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https://mathoverflow.net/questions/447874 | 0 | Consider the simplest SIR model:
$$S'=-a SI$$
$$I'=a SI - b I$$
$$R'=b I$$
It is known that the waiting time of an infeccious person in the compartment $I$ follows an exponential behavior with rate $b$.
I was trying to figure out how this relates with a Poisson process, but I am not finding the "hook".
For instan... | https://mathoverflow.net/users/172600 | Poisson Process x SIR model | The solution of this ODE defined over a compact interval $\left[0,T\right]$ for each initial condition can be formally cast as the limit (as the number of individuals $N$ scales to infinite) of a stochastic process modelling the peer-to-peer spread of a virus with Poisson clocks.
**Remark.** This answer is just inten... | 3 | https://mathoverflow.net/users/138242 | 447904 | 180,352 |
https://mathoverflow.net/questions/447892 | 0 | I am interested in the exponential sum
$$\sum\_{n=1}^X \frac{e(c\_1n^2+c\_2 n)}{1-e(c\_1n)}$$
where $c\_2$ is irrational and $e(x)=e^{2\pi i x}$. I know if the denominator is not there, this is a Weyl sum with a lot of bounds.
Is there anything that can be done to relate this sum to a Weyl sum or get some other b... | https://mathoverflow.net/users/479223 | Exponential sum with weight in bottom | Under your assumption that no $n$ from $1$ to $X$ has $|1 -e(c\_1n)|<\epsilon$, we have an upper bound for your sum of the form $2 \epsilon^{-1} \log X + O(X)$.
This is the "trivial bound" in that it just comes from estimating $$\sum\_{n=1}^X \frac{1}{ |1 - e(c\_1 n )|}.$$
By the pidgeonhole principle, the number o... | 3 | https://mathoverflow.net/users/18060 | 447907 | 180,354 |
https://mathoverflow.net/questions/447888 | 6 | We know several amazing techniques about the derived category $Perf (X)$ of a smooth projective variety such as the whole theory of Fourier-Mukai transforms. On the other hand, from a dg-categorical point of view, it is natural to work with smooth proper varieties instead of smooth projective varieties. Indeed, (dg) de... | https://mathoverflow.net/users/177839 | Derived categories of smooth proper varieties? | The study of derived categories is a special case of the more general study of semiorthogonal components of derived categories. By Chow lemma for any proper variety $X$ there is a blow up $\pi \colon X' \to X$ such that $X'$ is projective, and by Orlov's blowup formula the derived category of $X$ is a semiorthogonal co... | 7 | https://mathoverflow.net/users/4428 | 447914 | 180,356 |
https://mathoverflow.net/questions/447470 | 3 | Let $P$ be a polarization of a Hilbert space $\mathcal{H}$, i.e. a bounded idempotent: consider a group $G=GL\_{res}(\mathcal{H}):=\{g \in GL(\mathcal{H}): [g,P] \in HS\}$ (where $HS$ is the set of all Hilbert Schmidt operators). For $g \in G$ let $P\_g:=gPg^{-1}$ and define $F\_{g,h}=P\_gP\_h-P\_h+I$. One can check th... | https://mathoverflow.net/users/24078 | Determinant line of Fredholm operators and composition of morphisms | As mentioned in the comments the first part of the question is in Abbonandolo and Majers "Infinite dimensional Grassmannians". My Hilbert spaces are real and separable and infinite dimensional.
I want to state that I basically know next to nothing about trace class operators, so take the rest with a grain of salt.
... | 1 | https://mathoverflow.net/users/12156 | 447916 | 180,357 |
https://mathoverflow.net/questions/447920 | 8 | In Lurie's treatment of the cobordism hypothesis, the domain is $\mathsf{Bord}^{fr}\_n$, the symmetric monoidal $(\infty,n)$-category of $k$-bordisms with $n$-framing for $0\leq k\leq n$.
If I understand his definitions correctly, for $n\neq 0,1,3,7$, $\mathsf{Bord}^{fr}\_n$ does not contain $S^n$ since $S^n$ does not ... | https://mathoverflow.net/users/472749 | Stably-framed cobordism $(\infty,n)$-category | Yes, there is a stably-framed bordism category. Recall that tangential structures on smooth $n$-manifolds can be parameterized by maps $X \to BO(n)$; an $X$-structure on $M$ is then by definition a lift of the classifying map for the tangent bundle $T\_M : M \to BO(n)$ through $X$. In the stably-framed case, you take $... | 8 | https://mathoverflow.net/users/78 | 447923 | 180,360 |
https://mathoverflow.net/questions/447890 | 1 | Let $X$ be a smooth connected proper scheme over field $k$. It is known that
correspondences $\alpha \subset X \times X$ regarded as
objects in Chow groups $\text{CH}^\*(X \times X)$
act on cohomology $H^\*(X, \mathbb{Z})$ via "push and pull",
namely we have a cycle class map
$$ \text{cl}: \text{CH}^\*(X) \to H^\*(X,... | https://mathoverflow.net/users/501436 | Correspondences acting on cohomology groups $H^*(X)$ & splittings | Composition of a correspondence in $X \times Y$ and a correspondence in $Y \times Z$ is given by pulling both back to $X\times Y\times Z$, intersecting them, and pushing forward to $X \times Z$.
To check that $ D\times X \subset X\times X$ is idempotent under composition for $D$ of degree $1$, we pull back along two ... | 3 | https://mathoverflow.net/users/18060 | 447932 | 180,365 |
https://mathoverflow.net/questions/447925 | 12 | This was previously posted to Math StackExchange. I was originally unsure whether it is suitable for posting here, but I've yet to get an answer there, so I'm just trying to see if people here can help.
As in the question title, let $A, B$ be a partition of the unit circle $S^1$, equipped with the Haar measure. Here,... | https://mathoverflow.net/users/504602 | If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$? | Assuming the Continuum Hypothesis, the answer is yes. On CH, the $\mathbf Q$-dimension of $\mathbf R$ has the cardinality of the first uncountable ordinal $\omega\_1$, so we can find a $\mathbf Q$-linear basis $e\_\alpha, \alpha \in \omega\_1$ of ${\bf R}$, and we can normalize $e\_0=1$. Thus every irrational real numb... | 13 | https://mathoverflow.net/users/766 | 447940 | 180,367 |
https://mathoverflow.net/questions/313516 | 16 | In his famous paper ["On a problem of Kurosh, Jonsson groups, and applications"](https://www.sciencedirect.com/science/article/pii/S0049237X08713466) of 1980, Shelah constructed a CH-example of an uncountable group $G$ equal to $A^{6640}$ for any uncountable subset $A\subset G$.
Let us call a group $G$
$\bullet$ *$... | https://mathoverflow.net/users/61536 | A Shelah group in ZFC? | In [this preprint](https://arxiv.org/pdf/2305.11155.pdf), Mark Poor and Assaf Rinot construct a ZFC-example of a $10120$-Shelah group $G$ of cardinality $|G|=\lambda^+$, where $\lambda$ is an arbitrary regular сardinal.
| 4 | https://mathoverflow.net/users/61536 | 447941 | 180,368 |
https://mathoverflow.net/questions/115982 | 9 | I am searching for R. Bott's lectures on characteristic classes and Gel'fand Fuks cohomology (New Mexico State Univ. 1973), apparently there are notes of these lectures taken by Mostow and Perchik which were published in some proceedings. But I was not able to find these proceedings.
Thank you in advance for your hel... | https://mathoverflow.net/users/27816 | R. Bott's lectures on characteristic classes | It turns out that what you are looking for is [here](https://math.ucr.edu/%7Eres/bott-nmsu/Bott%20-%20Lectures%20on%20Gelfand-Fuks%20Cohomology%20and%20Foliations.pdf)
which was noted by Mostow and Perchik.
| 4 | https://mathoverflow.net/users/136661 | 447947 | 180,370 |
https://mathoverflow.net/questions/447942 | 7 | Is the following statement (†) consistent with ZFC?
* If $E \subseteq [0,1]^2$ is such that $E\_x := \{y\in[0,1] : (x,y)\in E\}$ has measure zero for almost all $x$, then $E^y := \{x\in[0,1] : (x,y)\in E\}$ has measure zero for almost all $y$.
Of course I'm not assuming $E$ to be measurable, because that case follo... | https://mathoverflow.net/users/17064 | Consistency of a strong Fubini type theorem for measure zero sets | ZFC refutes this principle. Let $\kappa=\text{non}(\mathcal{L}),$ i.e. the least cardinality of a set of reals of positive outer measure. Let $X \subset [0,1]$ be such that $|X|=\kappa$ and $\lambda^\*(X)>0,$ and let $\prec$ be a well-ordering of $X$ of order type $\kappa.$ Then $E:=\{(x,y) \in X^2: y \prec x\}$ is a c... | 12 | https://mathoverflow.net/users/109573 | 447952 | 180,372 |
https://mathoverflow.net/questions/447950 | 6 | Let $N$ be a finite subset of $\mathbb{N}$, where $|N|>1$. For $i\in\mathbb{N}$, let $a\_i$ and $b\_i$ be chosen uniformly at random from $N$. Is it true that $\mathbb{P}[\sup\_{n\in\mathbb{N}}\{\frac{a\_1\cdots a\_n}{b\_1\cdots b\_n}\}=\infty]=1$? Note that given the symmetry involved, I believe this should be the sam... | https://mathoverflow.net/users/505965 | Probability that the ratio of products of randomly chosen natural numbers is unbounded | Yes, this is true. Indeed, let
$$S\_n:=\sum\_{i=1}^n Y\_i=\frac1s\,\ln\frac{a\_1\cdots a\_n}{b\_1\cdots b\_n},$$
where $Y\_i:=X\_i/s$, $X\_i:=\ln\frac{a\_i}{b\_i}=\ln a\_i-\ln b\_i$, and $s:=\sqrt{Var\, X\_i}\in(0,\infty)$. Note that the $Y\_i$'s are i.i.d. zero-mean unit-variance random variables. So,
$$P\Big(\sup\_{n... | 8 | https://mathoverflow.net/users/36721 | 447953 | 180,373 |
https://mathoverflow.net/questions/447848 | 7 | We say that $\alpha$ is $\Sigma\_n$-extendable (to $\beta$), if there is $\beta>\alpha$ such that $L\_\alpha$ is a $\Sigma\_n$ elementary submodel of $L\_\beta$.
First ordinal: the least $\alpha\_0$ such that there are ordinals $\alpha\_0<\alpha\_1<...<\alpha\_{\omega^2}$, such that each $\alpha\_i$ is $\Sigma\_2$ ex... | https://mathoverflow.net/users/170286 | Which one of the following two ordinals is larger? | The second one is larger.
Because of the conflict of terminology with "extendible", I will say that an ordinal $\alpha$ is
*somewhere $\Sigma\_n$-stable*
if there is $\beta>\alpha$ such that $L\_\alpha\preccurlyeq\_n L\_\beta$.
Given a somewhere $\Sigma\_2$-stable $\alpha$,
let $\beta\_2(\alpha)$ be the least $\beta>... | 6 | https://mathoverflow.net/users/160347 | 447969 | 180,379 |
https://mathoverflow.net/questions/447980 | -2 | Show that the series
$$\sum\_{n=2}^{\infty}\frac{1}{[\frac{(1+\epsilon)\log n}{\log\log n}]!}$$converges for $\epsilon>0$.
