a set $M$, we can assign to every word $w\in WM$ a subset of $M$:
$$
\mathrm{supp}(w):=\{\textrm{ all elements of $M$ occurring in $w$ }\} \subseteq M
$$
that we could call the ... | https://mathoverflow.net/users/1261 | Finitary endofunctors: "Support" of elements of $TM$ where $T:\mathrm{Set}\to\mathrm{Set}$ is finitary | This seems like too much to hope for at this level of generality. For example take $T$ to be the "(-1)-truncation" functor
$$
T(\varnothing) = \varnothing, \qquad T(M) = \* \,\, \mbox{for $M \ne \varnothing$}.
$$
Then let $M$ be an arbitrary infinite set. The support map would have to take the element of $TM$ to a choi... | 5 | https://mathoverflow.net/users/126667 | 392764 | 162,472 |
https://mathoverflow.net/questions/392747 | 4 | I'd like to do the following: I consider a separable Banach space $X$ with a probability measure $\mu$ on the Borel $\sigma$-algebra $\mathcal B(X)$. Additionally, I have a sequence of measurable, linear and bounded operators $T\_n: X\to X$ which converges $\mu$-almost-surely, i.e. there is a set $A\in \mathcal B(X)$ w... | https://mathoverflow.net/users/88505 | Uniform boundedness principle for almost surely converging sequence of operators | No, it's not true.
Let's take $X$ to be a Hilbert space for simplicity. I will produce a non-degenerate Gaussian measure $\mu$ on $X$, and a sequence of bounded linear functionals $f\_n \in X^\*$ such that $f\_n(x) \to 0$ for $\mu$-a.e. $x$, but $\|f\_n\|\_{X^\*} \to \infty$. You can turn this into the desired counte... | 4 | https://mathoverflow.net/users/4832 | 392772 | 162,473 |
https://mathoverflow.net/questions/392771 | 4 | Let $\pi\_1: A\_1 \to B\_1$ and $\pi\_2: A\_2 \to B\_2$ be positive linear maps between complex $\*$-algebras. Is the mapping
$$\pi\_1 \otimes \pi\_2: A\_1 \otimes A\_2 \to B\_1 \otimes B\_2$$
again positive?
I.e., if $\sum\_{i=1}^n x\_i \otimes y\_i \in A\_1 \otimes A\_2$, do we have
$$\sum\_{i,j=1}^n \pi\_1(x\_i^\*... | https://mathoverflow.net/users/216007 | Tensor product of positive linear maps is positive | No. A standard example is given by $A\_1 = A\_2 = B\_1 = B\_2 = M\_2(\mathbb{C})$, where we choose $\pi\_1$ to be the identity map and $\pi\_2$ to be the transpose map. These maps are positive, but $\pi\_1 \otimes \pi\_2$ is not positive since, for example
$$
(\pi\_1 \otimes \pi\_2)\left(\begin{bmatrix}
1 & 0 & 0 & 1 ... | 7 | https://mathoverflow.net/users/11236 | 392777 | 162,475 |
https://mathoverflow.net/questions/392773 | 0 | As is known, the rank-1 PCA aims to solve the following optimization problem
$$\min\_{x\in\mathbb{R}^d}\quad -x^T \Sigma x\quad\quad\quad \text{s.t.}\quad \Vert x\Vert\_{2}=1,$$
where $\Sigma\in\mathbb{S}^{d}$ is the covariance matrix. Thus the optimum $x^\*$ of the PCA problem is the top unit eigenvector of $\Sigma$. ... | https://mathoverflow.net/users/207719 | PCA, relation between the error and variance | $\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}$Let us work in an orthonormal eigenbasis of $\Si$. Then without loss of generality $\Si$ is the diagonal matrix with diagonal entries $\la\_1\ge\la\_2\ge\cdots\ge\la\_d\ge0$, $x\_\*:=x^\*=[1,0,\dots,0]^T$, $x:=\tilde x=[x\_1,x\_2,\dots,x\_d]\i... | 2 | https://mathoverflow.net/users/36721 | 392778 | 162,476 |
https://mathoverflow.net/questions/392767 | 1 | Let $G$ be a $\{2,3\}$-group and $\lvert G\rvert=2^\alpha\cdot3^\beta$. For $p\in\{2,3\}$, define
$$
\nu\_p(G)\mathrel{:=}\min\left\{\log\_p\left(\frac{\lvert G\rvert}{\chi(1)}\right)\_p \mathrel{\bigg\vert} \chi\in\operatorname{Irr}(G)\right\},
$$
where $\operatorname{Irr}(G)$ is the set of all irreducible $\mathbb{C... | https://mathoverflow.net/users/99750 | The $\{2,3\}$-groups with a condition about $\mathbb{C}$-characters | If I understand your conditions correctly, you are assuming that there is a $3$-block of $G$ of defect zero (this corresponds to an irreducible character $\chi$ with $\chi(1)\_{3} = |G|\_{3}$) and a $2$-block of $G$ of defect $1$ (for in general if a finite group $G$ has order divisibly by $p^{a}$ (but by no higher pow... | 4 | https://mathoverflow.net/users/14450 | 392790 | 162,481 |
https://mathoverflow.net/questions/392785 | 5 | I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem.
[Link to the statement of the theorem](https://i.stack.imgur.com/LKzph.png)
Basically we want to prove that the Morse homology is independent of the pseudo-gradient field an... | https://mathoverflow.net/users/220552 | Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology" | Oh, I happen to know the guy who wrote that PDF. As to your questions.
1. Yes, that's the idea. In $V \times A$, $F = f\_0$ so the critical points are in one-to-one correspondence with those of $f\_0$. The shift in degree comes from this $g$ which has a maximum at $0$. Similarly in $V \times B$ but there is no shift ... | 6 | https://mathoverflow.net/users/121144 | 392791 | 162,482 |
https://mathoverflow.net/questions/392793 | 7 | Let $A$ be a local integral domain of characteristic $p$. Let $K$ be the fraction field and let $k$ be the residue field of $A$. If $K$ is perfect, is $k$ necessarily perfect?
*Thoughts*:
* If $A$ is normal, then $K$ perfect implies that in fact $A$ itself is perfect, hence $k$ is perfect.
* If $A$ is essentially o... | https://mathoverflow.net/users/112809 | Does perfect fraction field imply perfect residue field? | The answer is no. Let $F$ be any imperfect subfield of a perfect field $F'$. Let $B$ be the ring of integral [Puiseux series](https://en.wikipedia.org/wiki/Puiseux_series) over $F'$ (integral meaning ones involving only nonnegative powers of $T$) and let $A$ be the subring consisting of those series whose coefficient o... | 12 | https://mathoverflow.net/users/30186 | 392795 | 162,483 |
https://mathoverflow.net/questions/392769 | 10 | $\renewcommand\Im{\operatorname{\mathcal{Im}}}\newcommand\Ker{\operatorname{\mathcal{Ker}}}$I was sure that this is a trivial question and placed it on Math Stackexchange
<https://math.stackexchange.com/questions/4136830/a-detail-in-the-proof-of-schurs-lemma-the-closures-of-the-cal-ker-and-cal>
Surprisingly, no one a... | https://mathoverflow.net/users/137120 | A detail in the proof of Schur's lemma: the closures of the $\mathcal{Ker}$ and $\mathcal{Im}$ of the intertwiner | Let $V$ and $W$ be Hilbert spaces with irreducible unitary $G$-actions and let $T:V \to W$ be a bounded intertwiner. Then the adjoint is an intertwiner as well and hence so are $T^\ast T$ and $TT^\ast$. Claim: These two are multiples of the identity. It follows that $T$ is zero or a multiple of an isometric isomorphism... | 6 | https://mathoverflow.net/users/9928 | 392797 | 162,485 |
https://mathoverflow.net/questions/392776 | 2 | Let $\mathcal A$ be a central hyperplane arrangement in $\mathbb R^d$ and let's assume that it is essential, meaning the hyperplanes in $\mathcal A$ intersect in the origin. The intersection lattice $L(\mathcal A)$ consists of those subspaces of $\mathbb R^d$ that are intersections of hyperplanes from $\mathcal A$ (inc... | https://mathoverflow.net/users/15934 | Seeking combinatorial interpretation of a quantity that comes up from central hyperplane arrangements | Let $\Delta$ be a matroid complex, i.e., an abstract simplicial complex whose faces are the independent sets of a matroid $M$. Let $K[\Delta]$ denote the face ring (aka "Stanley-Reisner ring") of $\Delta$ over a field $K$. Let $\beta\_i(K[\Delta])$ denote the Betti numbers of a minimal free resolution of $K[\Delta]$, r... | 7 | https://mathoverflow.net/users/2807 | 392799 | 162,486 |
https://mathoverflow.net/questions/392779 | 14 | Given a line bundle $L$ on an abelian variety $A/k$, there is an associated Weil pairing $e\_L\colon\bigwedge^2V\_pA\to\mathbb Q\_p(1)$, where $p$ is a prime different from the residue characteristic of the base field $k$ and $V\_pA$ is the $\mathbb Q\_p$-linear Tate module of $A$. This is usually constructed by explic... | https://mathoverflow.net/users/126183 | Weil pairing on abelian varieties and etale Chern classes | It seems that one of the pairings is the negative of the other (in char 0 this assertion is actually Lemma 2.6 of <https://arxiv.org/pdf/1809.01440.pdf> ).
| 8 | https://mathoverflow.net/users/9658 | 392811 | 162,490 |
https://mathoverflow.net/questions/392820 | 7 | In the usual $\mathsf{ZFC}$, we know that there are $2^\mathfrak{c}$ many subsets of $\mathbb{R}$ that are not (Lebesgue) measurable. On the other hand, the [Solovay model](https://en.wikipedia.org/wiki/Solovay_model) also provides us a model of $\mathsf{ZF}$ which has $0$ non-measurable subsets of $\mathbb{R}$.
I wo... | https://mathoverflow.net/users/146831 | Models with fixed cardinality of non-Lebesgue measurable sets | Actually, $\mathsf{ZF}$ easily proves that if there exists at least one non-measurable subset of $\mathbb{R}$, then there must be $2^\mathfrak{c}$ non-measurable subsets of $\mathbb{R}$.