Stirling's approximation gives that the convergence for the series is equivalent to the series
$$\sum\_{n=2}^{\infty}\frac{1}{\sqrt{2\pi [\frac{(1+\epsilon)\log n}{\log\log n}]}}\left(\frac{e}{[\fr... | https://mathoverflow.net/users/484728 | convergence for a series | This is not a research level question, but I feel like answering it. Simply, use the elementary estimate
$$\log(m!)\sim m\log m.$$
The symbol $\sim$ means that the ratio of the two sides tends to $1$. For
$$m=\left[\frac{(1+\epsilon)\log n}{\log\log n}\right],$$
this becomes
$$\log(m!)\sim\frac{(1+\epsilon)\log n}{\log... | 2 | https://mathoverflow.net/users/11919 | 447982 | 180,382 |
https://mathoverflow.net/questions/447987 | 2 | Is the following embedding possible?
$\mathrm{Sp}\_{2m}(p)\leqslant S\_{p^m-1}$ where $S\_{p^m-1}$ is a symmetric group and $p$ is prime. I see that when $p=3$ and $m=3$, the order of the former does divide the order of the latter. I was also thinking along the lines of $\mathrm{PSp}$ being simple except for finite c... | https://mathoverflow.net/users/488802 | is the embedding $\mathrm{Sp}_{2m}(p)\leqslant S_{p^m-1}$ possible? | No. The smallest degrees of the faithful permutation representations of the finite simple groups are listed in Table 4.5 of [On the maximum orders of elements of finite almost simple groups and primitive permutation groups](https://arxiv.org/abs/1301.5166) by Guest, Morris, Praeger, and Spiga.
$\DeclareMathOperator\P... | 9 | https://mathoverflow.net/users/35840 | 447991 | 180,385 |
https://mathoverflow.net/questions/447951 | 3 | $\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Proj{\mathbf{Proj}}$Let $S$ be a Noetherian scheme and $X$ finite $S$-scheme. The finite morphism $X \to S$ is projective in the sense of the definition in the Stacks Project (tag [01W8](https://stacks.math.columbia.edu/tag/01W8), part (1)).
This is proved in tag [... | https://mathoverflow.net/users/nan | Finite subschemes of projective bundles | Welcome new contributor. There is no such morphism from the relative Proj to $X$: it would contradict Stein factorization. However, there is a simple proof that $\sigma$ is a closed immersion that does not need this.
For every quasi-coherent $\mathcal{O}\_S$-module $\mathcal{F}$, denote by $$(p\_\mathcal{F}:\mathbb{P... | 1 | https://mathoverflow.net/users/13265 | 447997 | 180,386 |
https://mathoverflow.net/questions/447949 | 18 | The Robertson–Seymour theorem concerns downwardly closed classes of isomorphism classes of finite undirected graphs. (Am I committing some sin by referring to a class of classes? An isomorphism class is a proper class, and a set, as opposed to a proper class, is a class that is a member of some other class. But ignore ... | https://mathoverflow.net/users/6316 | Is the Robertson–Seymour theorem equivalent to the compactness of some topological space? | The Robertson–Seymour graph minor theorem states that the set of all (isomorphism classes of) finite undirected graphs under the graph minor relation is a [well-quasi-ordering](https://en.wikipedia.org/wiki/Well-quasi-ordering) (or wqo for short). It is a theorem that a quasi-ordering $Q$ is a wqo if and only if the Al... | 25 | https://mathoverflow.net/users/3106 | 448004 | 180,389 |
https://mathoverflow.net/questions/447979 | 0 | Is there a characterization of all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying:
* $f(0)=0$
* $f$ is monotonically increasing
* $f$ is concave
My intuition is that $f$ should admit an integral representation:
$$
f(t) = \int\_0^t\, g(s)\,ds
$$
for some positive continuous function to satisfy ... | https://mathoverflow.net/users/36886 | Representation of continuous, monotone, concave functions | $\newcommand\R{\mathbb R}$Since $f$ is nondecreasing, concave, and continuous on $[0,\infty)$, it has a nonnegative nonincreasing right derivative $g=f'\_+$ on $[0,\infty)$, which is of course right continuous. So, there is a limit $a\_g:=\lim\_{u\to\infty}g(u)\in[0,b\_g]$, where $b\_g:=g(0)<\infty$.
Let $\mu\_g$ be ... | 2 | https://mathoverflow.net/users/36721 | 448012 | 180,393 |
https://mathoverflow.net/questions/448006 | 4 | Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}\_p$ such that
$$
(1+x^2)^3-1
$$
is not a square in $\mathbb{Z}\_p$? In particular, when $-1$ is not a square in $\mathbb{Z}\_p$, can we show that the equation
$$
-y^2 = (1+x^2)^3-1
$$
always has non-trivial solutions $(x,y) \ne (0,0)$ in $\mathbb{Z}\_... | https://mathoverflow.net/users/506018 | Polynomial that is not always a square over $\mathbb{Z}_p$ | Yes, this is true.
We have $(1+x^2)^3-1 = x^2 (x^4 + 3x^2 + 3)$, so we're asking for
nonzero x such that $x^4 + 3x^2 + 3$ is not a square. If none exist then
the genus-1 curve $y^2 = x^4 + 3x^2 + 3$ has at least $2p-4$ rational points
over the $p$-element field (including the two points at infinity, and
subtracting a... | 10 | https://mathoverflow.net/users/14830 | 448017 | 180,396 |
https://mathoverflow.net/questions/448013 | 0 | Let $f : \mathbb{R} \to \mathbb{R}$ be a smooth function which has a smooth inverse and satisfies the estimate
\begin{equation}
\lvert f(x) \rvert \leq \lvert x \rvert.
\end{equation}
Also, let $d\mu$ be the standard normal Gaussian measure on $\mathbb{R}$.
Assume that
\begin{equation}
\int\_{\mathbb{R}} \lvert f(x... | https://mathoverflow.net/users/56524 | Estimating the bound of the integral over whole $\mathbb{R}$ of the Taylor remainder term? | $\newcommand\R{\mathbb R}$We want to bound the ratio
\begin{equation\*}
R:=\frac ND,
\end{equation\*}
where
\begin{equation\*}
N:=\iint\_{\R^2}(F(x,w)-F(y,w))^2 \mu(dx)\mu(dy),\quad
D:=\iint\_{\R^2}(x-y)^2 \mu(dx)\mu(dy).
\end{equation\*}
We have $D=2$. Next,
\begin{equation\*}
A:=(F(x,w)-F(y,w))^2=(f(x)-f(y)-... | 3 | https://mathoverflow.net/users/36721 | 448023 | 180,400 |
https://mathoverflow.net/questions/448015 | 0 | Let $k$ be an algebraically closed field of characteristic zero. Let $\Sigma$ be the fan in $\mathbb{R}^2$ consisting of three cones, cone generated by $e\_1,e\_2$,cone generated by $e\_2,-e\_1-2e\_2$ and the cone generated by $-e\_1-2e\_2,e\_1$.Let $U$ be the toric variety associated to the cone in $\mathbb{R}^2$ whic... | https://mathoverflow.net/users/477848 | How to compute the $G$-theory of the weighted projective space $\mathbb{P}(1,1,2)$? | If $X$ is a projective rational normal surface then
$$
G\_0(X) \cong \mathbb{Z} \oplus \mathrm{Cl}(X) \oplus \mathbb{Z},
$$
see, e.g., Lemma 4.2 in [Karmazyn, Joseph; Kuznetsov, Alexander; Shinder, Evgeny. Derived categories of singular surfaces. J. Eur. Math. Soc. (JEMS) 24 (2022), no. 2, 461--526]. Since
$$
\mathrm{C... | 3 | https://mathoverflow.net/users/4428 | 448031 | 180,405 |
https://mathoverflow.net/questions/446397 | 4 | Adapted from [math stack exchange](https://math.stackexchange.com/questions/4694934/just-how-regular-are-the-sample-paths-of-1d-white-noise-smoothed-with-a-gaussian).
**Background**: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian... | https://mathoverflow.net/users/121501 | Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel? | Yes, $t \mapsto w\_t$ extends to an entire function $\mathbb{C} \to \mathbb{C}$ with probability $1$.
Here is an explicit construction for $(w\_t)\_{t \in \mathbb{R}}$.
Let $(X\_n)\_{n \ge 0}$ be independent real standard normal random variables, and let
$$w\_t := (\frac{\pi}{2 \beta})^\frac{1}{4} e^{-\frac{\beta}{2} t... | 1 | https://mathoverflow.net/users/42355 | 448039 | 180,407 |
https://mathoverflow.net/questions/448035 | 5 | Let $f\_1(x)\in \mathbb{Z}[x]$ be a fixed irreducible degree 4 polynomial such that its splitting field $F\_1$ is an $S\_4$-Galois extension over $\mathbb{Q}$ and the discriminant of $F\_1$ is of the form $-k^2$ for some integer $k$. It is possible to show that $F\_1$ contains $\mathbb{Q}(\sqrt{-1})$.
Does there alwa... | https://mathoverflow.net/users/116598 | Splitting fields of degree 4 irreducible polynomials containing a fixed quadratic extension | Yes, this can be done. Let $K$ be a $S\_4$-quartic field, $C$ its cubic resolvent field, and $L$ the Galois closure of $K$. By Galois correspondence, $L$ contains a unique quadratic subfield $Q$ which corresponds to the alternating group $A\_4$; by our hypothesis, this quadratic subfield is equal to $\mathbb{Q}(\sqrt{-... | 6 | https://mathoverflow.net/users/10898 | 448054 | 180,409 |
https://mathoverflow.net/questions/448051 | 1 | ### Question
I am making a tree using the following two functions:
$$f(x)=\frac{x}{r},\quad g(x)=\frac{x+b}{r}$$
where $1<r<2$ and $0<b$ are rationals. Everything is a real number here.
The starting point is $x=0$. From here, we will compute $f(0)$ and $g(0)$ which will be the branches of $x=0$. We keep on repe... | https://mathoverflow.net/users/173974 | Will this "tree" cover all rational numbers in a range? | The main Question is answered in the negative by @Saúl RM in the comments.
We answer the sub-question
"can there even be branches that have the same value?":
not for any rational $r$ other than $1$ and $-1$
(neither of which is in the allowed range $1 < r < 2$).
Indeed the "branches" are precisely the sums of finite ... | 4 | https://mathoverflow.net/users/14830 | 448055 | 180,410 |
https://mathoverflow.net/questions/448057 | 4 | I want to calculate the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$ when its generators $G\_i$ (whose number is finite) are known.
For any matrix $S$ that commutes with the group: $G\_iS$ = $SG\_i$, and I get a system of linear equations. Any commuting element then can be written as $S = \sum\_k x\_k S\... | https://mathoverflow.net/users/173855 | Calculating the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$ | There is an algorithm to do this (and also to test two matrices in ${\rm GL}(n,{\mathbb Z})$ for conjugacy) described in [the paper](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12246):
The conjugacy problem in ${\rm GL}(n,{\mathbb Z})$
Bettina Eick, Tommy Hofmann, E. A. O'Brien, Journal of the Lo... | 8 | https://mathoverflow.net/users/35840 | 448061 | 180,413 |
https://mathoverflow.net/questions/448046 | 2 | I'm trying to structure a proof where there are several algebras instantiated over sets, where the properties that you get from the algebraic theories are important, but the properties of the sets themselves are also important.