Let $X \subseteq \mathbb{R}$ be non-measurable. Then $X = (X \cap [0,\infty)) \cup (X \cap (-\infty,0)) =: A \cup B$. If both $A$ a... | 6 | https://mathoverflow.net/users/146831 | 392821 | 162,492 |
https://mathoverflow.net/questions/392746 | 2 | Here are some of the classical density functions for spherical distributions (on the $\mathcal{S}^{d-1}$ sphere, living in the Euclidean space $\mathbb{R}^d$):
1. $$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/2-1}}{2 \pi^{d/2} I\_{d/2-1}(\kappa)} \exp(\kappa \mathbf{x}^{\top} \boldsymbol{\mu}), \qquad (\text{called the Fis... | https://mathoverflow.net/users/129612 | Local limit theorems for circular/spherical distributions | $\textbf{Partial answer:}$ Note that
$$\boldsymbol{x}^{\top} \boldsymbol{\mu} = -\frac{1}{2} (\boldsymbol{x} - \boldsymbol{\mu})^{\top}(\boldsymbol{x} - \boldsymbol{\mu}) + 1$$ since $\boldsymbol{x}^{\top}\boldsymbol{x} = \boldsymbol{\mu}^\top\boldsymbol{\mu} = 1$, and it is possible to show (using the integral represe... | 0 | https://mathoverflow.net/users/129612 | 392823 | 162,493 |
https://mathoverflow.net/questions/392760 | 6 | Let $[\omega]^\omega$ denote the collection of infinite subsets on $\omega$, and let $$E=\big\{\{a, b\}:a,b\in [\omega]^\omega \text{ and } |a\cap b| \text{ is finite}\big\}.$$
Is every simple, undirected graph $G=(V,E)$ with $V\leq 2^{\aleph\_0}$ isomorphic to an induced subgraph of $([\omega]^\omega, E)$? If not, wha... | https://mathoverflow.net/users/8628 | Induced subgraphs of the almost-disjointness graph | I claim that every graph with $\leq \aleph\_1$ vertices can be embedded in $[\omega]^\omega$ in the manner Dominic described. This means that we have a consistent answer to Dominic's question: the answer is *yes* assuming that the Continuum Hypothesis holds.
Recall that $\mathcal P(\omega) / \mathrm{fin}$ denotes the... | 4 | https://mathoverflow.net/users/70618 | 392836 | 162,497 |
https://mathoverflow.net/questions/324251 | 9 | $\newcommand{\k}{\mathbf k}$
$\newcommand{\A}{\mathcal A}$
$\newcommand{\B}{\mathcal B}$
$\newcommand{\C}{\mathcal C}$
I'm wondering if anyone knows a reference for the following construction: let $\k$ be a field, say, and assume for convenience everything below is $\k$-linear, and that every category is essentially sm... | https://mathoverflow.net/users/13552 | Relative cocompletion of a category | This is a special case of the general construction of cocompletions that preserve existing colimits. The general statement can be found as Theorem 6.23 of Kelly's *Basic Concepts of Enriched Category Theory*, and more explicitly as Proposition 11.4 and Theorem 11.5 of Fiore's *Enrichment and Representation Theorems for... | 6 | https://mathoverflow.net/users/152679 | 392842 | 162,498 |
https://mathoverflow.net/questions/392825 | 2 | I have derived an optimization objective of the form
$$
f(x) = \sum\_{i,j} C\_{ij}\min(x\_i, x\_j), s.t. g(x) \geq 0
$$
where $C \in \mathcal{R}^{N \times N}$ is a positive definite matrix, and $x \in \mathcal{R}^{N}$ is a vector where each element $ x\_i \geq 1 $. Additionally, $ g(x) \geq 0 $ constrains the solution ... | https://mathoverflow.net/users/221614 | Convexity of a positive definite objective with min(x,y)-nonlinearity | $f(x)$ is not convex. Here is a counterexample to its convexity in MATLAB notation.
```
C = [2 1;1 2]
x1 = [1 2]'
x2 = [2 1]'
x3 = 0.5*(x1 + x2)
```
Then
```
f(x1) = f(x2) = 8
f(x3) = 9 > 0.5*(f(x1) + f(x2))
```
| 1 | https://mathoverflow.net/users/75420 | 392843 | 162,499 |
https://mathoverflow.net/questions/392808 | 0 | I found this problem I have tried but it has been a bit complicated for me,
Let $f:\mathbb{R}\to\mathbb{R}$ a bounded function. For each $\epsilon>0$, let $f\_\epsilon (x)=\inf\{f(y):|y-x|<\epsilon\}$. They ask us to prove that:
1. For each $\epsilon>0$, the function $f\_\epsilon$ is measurable Borel.
>
> I tri... | https://mathoverflow.net/users/171387 | $f_\epsilon=\inf\{f(y):|y-x|<\epsilon\}$ is measurable Borel | As noted, your idea is right. Let me try to do the proof with greater clarity; to make it easier for the reader to understand.
---
1.
$f\_\epsilon (x)=\inf\{f(y):|y-x|<\epsilon\}$.
Fix $\epsilon>0$. We claim that $f\_\epsilon$ is Borel measurable. [In fact, even more is true: $f\_\epsilon$ is upper semiconti... | 1 | https://mathoverflow.net/users/454 | 392849 | 162,502 |
https://mathoverflow.net/questions/392833 | 24 | Two closely related, but different tasks in combinatorics are
1. determining the *number* of elements in some set $A$, and
2. presenting all its *elements* one by one.
**Question:** What are some works in combinatorics literature that explicitly
consider the naming of these different tasks?
---
As a backgroun... | https://mathoverflow.net/users/171662 | The verbs in combinatorics: Enumerating, counting, listing and all that | I'm not sure if this is exactly what you're looking for, but the main topic of Herb Wilf's article [What is an Answer?](https://www.jstor.org/stable/2321713) is how to answer the question "How many \_\_\_\_\_\_ are there?" His basic thesis is that an alleged answer to such a question is satisfactory only if it provides... | 24 | https://mathoverflow.net/users/3106 | 392861 | 162,504 |
https://mathoverflow.net/questions/392850 | 0 | This is from Silverman's 'the arithmetic of elliptic curves', exercise 5.5.
Let $K$ be an imaginary quadratic field, and let $R\_1...R\_n$ be orders in $K$.
I would like to prove that there are more than 2 prime numbers $p$ such that $pR\_i$ is a prime ideal of $pR\_i$ for all $i$.
By using this, we can characteriz... | https://mathoverflow.net/users/144623 | Common prime of the finite number of order of imaginary quadratic field | If $R$ is a (nonmaximal) order in $O\_K$, then it has finite index in $O\_K$. If $p$ is a prime not dividing this index indices, the obvious map $R\to O\_K/pO\_K$ is surjective and its kernel is $pR$, so $R/pR\cong O\_K/pO\_K$. Hence if $p$ stays inert in $O\_K$, $pR$ will be prime in $R$.
Therefore, given a finite n... | 3 | https://mathoverflow.net/users/30186 | 392866 | 162,506 |
https://mathoverflow.net/questions/392858 | 3 | Let $G$ be a connected reductive group over a field $k$ (not necessarily algebraically closed). Let $S$ be a maximal split torus in $G$ with relative root system $\Phi = \Phi\_k(S,G)$. Let $\Phi^+$ be a choice of positive roots. A one-parameter subgroup $G\_m \rightarrow S$ acts on each root space as multiplication by ... | https://mathoverflow.net/users/222396 | Action of split torus on positive root spaces | Yes: take the one-parameter subgroup that is the sum of positive coroots. It acts on each simple root space by $t \mapsto t^2$, so by a positive integer power of $t$ on each positive root space.
Incidentally, this is a particularly natural choice, but, even if you didn't know that it existed, you could construct an o... | 4 | https://mathoverflow.net/users/2383 | 392873 | 162,509 |
https://mathoverflow.net/questions/392871 | 8 | Following is an experimental math claim.
We denote $(a,b)=\gcd(a,b)$.
Let $$G(a)=\sum\_{i=1}^{a-1}(-1)^i(a,i).$$
Note:
$$ G(a) =
\begin{cases}
0, & \text{if $a\equiv 1\pmod4$} \\
\text{odd}, & \text{if $a\equiv 2\pmod4$} \\
0, & \text{if $a\equiv 3\pmod4$} \\
\text{even}, & \text{if $a\equiv 0\pmod4$.}
\end{ca... | https://mathoverflow.net/users/149083 | Is it always true that $\sum_{i=1}^{a-1}(-1)^i(a,i)\ge-1$? | Yes, it is true. Further we suppose that $a=2b$ is even, as for odd $a$ the sum $G(a)$ equals 0 due to pairing $\{i,a-i\}$.
We start with $$(a,i)=\sum\_{d|(a,i)}\varphi(d)=\sum\_{d|a} \varphi(d)\cdot \mathbf{1}\_{d|i}.$$ Therefore
$$
G(a)=\sum\_{i=1}^{a-1}(-1)^i(a,i)=\sum\_{d|a}\varphi(d)\sum\_{i=1}^{a-1}(-1)^i \math... | 14 | https://mathoverflow.net/users/4312 | 392875 | 162,510 |
https://mathoverflow.net/questions/392835 | 6 | In a concrete category (i.e., where the morphisms are functions between sets), I define a **base** of an object $A$ to be a set of elements $M$ of $A$ such that for any morphisms $F,G:A\to B$ that coincide on $M$, we have $F=G$.
**Question:** Is there an established name for a **base** in that sense?
**Examples:** ... | https://mathoverflow.net/users/101775 | Name for a set of elements that fully determine a morphism | The term "base" should not be used, since, as you say, you are actually generalizing the notion of a generating set.
It is an **epi-sink**, also known as **jointly epimorphic family**. See [Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf), Definition 10.62 and (dual of) Definition 10.5. A family of morphism... | 15 | https://mathoverflow.net/users/2841 | 392876 | 162,511 |
https://mathoverflow.net/questions/392661 | 1 | Suppose we have a sequence of containers each of which contains multiple items. Each item $I\_i$ is associated with an nonnegative weight $w\_i$, a nonnegative value $v\_i$, and $I\_i(C)$ denotes the ID of the container item $I\_i$ belongs to. Given a bag with capacity $B$, the problem aims to select items into the bag... | https://mathoverflow.net/users/168850 | Is this variant of knapsack problem strongly NP-hard? | The same dynamic programming solution works for your instance as for knapsack.
Just order the elements according to which container contains them.
Then for each weight/value/container\_of\_last\_added\_element triple, you can compute the optimum by dynamic programming.
| 0 | https://mathoverflow.net/users/955 | 392880 | 162,512 |
https://mathoverflow.net/questions/392881 | 3 | Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that:
* $f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial),
* $g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=1$ (i.e. $g$ is a strictly positive density), and
* $f\*g=0$.