This is why I want to ask whether extending a proof system with models of first order theo... | https://mathoverflow.net/users/324769 | Extending a first-order deductive system with satisfaction relation | Given that your proof system includes ZFC, the standard way to handle algebraic structure like this in a set-theoretic context is simply to interpret those concepts in set theory. In the language of set theory we can interpret the notions of signature, model, language, and it is simply a ZFC theorem that every model $\... | 2 | https://mathoverflow.net/users/1946 | 448063 | 180,415 |
https://mathoverflow.net/questions/448008 | 5 | In Libgober's paper [Alexander polynomial of plane algebraic curves and cyclic multiple planes](http://homepages.math.uic.edu/%7Elibgober/otherpapers/export/1982alexanderduke.pdf), Example 2 (p.850), Libgober claims that the complement to this curve (i.e. $x^2u=y^3$ relative to the line in infinity $u=0$) is a retract ... | https://mathoverflow.net/users/100624 | Complement of plane curve and knot | If you have a weighted homogeneous polynomial $f(z\_1,z\_2)$ then that
means there's a $\mathbb{C}^\times$ action which preserves both
the curve $C=\{f=0\}$ and its complement. Take a small sphere
$S$ centred at the origin which is transverse to the orbits of
this group action. Its intersection with the curve is a link... | 4 | https://mathoverflow.net/users/10839 | 448065 | 180,416 |
https://mathoverflow.net/questions/448044 | 2 | Is [Scholl, *Motives for modular forms*, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight?
Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing the level of $f$. Then the $\pi$-adic ($\pi \mid p$) realization is crystalline at $p$ and the characteristic polynomi... | https://mathoverflow.net/users/471019 | $\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline | Blasius and Rogawski's paper "Motives for Hilbert modular forms" (1993) proves a more general result for Hilbert modular forms over any totally-real field, which includes this as a special case.
| 4 | https://mathoverflow.net/users/2481 | 448067 | 180,417 |
https://mathoverflow.net/questions/448056 | -2 | Consider a simple branching process $Z\_0,Z\_1,Z\_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$.
From the theory of branching processes, a basic result is that the average number of particles at the $n$th step i... | https://mathoverflow.net/users/481145 | Branching process with varying offspring distribution at each step | As far as terminology goes, I have seen this called "Branching process in a varying environment" and also "inhomogeneous Galton-Watson process".
Starting from one individual, the mean of $Z\_n$ is simply $\prod\_{i=1}^n m\_i$ where $m\_i$ is the mean of the offspring distribution for the $i$th generation. For example... | 1 | https://mathoverflow.net/users/5784 | 448069 | 180,418 |
https://mathoverflow.net/questions/448049 | 5 | Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the **alphabet** of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to denote the length of $g$. The [**Gromov boundary**](https://en.wikipedia.org/wiki/Gromov_boundary) of $F$, denoted by $\p... | https://mathoverflow.net/users/151332 | Cancellation of elements in the Gromov boundary of a free group | It is not true. Consider the following point at infinity:
$
\gamma = a \cdot b \cdot a \cdot bb \cdot a \cdot b \cdot a \cdot bbb \cdot a \cdot b \cdot a \cdot bb \cdot a \cdot b \cdot a \cdot bbbb \cdot a \cdot b \cdot a \cdot bb \cdot a \cdot b \cdot a \cdot bbb \cdot a \cdot b \cdot a \cdot bb \cdot a \cdot b \cdo... | 2 | https://mathoverflow.net/users/1650 | 448079 | 180,420 |
https://mathoverflow.net/questions/448087 | 1 | I am starting to learn the $K$-theory of triangulated categories and is stuck with the following.
Let $\mathcal{T}$ be a triangulated category having a semi-orthogonal decomposition $\langle \mathcal{A\_1},\mathcal{A\_2},…,\mathcal{A\_n}\rangle$.
I saw in the paper of Orlov titled Smooth and Proper Noncommutative Schem... | https://mathoverflow.net/users/477848 | How to compute the higher $K$-theory of a triangulated category having a semi-orthogonal decomposition? | Assume $\mathcal{T} = D^b(X)$ and $X$ is smooth. Then $K\_0(X \times X)$ has an algebra structure (with the convolution product) and $K\_\bullet(X)$ is a graded module over this algebra. So, the idea is to use a decomposition of $1 \in K\_0(X \times X)$ (the class of the diagonal $\Delta\_X$) into a sum of idempotents,... | 6 | https://mathoverflow.net/users/4428 | 448090 | 180,425 |
https://mathoverflow.net/questions/431926 | 14 | Wikipedia seems to have an answer
"The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was discovered in 1968 by Jean Bénab... | https://mathoverflow.net/users/429204 | Who introduced the notion of 2-categories? | It appears that the definition of 2-category was introduced independently by two authors, both of whom independently introduced the modern notion of enriched category, for which 2-categories appeared as an example.
* Jean Bénabou gives 2-categories as example (5) in the 1965 paper [Catégories relatives](https://galli... | 9 | https://mathoverflow.net/users/152679 | 448093 | 180,427 |
https://mathoverflow.net/questions/448095 | 6 | Let $\epsilon\_1,\ldots,\epsilon\_n$ be a sequence of signs and $M(t)$ be the tridiagonal matrix whose diagonal entries are $\epsilon\_1 t,\ldots, \epsilon\_n t$ and off-diagonal entries equal to $1$. Is it true that the number of real zeroes of $P(t)=\det M(t)$ is equal to the absolute value of $\epsilon\_1+\dots+\eps... | https://mathoverflow.net/users/14547 | Real zeroes of the determinant of a tridiagonal matrix | For $\epsilon\_1=\epsilon\_2=-1$ and $\epsilon\_3=\epsilon\_4=\epsilon\_5=1$ you get the counterexample $\operatorname{det}M(t)=t(t - 1)^2(t + 1)^2$.
Another example, with simple real roots, is $\epsilon\_1=\epsilon\_5=-1$ and $\epsilon\_2=\epsilon\_3=\epsilon\_4=1$ with $\operatorname{det}M(t)=(t - 1)t(t + 1)(t^2 + ... | 6 | https://mathoverflow.net/users/18739 | 448098 | 180,429 |
https://mathoverflow.net/questions/448101 | 0 | Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta\_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, $f\_a(y):= f(a,y)$ is not a continuous function in $y$. However, $E(f(X,y))$ is a continuous function in $y$. Example:... | https://mathoverflow.net/users/506211 | Convergence in expectation of a discontinuous function | Without more assumptions, it's not true.
Take $d=1$ and let
$$f(x,y) = \begin{cases} x, & \text{if } y=0 \\
0, & \text{otherwise}
\end{cases}$$
Let $X$ be any nontrivial random variable with mean zero, such as $X= \pm 1$ with probability $1/2$ each. Then $E[f(X, y)]=0$ for all $y$, which is certainly continuous in $y... | 1 | https://mathoverflow.net/users/4832 | 448104 | 180,431 |
https://mathoverflow.net/questions/448103 | 4 | Suppose that $G=(G\_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum\_{i,j}(G^5)\_{ij}\le\sum\_{i,j}(G^3)\_{ij}\,?$$
This is true if it is additionally assumed that all off-diagonal entries of $G... | https://mathoverflow.net/users/36721 | An inequality for certain positive-semidefinite matrices | If I have not committed any mistake, please, find below a counter-example.
>
> **Counter-example.** Let $G\in \mathbb{S}^3\_{+}$ be defined by
> $$
> G = \begin{pmatrix}
> 1 & -\frac{2}{5} & 0 \\
> -\frac{2}{5} & 1 & -\frac{2}{5}\\
> 0 & -\frac{2}{5} & 1
> \end{pmatrix}.
> $$
>
>
>
| 9 | https://mathoverflow.net/users/138242 | 448112 | 180,433 |
https://mathoverflow.net/questions/448110 | 2 | Can a computable partial order have a maximal chain of order-type $\omega\_1^{ck}$? My instinct is to say no, of course not, but I can't actually make the argument. If the p.o. also has chains of Harrison type, there seems to be no violation of $\Sigma^1\_1$-bounding.
*Edit:* The answer is yes. Let $T$ be the tree of... | https://mathoverflow.net/users/32178 | Maximal chains of order type $\omega_1^{ck}$ in computable partial orders? | This is perhaps a simplification of the example you give: let $H$ be a Harrison order and let $P$ be the poset of pairs $(a,b)\in H$ with $a<b$, ordered by $$(a,b)\le (a',b') \quad\iff\quad a\le a'\mbox{ and }b\ge b'.$$ Note that in a chain in $P$, we do *not* require both coordinates to simultaneously change.
Let $(... | 2 | https://mathoverflow.net/users/8133 | 448116 | 180,436 |
https://mathoverflow.net/questions/447986 | 1 | Given a partial order $P$ on a set $S$ does the set of ordered pairs $(x,y)$ in $S\times S\setminus P$ such that $P\cup\{(x,y)\}$ is a partial order have a name? (If so then it would apply to all sorts of orders not just partial orders.)
The answer was "no" in this crosspost: <https://cs.stackexchange.com/questions/1... | https://mathoverflow.net/users/5090 | does this relation associated with a poset have a name? | It appears that there is no established name for this concept, but if you are looking for a suggestion, **"potential covers"** might be a reasonable name, since these are precisely the pairs $(x,y)$ which are not in the partial order $P$ but would be a cover in $P\cup \{(x,y)\}$.
| 5 | https://mathoverflow.net/users/25028 | 448121 | 180,438 |
https://mathoverflow.net/questions/447993 | 2 | A Banach space $E$ is called *Grothendieck* if every weak\* convergent sequence in the dual space $E^\*$ is weakly convergent. A typical example of a Grothendieck space is $\ell\_\infty$. Diestel proved the following characterization of Grothendieck Banach spaces:
>
> For every Banach space $E$, TFAE:
>
>
> 1. $E... | https://mathoverflow.net/users/15860 | Weakly compact operators into $c_0$ and other separable spaces | In the light of Professor Johnson's comment, we cannot replace $c\_0$ with another $X$ unless $X$ contains a copy of $c\_0$.
Indeed, let $X$ be a Banach space that contains no isomorphic copy of $c\_0$. Consider a $C(K)$ space that is not a Grothendieck space (e.g., $C([0,1])$ ). Any bounded linear $T:C(K)\to X$ is u... | 3 | https://mathoverflow.net/users/164350 | 448127 | 180,440 |
https://mathoverflow.net/questions/447894 | 2 | Dirichlet's theorem says that all numbers $x\in [0,1]$ can be approximated by infinitely many fractions $p/q \in \mathbb{Q}$ with error $|x - p/q| \le 1/q^2$.
I am interested in the following question:
Fix $m>0$. Let $S(m)$ denote the set of $x\in [0,1]$ for which no fraction $p/q \in \mathbb{Q}$ with $q<100\sqrt{m}$... | https://mathoverflow.net/users/155604 | Simultaneously approximating all $x \in [0,1]$ with fractions of bounded denominator | Fix $T$ ($T=100$ in your example) and denote by $S\_T(n)$ the set of numbers $x\in [0,1]$ such that $|x-p/q|>1/n^2$ whenever $p,q$ are integers and $1\leqslant q\leqslant Tn$ (so, my $n$ is your $\sqrt{m}$).
Let $0=r\_0<r\_1<\ldots<r\_K=1$ be Farey sequence of all irreducible fractions with denominator at most $Tm$. ... | 3 | https://mathoverflow.net/users/4312 | 448129 | 180,442 |
https://mathoverflow.net/questions/448130 | 3 | Lets take the intersection of the theory of $L\_{\omega\_1^{CK}}$ and $\sf ZF + [V=L]$, this is [equivalent](https://mathoverflow.net/a/316109/95347) to the theory of [constructability from below + limit stages](https://mathoverflow.net/questions/316099/what-is-the-strength-of-this-strict-constructible-iterative-hierar... | https://mathoverflow.net/users/95347 | Are all constructible from below sets parameter free definable? | There are two issues with your question.