If we remove the condition that $g$ is a strictly posi... | https://mathoverflow.net/users/174195 | Vanishing convolution between density and compactly supported function | With your hypotheses above, $\widehat{g}:\mathbb{R}\to\mathbb{R}$ is a uniformly continuous function such that $\displaystyle \lim\_{|\gamma|\to\infty} \widehat{g}(\gamma) = 0$, and $$\widehat{f}(z) := \int f(t)e^{-2\pi itz} dt \hspace{28mm} (z\in\mathbb{C})$$ is an analytic function, for which the set $\{z\in\mathbb{C... | 3 | https://mathoverflow.net/users/164350 | 392891 | 162,514 |
https://mathoverflow.net/questions/392895 | 2 | I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem:
Recall that the Holder norm of a function is defined as follows:
$$[f]\_{C^{\alpha}} \mathrel{:=} \lVert f\rVert\_{L^{\infty}} + \... | https://mathoverflow.net/users/176086 | Estimate of Hölder Norms (Littlewood–Paley theory) | Let $a:=\alpha$, $[h]\_a:=[h]\_{C^a}$, and $\|h\|\_\infty:=\|h\|\_{L^\infty}$. For any distinct $x$ and $y$,
\begin{align\*}
|f(x)g(x)-f(y)g(y)|&=|f(x)g(x)-f(y)g(x)+f(y)g(x)-f(y)g(y)| \\
&\le|f(x)g(x)-f(y)g(x)|+|f(y)g(x)-f(y)g(y)| \\
&=|f(x)-f(y)|\,|g(x)|+|g(x)-g(y)|\,|f(y)| \\
&\le[f]\_a|x-y|^a\,\|g\|\_\infty+[... | 7 | https://mathoverflow.net/users/36721 | 392898 | 162,516 |
https://mathoverflow.net/questions/392885 | 3 | As far as I understand it, in recent years there has been a lot of progress on generalizations of classical deformation theory in characteristic 0 using tools such as simplicial deformation functors or ∞-categories, generalizing from characteristic 0 to positive characteristic and from deformations over local Artinian ... | https://mathoverflow.net/users/11084 | DG Lie algebras and derived deformation theory | Once you have your DGLA $L$, the simplest description is consider the functor from commutative dg Artinian local $k$-algebras $A= k \oplus \mathfrak{m}\_A$ in non-positive cochain degrees to simplicial sets given by Hinich's simplicial nerve. Explicitly, in simplicial level $n$, you take Maurer-Cartan elements $\mathrm... | 3 | https://mathoverflow.net/users/103678 | 392909 | 162,522 |
https://mathoverflow.net/questions/392917 | 7 | Let's call a smooth projective vartiety $M$ as Calabi-Yau (CY) manifold if it has trivial canonical class and
$h^i(M, \mathcal O\_M ) = 0$ for $0 < i < \dim(M)$.
In this definition, a CY 1-fold is an elliptic curve, a CY 2-fold is a projective $K3$ surface and etc.
For each $n$, I looking for a smooth projective vari... | https://mathoverflow.net/users/69559 | CY fibration over $\mathbb P^1$ without any singular fibers | No such variety exists: first, you can use Remark 3.2 [here](https://arxiv.org/pdf/1309.3773.pdf) to see that your map $\pi$ must be a holomorphic fiber bundle, and then Lemma 17 [here](https://arxiv.org/pdf/math/0701466.pdf) gives you that this bundle becomes a trivial product after pulling back your family via a fini... | 6 | https://mathoverflow.net/users/13168 | 392919 | 162,523 |
https://mathoverflow.net/questions/392834 | 3 | I asked this earlier on math.stackexchange but I think this is a better place for this question.
Computing the intersection of ideals belonging to the same maximal order of a number field $K$ can be reduced to computing the intersection of lattices of the same dimension.
How can I compute the intersection of an ide... | https://mathoverflow.net/users/106850 | How to compute the intersection of an ideal with the maximal order of a subfield? | This reduces easily to computing the intersection of two $\mathbf{Z}$-lattices (not necessarily of full rank) inside $\mathbf{Q}^n$ for some $n$. If you have two lattices $L, M$ of ranks $r$ and $s$, and you let $A$, $B$ be the $r \times n$, resp. $s \times n$, matrices whose rows are bases of $L$ and $M$ respectively,... | 3 | https://mathoverflow.net/users/2481 | 392928 | 162,525 |
https://mathoverflow.net/questions/392914 | 5 | Suppose you have two $k$ algebras $A, B$ (say also finitely generated if this helps) and a functor $F: A-mod \to B-mod $ such that $| F(M) |= |M|$. Here $|U|$ denotes the underlying $k$ vector space.
Can you find sufficient conditions so that $F$ is the restriction functor relative to $f:B \to A$?
I suspect that a ... | https://mathoverflow.net/users/140013 | Functors between module categories that comes from restriction | Let's assume that we are using right modules and that the isomorphism of underlying vector space functors is a natural isomorphism because we will just work up to isomorphism of functors. Note that $F$ is exact. To come from restriction $F$ should preserve all limits and colimits. Maybe this follows by functorially pre... | 3 | https://mathoverflow.net/users/15934 | 392934 | 162,526 |
https://mathoverflow.net/questions/392940 | 1 | Suppose $(X,\mu)$ is a probability space, and $T\_n, n \in \mathbb N$, is a sequence of *periodic* measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}\_{f,n}(x)$ be the average value of the finite set $\{ f(T\_n^0(x)), f(T\_n^1(x)), f(T\_n^2(x)),\dots\}$ (finite because of per... | https://mathoverflow.net/users/11145 | Ergodic theorem on limit of periodic transformations? | In fact, for the example you give, there is a failure of pointwise convergence. That was established in a short 1964 paper of [Rudin](https://www.ams.org/journals/proc/1964-015-02/S0002-9939-1964-0159918-8/home.html) in Proc. Amer. Math. Soc. A more detailed look at this example appears in "The strong sweeping out prop... | 3 | https://mathoverflow.net/users/11054 | 392959 | 162,534 |
https://mathoverflow.net/questions/392931 | 2 | I found this article
* J. A. Makowsky and U. Rotics, *On the clique-width of graphs with few $P\_4$s*, International Journal of Foundations of Computer Science Vol. 10, No. 03 (1999) pp. 329-348, doi:[10.1142/S0129054199000241](https://doi.org/10.1142/S0129054199000241),
[ResearchGate](https://www.researchgate.net/pu... | https://mathoverflow.net/users/224082 | Clique width of C_n | The upper bound in Makowsky and Rotics says that with 4 labels, you can build $P\_{n-1}$ in such a way that the two leaves have unique labels. From there you make all the non-leaves the same label, leaving you with an unused label. Then you insert a new vertex with this unused label and make it adjacent to the two leav... | 8 | https://mathoverflow.net/users/2663 | 392962 | 162,535 |
https://mathoverflow.net/questions/392936 | 6 | Let $\mathcal{X}$ be an algebraic stack of finite type over a (separably closed) field $ k$. Let's say that $\mathcal{X}$ has finite dimension $d \in \mathbb{Z}$. Is it still true that the number of irreducible components of dimension $d$ of $\mathcal{X}$ is the dimension of $H^{2d}\_c(\mathcal{X},\bar{\mathbb{Q}}\_{\e... | https://mathoverflow.net/users/146464 | Irreducible components of an algebraic stack | I will say **yes**, although the level of generality is a bit scary and I hope I am not missing some stacky subtlety. I just took the standard argument for schemes, stared at it, and couldn't see anything that wouldn't work in the general case.
Claim: if $X$ is an equidimensional finite type algebraic stack, and $d=\... | 4 | https://mathoverflow.net/users/1310 | 392965 | 162,536 |
https://mathoverflow.net/questions/392969 | 1 | **Definitions:**
Two real valued random variables $X\_0$ and $X\_1$ are called a *one step martingale* if $E[X\_1| X\_0] = X\_0$.
We say the one step martingale is in $L^2$ if both $X\_0$ and $X\_1$ are in $L^2(P)$.
**Question:**
Given an $L^2$ one step martingale $(X\_0, X\_1)$ does there exist a sequence $(Y\... | https://mathoverflow.net/users/173490 | Discrete approximation of one step martingale | Let $\mathcal F\_i^n$ be a sequence of $\sigma$-algebras increasing to $\mathcal F(X\_i)$, the $\sigma$-algebra generated by $X\_i$ and let $Y\_1^n=\mathbb E(X\_1|\mathcal F^n\_1\vee \mathcal F^n\_0)$ and $Y\_0^n=\mathbb E(X\_1|\mathcal F^n\_0)$.
Then $Y^n\_1$ and $Y^n\_0$ converge pointwise a.s. to $X\_1$ and $X\_0$... | 1 | https://mathoverflow.net/users/11054 | 392975 | 162,540 |
https://mathoverflow.net/questions/392973 | 11 | Let $M\_{n}(\mathbb{Q}) $ denote the $n$ times $n$ matrices over the rational number field. $N$ be a subspace of $M\_{n}(\mathbb{Q}) $.Then if all the non-zero matrices in $N$ are invertible, what is the maximum the dimension of $N$ can be?
We already know that if we take $M\_{n}(\mathbb{R}) $
instead of $M\_{n}(\mat... | https://mathoverflow.net/users/215016 | Problems concerning subspaces of $M_{n}(\mathbb{Q}) $ | Let's call this maximal dimension function $\rho\_{\mathbb{Q}}:\mathbb{N}\to\mathbb{N}$, i.e., $\rho\_{\mathbb{Q}}(n)$ is the largest possible dimension of a subspace $N\subset M\_n(\mathbb{Q})$ such that all of the nonzero elements of $N$ are invertible.
Then $\rho\_{\mathbb{Q}}(n)\ge n$, as the following constructi... | 21 | https://mathoverflow.net/users/13972 | 392984 | 162,542 |
https://mathoverflow.net/questions/392926 | 0 | I'd like to have a definition of intersection on non compact complex surfaces because all i have found so far is only about projective surfaces. for example how can i define the self intersection of curves on a complex surfaces which is not projective. and are there any references which clarify this.
thanks for your ... | https://mathoverflow.net/users/183596 | intersection and self intersection on non compact complex surfaces | Specifically for curves on surfaces, you can look at Barth-Hulek-Peters-Van de Ven Chapter II, Section 10 ("Intersection numbers") for a quick summary. The idea is as follows. For noncompact spaces, you can still define a nondegenerate pairing between $H^k(X)$ and $H^{dim(X)-k}\_c(X)$ where the subscript c means compac... | 1 | https://mathoverflow.net/users/10839 | 392987 | 162,543 |
https://mathoverflow.net/questions/392901 | 3 | While considering the zero curvature equation $U\_t - V\_x + [U, V] = 0$, I developed a similar problem, albeit one that discards time dependence entirely. For a given $U(x)$ and $C(x)$, find $V(x)$ such that $[U, V] - V\_x = C(x)$ holds. After considering this [question](https://math.stackexchange.com/questions/130709... | https://mathoverflow.net/users/170939 | On the equation $[U, V] - V_x = C(x)$ | Yes, you can always do this, as follows:
First, consider the equation $M\_x = -M U$ with the initial condition $M(0) = I\_n$. This linear equation with initial condition has a unique solution and $M(x)$ will be invertible for all $x$. Now, set $V = M^{-1}AM$ for some matrix $A$ and consider the equation
$$
V\_x - UV ... | 4 | https://mathoverflow.net/users/13972 | 392988 | 162,544 |
https://mathoverflow.net/questions/392303 | 6 | I have been reading [Asok and Doran - $\mathbb A^1$-homotopy groups, excision, and solvable quotients](https://reader.elsevier.com/reader/sd/pii/S0001870809000450). In the Example 2.17, page 1155, there is a claim that for $m>1$ and $n\geq 1$, $\pi\_0^{\mathbb{A}^1}(\mathbb{A}^m-0/\mu\_n)\simeq\mathbb{G}\_m/\mathbb{G}\... | https://mathoverflow.net/users/157738 | $\pi_0^{\mathbb{A}^1}(\mathbb{A}^m-0/\mu_n)\simeq\mathbb{G}_m/\mathbb{G}_m^n$ | Over a field of characteristic coprime to $n$, the classifying space ${\rm B}\_{\rm et}\mu\_n$ is $\mathbb{A}^1$-local. In this situation, any $\mu\_n$-covering $\widetilde{X}\to X$ is an $\mathbb{A}^1$-covering space and provides a classifying morphism $X\to {\rm B}\_{\rm et}\mu\_n$. From the Kummer sequence $\mu\_n\t... | 3 | https://mathoverflow.net/users/50846 | 392989 | 162,545 |
https://mathoverflow.net/questions/392993 | 8 | Do there exist integers $x,y,z$ such that
$$
x^2+y^2-z^2 = xyz -2 \quad ?
$$
Why this is interesting? First, this equation arose in an answer to the previous Mathoverflow question [What is the smallest unsolved diophantine equation?](https://mathoverflow.net/questions/316708/what-is-the-smallest-unsolved-diophantine-... | https://mathoverflow.net/users/31472 | On Markoff-type diophantine equation | There is no solution.
Fix a solution $(x,y,z)$ with $|x|+|y|+|z|$ minimal. We will show a contradiction.
We can't have $xyz=0$ as we would then obtain one of the unsolvable equations $x^2+y^2= -2$, $x^2-z^2=-2$, $y^2-z^2=-2$.
If $xyz>0$, then by swapping the signs of two of $x,y,z$ if necessary we can assume $x,y... | 14 | https://mathoverflow.net/users/18060 | 392995 | 162,546 |
https://mathoverflow.net/questions/392282 | 12 | Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d\_K(x):=\min\_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$.
Suppose that:
* $K$ is *regular* : it has a non-empty interior $\overset{\circ}{K}$, and the closure of $\overset{\circ}{K}$ is $K$; in particula... | https://mathoverflow.net/users/36886 | Smoothness of distance function to a compact set | If a domain $\Omega$ has boundary of class $C^k$, $k\geq 2$, then in fact the distance function $d$ to the boundary of $\Omega$ is of class $C^k$ in a neighborhood of the boundary. This is exactly what is proved in Lemma 14.16 in [1] mentioned by OP.
In the case of a convex set we have the following result. Note that... | 4 | https://mathoverflow.net/users/121665 | 393000 | 162,547 |
https://mathoverflow.net/questions/392922 | 0 | I asked the [following](https://math.stackexchange.com/questions/4136248/bounding-the-nth-term-of-a-sequence-given-a-recursive-non-linear-bound) question in MSE:
>
> Let $a,b\in\mathbb{R}^+$.
> Suppose that $\{x\_n\}\_{n=0}^\infty$ is a sequence satisfying
> $$|x\_n|\leq a|x\_{n-1}|+b|x\_{n-1}|^2, $$
> for all $n\i... | https://mathoverflow.net/users/105925 | Bounding the $n$-th term of a sequence, given a non-linear recursive bound | Let us define the sequence $y\_n$ by $y\_0=|x\_0|$, $y\_{n+1}=ay\_n+by\_n^2$. Then we have
$|x\_n|\leq y\_n$ for all $n$, since $ay+by^2$ is increasing for positive $a,b$,
and it is enough to estimate the positive sequence $y\_n$. Setting $z\_n=by\_n$, we obtain a simpler recurrent relation
$z\_{n+1}=az\_n+z\_n^2.$ For... | 1 | https://mathoverflow.net/users/25510 | 393005 | 162,550 |
https://mathoverflow.net/questions/393018 | 4 | I am faced with an operation $\otimes$ on a strict 2-category $C$ which walks and talks like a tensor, except that it only satisfies a "lax" interchange law. To be precise: for any $f,g,h,k\in C\_1$ with $f\circ h$ defined and $g\circ k$ defined, there is an "interchanger" 2-cell $I\_{f,g,h,k}:(f\otimes g)\circ(h\otime... | https://mathoverflow.net/users/30392 | Strict 2-Category with Lax Tensor? | This suggests that the monoidal product $\otimes\colon \mathbf{C}^2\to \mathbf{C}$ is a lax functor (see e.g. [here](https://kerodon.net/tag/008K) or [section 4.1 here](https://arxiv.org/abs/2002.06055)) instead of a strict 2-functor. If so, this determines which coherence laws you should expect. You don't mention iden... | 2 | https://mathoverflow.net/users/136562 | 393021 | 162,553 |
https://mathoverflow.net/questions/393020 | 4 | Let $(E,d),(F,d')$ be separable metric spaces endowed with their Borel algebra, $f:E\rightarrow F$ a continuous surjective function, and $Q$ a probability measure on $F$ with separable support.
**Question:** Does there exist a probability measure $P$ on $E$ such that $Q$ is the pushforward measure of $P$ by $f$?
| https://mathoverflow.net/users/159940 | Sufficient condition for a probability measure to be a pushforward measure | $\newcommand\C{\mathscr C}\newcommand\de{\delta}$A sufficient condition is that $E$ be compact. Indeed, since $Q$ has a separable support, without loss of generality $F$ is separable.
So, for each natural $n$ there is an (at most) countable set $\C\_n$ of nonempty pairwise disjoint Borel subsets of $F$ of diameter $\le... | 2 | https://mathoverflow.net/users/36721 | 393027 | 162,555 |
https://mathoverflow.net/questions/393046 | 3 | I was wondering if somebody could tell me the definition of homotopy pullback. More precisely, is there a description of the homotopy pullback in the category of (graded-commutative) DG algebras over a commutative ring $R$? What properties does this homotopy pullback satisfy?
| https://mathoverflow.net/users/226648 | Homotopy pullback in the category of DG algebras |
>
> I was wondering if somebody could tell me the definition of homotopy pullback.
>
>
>
Suppose $\def\C{{\cal C}}\C$ is a [relative category](https://ncatlab.org/nlab/show/relative+category), i.e., a category equipped with a subcategory,
morphisms in which are known as weak equivalences.
For example, take the c... | 5 | https://mathoverflow.net/users/402 | 393048 | 162,561 |
https://mathoverflow.net/questions/392980 | 5 | Originally formulated for elliptic curves, the Sato-Tate conjecture regarding the equidistribution of Frobenius trace values according to the Haar measure on a certain compact group (the Sato-Tate group) was generalised in 1994 by Serre to any motive over a number field - see 13.5? in [5]. As is now well known, the Sat... | https://mathoverflow.net/users/5744 | Is the Sato-Tate conjecture known for Bianchi modular forms? |
>
> Does it follow from the work of [1] that the Sato-Tate conjecture is
> known for some class of cuspidal automorphic forms for GL2 over CM
> fields?
>
>
>
Tautologically: "yes, those which correspond to modular elliptic curves". These are a (strict) subset of the forms which are cohomological of trivial weigh... | 5 | https://mathoverflow.net/users/2481 | 393060 | 162,567 |
https://mathoverflow.net/questions/392974 | 3 | For a bounded function $\operatorname{F}: \mathbb{R}\_{\,\ge\ 0} \to \mathbb{R}$ (*not necessarily non-negative*), if
$$
\int\_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \operatorname{F}\left(x\right)\,{\rm d}x = 0\quad
\forall s > 0
$$
where $k \in \mathbb{N}$ is a positive constant,
is... | https://mathoverflow.net/users/110835 | An integral transform and the Stone-Weierstrass theorem | Method 1: There is a clear harmonic-analytic interpretation: if
\[ u(x, s) = C\_k \int\_{\mathbb R^{k+2}} \frac{s}{(s^2 + |x - y|^2)^{(k+3)/2}} \times |y|^{-1} F(|y|) dy , \]
then $u$ a harmonic function in the half-space $x \in \mathbb R^{k+2}$, $s > 0$, and $u(0, s) = 0$ for all $s > 0$. We claim that $u$ is identica... | 3 | https://mathoverflow.net/users/108637 | 393062 | 162,568 |
https://mathoverflow.net/questions/393077 | 4 | Let $(R, m)$ be a commutative local ring (it is not Noetherian in general) and $F$ be a free $R$-module. Under what conditions every proper submodule of $F$ is contained in a maximal submodule.
| https://mathoverflow.net/users/176728 | When every proper submodule of a free module is contained in a maximal submodule | I believe that the answer is that every submodule of a free $R$-module is contained in a maximal submodule if and only if $R$ is a [perfect ring](https://en.wikipedia.org/wiki/Perfect_ring) (for a commutative ring, this is equivalent to dcc on principal ideals).
You can weaken local to the condition that $R$ has no ... | 6 | https://mathoverflow.net/users/15934 | 393088 | 162,573 |
https://mathoverflow.net/questions/393067 | 1 | [Algebras for endofunctors](https://ncatlab.org/nlab/show/algebra+for+an+endofunctor) bridge the gap between functors acting *on* a category and structures defined *in* it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more crucially a *morphism* of algebras is a map $a\to b$ betwee... | https://mathoverflow.net/users/84398 | Algebras for general transfors | The category of algebras for an endofunctor $F:C\to C$ is the lax limit of the 2-functor $[F]:D\to Cat$, where $D$ is the 2-category freely generated by one object and one endomorphism. Accordingly, it seems natural to me that more general lax limits could be regarded as categories of "algebras for transfors".