First, your statement "in other words" is not correct, since there are theories whose models have the property that whenever a statement holds of every parameter-free definable element, then it holds of every element, but not all models are pointwise definable.
**Theorem.** ... | 7 | https://mathoverflow.net/users/1946 | 448132 | 180,443 |
https://mathoverflow.net/questions/448142 | 2 | I am trying to understand paragraph 1.6 of Lusztig's paper "Character Sheaves I". The basic setup is that $X$ is a smooth irreducible variety over a field $k=\overline{k}$, $D\_i, i=1,...,r$ are smooth divisors with normal crossings and $\mathcal{L}$ is $\overline{\mathbb{Q}}\_l$-local system of rank one such that the ... | https://mathoverflow.net/users/173314 | Extending IC sheaves across smooth divisors with normal crossings | The local monodromy around $D\_i$ can be obtained by taking a $\eta$ a geometric generic point of $D\_i$, $R$ the etale local ring of $X$ at $\eta$ with uniformizer $\pi$, then pulling $\mathcal L$ back to $\operatorname{Spec} R[ \pi^{-1} ]$. We then obtain a representation of the étale fundamental group of $\operatorn... | 7 | https://mathoverflow.net/users/18060 | 448147 | 180,446 |
https://mathoverflow.net/questions/448133 | 5 | Recently I'm reading the paper [Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups](https://arxiv.org/pdf/math/0004010.pdf) by Pestov. When it comes to the definition of an extremely amenable topological group, it claims (without proof) that
>
> (the extreme amenability) is equivalent to... | https://mathoverflow.net/users/140578 | Extreme amenability of topological groups and invariant means | The action $G\curvearrowright\beta G$ is continuous iff $G$ is discrete, so for nondiscrete groups it is not true that $G$ is extremely amenable iff this action has a fixed point.
What one should look at instead, is the biggest compactification on which $G$ acts continuously, this is known as the Samuel compactificat... | 7 | https://mathoverflow.net/users/49381 | 448150 | 180,447 |
https://mathoverflow.net/questions/448107 | 2 | Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R\_{\mathfrak{p}}$ is a regular local ring? In general, can the dimension of the smooth locus be as low as possible or can the codimension of the singular locus be exactly ... | https://mathoverflow.net/users/130925 | Krull dimension of the smooth locus | $\DeclareMathOperator\Spec{Spec}$
Maybe I'm misreading this, but I don't see why you need dimension $\geq 4$, normal, domain, etc.
**EDIT:** I originally wrote this requiring R1 but I don't think we need R1 (regular in codimension 1). Instead, we should just require R0, regular at the minimal primes (ie, $R\_Q$ is ... | 4 | https://mathoverflow.net/users/3521 | 448155 | 180,449 |
https://mathoverflow.net/questions/447595 | 4 | Let $R$ be a ring. An $R$-module $M$ is called FL (FP) if it has a finite resolution consisiting of finitely generated free (projective) modules.
Given an exact sequence of $R$-modules, $0\to M\_1\to M\_2\to M\_3\to 0$, if two of the modules $M\_1$, $M\_2$ or $M\_3$ are FP, then the third is FP as well. This follows,... | https://mathoverflow.net/users/10482 | Exact sequences with two FL-modules | Yes, this is true. See Bourbaki, N. Éléments de mathématique. Algèbre. Chapitre 10. Algèbre homologique. (French) [Elements of mathematics. Algebra. Chapter 10. Homological algebra], Theorem 3.9.1
Let $K\_0(R)$ be the Grothendieck group of $R$ and $\widetilde K\_0(R)=K\_0(R)/\langle [R]\rangle$.
Then $[P]=0$ in $\wid... | 1 | https://mathoverflow.net/users/10482 | 448156 | 180,450 |
https://mathoverflow.net/questions/448153 | 2 | The question is as in the title.
I know that a traceless matrix can be written as a commutator of two matrices.
Then, let $v : \mathbb{R}^3 \to \mathbb{R}^3$ be a divergence-free smooth vector field. That is, $\nabla \cdot v=0$.
Then, the matrix $A(x)=\bigl[\partial\_i v\_j(x) \bigr]$ is smooth as a mapping from ... | https://mathoverflow.net/users/56524 | For a divergence free smooth vector field $v : \mathbb{R}^3 \to \mathbb{R}^3$, how to find the commutator form of the matrix $A=(\partial_i v_j)$? | You might want to first carry out a unitary transformation $A(x)\mapsto U(x)A(x)U^\top(x)$, such that all diagonal elements are zero.$^\ast$
Then $A(x)=BC(x)-C(x)B$ with
$$B=\begin{pmatrix}
1&0&0\\
0&2&0\\
0&0&3
\end{pmatrix},\;\;
C\_{ij}(x)=\begin{cases}
0&\text{if}\;i=j\\
\frac{A\_{ij}(x)}{B\_{ii}-B\_{jj}}&\text{i... | 2 | https://mathoverflow.net/users/11260 | 448163 | 180,454 |
https://mathoverflow.net/questions/448152 | 4 | Let $A$ be a densely-defined, positive, self-adjoint operator with compact resolvent on a Hilbert space $H$. Then, $\text{Range}(1+A)=H$ and there is a basis for $H$ consisting of eigenvectors of $1+A$. Assume also that $D\subset H$ is a core domain for $A$; that is, $D$ is dense in $\text{Dom(A)}$ with respect to the ... | https://mathoverflow.net/users/161393 | Diagonalizing selfadjoint operator on core domain | This sounds suspicious right away since the eigenvectors are what they are (nothing to choose here, unless you have degeneracies), but there is much choice for $D$ and we should be able to avoid eigenvectors.
For a concrete example, you can take $H=\ell^2$, $Ae\_n=ne\_n$, which is self-adjoint on its natural domain $... | 9 | https://mathoverflow.net/users/48839 | 448172 | 180,460 |
https://mathoverflow.net/questions/448185 | -1 | Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are identified. For each $t \in [0, 1]$, let $\mathcal{D}\_t$ be the filter on $\mathcal{B}([0, 1])$ generated by open neighborhoods... | https://mathoverflow.net/users/504602 | The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence | No. Suppose $\mathcal{F}$ is such a filter. Clearly each $X \in \mathcal{F}$ has positive measure intersection with every positive-length interval. Then neither $[0,1/2]$ nor $[1/2, 1]$ are in the filter generated by $\mathcal{D}\_{1/2}$ and $\mathcal{F},$ contradiction.
| 1 | https://mathoverflow.net/users/109573 | 448188 | 180,464 |
https://mathoverflow.net/questions/447937 | 3 | Suppose that $X$ is a standard Borel space (meaning it is endowed with a $\sigma$-algebra coming from some Polish topology on $X$) and $G$ is a Polish group acting in a Borel way on $X$. Denote by $E\_G$ the resulting orbit equivalence relation on $X$.
There are (at least) three competing definitions, that I know of,... | https://mathoverflow.net/users/16107 | Competing definitions of smooth orbit equivalence relation | Let me turn my comment into an answer now that I have time to write some details. The short answer is that yes, those three definitions are equivalent, the long answer is below.
Let's forget about orbit equivalence relations and let's look at an arbitrary Borel equivalence relations $E$ on a Polish space $X$ for a se... | 1 | https://mathoverflow.net/users/49381 | 448194 | 180,468 |
https://mathoverflow.net/questions/448193 | 3 | I have been working on minimal surfaces, only recently learnt about maximal surfaces and "maxfaces" in Lorentz spaces.
I can clearly see the mathematical motivations. But I wonder if zero-mean-curvature hypersurfaces in Lorentz spaces ($\mathbb{R}^2\_1$ or $\mathbb{R}^3\_1$), like minimal surfaces in Euclidean spaces... | https://mathoverflow.net/users/20595 | Applications of maximal surfaces in Lorentz spaces | Maybe you already encountered such maximal surfaces in the context of General Relativity. Still, the one application of spacelike maximal surfaces that I am aware of is as special kinds of initial data (Cauchy) surfaces for the Einstein equations (a popular broader class are constant mean curvature (CMC) surfaces, of w... | 5 | https://mathoverflow.net/users/2622 | 448196 | 180,469 |
https://mathoverflow.net/questions/195267 | 7 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$EDITED Let $G=\SL\_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$.
Let $H\subset G$ be a finite subgroup.
Set $X=G/H$ be the corresponding homogeneous space, it is a complex variety.
Let $\tau\in \Aut({\mathbb{C}})$ be an automorphism of the fie... | https://mathoverflow.net/users/4149 | Conjugation of the quotient of $\mathrm{SL}(n,\mathbb{C})$ by a finite subgroup | $\DeclareMathOperator\SL{SL}
\DeclareMathOperator\GL{GL}
\DeclareMathOperator\PSL{PSL}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Out{Out}
\DeclareMathOperator\Ad{Ad}
$I can only address Question 1/Question 4. Let me replace $\SL\_n$ with $\GL\_n$; our $H$ will be simple, so it will provide an example for $\SL\... | 3 | https://mathoverflow.net/users/125523 | 448202 | 180,470 |
https://mathoverflow.net/questions/448198 | 4 | What is the complexity (e.g. is it $\Sigma^0\_1$, arithmetic, fully $\Pi^1\_1$) of the relation $|a| < |b|$ given two notations $a, b \in \mathscr{O}$ (Kleene's O)?
What about the case where only one of the notations must be in $\mathscr{O}$ (where $b \not\in \mathscr{O} \implies |b| = \infty$)?
For instance, what ... | https://mathoverflow.net/users/23648 | Complexity of |a| < |b| for ordinal notations? | For your first question, Spector showed that the relation $|a| < |b|$ is uniformly computable from $\emptyset^{|b|+1}$. It's not $\Delta^1\_1$, but it is both the restriction of a $\Sigma^1\_1$ relation and of a $\Pi^1\_1$ relation to $\mathcal{O}$. The $\Sigma^1\_1$ relation is "there exists an embedding of $|a|+1$ in... | 7 | https://mathoverflow.net/users/32178 | 448205 | 180,471 |
https://mathoverflow.net/questions/448197 | 1 | Let $ K $ be finite degree extension of $ \mathbb{Q} $ such that $ -1 $ is not a square in $ K $. Let $ L = \frac{K[x]}{\langle x^2 +1\rangle}$. Thus every element of $ L $ is of the form $ a + ib $ where $ i^2 = -1 $ and $ a,b \in K $. Let $ \eta\_{2^i} $ is the primitive $ 2^i $-th root of unity contained in $ L $ wh... | https://mathoverflow.net/users/215016 | Norm of $2^{i}$-th primitive root | Let $K=\mathbb{Q}[\sqrt{-2}]$. Then $\frac{1}{2}\sqrt{-2}(1-i)$ is an eighth root of unity in $L$ whose norm is $\frac{1}{2}\sqrt{-2}(1-i)\frac{1}{2}\sqrt{-2}(1+i)=\frac{1}{4}(-2)(1-i^2)=-1\in K$. [I'm assuming that when you write $2^i$ it's a different $i$]
| 4 | https://mathoverflow.net/users/460592 | 448208 | 180,473 |
https://mathoverflow.net/questions/448186 | 2 | I'm trying to understand Bourgain's paper "Besicovitch type maximal operators and applications to Fourier analysis". Let $\xi\in S^2\subset\mathbb{R}^3$ be a unit vector and $\delta>0$, by a $(\xi,\delta)$ tube in $\mathbb{R}^3$, we mean a cylinder $\tau$ of unit length in direction $\xi$ and of radius $\delta$. For $f... | https://mathoverflow.net/users/147078 | Use of Stein's maximal principle in Bourgain's paper on Besicovitch sets | This may be a slight misattribution. It is the implication $(1.22) \implies (1.21)$ which is essentially in Stein's paper; the implication $(1.21) \implies (1.9)$ is much simpler, following from Marcinkiewicz interpolation. Roughly speaking, if one has a weak-type $L^p$ estimate on the Kakeya maximal operator, one can ... | 7 | https://mathoverflow.net/users/766 | 448209 | 180,474 |
https://mathoverflow.net/questions/448157 | 3 | Let $Q$ be a probability measure on $\mathbb{R}$. Let $$Q\_h(dy) = e^{y \cdot h} Q(dy) / M(h) \quad \text{where} \quad M(h) = \int e^{y \cdot h} Q(dy)$$ defined for $h \in (-c,\infty)$ with some $c > 0$. Let $Y\_h \sim Q\_h$ and $V(h) := EY\_h^2 - (EY\_h)^2$. Then, does it follow that $$\exists c' > 0,\ h\_0 > -c\ \tex... | https://mathoverflow.net/users/146981 | Variance lower bound for natural exponential family | The desired result obviously fails to hold if the probability measure $Q$ is degenerate, that is, supported on a singleton set. Indeed, then $V(h)=0$ for all $h$.