For in... | 3 | https://mathoverflow.net/users/49 | 393096 | 162,574 |
https://mathoverflow.net/questions/393101 | 2 | Let $(X, d)$ be a metric space and $F\_\varepsilon, F\colon X\to [-\infty, \infty]$. Suppose $F\_\varepsilon$ is an equicoercive sequence of functions on $X$, i.e. for all $t\in\mathbb{R}$ there exists a compact set $K\_t$ such that $\{ x\colon F\_\varepsilon(x)\le t \} \subset K\_t$ for every $\varepsilon > 0$. Suppos... | https://mathoverflow.net/users/98946 | Fundamental Theorem of Gamma-Convergence | **Yes.** Take any sequence $\{x\_i\}$ for which $F(x\_i)$ is bounded. For each $x\_i$ consider the sequence satisfying (2), $\{x^\epsilon\_i\}$ s.t. $x^\epsilon\_i\to x\_i$. Consider $x\_i^{\epsilon\_i}$ such that $F\_{\epsilon\_i}(x\_i^{\epsilon\_i})$ is bounded. Due to equicoercivity, $x\_i^{\epsilon\_i}$ has to be i... | 3 | https://mathoverflow.net/users/219013 | 393107 | 162,577 |
https://mathoverflow.net/questions/392933 | 4 | Take any family $(S\_i)\_{i∈I}$ such that each $S\_i$ is a convex set of functions $f : ℕ→[0,1]$ where $\sum\_{k∈ℕ} f(k) = 1$. By "convex" we mean that for any $f,g∈S\_i$ and any $a,b∈[0,1]$ such that $a+b=1$ we have $a·f+b·g∈S\_i$. By AC (axiom of choice) there is a choice function for $S$. But I get a feeling that AC... | https://mathoverflow.net/users/50073 | Is Axiom of Choice for convex sets of distributions on naturals necessary? | The existence of a choice function for the set of non-empty convex sets of distributions on $\mathbb N$ is equivalent to the existence of a well-ordering of $\mathbb R.$ This is true even if we strengthen convexity by allowing weighted combinations. The well-orderability of $\mathbb R$ is Form 79 in Howard-Rubin’s “Con... | 3 | https://mathoverflow.net/users/164965 | 393110 | 162,579 |
https://mathoverflow.net/questions/393120 | 4 | I recently came across the following characterization of unit regular elements of an endomorphism ring (Corollary to Theorem 1 in [this article](https://www.ams.org/journals/tran/1976-216-00/S0002-9947-1976-0387340-0/S0002-9947-1976-0387340-0.pdf)).
>
> Let $M$ be a vector space over the division ring $D$, and let ... | https://mathoverflow.net/users/98794 | Ehrlich's Characterization of Unit Regular Elements | Manny's answer is absolutely correct. I wanted to add that Ehrlich's argument, which in its original formulation requires the endomorphism ring to be von Neumann regular, actually characterizes unit-regular elements in the endomorphism ring of *arbitrary* modules, as noted by T. Y. Lam and others (in numerous places, b... | 3 | https://mathoverflow.net/users/3199 | 393124 | 162,585 |
https://mathoverflow.net/questions/393118 | 2 | Consider the closed convex subset $\mathcal{F} = \{f \in C[0,1] : 0 \leq f \leq 1, f(0)=0, f(1)=1\}$. Consider the polynomial class $\mathcal{P} = \{p \text{ is a polynomial} : p(0)=0, p(1)=1, 0 \leq p \leq 1\}$. Is $\mathcal{P}$ dense in $\mathcal{F}$ in the sup norm?
Is anything known about Weierstrass theorem gene... | https://mathoverflow.net/users/151406 | Polynomial approximation of continuous function with constraints | Indeed, $\mathcal{P}$ is dense in $\mathcal{F}$. The Bernstein polynomials approximating $f\in\mathcal{F}$ belong to $\mathcal{P}$, see Theorem 11.68 in:
[http://www.pitt.edu/~hajlasz/Teaching/Math1530Fall2018/selection.pdf](http://www.pitt.edu/%7Ehajlasz/Teaching/Math1530Fall2018/selection.pdf)
| 3 | https://mathoverflow.net/users/121665 | 393127 | 162,586 |
https://mathoverflow.net/questions/393119 | 7 | The following identity is well-known and there are a few proofs to it (see [Bijective proof problems](http://www-math.mit.edu/%7Erstan/bij.pdf), by R P Stanley, for this and similar formulae):
$$\sum\_{k=0}^n\binom{2k}k\binom{2n-2k}{n-k}=4^n \qquad \iff \qquad
\sum\_{j+i=n}\binom{2j}j\binom{2i}i\frac1{4^n}=1.\tag1$$
... | https://mathoverflow.net/users/66131 | Looking for a $q$-analogue of a binomial identity | We start with $q$-binomial theorem
$$
(x+y)(x+qy)\cdots(x+q^{n-1}y)=\sum q^{k\choose 2}{n\choose k}\_qy^k x^{n-k}.\quad\quad\quad(\heartsuit)
$$
Put $n=a+b$ in $(\heartsuit)$ and consider separately the first $a$ multiples in LHS and the last $b$ multiples. We get
$$
\left(\sum q^{j\choose 2}{a\choose j}\_qy^j x^{a-j}\... | 4 | https://mathoverflow.net/users/4312 | 393131 | 162,587 |
https://mathoverflow.net/questions/393053 | 7 | Here is a question I heared from others:
Given four distinct positive real numbers $a\_1,a\_2,a\_3,a\_4$ and set $$a:=\sqrt{\sum\_{1\leq i\leq 4}a\_i^2}$$
$A=(x\_{i,j})\_{1\leq i\leq3,1\leq j\leq4}$ is a $3\times4$-matrix specified by $$ x\_{i,j}=a\_i\delta\_{i,j}+a\_j\delta\_{4,j}-\frac{1}{a^2}(a\_i^2+a\_4^2)a\_j $$... | https://mathoverflow.net/users/nan | How to show a $3\times3$ matrix has three distinct eigenvalues? | To answer on methods applicable here (and elaborate on comments I made). The most promising is to use a surprisingly little-known theorem that says that the discriminant $D$ of a symmetric $n\times n$ matrix $A=(a\_{ij})$ with eigenvalues $\lambda\_1,\dots,\lambda\_n$, i.e.
$$ D\_A=\prod\_{1\leq i<j\leq n} (\lambda\_i-... | 11 | https://mathoverflow.net/users/11100 | 393132 | 162,588 |
https://mathoverflow.net/questions/393128 | 2 | This came up when thinking about this [question](https://mathoverflow.net/questions/393090/does-an-interior-point-necessarily-pass-through-the-boundary-under-a-homotopy).
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of whi... | https://mathoverflow.net/users/53155 | Approximate Jordan-Brouwer theorem | Yes, if $x$ is in the unbounded component of $\mathbb{R}^{n+1}\setminus f(S^n)$, then $x$ is also in the unbounded complement of $\mathbb{R}^{n+1}\setminus g(S^n)$, for any continuous $g$, provided $g$ is close enough to $f$. Indeed, domains in $\mathbb{R}^{n+1}$ are path connected so there is path $\gamma$ connecting ... | 4 | https://mathoverflow.net/users/121665 | 393134 | 162,589 |
https://mathoverflow.net/questions/392939 | 1 | I wonder if the convolution
\begin{equation}
f(y)=\int\_{-\infty}^{+\infty} \mathrm{Airy}(a\cdot x)\cdot e^{-b(y-x)^2} dx
\end{equation}
can be solved analytically. Or in case not, if there is an analytic expression for the zeros of $f(y) = 0$ and for which values of $a$ and $b$ they exist.
edit: $a>0$ and $b>0$
... | https://mathoverflow.net/users/224204 | Convolution of an Airy function with a Gaussian | Here's a proof sketch of
$$(\*) \quad \int\_{-\infty}^\infty e^{-b \ u^2}\text{Ai}(a(u+y))du = \frac{\sqrt{\pi}}{\sqrt{b}}
\exp{\big(\frac{a^6}{96b^3} + \frac{a^3y}{4b}}\big)\text{Ai}\big(a(y+\frac{a^3}{16b})\big).$$
The following formula will be used twice:
$$ (1) \quad \frac{1}{2\pi} \int\_{-\infty}^\infty \exp{\b... | 5 | https://mathoverflow.net/users/121836 | 393135 | 162,590 |
https://mathoverflow.net/questions/393136 | 3 | Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$
Consider the block matrix
$$A = \begin{pmatrix} A\_1 & A\_2 \\ A\_2^\* & -A\_1\end{pmatrix}$$
where $A\_1 = \operatorname{diag}(\lambda\_1,\lambda\_2)$ for $\lambda\_i \in \mathbb C$ and $A\_2: V^2 \to V^2.$
We th... | https://mathoverflow.net/users/150549 | Inverse of block matrix | $\newcommand{\la}{\lambda}$The answer is no. Indeed, let
\begin{equation}
A=\left(
\begin{array}{cccc}
1 & 0 & 2 & -1 \\
0 & -2 & 1 & 3 \\
2 & 1 & -1 & 0 \\
-1 & 3 & 0 & 2 \\
\end{array}
\right).
\end{equation}
Suppose that the desired result holds for some matrices $T\_1,\dots,T\_4$.
Then, letting $L$ denote t... | 3 | https://mathoverflow.net/users/36721 | 393143 | 162,591 |
https://mathoverflow.net/questions/393142 | 8 | Suppose $S$ is an associative semiring whose underling commutative monoid is free (in particular, cancellative) and that its Grothendieck ring $G(S)$ is a unital ring. Can we conclude that $S$ must be unital, and if not, is there a nice counter-example?
Alternatively, are there additional suppositions we can put on $... | https://mathoverflow.net/users/132371 | If the Grothendieck ring of a semiring on a free commutative monoid is unital, is the original semiring unital? | The answer is no. Let $S$ be a finite meet semilattice without maximum. For concreteness, take $S$ to be the proper subsets of $\{1,2\}$ under intersection. Let $\mathbb NS$ be the semigroup semiring of $S$. Then the underlying additive monoid is free on $S$. The Grothendieck ring is the semigroup ring $\mathbb ZS$ and... | 8 | https://mathoverflow.net/users/15934 | 393148 | 162,592 |
https://mathoverflow.net/questions/393108 | 2 | Let $p(x)\in\mathbb{R}[x]$ be a polynomial in $n$ variables, where we let $x=(x\_1,\dots,x\_n)$. It is known that $p(x)$ can be written as a linear combination of powers of linear forms, this is, there exist linear forms
$L\_1,\dots,L\_N$, real numbers $a\_1,\dots,a\_N$, and nonnegative integers $k\_1,\dots,k\_N$, such... | https://mathoverflow.net/users/109085 | Polynomial as a sum of powers of linear forms (with restrictions) | Yes, it is true. Every polynomial does have a decomposition like you ask for, with restrictions.
To explain why this is the case, first suppose that $p$ is homogeneous of degree $d$, and we seek a decomposition where each linear form $L\_i$ is a homogeneous linear form (no constant term) and each power $k\_i = d$. Th... | 4 | https://mathoverflow.net/users/88133 | 393151 | 162,593 |
https://mathoverflow.net/questions/393145 | 1 | I have recently been stuck trying to understand how game theorists extend a normal form game (matrix game) into a game with mixed strategies (so called mixed extension). I feel like I am missing something obvious, perhaps an implicit identification.
Take for example these [notes](https://ocw.mit.edu/courses/electrica... | https://mathoverflow.net/users/114734 | Should mixed strategies in normal form games be interpreted as measurable functions or probability vectors? | This could be a comment but it might clear things up. In short, a mixed strategy is a probability measure over a set of pure strategies (also called actions). If the set of actions is finite, we can represent the probability measure as a vector of probabilities, i.e. an element of the simplex.
I don't see strategies ... | 3 | https://mathoverflow.net/users/29697 | 393155 | 162,595 |
https://mathoverflow.net/questions/393076 | 14 | This is a follow-up to [this question](https://mathoverflow.net/questions/392820/). We say that a set $A \subseteq \mathbb{R}$ is bounded if there exists a finite interval $(a,b)$ such that $A \subseteq (a,b)$.
Working in $\mathsf{ZFC}$, the existence of a (Lebesgue) non-measurable set (of $\mathbb{R}$) easily implie... | https://mathoverflow.net/users/146831 | Is "There exists an unbounded non-measurable set but no bounded non-measurable set" consistent with $\mathsf{ZF}$? | The answer to your first question is yes, and the answer to your second question is no, under any of the multiple definitions of "measurable" in choiceless contexts.
We will prove a theorem relating various measure-theoretic consequences of countable choice.
(ZF) The following are equivalent. Note that (1)-(6) are ... | 9 | https://mathoverflow.net/users/109573 | 393162 | 162,598 |
https://mathoverflow.net/questions/393159 | 3 | We know that the number of squarefree integers $\le x$ that are coprime to $A$ is
$$
Q\_A(x) = x \prod\_{p|A} \left(1-\frac{1}{p}\right) \prod\_{p \nmid A} \left(1-\frac{1}{p^2}\right) + O(\sqrt{x}).
$$
Do we have explicit upper and lower bounds on $Q\_A(x)$?
| https://mathoverflow.net/users/73880 | Explicit bounds on number of squarefree numbers coprime to a certain number | I am assuming that explicit refers to the error term? In this case you can write $$Q\_A(x)=\sum\_{d\mid A}\mu(d)\sum\_{k\leq \sqrt x \atop \gcd(A,k)=1}\mu(k)\left[\frac{x}{dk^2}\right],$$ where $[t]$ denotes the largest integer $\leq t $. Note that $[t]=0$ when $0\leq t<1$ hence only the terms with $dk^2 \leq x $ make ... | 5 | https://mathoverflow.net/users/9232 | 393183 | 162,605 |
https://mathoverflow.net/questions/393188 | 2 | Let $G=(V(G), E(G))$ be a graph on $n$ vertices and let $S$ be a subset of $V(G)$. The boundary of $S$, denoted by $\partial S$, is the set of edges $(i, j)$ such that $i \in S$ and $j \in V(G) \setminus S$.
The expansion ratio of $G$, denoted by $h(G)$, is
\begin{equation}
h(G)= \min\_{S\subset V, 0<|S|\leq \frac{n... | https://mathoverflow.net/users/91089 | The complexity of expansion ratio (Cheeger constant) of a graph | [This paper](https://arxiv.org/pdf/2005.05812.pdf) says it is NP-hard and gives three references.
| 2 | https://mathoverflow.net/users/9025 | 393193 | 162,609 |
https://mathoverflow.net/questions/393129 | 3 | Suppose $G$ is a finite Galois group, and $M$ is an infinite $G$-module. When can I say that $H^1(G, M)$ is finite?
I know this not true in general. Is it true under certain assumptions on $M$?
To be more precise, can I say that $H^1(G, V/T)$ is finite? where $V$ is finite-dimensional $\mathbb{Q}\_p$-representation o... | https://mathoverflow.net/users/93778 | Finiteness of cohomology group | To summarize the discussion from the comments as an answer:
For a finite group and a finitely generated $R[G]$-module $M$, the groups $H^\*(G;M)$ are computed by a chain complex of finitely generated $R$-modules, so under reasonable assumptions on $R$ the cohomology groups are finitely generated as $R$-modules. Also,... | 4 | https://mathoverflow.net/users/39747 | 393202 | 162,613 |
https://mathoverflow.net/questions/393192 | 4 | Consider the set
$$C=\left\{\log\_\frac{a+1}{b+1}\frac{a}{b} : a\ne b\in\mathbb{Z}^+\right\}$$
The set $C$ cannot contain all real numbers in $[1,\infty)$ because it is a countable set. But is it dense in $[1,\infty)$, or in some subinterval of it (of positive length)?
To prove that it is dense, we would need that ... | https://mathoverflow.net/users/229462 | Density of logarithm with fractional base | $\newcommand{\ep}{\epsilon}\newcommand{\N}{\mathbb N}$The answer is no. Indeed, suppose the contrary: that the set of all values of
\begin{equation\*}
l(a,b):=\log\_\frac{a+1}{b+1}\frac ab \tag{0}
\end{equation\*}
for distinct natural numbers $a$ and $b$ is dense in the interval $[c,d]$ for some real $c$ and $d$ such ... | 3 | https://mathoverflow.net/users/36721 | 393203 | 162,614 |
https://mathoverflow.net/questions/393139 | 2 | What kind of jobs are there for someone with a strong, research-level theoretical background in the topic? I'm especially interested in the industry rather than academic jobs.
| https://mathoverflow.net/users/163629 | What kind of jobs are available for a quantum logician? | Let me try and give you a more focused answer. The good/promising news is this: there are many interesting and rewarding jobs available in the quantum technology industry. These jobs require a new type of skills, familiarity with quantum information processing, and these skills are in short supply. So you are not compe... | 14 | https://mathoverflow.net/users/11260 | 393210 | 162,617 |
https://mathoverflow.net/questions/392996 | 2 | I was recently trying taking a look at the paper "Some two-generator one-relator non-hopfian groups" by Baumslag and Solitar where they introduce the groups now known as the Baumslag-Solitar groups given by the presentation
$$
G = \left< a,b \mid a^{-1}b^la = b^m \right>
$$
The paper is only 3-pages long and a bit skim... | https://mathoverflow.net/users/99414 | Some questions on a paper of Baumslag and Solitar | I'll write out the answer to your first question (solving the word problem in one-relator groups) for how one might do this in practice. I'll focus on the case $\ell = 2, m=3, p=2$, but you'll hopefully be satisfied that the methods are sufficiently general!
Let $G = \langle a, b \mid a^{-1} b^2 a b^{-3} =1 \rangle$.... | 1 | https://mathoverflow.net/users/120914 | 393211 | 162,618 |
https://mathoverflow.net/questions/393214 | 11 | I have a formal power series in one variable that I think might be algebraic (or perhaps just D-finite). Is there software that could help me explore this?
By way of comparison, there’s a very simple way to see if a formal power series appears to be rational: for small values of $n$, compute the determinant of the $(... | https://mathoverflow.net/users/3621 | Software for recognizing algebraic or D-finite formal power series | Fricas is good at that. It can be accessed via sage, once installed.
```
sage: L=[catalan_number(i) for i in range(20)]
sage: fricas.guessHolo(L)
[
n 2 ,
[[x ]f(x): (4 x - x)f (x) + (2 x - 1)f(x) + 1 = 0,
... | 9 | https://mathoverflow.net/users/10881 | 393221 | 162,621 |
https://mathoverflow.net/questions/392813 | 5 | Throw $m$ balls into $n$ bins independently, each ball selecting a bin from the distribution $A \in \Delta\_n$. This question is about **lower-bounding the max-loaded bin**.
**Background.** In [this MO answer](https://mathoverflow.net/a/323756/29697) I wrote about an upper bound based on collisions. Let $Z\_k$ be all... | https://mathoverflow.net/users/29697 | Bounding the max-loaded bin using${m \choose k} \|A\|_k^k$ | The following inequality holds:
$$\mathbb{P}(C\_k(m)\geq 1)\geq \mathbb{P}(\mathrm{Bin}(m, \lVert A\rVert\_k)\geq k)$$
where here and in the sequel $\mathrm{Bin}(n,p)$ denotes a binomially distributed random variable with parameter $n$ and $p$.
(I now change notation so I can use my old notes. In the sequel $r\ge... | 3 | https://mathoverflow.net/users/48831 | 393224 | 162,623 |
https://mathoverflow.net/questions/393229 | 4 | It is known (cf. e.g. Theorem C.2.2.13 in the Elephant) that any (locally small) cocomplete quasitopos with a strong generator is locally presentable. In this question I am interested in cocomplete quasitoposes with just ordinary (not necessarily strong) generators, and in particular in cocomplete quasitoposes which ar... | https://mathoverflow.net/users/64128 | Cocomplete quasitoposes that are not locally presentable | The category of [pseudotopological spaces](https://ncatlab.org/nlab/show/pseudotopological+spaces) should be a good example. It is a locally small complete and cocomplete quasitopos (indeed, it is topological over Set), the one-point space is a non-strong generator, and it is surely not locally presentable (although pr... | 6 | https://mathoverflow.net/users/49 | 393230 | 162,625 |
https://mathoverflow.net/questions/393225 | 3 | Let $C\_n=\frac1{n+1}\binom{2n}n$ denote the Catalan numbers.