It is much harder to construct a counterexample with a non-degenerate $Q$. The idea of such a counterexample, given below, is to make the distribution $Q$ ... | 1 | https://mathoverflow.net/users/36721 | 448211 | 180,475 |
https://mathoverflow.net/questions/448182 | 6 | Suppose that $G=(G\_{ij})$ is an $n\times n$ positive-definite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Let $a$ be a column $n\times1$ matrix with nonnegative entries such that $a^\top G^{-1}a=1$. Does it then necessarily follow that $a^\top Ga\le1$?
Certain numeri... | https://mathoverflow.net/users/36721 | An inequality for certain positive-definite matrices | The answer seems to be **yes**.
Let $G$ be the Gram matrix of a base $(e\_1,\dots,e\_n)$ in some Euclidean space. Then $G^{-1}$ is the Gram matrix of the dual base $(f\_1,\dots,f\_n)$, i.e., the one satisfying $\langle e\_i,f\_j\rangle=\delta\_{ij}$.
Denote $u=\sum\_i a\_ie\_i$ and $v=\sum\_j a\_jf\_j$; then $\|u\|... | 5 | https://mathoverflow.net/users/17581 | 448225 | 180,479 |
https://mathoverflow.net/questions/248337 | 6 | Consider a Wang tiling (given a subset of $C^4$ for a finite set $C$ of colours, e.g.). It's well-known to be undecidable whether there exists a tiling, and also whether there exists a periodic tiling. It's also well-known that there exists a periodic tiling iff there exists a doubly periodic tiling.
On the other han... | https://mathoverflow.net/users/10481 | Decidability of (restricted) periodicity of Wang tilings | This is undecidable. I refer to the proof of the periodic tiling problem which is Theorem 5.7 in the lecture notes of Jarkko Kari
<https://users.utu.fi/jkari/wp-content/uploads/sites/1251/2021/10/part2.pdf>
but I give a construction that works with any such proof where the period is suitably marked.
In the proof I ci... | 2 | https://mathoverflow.net/users/123634 | 448239 | 180,483 |
https://mathoverflow.net/questions/448204 | 2 | I am quite confused between Helmholtz decomposition and Laplacian vector fields in the periodic case.
Let $\mathbb{T}^3$ be the $3$-dimensional torus. Then, I thought any divergence-free smooth vector field $v : \mathbb{T}^3 \to \mathbb{R}^3$ can be expressed as a curl of another smooth periodic vector field $V : \ma... | https://mathoverflow.net/users/56524 | Helmholtz decomposition vs Laplacian vector fields on $\mathbb{T}^3$? | The correct generalization to $\mathbb{T}^3 = (\mathbb{R}/\mathbb{Z})^3$ is the [Hodge decomposition](https://en.wikipedia.org/wiki/Hodge_theory). Every vector field on $\mathbb{T}^3$ is uniquely expressible as $u = \nabla \times v + \nabla w + h$, where $\nabla \cdot h = \nabla \times h = 0$ is a Laplace vector field ... | 3 | https://mathoverflow.net/users/2622 | 448248 | 180,486 |
https://mathoverflow.net/questions/448245 | 5 | Let $k/\mathbb{Q}$ be a number field and $\mathbb{A}$ its ring of adèles. As usual $\mathbb{A} = \mathbb{A\_f} \times \mathbb{A\_{\infty}}$.
The standard definition of an *automorphic representation* $(\pi,V)$ for $\textrm{GL}\_n(\mathbb{A})$ is its realization as (irreducible) subquotient of the space of automorphic... | https://mathoverflow.net/users/484997 | On the notion of cuspidality | To every local admissible representation $\pi\_v$ of $\mathrm{GL}\_n(k\_v)$, there is a local $L$-function $L(s,\pi\_v)$. For a global admissible representation $\pi=\otimes\_v \pi\_v$ of $\mathrm{GL}\_n(\mathbb{A}\_k)$, the corresponding global $L$-function is defined as $L(s,\pi)=\prod\_v L(s,\pi\_v)$. However, unles... | 8 | https://mathoverflow.net/users/11919 | 448251 | 180,489 |
https://mathoverflow.net/questions/447827 | 7 | $\DeclareMathOperator\SL{SL}$ Let $p>3$ and $G$ be an open subgroup of the special linear group $\SL\_2(\mathbf{Z}\_p)$ over the ring $\mathbf{Z}\_p$ of $p$-adic integers. Suppose that $G$ is topologically generated by a finite set $S=\{g\_1,\cdots,g\_r\}$ such that all $g\_i$ have finite order. Is it true that $G=\SL\... | https://mathoverflow.net/users/504912 | Topological generators for $\mathrm{SL}_2(\mathbf{Z}_p)$ | $\def\ZZ{\mathbb{Z}}\def\SL{\text{SL}}\def\Id{\text{Id}}$This seems false to me.
**Lemma:** $e:=\left[ \begin{smallmatrix} 1&p\\0&1 \\ \end{smallmatrix} \right]$ and $f:=\left[ \begin{smallmatrix} 1&0\\p&1 \\ \end{smallmatrix} \right]$ topologically generate an open subgroup of $\SL\_2(\ZZ\_p)$.
**Proof:** Let $G$ ... | 8 | https://mathoverflow.net/users/297 | 448256 | 180,491 |
https://mathoverflow.net/questions/448261 | 2 | We consider the heat kernel
$$
g :\mathbb R\_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Let $0 < t\_1 < t\_2 <\infty$ and $1\le\lambda<\infty$ such that $\frac{t\_2}{t\_1} \le \lambda$. I would like to ask if there are constants $C\_... | https://mathoverflow.net/users/477203 | Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}{t_1} \le C_1 \frac{g(C_2 t_2, \cdot)}{\sqrt{t_2}}$? | The answer is no. E.g., let $t\_1\sim t\_2\downarrow 0$ and $|x|\sim\sqrt{t\_2}$.
| 6 | https://mathoverflow.net/users/36721 | 448262 | 180,492 |
https://mathoverflow.net/questions/448231 | 0 | During developing a new statistical estimator, I faced the following problem.
Let $\mathbf{x}\_i$ be a sequence of i.i.d. $d$-dimensional random vectors with
\begin{align\*}
\mathbf{x}\_i = \mathbf{O}\_i \mathbf{\mu} + \mathbf{\varepsilon}\_i,
\end{align\*}
where $\mathbf{\mu}$ is a mean vector, $\mathbf{O}\_i$ is a... | https://mathoverflow.net/users/159685 | Estimation on rotationally-disturbed random vectors | $\newcommand\ep\varepsilon$This is impossible to do even for $d=1$, as your [model is not identifiable](https://en.wikipedia.org/wiki/Identifiability#Definition), that is, the parameters of the model are not identifiable even if the distribution of the $X\_i$'s is fully known.
Indeed, let $R\_1,\dots,R\_n$'s be indep... | 1 | https://mathoverflow.net/users/36721 | 448264 | 180,493 |
https://mathoverflow.net/questions/447494 | 2 | Let $K/\mathbb Q\_p$ be a finite extension, and $\mathcal O\_K$ the ring of integers of $K$. I am asking for a reference for a structure theorem
of finitely generated modules over the completed group algebra $\mathcal O\_K[[\mathbb Z\_p]]$.
I am familiar with structure theorems for $\Lambda = \mathbb Z\_p[[\mathbb Z\... | https://mathoverflow.net/users/161063 | Structure theorem for Iwasawa modules over $p$-adic rings of integers | Serre, Jean-Pierre, *Classes des corps cyclotomiques* (Séminaire Bourbaki décembre 1958) lemme 5 p. 90 contains a statement and proof of a structure theorem (up to pseudo-isomorphism) for finitely generated modules over a noetherian integrally closed domain $A$, of which $\mathcal O\_{K}[[\mathbb Z\_{p}]]$ is a very pa... | 3 | https://mathoverflow.net/users/2284 | 448276 | 180,497 |
https://mathoverflow.net/questions/448274 | 0 | Let $p>0$ and consider a metric space $(X,d)$. I have recently come across a problem where the space $(X,d^q)$ provides is natural; where $q>1$. However, the triangle inquality break (i.e. it is a ["semimetric space"](https://en.wikipedia.org/wiki/Metric_space#Semimetrics)).
I'm wondering, in this case, does $(X,d^p)... | https://mathoverflow.net/users/496781 | Generalized Triangle Inequality for Snowflakes | More generally, let $p>1$ and $a,b>0$, and set $\tilde{d}:X\times X\rightarrow [0,\infty)$ defined by
$$
\tilde{d}(x,z)\mapsto a \,d(x,z) + b\,d(x,z)^p \qquad (\text{ for all }x,z\in X)
$$
Then, for every $x,y,z\in X$ we have that
$$
\begin{aligned}
\tilde{d}(x,z) = &\, a \,d(x,z) + b\,d(x,z)^p \\
\le & a(d(x,y)+d(y,z... | 4 | https://mathoverflow.net/users/36886 | 448278 | 180,499 |
https://mathoverflow.net/questions/448235 | 11 | Say that a **long model** is an $\mathfrak{A}\models\mathsf{I\Sigma\_1}$ such that $\mathfrak{A}$ has strictly greater cardinality than each of its proper initial segments (in the case $\vert\mathfrak{A}\vert=\aleph\_1$ this is "$\omega\_1$-like"ness). If $\mathfrak{A}$ is a long model and $\vert\mathfrak{A}\vert$ is r... | https://mathoverflow.net/users/8133 | Can singular long models require less than PA? |
>
> The answer to the question is strongly in the positive, since the following theorem of Kaye implies that for *every* singular cardinal $\kappa$, and for any fixed natural number $n$, the common theory of $\kappa$-like models of $\mathrm{I}\Sigma\_{n}$ is strictly weaker than $\mathsf{PA}$.