This question is motivated by the (unanswered) [MO post by Alexander Burstein](https://mathoverflow.net/questions/391533/a-bijective-proof-for-the-odd-companion-to-shapiros-catalan-convolution) and [my own (answered by Fedor Petrov) MO post](https://mathov... | https://mathoverflow.net/users/66131 | Is there a $q$-analogue to Shapiro's convolution identity? | George E.Andrews gives a $q$-analog for Shapiro's convolution identity in the article [On Shapiroʼs Catalan convolution](https://doi.org/10.1016/j.aam.2010.07.003). The $q$-analog is
$$\sum\_{j=0}^n q^{2j} C\_{2j+1}(1,-q) C\_{2n+1-2j}(1,-q) = \frac{-q^{2n+1}(-q^2 : q^2)\_{n-1} C\_{n+1}(1,-q)}{(-q:q^2)\_{n+1}}$$
where... | 4 | https://mathoverflow.net/users/51668 | 393235 | 162,626 |
https://mathoverflow.net/questions/393207 | 6 | Let $X$ and $Y$ be smooth manifolds. The map $\mathcal{D}'(X)\times\mathcal{D}'(Y)\to\mathcal{D}'(X\times Y)$ given by $(S,T)\mapsto S\boxtimes T$ is continuous with respect to the strong topology. Is it continuous with respect to the weak-\* topology?
| https://mathoverflow.net/users/125956 | Is the tensor product of distributions a continuous bilinear map with respect to the weak topology? | I will use the convention $\mathbb{N}=\{1,2,\ldots\}$ and denote by $s(\mathbb{N})$ the space of (real) sequences $(\mu\_i)\_{i\ge 1}$ of rapid decay, i.e., such that for all integer $k\ge 0$,
$$
\|\mu\|\_k:=\sup\_{i}i^k|\mu\_i|\ <\infty\ .
$$
We give it the locally convex topology defined by the norms $\|\cdot\|\_k$, ... | 7 | https://mathoverflow.net/users/7410 | 393236 | 162,627 |
https://mathoverflow.net/questions/393233 | 3 | Let $A$ and $B$ be C\*-algebras, let $I\subseteq A$ and $J\subseteq B$ be closed, two-sided ideals, and let $\pi\colon A\to B$ be a bijective, linear map satisfying $\pi(I)=J$. Assume that both the restriction of $\pi$ to $I$ and the induced map $A/I\to B/J$ are continuous.
>
> Is $\pi$ continuous?
>
>
>
| https://mathoverflow.net/users/24916 | Continuity of linear bijection that is continuous on ideal and quotient | No.
**Counterexample.** Let $C$ be an infinite-dimensional $C^\*$-algebra and let $A = B = C \oplus C$. We set $I = J = C \oplus \{0\}$.
Let $\varphi: C \to C$ be a non-continuous linear mapping and define $\pi: A \to A$ by means of the operator matrix
$$
\begin{pmatrix}
\operatorname{id} & \varphi \\
0 & \opera... | 6 | https://mathoverflow.net/users/102946 | 393237 | 162,628 |
https://mathoverflow.net/questions/392837 | 36 | If $S⊂[0,1]^2$ intersects every connected subset of $[0,1]^2$ with a full projection on the $x$-axis, must $S$ have a connected component with a full projection on the $y$-axis?
**An equivalent form：**
If $S⊂[0,1]^2$ intersects every connected subset of $[0,1]^2$ with a full projection on the $x$-axis and $T⊂[0,1]^... | https://mathoverflow.net/users/221921 | A question about connected subsets of $[0,1]^2$ | A counterexample to this statement was posted as a [comment](https://math.stackexchange.com/questions/2824227/do-partitions-of-a-square-into-two-sets-always-connect-one-pair-of-opposite-edge#comment5823373_2824227) by Dejan Govc to the Math StackExchange question, [Do partitions of a square into two sets always connect... | 25 | https://mathoverflow.net/users/6514 | 393240 | 162,631 |
https://mathoverflow.net/questions/393241 | 5 | The following claim is from a paper [On the moduli spaces of bundles on K3 surfaces, I, p. 358] of Mukai. Consider an artinian module $\mathrm{M}$ over a local ring, and let $\mathrm{M}\_0$ be the submodule of all $x\in\mathrm{M}$ annhilated by the maximal ideal of the local ring. Then every endomorphism of $\mathrm{M}... | https://mathoverflow.net/users/104669 | Endomorphisms of Artinian modules | This is false. Take $A=k[x,y]/(x^2,y^2,xy)$ and $M=\omega\_A$. Then $\operatorname{Hom}(\omega,\omega)$ has length 3 and $\operatorname{Hom}(M/M\_0,M/M\_0)$ has length 4, so the natural map cannot be surjective.
| 10 | https://mathoverflow.net/users/9502 | 393248 | 162,633 |
https://mathoverflow.net/questions/393249 | 10 | Let $\mathcal{A}$ be the Steenrod Algebra and $\mathcal{A}(n)$ be the subalgebra generated by $Sq^1, Sq^{2}, Sq^{2^2},\ldots, Sq^{2^n}$.
It is known that
* $H^\*(H\mathbb{Z},\mathbb{Z}/2)=\mathcal{A}//\mathcal{A}(0)$,
* $H^\*(ko,\mathbb{Z}/2)=\mathcal{A}//\mathcal{A}(1)$,
* $H^\*(tmf,\mathbb{Z}/2)=\mathcal{A}//\mat... | https://mathoverflow.net/users/5420 | Spectrum $E$ with $H^\bullet(E,\mathbb{Z}/2)=\mathcal{A}//\mathcal{A}(n)$ | The first two nonzero elements of the $A$-module $A /\!/ A(n)$ are the generator $e$ (in degree 0) and $Sq^{2^{n+1}} e$ (in degree $2^{n+1}$): all other elements in this degree and lower are a sum of products of lower squares, as a consequence of the Adem relations. When $n=3$ this bottom class is $Sq^{16}$.
However,... | 11 | https://mathoverflow.net/users/360 | 393253 | 162,635 |
https://mathoverflow.net/questions/393250 | 9 | Let $G$ be a finitely generated residually finite group and let $M$ be a finitely generated $\mathbb{Z}[G]$-module.
**Question**: Must $M$ be residually finite in the sense that for all nonzero $x \in M$, there exists some submodule $N$ of $M$ such that $x \notin N$ and $M/N$ is finite?
If this is not true in gener... | https://mathoverflow.net/users/230596 | Residual finiteness for modules over group rings | It's true, and due to Ph. Hall, when $G$ is virtually nilpotent, and more generally (Roseblade) when $G$ is virtually polycyclic.
When $G=\mathbf{Z}\wr\mathbf{Z}$ there exists an infinite simple $\mathbf{Z}G$-module, so it's not residually finite.
---
Added: The counterexample is due to Ph. Hall. Since I alread... | 10 | https://mathoverflow.net/users/14094 | 393255 | 162,636 |
https://mathoverflow.net/questions/393265 | -1 | Let $A$ be a unital C\*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
| https://mathoverflow.net/users/120865 | Density of normal elements in a C*- algebra | The normal elements of any $C^\*$-algebra $A$ are norm-closed in $A$. Indeed, if $\{a\_n\}\_n$ is a sequence of normal elements with $a= \lim\_n a\_n$, then by joint continuity of multiplication and involution,
$$aa^\* = \lim\_n a\_n a\_n^\* = \lim\_n a\_n^\* a\_n = a^\*a.$$
Hence, your question is equivalent with: w... | 3 | https://mathoverflow.net/users/nan | 393269 | 162,642 |
https://mathoverflow.net/questions/393010 | 1 | Let $K$ be a finite extension of $\mathbb{Q}\_p$ and let $G$ be a 1-dimensional formal group defined over $\mathcal{O}\_K$. Consider the field $K\_\infty$ obtained by adjoining to $K$ all the solutions to $[p]^n(x) = 0$ with $x \in \mathbb{C}\_p$ with $|x|<1$ and $n\geq1$. (These torsion points depend on a choice of co... | https://mathoverflow.net/users/143589 | Is the completion of the field generated by torsion points of a 1-dimensional formal group perfectoid? | Yes it is. The extension $K\_\infty/K$ is deeply ramified. Now apply proposition 6.6.6 of Gabber-Ramero's "Almost ring theory".
| 3 | https://mathoverflow.net/users/5743 | 393278 | 162,644 |
https://mathoverflow.net/questions/393279 | 1 | Let $p\_1, ... ,p\_n$ be chosen independently from the uniform distribution on the unit torus $[0,1]^2$.
I want to prove a theorem of the form: "With high probability, every circle of radius $r$ contains approximately $\pi r^2 n$ points".
The problem is that there are infinitely many circles, so proving concentrati... | https://mathoverflow.net/users/17599 | Union bound over infinitely many events | $\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}$Let $\C$ denote the set of all disks of a radius $r\in(0,\infty)$ contained in the unit square. Using a rectangular grid of centers of disks in $\C$, we can cover $\C$ by $N=O(r^2/\ep^2)$ balls of radius $\ep\in(0,\infty)$, where the balls are considered with respec... | 2 | https://mathoverflow.net/users/36721 | 393299 | 162,651 |
https://mathoverflow.net/questions/392770 | 12 | Define the theta function as
$$
\theta(x) = \sum\_{n=-\infty}^\infty e^{-\gamma(x+n)^2}
$$
where $\gamma>0$. Clearly, $\theta$ is 1-periodic, non-zero and smooth. Therefore, the reciprocal map $x \mapsto \frac{1}{\theta(x)}$ is 1-periodic and smooth as well and can be expanded as a Fourier series:
$$
\frac{1}{\theta(x)... | https://mathoverflow.net/users/170539 | Does there exist an upper bound on the Fourier coefficients of the reciprocal theta function $\frac {1}{\theta}$? | As [@TerryTao](https://mathoverflow.net/users/766/terry-tao) stated correctly in the comments, a bound on the Fourier coefficients can be obtained by the saddle point method: The extension of $\frac 1 \theta$ to a meromorphic function on $\mathbb C$ has poles which are bounded away from the real axis. This implies that... | 7 | https://mathoverflow.net/users/223636 | 393305 | 162,652 |
https://mathoverflow.net/questions/393304 | 1 | Let $\phi\in C^s,0<\alpha\leq s<1$, where $C^s(0,T]$ is Holder continuous functions. Is it possible to show the following inequality
$$|\frac{\phi(x)}{x^\alpha}-\frac{\phi(y)}{y^\alpha}|\leq \frac{|\phi(x)-\phi(y)|}{|x-y|^\alpha}?$$
| https://mathoverflow.net/users/172842 | Is it possible to show the following inequality? | The answer is no. E.g., let $\phi(u)=1$ for all $u$, and let $x\ne y$. Then the inequality fails to hold.
| 2 | https://mathoverflow.net/users/36721 | 393313 | 162,654 |
https://mathoverflow.net/questions/393306 | 0 | Let $A$ be a unital C\*-algebra. Let $x,y\in A$ be self adjoint elements in $A$, with $x$ being invertible. Can we say that the spectrum of $x^{-1}y$ is a subset of the real line? I know this is true if $x$ was also positive.
| https://mathoverflow.net/users/120865 | Regarding an element being self adjoint | The golden rule for conjectures in operator theory:
**Every ad-hoc conjecture is most likely false for $2 \times 2$-matrices.** :-)
So here's a $2 \times 2$-counterexample for the question:
Let $A = \mathbb{C}^{2 \times 2}$ and
$$
x =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\quad \text{and} \quad
y =
\be... | 6 | https://mathoverflow.net/users/102946 | 393320 | 162,656 |
https://mathoverflow.net/questions/392990 | 14 | We all know and love Cohen reals, and we can (and often do) define the Cohen forcing as partial functions $p\colon\omega\to 2$ with finite domain. The Prikry–Silver forcing is defined as partial functions $p\colon\omega\to 2$ with co-infinite domain.
These two couldn't be any more different. For example, Cohen reals ... | https://mathoverflow.net/users/7206 | What is the "Prikry–Silver collapse" when CH fails? | If $\mathrm{CH}$ fails then $\mathrm{Col}(\omega, \omega\_1)$ does not add a generic for the "Prikry-Silver collapse" $\mathbb P$: Let $\mathbb U$ be $(\mathcal{P}(\omega)/I)^+$ where $I$ is the ideal of finite subsets of $\omega$. The map
$$\pi:\mathbb P\rightarrow \mathbb U,\ p\mapsto [\omega\setminus\mathrm{dom}(p)]... | 6 | https://mathoverflow.net/users/125703 | 393328 | 162,660 |
https://mathoverflow.net/questions/393337 | 3 | I am looking for a nudge in the right direction on the derivation of a formula for the Total Curvature of the Caustics to a manifold (a caustic is a planar family of curves reflected by a manifold).
Background - if the Fundamental Form is positive definite, the second fundamental form can be diagonalized with respect... | https://mathoverflow.net/users/85716 | Total curvature of a Caustic | **Q:** *"I am asking for a reference to a formula I have been given anecdotally".*
**A:** This is equation (10) on page 340 in volume III of the classic text by Gaston Darboux, [Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal](https://gallica.bnf.fr/ark:/12148/bpt6... | 6 | https://mathoverflow.net/users/11260 | 393340 | 162,663 |
https://mathoverflow.net/questions/94968 | 8 | Any small [pretopos](http://ncatlab.org/nlab/show/pretopos) $C$ can be embedded into a Grothendieck topos by a fully faithful functor that preserves all the pretopos structure (limits, images, finite unions of subobjects, disjoint coproducts, and quotients of equivalence relations). Namely, we may consider the topos of... | https://mathoverflow.net/users/49 | Sheaf embedding preserving initial algebras? | Sometimes, but not always.
One case when this is possible is given by Proposition D5.1.8 in *Sketches of an Elephant*. Namely, if $C$ is a small elementary topos with an NNO that is *standard*, meaning that the family of all numerals $s^n o : 1\to N$ (for external natural numbers $n\in \mathbb{N}$) is epimorphic, the... | 5 | https://mathoverflow.net/users/49 | 393343 | 162,665 |
https://mathoverflow.net/questions/393330 | 3 | Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an *operator range* if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient argument, this is equivalent to the existence of a complete norm on $U$ such that the injection $U \hookrightarrow F$ is... | https://mathoverflow.net/users/102946 | Reference request: Baire's theorem for operator ranges | (4) follows immediately from a version of the open mapping theorem: If a continuous linear operator between between Banach (or Fréchet or even more general) spaces has non-meager range, then it is open and, in particular, surjective.
This is theorem 2.11 in Rudin's *Functional Analysis*. As you said, (5) is an easy c... | 3 | https://mathoverflow.net/users/21051 | 393345 | 162,666 |
https://mathoverflow.net/questions/393319 | 12 | In Descriptive Set Theory we often see the notion of encoding a real as a sequence of integers or natural numbers -- i.e. there obviously is a bijection according to ZF axioms. But how does it look like concretely? Anybody has seen a simple construction?
My own approach is by chain-fractions:
Let $q\in\mathbb{R}$ b... | https://mathoverflow.net/users/152241 | What's the bijection between reals and infinite sequences of integers? | [**Note:** this answer uses the convention where $\mathbb{N} := \{ 0, 1, 2, \dots \}$ contains zero.]
There's an elegant explicit order-preserving bijection between the [Baire space](https://en.wikipedia.org/wiki/Baire_space_(set_theory)) $\mathbb{N}^{\mathbb{N}}$ (under lexicographical order) and $\mathbb{R}\_{\geq ... | 44 | https://mathoverflow.net/users/39521 | 393348 | 162,667 |
https://mathoverflow.net/questions/393352 | 1 | Let $A$ be a unital $C^\*$ algebra. Assume that $D:A\to A$ is a bounded derivation.
Can one say that $1$ can not be in the image of $D$?
If the answer is no:
What is a counter example? What kind of $C^\*$ algebra admits outer bounded derivation but stil they satisfy the above prevent property?
**Motivation:** I... | https://mathoverflow.net/users/36688 | On solvability of equation $D(x)=1$ where $D:A\to A$ is a bounded outer derivation on a $C^*$ algebra | $1$ cannot belong to the image of $D$.
Assume that $A\subset B(H)$. According to Theorem 4 from "Derivations of operator algebras" by Kadison any derivation $D$ is spatial, i.e. there is an operator $T \in B(H)$ such that $D(x) = [T,x]$. If $D(x) = 1$, then we would have represented the identity as a commutator of tw... | 3 | https://mathoverflow.net/users/24953 | 393375 | 162,672 |
https://mathoverflow.net/questions/393360 | 4 | Let $C = \operatorname{Cl}(V,q)$ be a Clifford algebra where $V$ is an $N$-dimensional space with basis $B = \{e\_1,e\_2, \dotsc, e\_N\}$. I'm looking for a way to invert elements.
What I've already worked out for myself is that if $x = \sum\_{I \subseteq B} \lambda\_I \hat{I}$ where $\hat{I}$ is the (ordered) produc... | https://mathoverflow.net/users/114960 | Finding inverses in Clifford Algebras | You should have a look at F. Reese Harvey's book "Spinors and Calibrations", where your question is answered for Clifford algebras of every signature.
The main point is that, when the ground field is $\mathbb{R}$, each Clifford algebra is actually isomorphic to a classical matrix algebra (or sum of two matrix algebra... | 5 | https://mathoverflow.net/users/13972 | 393381 | 162,673 |
https://mathoverflow.net/questions/393389 | 2 | **Definitions:**
Let $E$ be a measurable, bounded subset of $\mathbb R^n$ with nonzero Lebesgue measure.
Denote by $\partial E$ the measure theoretic boundary of $E$, defined as the set of points in $\mathbb R^n$ where the measure theoretic density of $E$ is not $0$ or $1$.
For $\varepsilon > 0$, write $\partial ... | https://mathoverflow.net/users/173490 | Growth and shrinking rate of measurable sets along the boundary | I think one of the classic counterexamples works here, to show that this is false: Let $\{q\_i\}\_{i\in\mathbb{N}}$ dense in $[0,1]^n$, $\delta >0$ and construct $$E = \bigcup\_{i\in\mathbb{N}} B\_{\delta 2^{-i}}(q\_i).$$
Then $\mu(E) \leq c\delta^n$, but $E$ is dense in $[0,1]^n$. If I am not completely mistaken (Yo... | 3 | https://mathoverflow.net/users/51695 | 393391 | 162,678 |
https://mathoverflow.net/questions/393293 | 1 | I was looking at this old [question](https://math.stackexchange.com/questions/2722110/an-analogue-of-the-poisson-bracket-in-contact-geometry) and thought it might get more attention at this site. In summary, the OP asks the following question:
>
> McDuff and Salamon define an analogue of the Poisson bracket in
> co... | https://mathoverflow.net/users/153228 | An analogue of the Poisson bracket in contact geometry? | This bracket is sometimes called the Lagrange bracket in the literature (although that terminology is unfortunately not universal, and sometimes refers to something different). It can be characterised as the map $\lbrace\cdot,\cdot\rbrace:C^\infty(M)\times C^\infty(M)\to C^\infty(M)$ that satisfies $[X\_F,X\_G] = X\_{\... | 2 | https://mathoverflow.net/users/17945 | 393410 | 162,686 |
https://mathoverflow.net/questions/393430 | 1 | **Question:** Is there a simple method for calculating the Fourier transform of a holomorphic complex function ${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$?
In order for my question to be well-posed I define a holomorphic function ${f}:\Omega\to{\mathbb{{{C}}}}$ to posses continuous first partial derivatives a... | https://mathoverflow.net/users/170939 | Fourier transform of a holomorphic function | It isn't clear what you mean by the Fourier transform of a holomorphic function. You would need to supply a definition of this concept before you can get coherent answers. Perhaps your question is about an attempt to define such a transform?
It seems like you are interested in expressing functions in some orthonormal... | 8 | https://mathoverflow.net/users/1106 | 393433 | 162,689 |
https://mathoverflow.net/questions/393415 | 2 | I have a question concerning the proof of Corollary 7.3.6.5 in Luries "Higher Topos Theory" (the same issue also occurs in the proof of 7.3.6.10, but it is clearer here). Given is a continuous map $p:X\rightarrow Y$ between paracompact topological spaces, where $Y$ has finite covering dimension, this induces $p\_\*: \o... | https://mathoverflow.net/users/156537 | (Local) Homotopy dimension of $\infty$-topoi on paracompact spaces | I think if $X$ is paracompact of covering dimension $\leq n$ then $\mathrm{Shv}(X)$ is also locally of homotopy dimension $\leq n$:
First, the $F\_\sigma$ open subsets of $X$ form a basis of the topology closed under finite intersections, so they generate the $\infty$-topos $\mathrm{Shv}(X)$ under colimits.
If $U\s... | 2 | https://mathoverflow.net/users/20233 | 393436 | 162,690 |
https://mathoverflow.net/questions/393435 | 1 | I am wondering if there is an extreme value distribution that is closed under both the minimum and the maximum operation.
For example, for there is a Gumbel maximum distribution closed under the maximum (provided $\beta$ is the same for both distributions). Also there is the Gumbel minimum distribution closed under min... | https://mathoverflow.net/users/171482 | Extreme value distribution for both minimum and maximum at the same time | First, a few comments to make sure I understand the question correctly:
* I believe the correct wording is "stabie under maxima/minima" or "max/min-stable" rather than "closed". (Similarly, I think "same kind of distribution" is not widely used, a more standard term would be "distribution of the same type".)
* Max-st... | 1 | https://mathoverflow.net/users/108637 | 393444 | 162,693 |
https://mathoverflow.net/questions/393445 | 0 |
>
> If a $\Pi\_1$ sentence is independent from PA, then it is true.
>
> CON(ZFC) is a $\Pi\_1$ sentence and independent from PA.
>
> Therefore, CON(ZFC).
>
>
>
If this is a valid argument in ZFC, it would violate Gödel's incompleteness theorem. What's wrong with it?
| https://mathoverflow.net/users/235026 | What's wrong with this argument for CON(ZFC)? | Well, if CON(FZC) is false, then it is not independent of PA - as PA can look at the ZFC-proof of falsity and confirm it.
| 5 | https://mathoverflow.net/users/15002 | 393446 | 162,694 |
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