>
>
>
**Theorem.**... | 11 | https://mathoverflow.net/users/9269 | 448280 | 180,500 |
https://mathoverflow.net/questions/448218 | 1 | Let $M$ be a simply connected Hadamard manifold. That is, $M$ is a complete Riemannian manifold with sectional curvature bounded above by 0. Let $f(x,y)=\frac{1}{2}d^2(x,y)$, where $x,y \in M$, and $d(.,.)$ is geodesic distance. I'm curious if $f:M^2 \to \mathbb{R}$ is a Morse-Bott function, and how to show it.
I kno... | https://mathoverflow.net/users/141449 | Is squared geodesic distance a Morse-Bott function on simply-connected Hadamard manifolds? | Choose $V,W \in T\_p M$ and note that
$$
D^2f((V,W),(V,W)) = \frac{d^2}{dt^2}\Big|\_{t=0} \frac 12 d(\exp\_p(tV),\exp\_p(tW))^2 .
$$
By the computation in [Section 1.3 here](https://www2.math.upenn.edu/%7Ewziller/math660/TopogonovTheorem-Myer.pdf)
$$
\frac 12 d(\exp\_p(tV),\exp\_p(tW))^2 = \frac{t^2}{2} |V-W|^2 + O(t^3... | 2 | https://mathoverflow.net/users/1540 | 448290 | 180,504 |
https://mathoverflow.net/questions/448286 | 2 | Let $a\_i (i \in\{1...k\})$ be $k$ IID standard Gaussian random variables, $P\_i$ are $d$-dimensional constant vectors. How to prove with a probability of at least $1/e$,
$$
\left\|\sum\_{i=1}^k a\_{i} P\_{i}\right\|\_2 \geq\left\|P\_{1}\right\|\_2 .
$$
holds.
| https://mathoverflow.net/users/501704 | prove with a probability of at least $1/e$: $\left\|\sum_{i=1}^k a_{i} P_{i}\right\|_2 \geq\left\|P_{1}\right\|_2$ holds | $\newcommand\si\sigma$Let
$$p:=P\Big(\Big\|\sum\_{i=1}^k a\_i P\_i\Big\| \ge\|P\_1\|\Big).$$
Here $\|\cdot\|:=\|\cdot\|\_2$.
The inequality $p\ge1/e=0.367\dots$ does not hold in general.
Actually, the best lower bound on the probability $p$
is
$$p\_\*:=P(|a\_1|\ge1)=0.317\dots,$$
which is $<1/e=0.367\dots$.
... | 8 | https://mathoverflow.net/users/36721 | 448295 | 180,506 |
https://mathoverflow.net/questions/448297 | 2 | For a non zero rational $r=p/q$ ($p,q\in\mathbb Z$ coprimes), define the height of $r$ by $\mathrm{ht}(r)=\max(|p|,|q|)$ (by convention $\mathrm{ht}(0)=0$). For a polynomial $P\in\mathbb Q[X]$, define the height of $P$ by the maximum of height of its coefficients. Let $A$ and $C$ be two non zero polynomials of $\mathbb... | https://mathoverflow.net/users/33128 | Bounds of heights of coefficients of rational polynomials | In general, no. Take $A=(x-1)^2$, $C=(x^n-1)^2$ for large $n$. Then $B=(1+x+\ldots+x^{n-1})^2$ has height $n$.
| 5 | https://mathoverflow.net/users/4312 | 448300 | 180,507 |
https://mathoverflow.net/questions/448288 | 3 | Let $p$ be an odd prime, $g$ a primitive root of $p$, and
$f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\_p$
is defined by
$$f(i,x) = g^{i+1} + g^{-i-1} - g^i x^2\mod p.$$
**Is it true that $f$ attains all values modulo $p$?**
It is not hard to see that $g^{i+1} + g^{-i-1}$ produces $(p+1)/2$ distinct values and $x^2... | https://mathoverflow.net/users/506363 | The range of values of $f(i,x) = g^{i+1} + g^{-i-1} - g^i x^2$ modulo a prime p | This is true.
Denote $g^{i+1}=y$, then $y$ takes any non-zero value mod $p$. Fix an element $a\in \mathbb{F}\_p$ and assume that the equation $y+1/y-g^{-1}yx^2=a$ does not have a solution. It yields that $y+1/y-a\ne g^{-1}yx^2$ for all $y\ne 0$ and all $x$. Thus, in particular, $y+1/y-a\ne 0$ for all $y\ne 0$, and fo... | 6 | https://mathoverflow.net/users/4312 | 448306 | 180,512 |
https://mathoverflow.net/questions/448279 | 5 | Let $S$ be a smooth closed connected hyperbolic surface. On $S$ we have the Laplace operator $\Delta$, whose eigenvalues form a discrete sequence
\begin{equation}
0=\lambda\_0<\lambda\_1\leq \lambda\_2\leq ... \leq \lambda\_k\nearrow\infty
\end{equation}
To my knowledge, a generic surface will not have eigenvalues of m... | https://mathoverflow.net/users/155336 | Multiplicity of Laplace eigenvalues and symmetry | Let me extend, and correct, the argument expressed in the comment made by user378654.
Let us start with a surface $S$ for which $\Delta$ admits a double eigenvalue $\lambda$. For instance, you may choose an $S$ with a non-trivial symmetry group. Denote $V=\ker(\lambda-\Delta)$. Consider now a smooth $m$-parameter fam... | 3 | https://mathoverflow.net/users/8799 | 448312 | 180,514 |
https://mathoverflow.net/questions/448314 | 2 | Let $\kappa:\lbrace 1,2,3,\ldots\rbrace\longrightarrow \mathbb Z$
be the convolutional inverse in the Dirichlet ring of $n\longmapsto {n+1\choose 2}$. It is defined by $\kappa(1)=1$ and
by the functional equation $\sum\_{d\vert n}{d+1\choose 2}\kappa(n/d)=0$ for $n\geq 2$.
It is not hard to show that $\kappa(3\cdot 5... | https://mathoverflow.net/users/4556 | Exceptional zeros of a convolutional inverse | There are many more examples. An infinite series is for $n=5^8\cdot7\cdot p^2$ for primes $p$ different from $5$ and $7$.
Another infinite series is $n=5\cdot11\cdot17^3\cdot p^2$ for primes $p$ different from $5,11,17$. And there are many more infinite series.
| 4 | https://mathoverflow.net/users/18739 | 448319 | 180,516 |
https://mathoverflow.net/questions/448266 | 4 | It seems to me from a quick glance at several sources describing the complex and modular irreducible representations of $\mathrm{GL}(2,p)$ that any field $K$ containing a primitive $(p-1)$-root of unity is a splitting field for $\mathrm{GL}(2,p)$ (but maybe I need to read the details more carefully). Is this correct? A... | https://mathoverflow.net/users/15934 | Splitting field for $\mathrm{GL}(2,p)$ - reference request | I realised that my previous version of this answer was irrelevant to the question (though what was said was accurate, as far as it went).
I think you can make the desired conclusion if you resort to modular representation theory, specifically, the theory of blocks with cyclic defect group (for this question we only n... | 4 | https://mathoverflow.net/users/14450 | 448320 | 180,517 |
https://mathoverflow.net/questions/448281 | 2 | Baur-Monk quantifier elimination theorem asserts that any formula in the language of modules is modulo the theory a boolean combination of BG-Invariants and positive primitive formulas. However, in p.54 Model theory of Modules by Prest, just right after proving the theorem, it was immediately concluded as Cor 2.15 that ... | https://mathoverflow.net/users/506361 | Question on Baur-Monk quantifier elimination for modules | By Baur-Monk QE, every sentence is equivalent to a boolean combination of BG-invariants and positive primitive *sentences* (which have no free variables).
A positive primitive sentence $\psi$ has the form $\exists x\_1\dots\exists x\_n\,\varphi(x\_1,\dots,x\_n)$, where $\varphi$ is a finite conjunction of equations b... | 3 | https://mathoverflow.net/users/2126 | 448337 | 180,521 |
https://mathoverflow.net/questions/448330 | 1 | This question is cross-posted from <https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials>
I am trying to study the asymptotic behavior of a stochastic process defined on the space of single variable polynomials whose coefficients are either $0$ or $1$.... | https://mathoverflow.net/users/143913 | Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials | Just a comment really, but too long to fit in a comment box! I think you can simplify the formulation considerably. Let $X\_t$ be your $\kappa^{t}a+b\_{t-1}(\kappa)(1-\kappa)$. Then you can describe the process $X\_t$ as a Markov chain in the following way: $X\_1=\kappa a$, and for $t\geq 1$,
\begin{equation\*}
X\_{... | 2 | https://mathoverflow.net/users/5784 | 448343 | 180,524 |
https://mathoverflow.net/questions/448360 | 5 | Hall's (weak) conjecture is the statement that for all $\varepsilon > 0$ there exists a positive number $c(\varepsilon) > 0$ such that for all $x,y \in \mathbb{Z}$ with $y^2 \ne x^3$, that
$$\displaystyle \left \lvert y^2 - x^3 \right \rvert \geq c(\varepsilon) |x|^{\frac{1}{2} - \varepsilon}.$$
My question concern... | https://mathoverflow.net/users/10898 | Upper bound for Hall's conjecture on separation of squares and cubes | The best $\theta$ is $0$. It is known that there are infinitely many
solutions of 0 < $|x^3 - y^2| \ll x^{1/2}$, parametrized by certain
"Pell equations"; indeed one such family attains $|x^3 - y^2| \sim C x^{1/2}$
with $C = 5^{-5/2} 54 \approx 0.966$. See [D].
The underlying polynomial identity
$$
(t^2 + 10t + 5)^3 - ... | 15 | https://mathoverflow.net/users/14830 | 448363 | 180,528 |
https://mathoverflow.net/questions/448258 | 9 | I have some basic question about real analytic sets which I've asked in MSE but couldn't get an answer there.
Let $M$ be a real analytic manifold, and let $X \subset M$ be an analytic set. We know by the definition of an analytic set that for every $x \in X$ there exists an open neighborhood $U$ of $x$ in $M$ and a r... | https://mathoverflow.net/users/30048 | Can a real analytic set be expressed as the zero set of a single real analytic function? | Let me expand on my comment. In the terrific book "Topics on real analytic spaces" by Guaraldo, Macri and Tancredi one can find a discussion of such questions. See page 64 & 65 and in particular Theorem 2.1, Definition 2.2 and Remark 2.3. Let me quote Theorem 2.1 and make some remarks.
**Theorem 2.1.** Let $Y$ be a c... | 11 | https://mathoverflow.net/users/109193 | 448372 | 180,530 |
https://mathoverflow.net/questions/448373 | 4 | Assume that $ (M,g) $ and $ (N,h) $ are two smooth closed manifold and $ N $ is embedded isometrically into $ \mathbb{R}^K $ for some $ K\in\mathbb{Z}\_+ $. Assume that $ u\in C^{\infty}(M\times\mathbb{R}\_+,N) $ satisfies the equation of harmonic heat flow
$$
\frac{\partial u}{\partial t}-\Delta\_g u=A(u)(\nabla u,\na... | https://mathoverflow.net/users/241460 | A formula in harmonic heat flow | Actually there is a typo in the formula, i.e.
$$
(\partial\_t-\Delta\_g)|\partial\_tu|^2=-2|\nabla\partial\_tu|^2+2R^N(\nabla u,\partial\_tu,\nabla u,\partial\_t u).
$$
Firstly for $ x\in M $, we can choose an orthorgonal basis $ \{e\_{\alpha}\} $. Then it follows from simple calculations that
$$
\begin{aligned}
\parti... | 1 | https://mathoverflow.net/users/241460 | 448386 | 180,534 |
https://mathoverflow.net/questions/448371 | 6 | We set :
* $\phi\_1, \phi\_2 : \mathbb{R} \to \mathbb{R}^M$ with compact support
* $\theta \in \mathbb{R}^M$ non-zero, $\theta\_{p-1} = (\operatorname{sign}(\theta\_i)|\theta\_i|^{p-1})\_{1 \leq i \leq M} \in \mathbb{R}^M$ with $p \geq 2$.
We assume that the matrix
$$ \int\_\mathbb{R} \phi\_1 \phi\_1^T - \int\_\m... | https://mathoverflow.net/users/402235 | Cauchy-Schwarz-like inequality with a power $p$ term | $\newcommand\th\theta\newcommand\R{\mathbb R}$This is false for any real $p>2$ (actually, this is false for any real $p>1$ such that $p\ne2$).
Indeed, if this were true, then, by continuity, we could replace "positive definite" with "positive semi-definite", at the same time replacing $<$ in the desired inequality by... | 4 | https://mathoverflow.net/users/36721 | 448402 | 180,537 |
https://mathoverflow.net/questions/448370 | 2 | I am reading O. Fujino's book [*Iitaka Conjecture*](https://link.springer.com/book/10.1007/978-981-15-3347-1). In page 42, Lemma 3.1.19 he restated one result due to Viehweg to use the base change arguments.
There exists some details in the proof of **Step 2** in Lemma 3.1.19 that I can't figure out as follows:
> ... | https://mathoverflow.net/users/141609 | Inclusion of (pulling back of) dualizing sheaves under normalization | $X$ is Gorenstein, so $\omega\_X$ is a line bundle and hence so is $\nu^\*\omega\_X$. In particular, it is torsion-free and hence $\mu(\mathcal T)=0$, so $\mu$ indeed factors through $\omega\_{\widetilde X}$.
As $X$ is reduced, $\nu$ is generically an isomorphism, so ${\rm Ker}\ \iota$ is a torsion sheaf, and then beca... | 4 | https://mathoverflow.net/users/10076 | 448425 | 180,540 |
https://mathoverflow.net/questions/448347 | 10 | Consider a set of $n$ elements $S=\lbrace 1,\dots,n\rbrace$ and $\mathcal{P}(S)$ to be the power set of $S$, which is a well-defined poset with respect to the inclusions. Now consider $\emptyset\neq T\varsubsetneq S$ and define $A=\lbrace \emptyset\neq U\in\mathcal{P}(S)\mid T\nsubseteq U\rbrace$. I want to compute the... | https://mathoverflow.net/users/482329 | Homotopy type of the geometric realization of a poset | First recall that geometric realisation of posets preserves products: the projections $P\xleftarrow{p}P\times Q\xrightarrow{q}Q$ give a map $(|p|,|q|)\colon |P\times Q|\to|P|\times|Q|$, and it is a standard fact that this is a homeomorphism.
Next, if $f\_0,f\_1\colon P\to Q$ are morphisms of posets and $f\_0(x)\leq f... | 6 | https://mathoverflow.net/users/10366 | 448433 | 180,541 |
https://mathoverflow.net/questions/448369 | 2 | This posting is generally related to a prior posting titled ["Are all constructible from below sets parameter free definable?"](https://mathoverflow.net/questions/448130/are-all-constructible-from-below-sets-parameter-free-definable)
If we work in infinitary language $\mathcal L\_{\omega\_1, \omega}$, then we can def... | https://mathoverflow.net/users/95347 | Does adding definability axiom expressed in infinitary language to ZF, let all models be pointwise definable? | Yes, the theory is consistent, if ZF is consistent, because there are pointwise definable models of ZF. Any such model is a model of your theory, which is therefore satisfiable and hence consistent.
And yes, clearly every model of your theory is pointwise definable (in the first-order language), because that is preci... | 3 | https://mathoverflow.net/users/1946 | 448434 | 180,542 |
https://mathoverflow.net/questions/448426 | 14 | Let $R$ be a commutative unital ring. Let $n\in\mathbb{N}\_+$. Consider a $n\times n$-matrix $A=(a\_{ij})\_{i,j=1}^n$ with entries in $R$. A diagonal matrix is defined obviously. We can define several notions about diagonalizability (for some prime $\mathfrak{p}\subseteq R$):
1. $A$ is *diagonalizable in $R$*: if the... | https://mathoverflow.net/users/105537 | Is diagonalizability a local property? | Here is a simple counterexample (simplified and generalised after Mohan's comment): let $R$ be a domain with a non-free finite projective module $P$, and let $Q$ be a finite projective module with $P \oplus Q \cong R^n$. This gives an idempotent matrix $A \in M\_n(R)$ corresponding to $(p,q) \mapsto (p,0)$.
For each ... | 18 | https://mathoverflow.net/users/82179 | 448447 | 180,547 |
https://mathoverflow.net/questions/448441 | 2 | I am looking at showing that a complex symmetric invertible matrix always has a complex symmetric square root and I refer to [this Q&A](https://mathoverflow.net/a/438232/497608) for the answer to this question. I am little confused at the reasoning given in the answer however, specifically in two places.
1. Firstly, ... | https://mathoverflow.net/users/497608 | Questions regarding answer to complex symmetric square root of a complex symmetric invertible matrix | $\newcommand\la\lambda$
1. Let $g(z)$ denote some branch of $\sqrt z$, so that $Q(z)=P(z)^2-g(z)^2$ and $P^{(k)}(\la\_j)=g^{(k)}(\la\_j)$ for each $j$ and all $k=0,\dots,m\_j-1$, where the $\la\_j$'s are the distinct eigenvalues of $A$ and the $m\_j$'s are their multiplicities. Then, by the [Leibniz rule](https://en.... | 3 | https://mathoverflow.net/users/36721 | 448449 | 180,549 |
https://mathoverflow.net/questions/448444 | 2 | Consider $\mathbb{R}\_t \times \mathbb{R}\_x ^n$ , let $b\_1(t,x)$ and $b\_2 (t,x)$ be two velocity fields with all the regularity you want and consider the flow of the point $(0,x\_0)$ for a time $T$. Now, I can view the trajectories of $(0,x\_0)$ as curves in $\mathbb{R}^{n+1}$ like $ \gamma\_1 =(t, \Phi\_1 (t,x\_0))... | https://mathoverflow.net/users/109382 | Plateau problem for fluxes of curves | Almgren (<https://mathscinet.ams.org/mathscinet-getitem?mr=855173>) proves that the Plateau solution will satisfy the 2-dimensional Euclidean isoperimetric inequality. So you can bound
$$
\textrm{Area(Plateau Solution)} \leq (4\pi)^{-1}(L(\gamma\_1)+L(\gamma\_2)+L(I\_T))^2.
$$
If you just want a coarse estimate you sho... | 4 | https://mathoverflow.net/users/1540 | 448453 | 180,550 |
https://mathoverflow.net/questions/448452 | 0 | I'm wondering about the following:
1. Every continuous map between smooth manifolds is homotopic to a smooth map.
2. By density of polynomials in space of continuous functions on [0,1], continuous functions can be approximated by smooth ones with respect to the maximum norm.
Now, assume $G\_1$ and $G\_2$ are Lie gr... | https://mathoverflow.net/users/506544 | Continuous map between Lie groups approximation | Let $G\_1=G\_2=U(1)$. Take $f$ such that its image is in an $1/2$-neighborhood of the identity. On the other hand any group homomorphism is either constant or surjective. So there is no chance of approximating continuous maps by homomorphisms.
| 4 | https://mathoverflow.net/users/13842 | 448455 | 180,551 |
https://mathoverflow.net/questions/448450 | 4 | This is closely related to the question [here](https://mathoverflow.net/questions/82613/riemann-mapping-theorem-and-smoothness-on-the-boundary). The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a biholomorphic mapping from $U... | https://mathoverflow.net/users/480683 | Riemann mapping theorem with smooth boundary | The main reference on this topic is the book "Boundary behavior of conformal maps" by Pommerenke.
If the curve is $C^\infty$, then the biholomorphic mapping extends to a smooth map on the closure of the unit disk, and the derivative is non-vanishing. This is implicit in Theorem 3.2 of Pommerenke's book, see also Theo... | 7 | https://mathoverflow.net/users/1162 | 448458 | 180,553 |
https://mathoverflow.net/questions/448309 | 2 | Let $X$ be a surface $P$ be a point on $X$. Let $Z$ be a subscheme supported on the single point $P$ ($Z$ not equal $P$), denote its local multiplicity by $\mu\_P(Z)$.
Now blow up $\pi: \tilde{X} \to X$ along $Z$, we get corresponding exceptional divisor $E$. I would like to know what is $E^2$. I made my guess that i... | https://mathoverflow.net/users/80167 | Self-intersection of exceptional divisor from blowing up a subscheme | I'm afraid it does not work the way you are hoping. At least not right out of the box. First of all, why do you think that $E^2$ is even defined? Also, what exactly is $E$? Is it the pre-image of $Z$, or the reduced exceptional divisor? I assume you are thinking of the former, because the latter could be rather tricky.... | 0 | https://mathoverflow.net/users/10076 | 448461 | 180,555 |
https://mathoverflow.net/questions/444841 | 9 | Namba forcing is stationary-preserving and forces $cf(\omega\_2^{\mathbf{V}}) = \omega$. Ronald Jensen used $\mathcal{L}$-forcing to iterate Namba posets in order to solve the extended Namba problem: for any strongly inaccessible $\kappa$, he constructed a stationary-preserving forcing notion $\mathbb{P}\_{\kappa}$ tha... | https://mathoverflow.net/users/29231 | Extending Namba forcing to arbitrary lengths | An alternative to Andreas’ reply in the comments; using side condition methods one can construct forcings that perform similar jobs as the forcings by Jensen that were mentioned. The following is provable in ZFC by such rather elementary tools making use of countable models as side conditions:
**Theorem.**
For every ... | 4 | https://mathoverflow.net/users/506456 | 448463 | 180,556 |
https://mathoverflow.net/questions/448462 | 2 | Let $M, N$ be monads of rank $\lambda$, where $\lambda$ is a regular cardinal (I'm primarily interested in the case of finitary monads). Is there a known characterization of functors $\mathrm{Alg}~N \to \mathrm{Alg}~M$ (of course, I mean the Eilenberg-Moore category) coming from monad morphisms $M \to N$ (that is, a mo... | https://mathoverflow.net/users/148161 | What functors between categories of algebras are induced by morphisms of monads on $\mathrm{Set}$? | The functor $U\_{({-})} : \mathrm{Mnd}(\mathrm{Set})^\circ \to \mathrm{CAT}/\mathrm{Set}$, sending each monad to the forgetful functor from its category of algebras, is fully faithful (Theorem 3 of Frei's [Some remarks on triples](https://link.springer.com/article/10.1007/BF01110118)). Therefore, a functor between cate... | 6 | https://mathoverflow.net/users/152679 | 448464 | 180,557 |
https://mathoverflow.net/questions/448226 | 9 | Let $r\leq n$ and $d$ be positive integers. A *probability vector* is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let
$$s(\lambda)=(\dim[\pi] \cdot s\_{\pi}(\lambda))\_{\substack{\pi \vdash d}},$$
where $\pi \vdash d$ means that $\pi$ is an integer partition ... | https://mathoverflow.net/users/150898 | The convex hull of Schur polynomial evaluations | If I understand the question correctly then this is false. Since you want $t$ to live in the same vector space as the $s(\lambda)$'s, I am assuming that the vector $t$ has zero coordinates at each $\pi \vdash d, \ell(\pi)> r$.
Taking $d=3, r=2, n=6$ and indexing the vectors in $\mathbb C^3$ as $(x\_{(3)}, x\_{(2,1)},... | 2 | https://mathoverflow.net/users/2384 | 448470 | 180,558 |
https://mathoverflow.net/questions/448471 | 2 | Working in [$\mathcal L\_{\omega\_1, \omega\_1}$](https://plato.stanford.edu/entries/logic-infinitary/), add symbol $=$ with its axioms; add symbol $\in$ and axiomatize:
$\textbf{Extensionality: } \forall x \forall y : \forall z (z \in x \leftrightarrow z \in y) \to x=y$
$\textbf{Foundation: } (\forall v\_n)\_{n \i... | https://mathoverflow.net/users/95347 | Is this theory of well founded countable sets formalized in infinitary logic, complete and categorical? | Yes, this is categorical and hence complete (with respect to any satisfactory notion of proof, that is). Specifically, I claim that any model $M$ of your theory is isomorphic to $\mathsf{HC}$, the set of hereditarily countable sets.
First, given $M$ we can construct recursively an embedding $i:\mathsf{HC}\rightarrow ... | 8 | https://mathoverflow.net/users/8133 | 448475 | 180,559 |
https://mathoverflow.net/questions/448405 | 5 | This question is an extension of my earlier question [here](https://mathoverflow.net/questions/448360/upper-bound-for-halls-conjecture-on-separation-of-squares-and-cubes), answered by Noam Elkies.
Let $A,B \in \mathbb{Z}$. Consider the inequality
$$\displaystyle |y^2 - x^3 - Ax - B| = O(|x|^{1/2 + \theta}).$$
For... | https://mathoverflow.net/users/10898 | Integral points near elliptic curves | You can take $\theta = 0$, even $\theta = -1/6$ works.
Fix an integer $A \ne 0$ and an integer $B$. If $r$ is an integer, the elliptic curve $E : y^{2} = x^{3} + Ax + r^{2} A^{2}$ has the obvious point $P = (0,rA)$. Somewhat miraculously, the point $3P$ is integral. Setting $y = 512 A^{3} r^{9} + 96A^{2} r^{5} + 3Ar$... | 10 | https://mathoverflow.net/users/48142 | 448479 | 180,560 |
https://mathoverflow.net/questions/448465 | 8 | Let $M$ be a closed smooth manifold of dimension $n$ and $z\in H\_l(M,\mathbb{Z})$ a $k$-dimensional integral homology class. Theorem II.4 of Thom's classical 1954 paper states that for $l< n/2$ or $n-l$ odd, some multiple of $z$ can be represented by a smoothly embedded submanifold with the trivial normal bundle.
I ... | https://mathoverflow.net/users/482183 | Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle | With the trivial normal bundle condition, it's fairly easy to produce non-realizable examples using Théorème II.2 of Thom's paper. Namely, a class $z\in H\_l(M^n;\mathbb{Z})$ is realizable by an embedding with trivial normal bundle if and only if the Poincaré dual class $y\in H^{n-l}(M;\mathbb{Z})$ is *spherical*, mean... | 9 | https://mathoverflow.net/users/8103 | 448487 | 180,563 |
https://mathoverflow.net/questions/448486 | 2 | Working in $\mathcal L\_{\omega\_1, \omega}$, can Foundation be captured?
My idea is to formalize a theory where all of its models are the well founded pointwise definable models of $\sf ZFC$. I attempt to formalize it as:
$\textbf{Foundation: } \\\forall x: \neg [ \bigwedge\_{n \in \omega} (\exists v\_0,..,\exists... | https://mathoverflow.net/users/95347 | Can Foundation be captured in $\mathcal L(\omega_1,\omega)$? | Your Foundation axiom does not assert that there is no infinite descending sequence, but rather merely rules out sets at infinite set-theoretic rank. For example, if $x=\omega$, then we can find for every $n$ a descending $\in$ sequence of length $n$.
You can formulate well-foundedness in $L\_{\omega\_1,\omega\_1}$ b... | 5 | https://mathoverflow.net/users/1946 | 448488 | 180,564 |
https://mathoverflow.net/questions/448438 | 6 | Let $f,g \in \mathbb{R}[x\_0,\dots,x\_k]$ be homogeneous polynomials and $X:=Z(f) \subset \mathbb{RP}^k$ be the projective variety defined by $f$.
Assume that $X$ is smooth and has codimension $1$.
Then it's a folklore statement that $X\_\epsilon:=Z(f+\epsilon g)$ is diffeomorphic to $X$ for small $\epsilon$.
>
> Q... | https://mathoverflow.net/users/164084 | How small need a perturbation be to not change the diffeomorphism type of a variety? | Let me prove $(1)$.
First of all, I guess that $f, \, g$ are homogeneous polynomials *of the same degree* $d$, otherwise $Z(f+ \varepsilon g)$ is not well-defined as a subvariety of $\mathbb{RP}^k$.
That said, note that the locus $S\_d$ of singular hypersurfaces of degree $d$ in $\mathbb{RP}^k$ is closed in the loc... | 6 | https://mathoverflow.net/users/7460 | 448490 | 180,565 |
https://mathoverflow.net/questions/448485 | 6 | Let $X$ and $Y$ be standard Borel measurable spaces. A **Markov kernel** $f : X \rightsquigarrow Y$ is a map $f(-|-) : \Sigma\_Y \times X \to [0,1]$ such that:
* $f(-|x)$ is a probability measure on $Y$ for every $x \in X$,
* $f(S|-)$ is a measurable function $X \to [0,1]$ for every $S \in \Sigma\_Y$.
The **set of ... | https://mathoverflow.net/users/27013 | Atoms for Markov kernels | Here probably a partial answer.
I claim that the set $A$ is an analytic subset of $Y$ and thus universally measurable.
If we assume that $X$ and $Y$ are uncountable standard Borel spaces, then wlog we can assume that $X=Y=[0,1]$.
There we can then consider the cumulative distribution functions:
\begin{align}
F(y... | 5 | https://mathoverflow.net/users/506586 | 448496 | 180,569 |
https://mathoverflow.net/questions/448494 | 6 | Let $R$ be a ring , $n \gt 1$, such that for all $x \in R$: $x^n-x \in Z(R)$, the center of $R$. Does it follow that $R$ is commutative?
For $n=2,3$ this is pretty straightforward to prove. But what about higher powers? This is a generalization of Nathan Jacobson's theorem that if $x^n=x$ for all $x \in R$, then $R$ ... | https://mathoverflow.net/users/11629 | Ring in which $x^n-x$ is central for every $x$ | Herstein, "A generalization of a theorem of Jacobson" Amer. J. Math. 73 (1951), 756–762 proves this. Also part III, Amer. J. Math. 75 (1953), 105–111 proves something a bit more general. Namely, the hypothesis can be weakened to $\forall x\ \exists\,n>1\ \cdots$
| 10 | https://mathoverflow.net/users/460592 | 448498 | 180,570 |
https://mathoverflow.net/questions/448499 | 3 | Let $\mathrm{AlgTh}$ be the category of one-sorted algebraic theories (synonym: Lawvere theories; morphisms are functors that are identical on objects and strictly preserve products). It is known that it is locally representable, hence it is bicomplete.
I wonder how the limits and colimits are described in it.
I kn... | https://mathoverflow.net/users/148161 | Limits and colimits in the category of algebraic theories | Typically, limits of multisorted algebraic theories (by which I mean a pair of a set $S$ and an $S$-sorted algebraic theory $\mathbb F(S) \to L$) are most easily described in terms of their presentation as categories with finite products (i.e. a [cartesian (monoidal) category](https://ncatlab.org/nlab/show/cartesian+mo... | 4 | https://mathoverflow.net/users/152679 | 448503 | 180,571 |
https://mathoverflow.net/questions/448491 | 0 | We consider the heat kernel
$$
g :\mathbb R\_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Then
$$
\partial\_t g(t, x) = \Delta g(t, x) = \left(\frac{|x|^2-2 d t}{4 t^2}\right) g(t, x).
$$
*Corollary 1.3* and *Theorem 1.2* in the pape... | https://mathoverflow.net/users/477203 | Is there $f$ such that $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(Ct, x)$ for all $t>0$ and $x \in \mathbb R^d$? | Such a function $f$ does not exist.
Indeed, if the inequality
$|\partial\_t g|(t,x)\le f(t)g(Ct,x)$ holds for all $t>0$ and $x\in\mathbb R^d$, then it holds for $x=0$, so that
$$f(t)\ge f\_\*(t):=\frac{|\partial\_t g|(t,0)}{g(Ct,0)}
=C^{d/2}\frac{d }{2 t}$$
for all real $t>0$.
So, $\int\_0^t f\ge\int\_0^t f\_\*=\inft... | 4 | https://mathoverflow.net/users/36721 | 448522 | 180,578 |
https://mathoverflow.net/questions/448385 | 2 | $\DeclareMathOperator\Inv{Inv}$Baur-Monk quantifier elimination implies that a sentence in the language of modules is a combination of BG invariant statements.
A BG invariant sentence is a boolean combination of $ \Inv(A,\phi, \psi) \* k$ where $\*$ is one of $=,>,<$ where $\phi$ and $\psi$ are positive primitive for... | https://mathoverflow.net/users/506361 | Baur-Monk quantifier elimination (BG-invariants in 1-free variable) | This doesn't follow directly from the statement of QE (and I don't think it has anything to do with the fact that pp formulas are conjunction-closed). To understand it, you really have to look at the proof of QE. For example, see the proof of Theorem 1.1 on p. 155 of Ziegler's paper *[Model Theory of Modules](https://d... | 3 | https://mathoverflow.net/users/2126 | 448528 | 180,581 |
https://mathoverflow.net/questions/448536 | 0 | In a closed model category, is the identity $\textrm{id}: A \to A$ a cofibration? Does it only hold on some special cases? Or is it never true?
| https://mathoverflow.net/users/492610 | Is the identity a cofibration? | Yes: it is always a cofibration. Fix a trivial fibration $f : X \to Y$. It suffices to show that $\hom(A,X) \to \hom(A,Y) \times\_{\hom(A,Y)} \hom(A,X)$ is surjective (informally: $\mathsf{id}$ lifts against $f$). Inspecting the codomain of this map, we see that it's isomorphic to $\hom(A,X)$ so the map is not only sur... | 5 | https://mathoverflow.net/users/76636 | 448537 | 180,583 |
https://mathoverflow.net/questions/448398 | 5 | The Giry monad $G : \textbf{Meas} \to \textbf{Meas}$ maps a measurable space $(X, \mathcal{F})$ to its set of probability measures. The $\sigma$-algebra of $G(X, \mathcal{F})$ is the smallest algebra such that the evaluation map $p \mapsto p(E)$ is measurable for all $E \in \mathcal{F}$. I am aware that this condition ... | https://mathoverflow.net/users/477321 | Intuitive meaning of Giry monad's $\sigma$-algebra | (Disclaimer: I’m not at all seriously experienced with the Giry monad; I’ve not read much beyond the original Giry paper and what I’ve picked up on the street.)
Here are two kinds of sets which typically aren’t in the Giry $\sigma$-algebra $G(X)$:
* $\{ p \mid p(E) \in A \}$, for some measurable $E \subseteq X$ and... | 3 | https://mathoverflow.net/users/2273 | 448553 | 180,588 |
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